UOR Foundation
type: Ontology
version info: 0.2.0
comment: Universal Object Reference Foundation Ontology — the complete vocabulary for the UOR kernel, type system, resolution pipeline, and state model.

space
type: AnnotationProperty
comment: Whether this module belongs to kernel-space (immutable, compiled), user-space (parameterizable, runtime), or bridge (kernel-computed, user-consumed). Values: 'kernel', 'user', 'bridge'.
range: string

UOR Content Addressing
type: Ontology
comment: Content-addressable identifiers for ring elements. Each element carries a unique content-derived identifier.
space: kernel

Element
type: Class
comment: A content-addressable ring element. Each Element uniquely identifies a piece of content via its hash-derived identifier.
subclass of: Thing

length
type: DatatypeProperty
type: FunctionalProperty
comment: The number of Braille glyphs in an address string.
domain: Element
range: nonNegativeInteger

addresses
type: ObjectProperty
type: FunctionalProperty
comment: The datum that this address references. Inverse of schema:glyph.
domain: Element
range: Thing

digest
type: DatatypeProperty
type: FunctionalProperty
comment: The content hash of this address. Format: (blake3|sha256) colon followed by 64 lowercase hex characters. The algorithm prefix must match u:digestAlgorithm.
domain: Element
range: string

digestAlgorithm
type: DatatypeProperty
type: FunctionalProperty
comment: The hash algorithm used to produce u:digest. Allowed values: 'blake3' (primary), 'sha256' (secondary).
domain: Element
range: string

canonicalBytes
type: DatatypeProperty
type: FunctionalProperty
comment: The canonical byte serialisation of the addressed datum, per Amendment 43 section 2: header(k) || le_bytes(x, k+1). This is the exact byte string hashed to produce u:digest.
domain: Element
range: hexBinary

wittLength
type: DatatypeProperty
type: FunctionalProperty
comment: The Witt level n of this element. The element encodes a datum in R_n = Z/(2^n)Z.
domain: Element
range: positiveInteger

UOR Schema
type: Ontology
imports: UOR Content Addressing
comment: Core value types and term language for the UOR ring substrate. Defines Datum (ring element), Term (syntactic expression), and the Ring container.
space: kernel

Datum
type: Class
disjoint with: Term
comment: An element of the ring Z/(2^n)Z at a specific Witt level n. The primary semantic value type. Disjoint from Term: datums are values, terms are syntactic expressions that evaluate to datums.
subclass of: Thing

Term
type: Class
disjoint with: Datum
comment: A syntactic expression in the UOR term language. Terms are evaluated to produce Datums. Disjoint from Datum.
subclass of: Thing

Triad
type: Class
comment: A three-component structure encoding an element's position in the UOR address space: stratum (ring layer), spectrum (bit pattern), and glyph (Braille address).
subclass of: Thing

Literal
type: Class
comment: A term that directly denotes a datum value. A Literal is a leaf node in the term language — it refers to a concrete Datum via schema:denotes without being a Datum itself.
subclass of: Term
subclass of: SurfaceSymbol

Application
type: Class
comment: A term formed by applying an operation to one or more argument terms. The application's value is the result of evaluating the operator on the evaluated arguments.
subclass of: Term

Ring
type: Class
comment: The ambient ring Z/(2^n)Z at a specific Witt level n. The Ring is the primary data structure of the UOR kernel. Its two generators (negation and complement) produce the dihedral group D_{2^n} that governs the invariance frame.
subclass of: Thing

W16Ring
type: Class
comment: The concrete ring Z/(2^16)Z at Witt level 16. Subclass of schema:Ring. Carries 65,536 elements. W16Ring is the first extension of the default Q0 ring and is the target of Amendment 26's universality proofs.
subclass of: Ring

WittLevel
type: Class
comment: A named Witt level Q_k at which the UOR ring operates. Level Q_k uses 8*(k+1) bits, 2^(8*(k+1)) states, and modulus 2^(8*(k+1)). The named individuals Q0-Q3 are the spec-defined reference levels. The class is open: Prism implementations operating at higher levels declare their own WittLevel individuals. The nextWittLevel property forms an unbounded chain Q0 -> Q1 -> Q2 -> ...
subclass of: Thing

TermExpression
type: Class
comment: Root AST node for parsed EBNF term expressions. Identity lhs/rhs values are instances of TermExpression subtypes. Maps to the `term` production in the EBNF grammar.
subclass of: Thing

LiteralExpression
type: Class
comment: A leaf AST node: an integer literal, variable reference, or named constant.
subclass of: TermExpression

ApplicationExpression
type: Class
comment: An AST node representing operator application: an operator applied to an argument list (e.g., add(x, y)).
subclass of: TermExpression

InfixExpression
type: Class
comment: An AST node for infix relations and logical connectives (e.g., x <= y, P -> Q, a = b).
subclass of: TermExpression

SetExpression
type: Class
comment: An AST node for set-builder notation (e.g., {x : P(x)}).
subclass of: TermExpression

CompositionExpression
type: Class
comment: An AST node for function composition (f compose g).
subclass of: TermExpression

ForAllDeclaration
type: Class
comment: A structured quantifier binding: typed variable declarations with a domain and quantifier kind (universal or existential). Replaces the string-valued op:forAll property.
subclass of: Thing

VariableBinding
type: Class
comment: A single variable binding: a variable name bound to a domain type (e.g., x in R_n).
subclass of: Thing

QuantifierKind
type: Class
comment: The kind of quantifier: Universal (forall) or Existential (exists). Controlled vocabulary with exactly 2 individuals.
subclass of: Thing

SurfaceSymbol
type: Class
comment: An abstract leaf value that a grounding map can accept as surface input. Has no direct instances: every SurfaceSymbol is either a Datum-denoting schema:Literal or an xsd-typed schema:HostValue, and the two cases are disjoint.
subclass of: Thing

HostValue
type: Class
disjoint with: Term
disjoint with: Datum
comment: An xsd-typed value that denotes a host datatype rather than a ring datum. Used in property-position slots whose range is xsd and as the host-side input of a grounding map.
subclass of: SurfaceSymbol

HostStringLiteral
type: Class
comment: A host string literal carrying an xsd:string value.
subclass of: HostValue

HostBooleanLiteral
type: Class
comment: A host boolean literal carrying an xsd:boolean value.
subclass of: HostValue

value
type: DatatypeProperty
type: FunctionalProperty
comment: The integer value of a datum element. For a Datum in Z/(2^n)Z, this is an integer in [0, 2^n).
domain: Datum
range: nonNegativeInteger

wittLength
type: DatatypeProperty
type: FunctionalProperty
comment: The Witt level n of a datum, where the datum's ring is Z/(2^n)Z. Determines the bit width and modulus of the datum.
domain: Datum
range: positiveInteger

stratum
type: DatatypeProperty
type: FunctionalProperty
comment: The ring-layer index of a datum, indicating its position in the stratification of Z/(2^n)Z.
domain: Datum
range: nonNegativeInteger

spectrum
type: DatatypeProperty
type: FunctionalProperty
comment: The bit-pattern representation of a datum, encoding its position in the hypercube geometry of Z/(2^n)Z.
domain: Datum
range: nonNegativeInteger

element
type: ObjectProperty
type: FunctionalProperty
comment: The content-addressable element associated with this datum, linking the algebraic value to its identifier.
domain: Datum
range: Element

operator
type: ObjectProperty
type: FunctionalProperty
comment: The operation applied in an Application term.
domain: Application
range: Operation

argument
type: ObjectProperty
comment: An argument term in an Application. The ordering of arguments follows rdf:List semantics.
domain: Application
range: Term

ringWittLength
type: DatatypeProperty
type: FunctionalProperty
comment: The bit width n of the ring Z/(2^n)Z. Distinct from schema:wittLength on Datum — ringWittLength is the container's bit width; datum wittLength is a membership property.
domain: Ring
range: positiveInteger

modulus
type: DatatypeProperty
type: FunctionalProperty
comment: The modulus 2^n of the ring. Equals 2 raised to the power of ringWittLength.
domain: Ring
range: positiveInteger

generator
type: ObjectProperty
type: FunctionalProperty
comment: The generator element π₁ (value = 1) of the ring. Under iterated successor application, π₁ generates all ring elements.
domain: Ring
range: Datum

negation
type: ObjectProperty
type: FunctionalProperty
comment: The ring reflection involution: neg(x) = (-x) mod 2^n. One of the two generators of the dihedral group D_{2^n}.
domain: Ring
range: Involution

complement
type: ObjectProperty
type: FunctionalProperty
comment: The hypercube reflection involution: bnot(x) = (2^n - 1) ⊕ x. The second generator of the dihedral group D_{2^n}.
domain: Ring
range: Involution

denotes
type: ObjectProperty
type: FunctionalProperty
comment: The datum value that a Literal term denotes. Bridges the Term/Datum disjointness: a Literal refers to a Datum without being one. Evaluation of a Literal produces its denoted Datum.
domain: Literal
range: Datum

bitsWidth
type: DatatypeProperty
type: FunctionalProperty
comment: The bit width 8*(k+1) of this Witt level.
domain: WittLevel
range: positiveInteger

cycleSize
type: DatatypeProperty
type: FunctionalProperty
comment: The number of distinct states 2^(8*(k+1)) at this Witt level.
domain: WittLevel
range: positiveInteger

nextWittLevel
type: ObjectProperty
type: FunctionalProperty
comment: The next Witt level in the chain: Q_k -> Q_(k+1). The chain is unbounded; Q3 points to Q4, which is not a named individual in the spec but may be declared by Prism implementations.
domain: WittLevel
range: WittLevel

wittLevelPredecessor
type: ObjectProperty
type: FunctionalProperty
comment: The predecessor Witt level in the chain: Q_(k+1) -> Q_k. Inverse of nextWittLevel. If nextWittLevel(Q_k) = Q_(k+1), then wittLevelPredecessor(Q_(k+1)) = Q_k. Formalizes the chain extension protocol (QL_8).
domain: WittLevel
range: WittLevel

atWittLevel
type: ObjectProperty
type: FunctionalProperty
comment: The Witt level at which this Ring instance operates. Links a concrete Ring individual to its WittLevel.
domain: Ring
range: WittLevel

W16bitWidth
type: DatatypeProperty
type: FunctionalProperty
comment: Bit width of the Q1 ring: 16.
domain: W16Ring
range: positiveInteger

W16capacity
type: DatatypeProperty
type: FunctionalProperty
comment: Carrier set size of the Q1 ring: 65,536 elements.
domain: W16Ring
range: positiveInteger

boundVariables
type: ObjectProperty
comment: The variable bindings in a quantifier declaration. Non-functional: a ForAllDeclaration may bind multiple variables.
domain: ForAllDeclaration
range: VariableBinding

variableDomain
type: ObjectProperty
type: FunctionalProperty
comment: The domain type of a variable binding (e.g., schema:Ring, type:ConstrainedType).
domain: VariableBinding
range: Class

variableName
type: DatatypeProperty
type: FunctionalProperty
comment: The name of a bound variable (e.g., 'x', 'y', 'n').
domain: VariableBinding
range: string

quantifierKind
type: ObjectProperty
type: FunctionalProperty
comment: The kind of quantifier: Universal or Existential.
domain: ForAllDeclaration
range: QuantifierKind

expressionOperator
type: ObjectProperty
type: FunctionalProperty
comment: The operator in an application expression (e.g., op:add, op:neg).
domain: ApplicationExpression
range: Operation

leftOperand
type: ObjectProperty
type: FunctionalProperty
comment: The left operand of an infix expression.
domain: InfixExpression
range: TermExpression

rightOperand
type: ObjectProperty
type: FunctionalProperty
comment: The right operand of an infix expression.
domain: InfixExpression
range: TermExpression

arguments
type: ObjectProperty
comment: The argument list of an application expression. Non-functional: an application may take multiple arguments.
domain: ApplicationExpression
range: TermExpression

literalValue
type: DatatypeProperty
type: FunctionalProperty
comment: The string representation of a literal expression value (e.g., '42', 'x', 'pi1').
domain: LiteralExpression
range: string

infixOperator
type: DatatypeProperty
type: FunctionalProperty
comment: The operator symbol in an infix expression (e.g., '=', '\u{2264}', '\u{2192}').
domain: InfixExpression
range: string

hostString
type: DatatypeProperty
type: FunctionalProperty
comment: The string value carried by a HostStringLiteral.
domain: HostStringLiteral
range: string

hostBoolean
type: DatatypeProperty
type: FunctionalProperty
comment: The boolean value carried by a HostBooleanLiteral.
domain: HostBooleanLiteral
range: boolean

Universal
type: NamedIndividual
type: QuantifierKind
comment: Universal quantification (forall).

Existential
type: NamedIndividual
type: QuantifierKind
comment: Existential quantification (exists).

π₁
type: NamedIndividual
type: Datum
comment: The unique generator of R_n under successor. Value = 1 at every Witt level. Under iterated application of succ, π₁ generates every element of the ring.
value: 1

zero
type: NamedIndividual
type: Datum
comment: The additive identity of the ring. Value = 0 at every Witt level. op:add(x, zero) = x for all x in R_n.
value: 0

W8
type: NamedIndividual
type: WittLevel
comment: Witt level 0: 8-bit ring Z/256Z, 256 states. The reference level for all ComputationCertificate proofs in the spec.
bitsWidth: 8
cycleSize: 256
nextWittLevel: W16

W16
type: NamedIndividual
type: WittLevel
comment: Witt level 1: 16-bit ring Z/65536Z, 65,536 states.
bitsWidth: 16
cycleSize: 65536
nextWittLevel: W24
wittLevelPredecessor: W8

W24
type: NamedIndividual
type: WittLevel
comment: Witt level 2: 24-bit ring Z/16777216Z, 16,777,216 states.
bitsWidth: 24
cycleSize: 16777216
nextWittLevel: W32
wittLevelPredecessor: W16

W32
type: NamedIndividual
type: WittLevel
comment: Witt level 3: 32-bit ring Z/4294967296Z, 4,294,967,296 states. The highest named level in the spec. nextWittLevel is absent — Prism implementations may extend the chain.
bitsWidth: 32
cycleSize: 4294967296
wittLevelPredecessor: W24

term_criticalIdentity_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_AD_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: addresses(glyph(d))

term_AD_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: d

term_AD_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: d ∈ R_n

term_AD_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: glyph(ι(addresses(a)))

term_AD_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ι_addr(a)

term_AD_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: a ∈ Addr(R_n), ι : R_n → R_{n'}

term_R_A1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: add(x, add(y, z))

term_R_A1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: add(add(x, y), z)

term_R_A1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y, z ∈ R_n

term_R_A2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: add(x, 0)

term_R_A2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: x

term_R_A2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_R_A3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: add(x, neg(x))

term_R_A3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 0

term_R_A3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_R_A4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: add(x, y)

term_R_A4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: add(y, x)

term_R_A4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_R_A5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: sub(x, y)

term_R_A5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: add(x, neg(y))

term_R_A5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_R_A6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: neg(neg(x))

term_R_A6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: x

term_R_A6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_R_M1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: mul(x, mul(y, z))

term_R_M1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: mul(mul(x, y), z)

term_R_M1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y, z ∈ R_n

term_R_M2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: mul(x, 1)

term_R_M2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: x

term_R_M2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_R_M3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: mul(x, y)

term_R_M3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: mul(y, x)

term_R_M3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_R_M4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: mul(x, add(y, z))

term_R_M4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: add(mul(x, y), mul(x, z))

term_R_M4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y, z ∈ R_n

term_R_M5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: mul(x, 0)

term_R_M5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 0

term_R_M5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_B_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: xor(x, xor(y, z))

term_B_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: xor(xor(x, y), z)

term_B_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y, z ∈ R_n

term_B_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: xor(x, 0)

term_B_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: x

term_B_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_B_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: xor(x, x)

term_B_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 0

term_B_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_B_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: and(x, and(y, z))

term_B_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: and(and(x, y), z)

term_B_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y, z ∈ R_n

term_B_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: and(x, 2^n - 1)

term_B_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: x

term_B_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_B_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: and(x, 0)

term_B_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 0

term_B_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_B_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: or(x, or(y, z))

term_B_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: or(or(x, y), z)

term_B_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y, z ∈ R_n

term_B_8_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: or(x, 0)

term_B_8_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: x

term_B_8_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_B_9_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: and(x, or(x, y))

term_B_9_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: x

term_B_9_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_B_10_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: and(x, or(y, z))

term_B_10_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: or(and(x, y), and(x, z))

term_B_10_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y, z ∈ R_n

term_B_11_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: bnot(and(x, y))

term_B_11_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: or(bnot(x), bnot(y))

term_B_11_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_B_12_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: bnot(or(x, y))

term_B_12_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: and(bnot(x), bnot(y))

term_B_12_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_B_13_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: bnot(bnot(x))

term_B_13_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: x

term_B_13_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_X_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: neg(x)

term_X_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: sub(0, x)

term_X_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_X_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: bnot(x)

term_X_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: xor(x, 2^n - 1)

term_X_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_X_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: succ(x)

term_X_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: add(x, 1)

term_X_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_X_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: pred(x)

term_X_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: sub(x, 1)

term_X_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_X_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: neg(x)

term_X_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: add(bnot(x), 1)

term_X_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_X_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: bnot(x)

term_X_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: pred(neg(x))

term_X_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_X_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: xor(x, y)

term_X_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: add(x, y) - 2 * and(x, y)

term_X_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ Z (before mod)

term_D_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: succ^{2^n}(x)

term_D_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: x

term_D_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_D_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: neg(succ(neg(x)))

term_D_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: pred(x)

term_D_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_D_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: bnot(neg(x))

term_D_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: pred(x)

term_D_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_D_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: D_{2^n}

term_D_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: {succ^k, neg ∘ succ^k : 0 ≤ k < 2^n}

term_D_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: n ≥ 1

term_U_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: R_n×

term_U_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Z/2 × Z/2^{n-2}

term_U_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: n ≥ 3

term_U_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: R_1× ≅ {1}; R_2× ≅ Z/2

term_U_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: special cases for small n

term_U_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: n ∈ {1, 2}

term_U_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ord(u)

term_U_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: lcm(ord((-1)^a), ord(3^b))

term_U_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: u = (-1)^a · 3^b ∈ R_n×

term_U_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ord_g(2)

term_U_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: divides φ(g)

term_U_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: g odd

term_U_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: step_g

term_U_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 2 * ((g - (2^n mod g)) mod g) + 1

term_U_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: g odd, g > 1

term_AG_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: μ_u

term_AG_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∉ D_{2^n}

term_AG_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: u ∈ R_n×, u ≠ ±1

term_AG_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Aff(R_n)

term_AG_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: R_n× ⋉ R_n

term_AG_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: n ≥ 1

term_AG_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: |Aff(R_n)|

term_AG_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 2^{2n-1}

term_AG_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: n ≥ 1

term_AG_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: D_{2^n}

term_AG_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ⊂ Aff(R_n), u ∈ {±1}

term_AG_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: n ≥ 1

term_CA_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: add(x,y)_k

term_CA_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: xor(x_k, xor(y_k, c_k(x,y)))

term_CA_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n, 0 ≤ k < n

term_CA_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: c_{k+1}(x,y)

term_CA_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: or(and(x_k,y_k), and(xor(x_k,y_k), c_k))

term_CA_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_CA_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: c(x, y)

term_CA_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: c(y, x)

term_CA_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_CA_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: c(x, 0)

term_CA_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 0

term_CA_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n, all positions

term_CA_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: c(x, neg(x))_k

term_CA_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 1

term_CA_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n, k > v_2(x)

term_CA_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_Δ(x, y) > 0

term_CA_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∃ k : c_k(x,y) = 1

term_CA_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_C_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: pins(compose(A, B))

term_C_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: pins(A) ∪ pins(B)

term_C_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraints A, B

term_C_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: compose(A, B)

term_C_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: compose(B, A)

term_C_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraints A, B

term_C_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: compose(compose(A,B), C)

term_C_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: compose(A, compose(B,C))

term_C_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraints A, B, C

term_C_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: compose(A, A)

term_C_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: A

term_C_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint A

term_C_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: compose(A, ε)

term_C_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: A

term_C_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint A, identity ε

term_C_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: compose(A, Ω)

term_C_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Ω

term_C_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint A, annihilator Ω

term_CDI_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: carry(residue(T))

term_CDI_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: depth(T)

term_CDI_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T ∈ T_n

term_CL_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Constraint/≡

term_CL_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 2^{[n]}

term_CL_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint equivalence classes

term_CL_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: A ∨ B

term_CL_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: compose(A, B)

term_CL_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraints A, B

term_CL_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: pins(A ∧ B)

term_CL_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: pins(A) ∩ pins(B)

term_CL_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraints A, B

term_CL_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: (A ∨ B) ∧ C

term_CL_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: (A ∧ C) ∨ (B ∧ C)

term_CL_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraints A, B, C

term_CL_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: A ∧ A̅ = ε, A ∨ A̅ = Ω

term_CL_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∃ A̅ (complement)

term_CL_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint A

term_CM_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: C_i redundant in {C_1,...,C_k}

term_CM_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: pins(C_i) ⊆ ∪_{j≠i} pins(C_j)

term_CM_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint set {C_1,...,C_k}

term_CM_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: minimal cover

term_CM_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: irredundant sub-collection (greedy removal)

term_CM_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CompositeConstraint

term_CM_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: min constraints to cover n sites

term_CM_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ⌈n / max_k pins_per_constraint_k⌉

term_CM_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: n sites, constraint set

term_CR_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: cost(ResidueConstraint(m, r))

term_CR_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: step_m = 2 × ((−2^n) mod m) + 1

term_CR_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ResidueConstraint

term_CR_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: cost(CarryConstraint(p))

term_CR_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: popcount(p)

term_CR_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CarryConstraint

term_CR_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: cost(DepthConstraint(d_min, d_max))

term_CR_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: cost(residue) + cost(carry)

term_CR_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: DepthConstraint

term_CR_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: cost(compose(A, B))

term_CR_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ cost(A) + cost(B)

term_CR_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraints A, B

term_CR_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: optimal resolution order

term_CR_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: increasing cost order

term_CR_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint set

term_F_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: pinned site

term_F_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: cannot be unpinned

term_F_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SiteIndex

term_F_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: pin operations to close

term_F_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ n

term_F_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: FreeRank

term_F_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: pinnedCount + freeRank

term_F_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: totalSites = n

term_F_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: FreeRank

term_F_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: isClosed

term_F_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRank = 0 ⇔ pinnedCount = n

term_F_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: FreeRank

term_FL_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ⊥

term_FL_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: all sites free (freeRank = n)

term_FL_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: FreeRank lattice

term_FL_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ⊤

term_FL_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: all sites pinned (pinnedCount = n)

term_FL_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: FreeRank lattice

term_FL_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: join(S₁, S₂)

term_FL_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: union of pinnings from S₁ and S₂

term_FL_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: FreeRank states S₁, S₂

term_FL_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: lattice height

term_FL_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: n

term_FL_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: FreeRank lattice

term_FPM_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: x ∈ Unit

term_FPM_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: site_0(x) = 1 (x is odd)

term_FPM_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_FPM_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: x ∈ Exterior

term_FPM_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: x ∉ carrier(T)

term_FPM_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n, type T

term_FPM_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: x ∈ Irreducible

term_FPM_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: x ∉ Unit ∪ Exterior AND no non-trivial factorization

term_FPM_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_FPM_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: x ∈ Reducible

term_FPM_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: x ∉ Unit ∪ Exterior ∪ Irreducible

term_FPM_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_FPM_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: x = 2^{v(x)} ⋅ u

term_FPM_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: u odd, v(x) = min position of 1-bit

term_FPM_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_FPM_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: |{x: v(x) = k}|

term_FPM_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 2^{n−k−1} for 0 < k < n; 1 for k = n

term_FPM_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: R_n

term_FPM_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: base type partition

term_FPM_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: |Unit| = 2^{n−1}; |Irr| = 2^{n−2}; |Red| = 2^{n−2}

term_FPM_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: R_n, n ≥ 3

term_FS_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: site_k(x)

term_FS_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: (x >> k) AND 1

term_FS_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n, 0 ≤ k < n

term_FS_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: site_0(x)

term_FS_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: x mod 2 (parity)

term_FS_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_FS_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: site_k(x) given sites 0..k−1

term_FS_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: determines x mod 2^{k+1}

term_FS_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_FS_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: sites 0..k together

term_FS_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: determine x mod 2^{k+1}

term_FS_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_FS_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: all n sites

term_FS_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: determine x uniquely

term_FS_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_FS_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: stratum(x)

term_FS_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: v_2(x) = min{k : site_k(x) = 1}

term_FS_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_FS_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: depth(x)

term_FS_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ v_2(x) for irreducible elements

term_FS_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n, base type

term_RE_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Π_D(T)

term_RE_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Π_C(T) = Π_E(T)

term_RE_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T ∈ T_n

term_IR_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: pinnedCount(state_{i+1})

term_IR_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≥ pinnedCount(state_i)

term_IR_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: resolution states

term_IR_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: iterations to converge

term_IR_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ n

term_IR_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: resolution loop

term_IR_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: convergenceRate

term_IR_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: pinnedCount / iterationCount

term_IR_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ResolutionState

term_IR_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: constraint set spans all sites

term_IR_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: loop terminates

term_IR_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: complete constraint set

term_SF_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: n ≡ 0 (mod ord_g(2))

term_SF_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: factor g has optimal resolution at level n

term_SF_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: factor g, quantum n

term_SF_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: constraints with smaller step_g

term_SF_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: are more constraining, apply first

term_SF_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint ordering

term_RD_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: same type T and constraint sequence

term_RD_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: unique resolved partition

term_RD_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T ∈ T_n, [C₁,...,Cₖ]

term_RD_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: complete constraint set, any order

term_RD_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: same final partition

term_RD_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: closing constraint set

term_SE_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: EvaluationResolver

term_SE_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: directly computes set-theoretic partition

term_SE_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T ∈ T_n

term_SE_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: DihedralFactorizationResolver

term_SE_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: orbit decomposition yields same partition

term_SE_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T ∈ T_n

term_SE_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: CanonicalFormResolver

term_SE_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: confluent rewrite → same partition

term_SE_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T ∈ T_n

term_SE_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Π_D(T) = Π_C(T) = Π_E(T)

term_SE_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: all compute same set-theoretic partition

term_SE_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T ∈ T_n

term_OO_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: benefit(C_i, S)

term_OO_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: |pins(C_i) ∖ S|

term_OO_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint C_i, pinned set S

term_OO_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: cost(C_i)

term_OO_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: step_{m_i} or popcount(p_i)

term_OO_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ResidueConstraint or CarryConstraint

term_OO_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: greedy selection

term_OO_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: argmax benefit(C_i, S) / cost(C_i)

term_OO_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: each resolution step

term_OO_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: greedy approximation

term_OO_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: (1 − 1/e) ≈ 63% of optimal

term_OO_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: weighted set cover

term_OO_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: equal cost tiebreaker

term_OO_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: prefer vertical (residue) before horizontal (carry)

term_OO_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: cost-tied constraints

term_CB_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: min convergenceRate

term_CB_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 1 site per iteration

term_CB_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: worst case

term_CB_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: max convergenceRate

term_CB_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: n sites in 1 iteration

term_CB_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: best case

term_CB_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: expected rate (residue)

term_CB_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ⌊log_2(m)⌋ sites per constraint

term_CB_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ResidueConstraint(m, r)

term_CB_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: convergenceRate < 1 for 2 iterations

term_CB_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: constraint set may be insufficient

term_CB_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: stall detection

term_CB_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∪_i pins(C_i) = {0,...,n−1}

term_CB_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: constraint set closes budget

term_CB_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: sufficiency criterion

term_CB_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: iterations for k constraints

term_CB_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ min(k, n)

term_CB_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: well-formed model

term_OB_M1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_R(x, z)

term_OB_M1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ d_R(x, y) + d_R(y, z)

term_OB_M1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y, z ∈ R_n

term_OB_M2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_H(x, z)

term_OB_M2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ d_H(x, y) + d_H(y, z)

term_OB_M2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y, z ∈ R_n

term_OB_M3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_Δ(x, y)

term_OB_M3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: |d_R(x, y) − d_H(x, y)|

term_OB_M3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_OB_M4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_R(neg(x), neg(y))

term_OB_M4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_R(x, y)

term_OB_M4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_OB_M5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_H(bnot(x), bnot(y))

term_OB_M5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_H(x, y)

term_OB_M5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_OB_M6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_R(succ(x), succ(y))

term_OB_M6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_R(x, y) but d_H may differ

term_OB_M6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_OB_C1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: [neg, bnot](x)

term_OB_C1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 2

term_OB_C1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_OB_C2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: [neg, add(•,k)](x)

term_OB_C2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: −2k mod 2^n

term_OB_C2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n, constant k

term_OB_C3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: [bnot, xor(•,k)](x)

term_OB_C3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 0

term_OB_C3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n, constant k

term_OB_H1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: closed additive path monodromy

term_OB_H1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: trivial (abelian ⇒ path-independent)

term_OB_H1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: additive group

term_OB_H2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: closed {neg,bnot} path monodromy

term_OB_H2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∈ D_{2^n}

term_OB_H2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: dihedral generators

term_OB_H3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: succ-only path WindingNumber

term_OB_H3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: path length / 2^n

term_OB_H3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: closed succ path

term_OB_P1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: PathLength(p₁ ⋅ p₂)

term_OB_P1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: PathLength(p₁) + PathLength(p₂)

term_OB_P1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: paths p₁, p₂

term_OB_P2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: TotalVariation(p₁ ⋅ p₂)

term_OB_P2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ TotalVariation(p₁) + TotalVariation(p₂)

term_OB_P2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: paths p₁, p₂

term_OB_P3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ReductionLength(c₁ ; c₂)

term_OB_P3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ReductionLength(c₁) + ReductionLength(c₂)

term_OB_P3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: reductions c₁, c₂

term_CT_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: catastrophe boundaries

term_CT_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: g = 2^k for 1 ≤ k ≤ n−1

term_CT_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: stratum transitions

term_CT_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: odd prime catastrophe

term_CT_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ResidueConstraint(p, •) transitions visibility

term_CT_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: odd prime p

term_CT_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: CatastropheThreshold(g)

term_CT_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: step_g / n

term_CT_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: factor g

term_CT_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: composite catastrophe g = p⋅q

term_CT_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: max(step_p, step_q) / n

term_CT_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: composite g

term_CF_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: CurvatureFlux(γ)

term_CF_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σ |d_R(x_i, x_{i+1}) − d_H(x_i, x_{i+1})|

term_CF_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: path γ

term_CF_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ResolutionCost(T)

term_CF_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≥ CurvatureFlux(γ_opt)

term_CF_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: type T

term_CF_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: CurvatureFlux(x, succ(x))

term_CF_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: trailing_ones(x) for t < n; n−1 for x = 2^n−1

term_CF_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_CF_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σ_{x ∈ R_n} CurvatureFlux(x, succ(x))

term_CF_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 2^n − 2

term_CF_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: n ≥ 1

term_HG_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: additive holonomy

term_HG_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: trivial (abelian ⇒ path-independent)

term_HG_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: additive group

term_HG_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: {neg, bnot, succ, pred} holonomy

term_HG_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: D_{2^n}

term_HG_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: dihedral generators

term_HG_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: {mul(•, u) : u ∈ R_n×} holonomy

term_HG_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: R_n× ≅ Z/2 × Z/2^{n−2}

term_HG_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: unit group

term_HG_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Hol(R_n)

term_HG_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Aff(R_n) = R_n× ⋉ R_n

term_HG_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: n ≥ 1

term_HG_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Hol(R_n) decomposition

term_HG_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: D_{2^n} ⋅ {mul(•,u) : u ∈ R_n×, u ≠ ±1}

term_HG_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: n ≥ 1

term_T_C1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: compose(id, f)

term_T_C1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: f

term_T_C1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: f ∈ Transform

term_T_C2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: compose(f, id)

term_T_C2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: f

term_T_C2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: f ∈ Transform

term_T_C3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: compose(f, compose(g, h))

term_T_C3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: compose(compose(f, g), h)

term_T_C3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: f, g, h ∈ Transform

term_T_C4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: f composesWith g

term_T_C4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: target(f) = source(g)

term_T_C4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: f, g ∈ Transform

term_T_I1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_R(neg(x), neg(y))

term_T_I1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_R(x, y)

term_T_I1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_T_I2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_H(bnot(x), bnot(y))

term_T_I2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_H(x, y)

term_T_I2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_T_I3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: succ = neg ∘ bnot

term_T_I3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: preserves d_R but not d_H

term_T_I3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_T_I4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ring isometries

term_T_I4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: form a group under composition

term_T_I4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Isometry

term_T_I5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: CurvatureObservable

term_T_I5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: measures failure of ring isometry to be Hamming isometry

term_T_I5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Isometry

term_T_E1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ι(x) = ι(y)

term_T_E1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: x = y

term_T_E1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n (injectivity)

term_T_E2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ι(add(x,y))

term_T_E2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: add(ι(x), ι(y)); ι(mul(x,y)) = mul(ι(x), ι(y))

term_T_E2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_T_E3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ι₂ ∘ ι₁ : R_n → R_k

term_T_E3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: is an embedding (transitivity)

term_T_E3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ι₁: R_n → R_m, ι₂: R_m → R_k

term_T_E4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: glyph ∘ ι ∘ addresses

term_T_E4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: well-defined

term_T_E4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: embedding ι

term_T_A1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: g ∈ D_{2^n} on Constraint C

term_T_A1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: g⋅C (transformed constraint)

term_T_A1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: g ∈ D_{2^n}, C ∈ Constraint

term_T_A2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: g ∈ D_{2^n} on Partition

term_T_A2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: permutes components

term_T_A2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: g ∈ D_{2^n}

term_T_A3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: D_{2^n} orbits on R_n

term_T_A3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: determine irreducibility boundaries

term_T_A3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: DihedralFactorizationResolver

term_T_A4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: fixed points of neg

term_T_A4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: {0, 2^{n−1}}; bnot has none (n > 0)

term_T_A4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: R_n

term_AU_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Aut(R_n)

term_AU_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: {μ_u : x ↦ mul(u, x) | u ∈ R_n×}

term_AU_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: n ≥ 1

term_AU_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Aut(R_n)

term_AU_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≅ R_n× ≅ Z/2 × Z/2^{n−2}

term_AU_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: n ≥ 3

term_AU_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: |Aut(R_n)|

term_AU_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 2^{n−1}

term_AU_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: n ≥ 1

term_AU_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Aut(R_n) ∩ D_{2^n}

term_AU_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: {id, neg}

term_AU_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: n ≥ 1

term_AU_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Aff(R_n)

term_AU_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ⟨D_{2^n}, μ_3⟩

term_AU_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: n ≥ 1

term_EF_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: F_ι(f)

term_EF_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ι ∘ f ∘ ι⁻¹ on Im(ι)

term_EF_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ι: R_n → R_m, f ∈ Cat(R_n)

term_EF_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: F_ι(f ∘ g)

term_EF_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: F_ι(f) ∘ F_ι(g)

term_EF_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ι: R_n → R_m

term_EF_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: F_ι(id_{R_n})

term_EF_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: id_{Im(ι)}

term_EF_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ι: R_n → R_m

term_EF_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: F_{ι₂ ∘ ι₁}

term_EF_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: F_{ι₂} ∘ F_{ι₁}

term_EF_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ι₁: R_n → R_m, ι₂: R_m → R_k

term_EF_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: F_ι(ring isometry)

term_EF_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ring isometry at level m

term_EF_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ι: R_n → R_m

term_EF_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: F_ι(D_{2^n})

term_EF_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ⊆ D_{2^m} as subgroup

term_EF_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ι: R_n → R_m

term_EF_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: F_ι(R_n×)

term_EF_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ⊆ R_m× as subgroup

term_EF_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ι: R_n → R_m

term_AA_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: glyph(x)

term_AA_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: [braille(x[0:5]), braille(x[6:11]), ...]

term_AA_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n (6-bit blocks)

term_AA_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: braille(a ⊕ b)

term_AA_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: braille(a) ⊕ braille(b)

term_AA_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: a, b ∈ {0,1}^6

term_AA_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: glyph(bnot(x))

term_AA_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: complement each Braille character of glyph(x)

term_AA_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_AA_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: glyph(add(x, y))

term_AA_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≠ f(glyph(x), glyph(y)) in general

term_AA_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_AA_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: liftable operations

term_AA_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: {xor, and, or, bnot}; NOT {add, sub, mul, neg, succ, pred}

term_AA_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: operations on R_n

term_AA_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: neg(x) = succ(bnot(x))

term_AA_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: bnot lifts, succ does not

term_AA_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_AM_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_addr(a, b)

term_AM_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σ_i popcount(braille_i(a) ⊕ braille_i(b))

term_AM_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: addresses a, b

term_AM_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_addr(glyph(x), glyph(y))

term_AM_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_H(x, y)

term_AM_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_AM_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_addr

term_AM_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: does NOT preserve d_R in general

term_AM_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: addresses

term_AM_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_Δ(x, y)

term_AM_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: |d_R(x,y) − d_addr(glyph(x), glyph(y))|

term_AM_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_TH_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: S(state)

term_TH_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRank × ln 2

term_TH_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: state ∈ FreeRank

term_TH_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: S(⊥)

term_TH_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: n × ln 2

term_TH_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: unconstrained type

term_TH_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: S(⊤)

term_TH_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 0

term_TH_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: fully resolved type

term_TH_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: total resolution cost

term_TH_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: n × k_B T × ln 2

term_TH_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Landauer bound

term_TH_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: β*

term_TH_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ln 2

term_TH_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: UOR site system

term_TH_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: constraint application

term_TH_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: removes entropy; convergenceRate = cooling rate

term_TH_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: resolution loop

term_TH_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: CatastropheThreshold

term_TH_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: temperature of partition phase transition

term_TH_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: partition bifurcation

term_TH_8_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: step_g

term_TH_8_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: free-energy cost of constraint boundary

term_TH_8_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint boundary g

term_TH_9_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: computational hardness

term_TH_9_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: type incompleteness (high temperature)

term_TH_9_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: type specification

term_TH_10_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: type resolution

term_TH_10_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: measurement; cost ≥ entropy removed

term_TH_10_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: resolution process

term_AR_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: adiabatic schedule

term_AR_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: decreasing freeRank × cost-per-site order

term_AR_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint ordering

term_AR_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Cost_adiabatic

term_AR_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σ_i cost(C_{σ(i)}) where σ is optimal

term_AR_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: optimal ordering

term_AR_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Cost_adiabatic

term_AR_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≥ n × k_B T × ln 2

term_AR_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Landauer bound

term_AR_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: η = (n × ln 2) / Cost_adiabatic

term_AR_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ 1

term_AR_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: adiabatic efficiency

term_AR_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: greedy vs adiabatic difference

term_AR_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ 5% for n ≤ 16

term_AR_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: empirical, Q0–Q4

term_PD_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: phase space

term_PD_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: {(n, g) : n ∈ Z₊, g constraint boundary}

term_PD_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: UOR phase diagram

term_PD_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: phase boundaries

term_PD_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: g | (2^n − 1) or g = 2^k

term_PD_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: (n, g) plane

term_PD_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: parity boundary

term_PD_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: |Unit| = 2^{n−1}, |non-Unit| = 2^{n−1}

term_PD_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: g = 2

term_PD_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: resonance lines

term_PD_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: n = k ⋅ ord_g(2)

term_PD_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: (n, g) plane

term_PD_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: phase count at level n

term_PD_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ 2^n (typical O(n))

term_PD_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: quantum level n

term_RC_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: reversible pinning of site k

term_RC_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: store prior state in ancilla site k'

term_RC_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SiteIndex k

term_RC_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: reversible pinning entropy

term_RC_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ΔS_total = 0

term_RC_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: reversible strategy

term_RC_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: deferred Landauer cost

term_RC_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: n × k_B T × ln 2 at ancilla erasure

term_RC_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ancilla cleanup

term_RC_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: reversible total cost

term_RC_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: = irreversible total cost (redistributed)

term_RC_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: reversible strategy

term_RC_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: quantum UOR

term_RC_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: superposed sites, cost ∝ winning path

term_RC_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: hypothetical quantum

term_DC_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∂_R f(x)

term_DC_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: f(succ(x)) - f(x)

term_DC_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: f : R_n → R_n, x ∈ R_n

term_DC_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∂_H f(x)

term_DC_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: f(bnot(x)) - f(x)

term_DC_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: f : R_n → R_n, x ∈ R_n

term_DC_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∂_H id(x)

term_DC_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: bnot(x) - x = -(2x + 1) mod 2^n

term_DC_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_DC_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: [neg, bnot](x)

term_DC_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∂_R neg(x) - ∂_H neg(x)

term_DC_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_DC_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∂_R f - ∂_H f

term_DC_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σ carry contributions

term_DC_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: f : R_n → R_n

term_DC_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: J_k(x)

term_DC_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∂_R f_k(x) where f_k = site_k

term_DC_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n, 0 ≤ k < n

term_DC_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: J_{n-1}(x)

term_DC_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: may differ from lower sites

term_DC_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_DC_8_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: rank(J(x))

term_DC_8_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: = d_H(x, succ(x)) - 1 for generic x

term_DC_8_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_DC_9_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: κ(x)

term_DC_9_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σ_k J_k(x)

term_DC_9_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n

term_DC_10_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: optimal next constraint

term_DC_10_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: argmax J_k over free sites

term_DC_10_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: resolution step

term_DC_11_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σ_{x} J_k(x)

term_DC_11_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≈ (2^n - 2)/n for each k

term_DC_11_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: 0 ≤ k < n

term_HA_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: N(C)

term_HA_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: simplicial complex on constraints

term_HA_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint set C

term_HA_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: resolution stalls

term_HA_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ⟺ H_k(N(C)) ≠ 0 for some k > 0

term_HA_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint set C

term_HA_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: S_residual

term_HA_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≥ Σ_k β_k × ln 2

term_HA_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint configuration C

term_IT_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: χ(N(C))

term_IT_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σ_k (-1)^k β_k

term_IT_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint nerve N(C)

term_IT_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: χ(N(C))

term_IT_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σ_k (-1)^k dim(H_k)

term_IT_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint nerve N(C)

term_IT_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: λ_1(N(C))

term_IT_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: lower bounds convergence rate

term_IT_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint nerve N(C)

term_IT_7a_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σ κ_k - χ(N(C))

term_IT_7a_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: = S_residual / ln 2

term_IT_7a_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint configuration C

term_IT_7b_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: S_residual

term_IT_7b_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: = (Σ κ_k - χ) × ln 2

term_IT_7b_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint configuration C

term_IT_7c_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: resolution cost

term_IT_7c_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≥ n - χ(N(C))

term_IT_7c_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint configuration C

term_IT_7d_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: resolution complete

term_IT_7d_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ⟺ χ(N(C)) = n and all β_k = 0

term_IT_7d_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint nerve N(C)

term_phi_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: φ₁(neg, ResidueConstraint(m,r))

term_phi_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ResidueConstraint(m, m-r)

term_phi_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ring op, constraint

term_phi_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: φ₂(compose(A,B))

term_phi_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: φ₂(A) ∪ φ₂(B)

term_phi_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraints A, B

term_phi_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: φ₃(closed site state)

term_phi_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: unique 4-component partition

term_phi_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: closed FreeRank

term_phi_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: φ₄(T, x)

term_phi_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: φ₃(φ₂(φ₁(T, x)))

term_phi_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T ∈ T_n, x ∈ R_n

term_phi_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: φ₅(neg)

term_phi_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: preserves d_R, may change d_H

term_phi_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: op ∈ Operation

term_phi_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: φ₆(state, observables)

term_phi_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: RefinementSuggestion

term_phi_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ResolutionState

term_psi_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: N(constraints)

term_psi_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: SimplicialComplex

term_psi_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint set

term_psi_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: C(K)

term_psi_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ChainComplex

term_psi_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: simplicial complex K

term_psi_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: H_k(C)

term_psi_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ker(∂_k) / im(∂_{k+1})

term_psi_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: chain complex C

term_psi_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: C^k

term_psi_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Hom(C_k, R)

term_psi_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: chain complex C, ring R

term_psi_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: H^k(C)

term_psi_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ker(δ^k) / im(δ^{k-1})

term_psi_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: cochain complex C

term_surfaceSymmetry_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: P(Π(G(s)))

term_surfaceSymmetry_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: s' where type(s') ≡ type(s) under F.constraint

term_surfaceSymmetry_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: G: GroundingMap, P: ProjectionMap, F: Frame, s: Literal, G.symbolConstraints = P.projectionOrder

term_CC_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: resolution(x, T)

term_CC_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: O(1) for CompleteType T

term_CC_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n, T: CompleteType

term_CC_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: completeness(T)

term_CC_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: implies completeness(T')

term_CC_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T, T': ConstrainedType, T ⊆ T'

term_CC_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: sitesClosed(W₁) + sitesClosed(W₂)

term_CC_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: = n for valid concat(W₁, W₂)

term_CC_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: W₁, W₂: CompletenessWitness

term_CC_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: CompletenessResolver

term_CC_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: fix(ψ-pipeline, CompletenessCandidate)

term_CC_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CompletenessCandidate

term_CC_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: cert.verified = true

term_CC_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: implies χ(N(C)) = n ∧ ∀k: β_k = 0

term_CC_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: cert: CompletenessCertificate

term_QL_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: neg(bnot(x))

term_QL_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: succ(x) in Z/(2ⁿ)Z

term_QL_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n, n ≥ 1

term_QL_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: |D_{2ⁿ}|

term_QL_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 2ⁿ⁺¹ for all n ≥ 1

term_QL_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: n ≥ 1

term_QL_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: P(j) = 2^{-j}

term_QL_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Boltzmann distribution at β* = ln 2, all n ≥ 1

term_QL_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: j ∈ R_n, n ≥ 1

term_QL_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: siteBudget(PrimitiveType, n)

term_QL_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: = n (one site per bit)

term_QL_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: PrimitiveType, n ≥ 1

term_QL_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: resolution(CompleteType, n)

term_QL_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: O(1) for all n ≥ 1

term_QL_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CompleteType, n ≥ 1

term_QL_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: contentAddress(x, n)

term_QL_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: bijection for all n ≥ 1

term_QL_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n, n ≥ 1

term_QL_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ψ-pipeline(ConstrainedType, n)

term_QL_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: valid ChainComplex for all n ≥ 1

term_QL_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ConstrainedType, n ≥ 1

term_GR_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRank(B_{i+1})

term_GR_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ freeRank(B_i)

term_GR_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: i in Session S

term_GR_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: b.datum resolves under b.constraint

term_GR_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: in O(1) iff Binding b is sound

term_GR_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: b: Binding

term_GR_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∃ i: freeRank(B_i) = 0

term_GR_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Session S converges

term_GR_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Session S

term_GR_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: bindings(C_fresh) ∩ bindings(C_prior)

term_GR_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: = ∅ after SessionBoundary

term_GR_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: C_prior, C_fresh: Context, SessionBoundary event

term_GR_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ContradictionBoundary

term_GR_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: iff ∃ b, b': same address, different datum, same constraint

term_GR_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: b, b': Binding in same Context

term_TS_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: nerve(T, target)

term_TS_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∃ ConstrainedType T over R_n realising target

term_TS_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: target: χ* ≤ n, β₀* = 1, β_k* = 0 for k ≥ 1

term_TS_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: basisSize(T, IT_7d target)

term_TS_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: n

term_TS_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: IT_7d target, n-site types

term_TS_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: χ(N(C + constraint))

term_TS_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≥ χ(N(C))

term_TS_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: C: synthesis candidate constraint set

term_TS_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: steps(TypeSynthesisResolver, target)

term_TS_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ n

term_TS_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: target: realisable n-site type synthesis goal

term_TS_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: SynthesizedType achieves IT_7d

term_TS_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: iff CompletenessResolver certifies CompleteType

term_TS_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T: SynthesizedType

term_TS_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: E[steps, Jacobian-guided synthesis]

term_TS_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: O(n log n) vs O(n²) uninformed

term_TS_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T: n-site type synthesis goal

term_TS_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: β₀(N(C)) for non-empty C

term_TS_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≥ 1

term_TS_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: C: non-empty constraint set

term_WLS_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: WittLift T' is CompleteType

term_WLS_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: iff spectral sequence collapses at E_2

term_WLS_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T: CompleteType at Q_n, T': WittLift to Q_{n+1}

term_WLS_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: non-trivial LiftObstruction location

term_WLS_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: specific site at bit position n+1

term_WLS_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: non-trivial LiftObstruction

term_WLS_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: basisSize(T') for trivial lift

term_WLS_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: basisSize(T) + 1

term_WLS_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T: CompleteType at Q_n with closed constraint set

term_WLS_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: spectral sequence convergence page

term_WLS_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ E_{d+2}

term_WLS_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: depth-d constraint configuration

term_WLS_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: universallyValid identity in R_{n+1}

term_WLS_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: holds with lifted constraint set

term_WLS_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: every op:universallyValid identity, WittLift T'

term_WLS_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ψ-pipeline ChainComplex(T')

term_WLS_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: valid and restricts to ChainComplex(T) on base nerve

term_WLS_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T': any WittLift of a CompleteType T

term_MN_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: HolonomyGroup(T)

term_MN_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ D_{2^n}

term_MN_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T: ConstrainedType over R_n

term_MN_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: HolonomyGroup(T) for additive constraints

term_MN_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: {id} (trivial: T is FlatType)

term_MN_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T: all ResidueConstraint or DepthConstraint

term_MN_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: HolonomyGroup(T) with neg + bnot in closed path

term_MN_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: D_{2^n} (full dihedral holonomy)

term_MN_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T: ConstrainedType with neg-related and bnot-related constraints in closed path

term_MN_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: HolonomyGroup(T) ≠ {id}

term_MN_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ⟹ β₁(N(C(T))) ≥ 1

term_MN_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T: ConstrainedType

term_MN_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: CompleteType (IT_7d) ⟹ β₁ = 0 ⟹ holonomy

term_MN_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: trivial ⟹ FlatType

term_MN_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T: CompleteType

term_MN_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: monodromy(p₁ · p₂)

term_MN_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: monodromy(p₁) · monodromy(p₂) in D_{2^n}

term_MN_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: p₁, p₂: ClosedConstraintPath

term_MN_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: TwistedType T ⟹ H²(N(C(T')); ℤ/2ℤ)

term_MN_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: non-zero class (non-trivial LiftObstruction)

term_MN_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T': any WittLift of TwistedType T

term_PT_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: siteBudget(A × B)

term_PT_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: siteBudget(A) + siteBudget(B)

term_PT_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: A, B: TypeDefinition

term_PT_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: partition(A × B)

term_PT_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: partition(A) ⊗ partition(B)

term_PT_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: A, B: TypeDefinition

term_PT_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: χ(N(C(A × B)))

term_PT_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: χ(N(C(A))) + χ(N(C(B)))

term_PT_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: A, B: TypeDefinition

term_PT_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: S(A × B)

term_PT_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: S(A) + S(B)

term_PT_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: A, B: TypeDefinition

term_ST_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: siteBudget(A + B)

term_ST_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: max(siteBudget(A), siteBudget(B))

term_ST_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: A, B: TypeDefinition

term_ST_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: S(A + B)

term_ST_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ln 2 + max(S(A), S(B))

term_ST_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: A, B: TypeDefinition

term_GS_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: T_ctx(C)

term_GS_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRank(C) × ln 2 / n

term_GS_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: C: Context, n = siteBudget

term_GS_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: σ(C)

term_GS_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: (n − freeRank(C)) / n

term_GS_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: C: Context, n = siteBudget

term_GS_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: σ(B_{i+1})

term_GS_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≥ σ(B_i)

term_GS_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: i in Session S

term_GS_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: σ(C) = 1

term_GS_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRank(C) = 0 ↔ S(C) = 0 ↔ T_ctx(C) = 0

term_GS_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: C: Context

term_GS_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: stepCount(q, C) at freeRank(C) = 0

term_GS_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 0

term_GS_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: q: Query, C: GroundedContext

term_GS_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: effectiveBudget(q, C)

term_GS_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: max(0, siteBudget(q.type) − |pinnedSites(C) ∩ q.siteSet|)

term_GS_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: q: Query, C: Context

term_GS_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Cost_saturation(C)

term_GS_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: n × k_B T × ln 2

term_GS_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: C: GroundedContext, n = siteBudget

term_MS_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: β₀(N(C))

term_MS_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≥ 1

term_MS_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: C: non-empty ConstrainedType

term_MS_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: χ(N(C))

term_MS_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ n

term_MS_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: C: ConstrainedType at quantum level n

term_MS_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: χ(N(C + c))

term_MS_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≥ χ(N(C))

term_MS_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: C: ConstrainedType, c: Constraint

term_MS_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: achievable(χ*, β_k*, n)

term_MS_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: → achievable(χ*, β_k*, n+1)

term_MS_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: (χ*, β_k*) achievable at level n

term_MS_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: verified SynthesisSignature set

term_MS_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: → exact morphospace boundary in the limit

term_MS_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: all quantum levels

term_GD_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: isGeodesic(T)

term_GD_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: AR_1-ordered(T) ∧ DC_10-selected(T)

term_GD_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T: ComputationTrace

term_GD_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ΔS_step(i) on geodesic

term_GD_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ln 2

term_GD_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: step i of GeodesicTrace T

term_GD_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Cost_geodesic(T)

term_GD_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRank_initial × k_B T × ln 2

term_GD_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T: GeodesicTrace

term_GD_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Cost(T) for geodesic T

term_GD_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: = Cost(T') for any geodesic T' on same type

term_GD_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T, T': GeodesicTrace on same ConstrainedType

term_GD_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: isSubgeodesic(T)

term_GD_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∃ i: J_k(step_i) < max_{free} J_k(state_i)

term_GD_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T: ComputationTrace

term_QM_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: S_vonNeumann(ψ)

term_QM_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Cost_Landauer(collapse(ψ))

term_QM_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ψ: SuperposedSiteState

term_QM_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: collapse(ψ)

term_QM_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: apply(ResidueConstraint, collapsed_site)

term_QM_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ψ: SuperposedSiteState

term_QM_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: S_vN(ψ)

term_QM_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∈ [0, ln 2]

term_QM_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ψ: single-site SuperposedSiteState

term_QM_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: collapse(collapse(ψ))

term_QM_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: collapse(ψ)

term_QM_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ψ: SuperposedSiteState

term_QM_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σᵢ |αᵢ|²

term_QM_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 1

term_QM_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SuperposedSiteState ψ

term_RC_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: normalize(ψ)

term_RC_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ψ / sqrt(Σ |αᵢ|²)

term_RC_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SuperposedSiteState ψ

term_FPM_8_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: |Irr| + |Red| + |Unit| + |Ext|

term_FPM_8_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 2ⁿ

term_FPM_8_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Partition P over R_n

term_FPM_9_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: x ∈ Ext(T)

term_FPM_9_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: x ∉ carrier(T)

term_FPM_9_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: TypeDefinition T, Datum x

term_MN_8_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: holonomyClassified(T)

term_MN_8_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: isFlatType(T) xor isTwistedType(T)

term_MN_8_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ConstrainedType T with holonomyGroup

term_QL_8_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: wittLevelPredecessor(nextWittLevel(Q_k))

term_QL_8_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Q_k

term_QL_8_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: WittLevel W_n with nextWittLevel defined

term_D_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: compose(rᵃ sᵖ, rᵇ sᵠ)

term_D_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: r^((a + (-1)ᵖ b) mod 2ⁿ) s^(p xor q)

term_D_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: DihedralElement rᵃ sᵖ, rᵇ sᵠ in D_{2ⁿ}

term_SP_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: resolve_superposition(classical(x))

term_SP_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: resolve_classical(x)

term_SP_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Datum x

term_SP_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: collapse(resolve_superposition(ψ))

term_SP_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: resolve_classical(collapse(ψ))

term_SP_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SuperposedSiteState ψ

term_SP_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: amplitudeVector(resolve_superposition(ψ))

term_SP_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: normalized

term_SP_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SuperposedSiteState ψ

term_SP_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: P(collapse to site k)

term_SP_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: |α_k|²

term_SP_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SuperposedSiteState ψ, site index k

term_PT_2a_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Π(A × B)

term_PT_2a_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: PartitionProduct(Π(A), Π(B))

term_PT_2a_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ProductType A × B

term_PT_2b_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Π(A + B)

term_PT_2b_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: PartitionCoproduct(Π(A), Π(B))

term_PT_2b_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SumType A + B

term_GD_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: isGeodesic(trace)

term_GD_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: isAR1Ordered(trace) ∧ isDC10Selected(trace)

term_GD_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ComputationTrace trace

term_WT_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: LiftChain(Q_j, Q_k) valid

term_WT_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: every chainStep has trivial or resolved LiftObstruction

term_WT_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: LiftChain from Q_j to Q_k

term_WT_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: obstructionCount(chain)

term_WT_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: <= chainLength(chain)

term_WT_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: LiftChain

term_WT_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: resolvedBasisSize(Q_k)

term_WT_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: basisSize(Q_j) + chainLength + obstructionResolutionCost

term_WT_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: LiftChain with source Q_j, target Q_k

term_WT_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: isFlat(chain)

term_WT_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: obstructionCount = 0 iff HolonomyGroup trivial at every step

term_WT_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: LiftChain

term_WT_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: LiftChainCertificate exists

term_WT_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: CompleteType at Q_k satisfies IT_7d with witness chain

term_WT_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Q_k for arbitrary k

term_WT_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: QT_3 with chainLength=1, cost=0

term_WT_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: reduces to QLS_3

term_WT_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Single-step chains

term_WT_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: flat chain resolvedBasisSize(Q_k)

term_WT_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: basisSize(Q_j) + (k - j)

term_WT_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: LiftChain with isFlat = true

term_CC_PINS_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: pinsSites(CarryConstraint(p))

term_CC_PINS_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: {k : p(k)=1} union {first-zero(p)}

term_CC_PINS_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: bit-pattern p in CarryConstraint

term_CC_COST_SITE_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: |pinsSites(CarryConstraint(p))|

term_CC_COST_SITE_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: popcount(p) + 1

term_CC_COST_SITE_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: bit-pattern p in CarryConstraint

term_jsat_RR_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: jointSat(Res(m1,r1), Res(m2,r2))

term_jsat_RR_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: gcd(m1,m2) | (r1 - r2)

term_jsat_RR_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ResidueConstraint pairs over R_n

term_jsat_CR_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: jointSat(Carry(p), Res(m,r))

term_jsat_CR_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: pin-site intersection residue-class compatible

term_jsat_CR_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CarryConstraint, ResidueConstraint pairs

term_jsat_CC_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: jointSat(Carry(p1), Carry(p2))

term_jsat_CC_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: p1 AND p2 conflict-free

term_jsat_CC_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CarryConstraint pairs over R_n

term_D_8_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: (r^a s^p)^(-1)

term_D_8_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: r^(-(−1)^p a mod 2^n) s^p

term_D_8_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: a in 0..2^n, p in {0,1}

term_D_9_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ord(r^k s^1)

term_D_9_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 2

term_D_9_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: k in Z/(2^n)Z

term_EXP_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: carrier(C) is monotone

term_EXP_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: all residues of C = modulus - 1, no Carry/Depth

term_EXP_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ConstrainedType C over R_n

term_EXP_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: count of monotone ConstrainedTypes at Q_n

term_EXP_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 2^n

term_EXP_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: WittLevel W_n, n >= 1

term_EXP_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: carrier(SumType(A,B))

term_EXP_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: coproduct(carrier(A), carrier(B))

term_EXP_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SumType A + B

term_ST_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: chi(N(C(A+B)))

term_ST_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: chi(N(C(A))) + chi(N(C(B)))

term_ST_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: disjoint SumType A + B

term_ST_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: beta_k(N(C(A+B)))

term_ST_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: beta_k(N(C(A))) + beta_k(N(C(B)))

term_ST_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: disjoint SumType A + B, k >= 0

term_ST_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: CompleteType(A + B)

term_ST_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: CompleteType(A) and CompleteType(B) and Q(A)=Q(B)

term_ST_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SumType A + B

term_TS_8_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: min constraints for beta_1 = k

term_TS_8_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 2k + 1

term_TS_8_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: first Betti number k >= 1, n-site type

term_TS_9_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: TypeSynthesisResolver terminates

term_TS_9_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: within 2^n steps

term_TS_9_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: WittLevel W_n, any target signature

term_TS_10_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ForbiddenSignature(sigma)

term_TS_10_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: no ConstrainedType with <= n constraints realises sigma

term_TS_10_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: topological signature sigma at Q_n

term_WT_8_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ObstructionChain length from Q_j to Q_k

term_WT_8_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: <= (k-j) * C(basisSize(Q_j), 3)

term_WT_8_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: LiftChain from Q_j to Q_k

term_WT_9_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: TowerCompletenessResolver terminates

term_WT_9_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: within QT_8 bound

term_WT_9_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: LiftChain of finite length

term_COEFF_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: standard coefficient ring for psi-pipeline

term_COEFF_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Z/2Z

term_COEFF_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CechNerve N(C) at any quantum level

term_GO_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: pinsSites(killing constraint for obstruction c)

term_GO_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: superset of pinsSites(C_i) cap pinsSites(C_j)

term_GO_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: GluingObstruction c, cycle pair (C_i, C_j)

term_GR_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRank(q) after grounding

term_GR_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: sites of q not in BindingAccumulator

term_GR_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: grounded Session, new RelationQuery q

term_GR_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: sigma after re-entry with query q

term_GR_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: min(sigma, 1 - freeRank(q)/n)

term_GR_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SessionResolver, new query q

term_QM_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: amplitude index set of SuperposedSiteState over T

term_QM_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: monotone pinning trajectories consistent with T

term_QM_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SuperposedSiteState over ConstrainedType T at Q_n

term_CIC_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: valid(T) ∧ T: Transform

term_CIC_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∃ c: TransformCertificate. certifies(c, T)

term_CIC_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Transform T

term_CIC_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: isometry(T) ∧ metric-preserving(T)

term_CIC_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∃ c: IsometryCertificate. certifies(c, T)

term_CIC_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Isometry T

term_CIC_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: f(f(x)) = x ∀ x ∈ R_n

term_CIC_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∃ c: InvolutionCertificate. certifies(c, f)

term_CIC_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Involution f

term_CIC_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: σ(C) = 1 ∧ freeRank = 0

term_CIC_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∃ c: GroundingCertificate. certifies(c, C)

term_CIC_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: GroundedContext C

term_CIC_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: AR_1-ordered ∧ DC_10-selected trace

term_CIC_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∃ c: GeodesicCertificate. certifies(c, trace)

term_CIC_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: GeodesicTrace

term_CIC_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: S_vN = L_cost at β* = ln 2

term_CIC_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∃ c: MeasurementCertificate. certifies(c, event)

term_CIC_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: MeasurementEvent

term_CIC_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: P(k) = |α_k|² verified

term_CIC_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∃ c: BornRuleVerification. certifies(c, event)

term_CIC_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: MeasurementEvent with amplitude vector

term_GC_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: shared-frame grounding of symbol s

term_GC_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∃ c: GroundingCertificate. certifies(c, map)

term_GC_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: GroundingMap with valid ProjectionMap

term_GR_8_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: compose(S_A, S_B) valid at Q_k

term_GR_8_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∀ j ≤ k: ∀ a ∈ pinnedSites(S_A, Q_j) ∩ pinnedSites(S_B, Q_j): datum(S_A, a, Q_j) = datum(S_B, a, Q_j)

term_GR_8_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: S_A, S_B: Session at quantum level Q_k (k ≥ 0)

term_GR_9_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: leasedSites(L_A) ∩ leasedSites(L_B)

term_GR_9_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: = ∅

term_GR_9_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: L_A, L_B: ContextLease on SharedContext C, L_A ≠ L_B

term_GR_10_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: finalState(R, P_1, Q)

term_GR_10_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: = finalState(R, P_2, Q) for any P_1, P_2: ExecutionPolicy

term_GR_10_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SessionResolver R with ExecutionPolicy P, pending query set Q

term_MC_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σᵢ freeRank(leasedSites(L_i))

term_MC_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: = freeRank(C)

term_MC_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SharedContext C; leaseSet {L_1, …, L_k} covering all sites of C

term_MC_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRank(B_{i+1} |_L)

term_MC_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ freeRank(B_i |_L)

term_MC_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ContextLease L held by Session S; binding step i within S restricted to leasedSites(L)

term_MC_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRank(compose(S_A, S_B))

term_MC_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRank(S_A) + freeRank(S_B) − |pinnedSites(S_A) ∩ pinnedSites(S_B)|

term_MC_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: S_A, S_B: Session; compose(S_A, S_B) valid (SR_8 satisfied)

term_MC_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRank(compose(S_A, S_B))

term_MC_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: = freeRank(S_A) + freeRank(S_B)

term_MC_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: S_A, S_B on ContextLeases L_A, L_B within SharedContext C; SR_9 holds

term_MC_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: finalBindings(R, P_1, Q)

term_MC_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: = finalBindings(R, P_2, Q)

term_MC_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SessionResolver R; pending query set Q; ExecutionPolicy P_1, P_2

term_MC_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: σ(compose(S_1, …, S_k))

term_MC_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: = 1 (FullGrounding)

term_MC_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SharedContext C; leases {L_1, …, L_k} pairwise disjoint (SR_9) and fully covering C; each S_i with freeRank = 0 within L_i

term_MC_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: stepCount(q, C*)

term_MC_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: = 0

term_MC_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: q: RelationQuery; C* = compose(S_1, …, S_k) with σ(C*) = 1 by MC_6

term_MC_8_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: max_i stepCount(S_i to convergence within L_i)

term_MC_8_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ ⌈n/k⌉

term_MC_8_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SharedContext C with totalSites = n; uniform partition into k leases

term_WC_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: a_k(x)

term_WC_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: x_k (k-th bit of x)

term_WC_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n, 0 ≤ k < n

term_WC_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: S_k − x_k − y_k (mod 2)

term_WC_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: c_k(x,y)

term_WC_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n, 0 ≤ k < n

term_WC_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: CA_2 recurrence

term_WC_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: S_{k+1} ghost equation at p=2

term_WC_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_WC_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: δ_k(x+y) correction

term_WC_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: c_{k+1}(x,y)

term_WC_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_WC_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: obstruction_trivial = false

term_WC_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: δ_k ≠ 0 for some pair

term_WC_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Q_k, k ≥ 1

term_WC_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_Δ(x,y) > 0

term_WC_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ghost defect nonzero

term_WC_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_WC_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: succ^{2ⁿ}(x) = x

term_WC_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: r^{2ⁿ} = 1 in Witt-Burnside ring

term_WC_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n, n ≥ 1

term_WC_8_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: neg(succ(neg(x)))

term_WC_8_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: srs = r⁻¹ relation

term_WC_8_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n, n ≥ 1

term_WC_9_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: bnot(neg(x)) = pred(x)

term_WC_9_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Product of Witt-Burnside reflections

term_WC_9_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n, n ≥ 1

term_WC_10_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: φ(x) on W_n(F_2)

term_WC_10_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: x (identity, F_2 perfect)

term_WC_10_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n, n ≥ 1

term_WC_11_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: V(x) on W_n(F_2)

term_WC_11_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: add(x, x) in Z/(2ⁿ)Z

term_WC_11_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n, n ≥ 1

term_WC_12_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: δ(x) on W_n(F_2)

term_WC_12_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: (x − mul(x,x)) / 2

term_WC_12_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x ∈ R_n, n ≥ 2

term_OA_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: |2|_2 · |2|_∞

term_OA_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 1 in Q×

term_OA_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: p = 2

term_OA_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: CrossingCost(p=2)

term_OA_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ln 2 = −ln|2|_2

term_OA_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: p = 2

term_OA_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: β* in Cost_Landauer

term_OA_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: CrossingCost(p=2)

term_OA_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: p = 2

term_OA_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: P(outcome k) = |α_k|_∞²

term_OA_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Archimedean image of 2-adic amplitude

term_OA_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: rational amplitudes

term_OA_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Information cost of δ (division by 2)

term_OA_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ln 2 nats

term_OA_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: p = 2

term_HT_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: N(C)

term_HT_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: KanComplex

term_HT_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint configuration C

term_HT_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: π₀(N(C))

term_HT_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Z^{β₀}

term_HT_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint configuration C

term_HT_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: π₁(N(C)) → D_{2^n}

term_HT_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: HolonomyGroup factorization

term_HT_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint configuration C

term_HT_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: π_k(N(C)) for k > dim(N(C))

term_HT_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 0

term_HT_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint configuration C, k > dim(N(C))

term_HT_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: τ_{≤1}(N(C))

term_HT_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: FlatType/TwistedType classification

term_HT_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint configuration C

term_HT_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: κ_k trivial for all k > d

term_HT_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: spectral sequence collapses at E_{d+2}

term_HT_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint configuration C, d = max simplex dim

term_HT_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: [α, β] ≠ 0 in π_{p+q−1}

term_HT_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: LiftObstruction non-trivial

term_HT_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: α ∈ π_p, β ∈ π_q

term_HT_8_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: π_k(N(C)) ⊗ Z

term_HT_8_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: H_k(N(C); Z) for smallest k with π_k ≠ 0

term_HT_8_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint configuration C

term_psi_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: KanComplex(N(C))

term_psi_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: PostnikovTower

term_psi_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint configuration C

term_psi_8_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: PostnikovTower(τ≤k)

term_psi_8_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: HomotopyGroups(π_k)

term_psi_8_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint configuration C

term_psi_9_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: HomotopyGroups(π_k)

term_psi_9_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: KInvariants(κ_k)

term_psi_9_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint configuration C

term_HP_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ψ_7 ∘ ψ_1

term_HP_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Kan promotion of nerve

term_HP_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint configuration C

term_HP_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ψ_8(τ≤k) restricted

term_HP_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ψ_3(C≤k)

term_HP_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint configuration C, truncation level k

term_HP_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ψ_9 detects convergence

term_HP_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: spectral sequence converges at E_{d+2}

term_HP_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint configuration C

term_HP_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: HomotopyResolver time

term_HP_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: O(n^{d+1})

term_HP_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: d = max simplex dimension

term_MD_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: dim(M_n)

term_MD_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: basisSize(T)

term_MD_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: T in M_n

term_MD_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: H^0(Def(T))

term_MD_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Aut(T) ∩ D_{2^n}

term_MD_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CompleteType T

term_MD_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: H^1(Def(T))

term_MD_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: T_T(M_n)

term_MD_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CompleteType T

term_MD_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: H^2(Def(T))

term_MD_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: LiftObstruction space

term_MD_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CompleteType T

term_MD_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: FlatType stratum codimension

term_MD_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 0

term_MD_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: M_n at any quantum level

term_MD_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: TwistedType stratum codimension

term_MD_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≥ 1

term_MD_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: M_n at any quantum level

term_MD_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: VersalDeformation existence

term_MD_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: guaranteed when H^2 = 0

term_MD_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CompleteType T

term_MD_8_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: familyPreservesCompleteness

term_MD_8_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: H^2(Def(T_t)) = 0 along path

term_MD_8_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: DeformationFamily {C_t}

term_MD_9_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: site(ModuliTowerMap, T) dimension

term_MD_9_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 1 when obstructionTrivial

term_MD_9_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CompleteType T, obstruction = 0

term_MD_10_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: site(ModuliTowerMap, T)

term_MD_10_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: empty iff TwistedType at every level

term_MD_10_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CompleteType T

term_MR_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ModuliResolver boundary

term_MR_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: MorphospaceBoundary

term_MR_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: M_n

term_MR_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: StratificationRecord coverage

term_MR_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: every CompleteType in exactly one stratum

term_MR_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: M_n

term_MR_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ModuliResolver complexity

term_MR_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: O(n × basisSize²)

term_MR_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CompleteType T

term_MR_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: achievabilityStatus = Achievable

term_MR_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: signature in some HolonomyStratum

term_MR_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: MorphospaceRecord

term_CY_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: generate(k)

term_CY_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: and(x_k, y_k) = 1

term_CY_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n, 0 ≤ k < n

term_CY_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: propagate(k)

term_CY_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: xor(x_k, y_k) = 1 ∧ c_k = 1

term_CY_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n, 0 ≤ k < n

term_CY_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: kill(k)

term_CY_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: and(x_k, y_k) = 0 ∧ xor(x_k, y_k) = 0

term_CY_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n, 0 ≤ k < n

term_CY_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_Δ(x, y)

term_CY_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: |carryCount(x+y) − hammingDistance(x, y)|

term_CY_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_CY_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: argmin_enc Σ d_Δ(enc(s_i), enc(s_j))

term_CY_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: enc where carry significance ≅ domain dependency

term_CY_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: finite symbol set S, observed pairs (s_i, s_j)

term_CY_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: min d_Δ site ordering

term_CY_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: high-significance sites → most informative observables

term_CY_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: EncodingConfiguration over ordered domain

term_CY_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: T(carry_chain(n))

term_CY_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: O(log n) via prefix computation on (g_k, p_k) pairs

term_CY_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CarryChain of length n

term_BM_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: σ(C)

term_BM_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: (n − freeRank(C)) / n

term_BM_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Context C with n sites

term_BM_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: χ(nerve(C))

term_BM_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σ(−1)^k β_k(nerve(C))

term_BM_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint set C

term_BM_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σκ_k − χ

term_BM_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: S_residual / ln 2

term_BM_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: computation state

term_BM_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: J_k (pinned site k)

term_BM_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 0

term_BM_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: pinned site k

term_BM_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_Δ(x, y) > 0

term_BM_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: carry(x + y) ≠ 0

term_BM_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: x, y ∈ R_n

term_BM_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: metric tower

term_BM_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_Δ → {σ, J_k} → β_k → χ → r

term_BM_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: computation state

term_GL_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: σ(T)

term_GL_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: lower_adjoint(T)

term_GL_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: type T in type lattice

term_GL_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: r(T)

term_GL_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 1 − upper_adjoint(T)

term_GL_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: type T in type lattice

term_GL_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: upper(lower(T))

term_GL_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: T (fixpoint: σ=1, r=0)

term_GL_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CompleteType T

term_GL_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: T₁ ≤ T₂ in type lattice

term_GL_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: site(T₂) ⊆ site(T₁)

term_GL_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: types T₁, T₂

term_NV_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: nerve(C₁ ∪ C₂)

term_NV_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: nerve(C₁) ∪ nerve(C₂)

term_NV_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: disjoint constraint domains C₁, C₂

term_NV_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: β_k(C₁ ∪ C₂)

term_NV_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: β_k(C₁) + β_k(C₂) − β_k(C₁ ∩ C₂)

term_NV_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint sets C₁, C₂

term_NV_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Δβ_k

term_NV_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∈ {−1, 0, +1}

term_NV_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: single constraint addition, dimension k

term_NV_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: β_k(C ∪ {c})

term_NV_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ β_k(C)

term_NV_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint set C, new constraint c

term_SD_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: quantize(value, range, bits)

term_SD_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ring element with d_R ∝ |v₁ − v₂|

term_SD_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: values v in ordered domain

term_SD_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: encoding(alphabet)

term_SD_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: argmin_{e} Σ d_Δ(e(a), e(b))

term_SD_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: symbol pairs (a,b) in alphabet

term_SD_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Seq(T)

term_SD_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Free(T) with backbone ordering

term_SD_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: element type T

term_SD_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: sites(Tuple(f₁,...,fₖ))

term_SD_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σᵢ sites(fᵢ)

term_SD_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: fields f_1,...,f_k of tuple type

term_SD_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: nerve(Graph(V,E))

term_SD_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: constraint_nerve(Graph(V,E))

term_SD_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: graph (V,E) with typed nodes

term_SD_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_Δ(Set(a,b,c))

term_SD_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_Δ(Set(σ(a,b,c)))

term_SD_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: permutation σ of set elements

term_SD_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: β_1(Tree(V,E))

term_SD_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 0

term_SD_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: tree (V,E) with beta_0=1

term_SD_8_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Table(S)

term_SD_8_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Seq(Tuple(S))

term_SD_8_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: schema S defining tuple fields

term_DD_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: δ(q, R)

term_DD_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: δ(q, R)

term_DD_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: query q, registry R

term_DD_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: dom(R)

term_DD_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: {q | ∃r. δ(q,R)=r}

term_DD_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: registry R

term_PI_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ι(ι(s,C),C)

term_PI_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ι(s,C)

term_PI_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: symbol s, GroundedContext C

term_PI_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: type(ι(s,C))

term_PI_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: consistent(C)

term_PI_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: symbol s, context C

term_PI_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ι(s,C)

term_PI_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: P(Π(G(s,C)))

term_PI_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: symbol s, context C

term_PI_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: complexity(ι(s,C))

term_PI_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: O(|C|)

term_PI_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: symbol s, context C

term_PI_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: roundTrip(P(Π(G(s))))

term_PI_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: s

term_PI_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: symbol s

term_PA_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: α(b₁,α(b₂,C))

term_PA_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: α(b₂,α(b₁,C))

term_PA_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: bindings b₁,b₂, context C at saturation

term_PA_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: sites(α(b,C))

term_PA_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: sites(C) ∪ {b.site}

term_PA_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: binding b, context C

term_PA_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: constraints(α(b,C))

term_PA_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: constraints(C) ∪ constraints(b)

term_PA_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: binding b, context C

term_PA_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: α(b,C)|_{pinned(C)}

term_PA_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: C|_{pinned(C)}

term_PA_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: binding b, context C

term_PA_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: α(∅, C)

term_PA_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: C

term_PA_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: context C

term_PL_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Lᵢ ∩ Lⱼ

term_PL_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∅

term_PL_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: leases Lᵢ, Lⱼ with i ≠ j

term_PL_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ⋃ᵢ Lᵢ

term_PL_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: S

term_PL_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: shared context S, leases Lᵢ

term_PL_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: |{i | f ∈ Lᵢ}|

term_PL_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 1

term_PL_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: site f in shared context S

term_PK_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: valid(κ(S₁,S₂))

term_PK_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: disjoint(S₁,S₂)

term_PK_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: sessions S₁,S₂

term_PK_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: resolve(s, κ(S₁,S₂))

term_PK_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: resolve(s, S₁) ∨ resolve(s, S₂)

term_PK_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: symbol s, sessions S₁,S₂

term_PP_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: κ(λₖ(α*(ι(s,·))), C)

term_PP_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: resolve(s, C)

term_PP_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: symbol s, context C

term_PE_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: state(ψ, 0)

term_PE_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 1

term_PE_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: reduction ψ

term_PE_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ψ_1(q)

term_PE_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: δ(q, R)

term_PE_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: query q, registry R

term_PE_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ψ_2(r)

term_PE_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: G(r)

term_PE_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: resolver r

term_PE_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ψ_3(a)

term_PE_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Π(a)

term_PE_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: address a

term_PE_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ψ_4(c)

term_PE_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: α*(c)

term_PE_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constraint set c

term_PE_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ψ_5(b)

term_PE_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: P(b)

term_PE_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: binding b

term_PE_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ψ_5 ∘ ψ_4 ∘ ψ_3 ∘ ψ_2 ∘ ψ_1 ∘ ψ_0

term_PE_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ψ

term_PE_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: reduction ψ

term_PM_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: phase(stage_k)

term_PM_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Ω^k

term_PM_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: stage index k in 0..5

term_PM_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: gate(k)

term_PM_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: phase(k) == Ω^k

term_PM_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: stage boundary k

term_PM_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: fail(gate(k))

term_PM_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: rollback(z → z̄)

term_PM_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: stage k with gate failure

term_PM_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: conj(conj(z))

term_PM_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: z

term_PM_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: complex value z

term_PM_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: sat(epoch_n)

term_PM_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: sat(epoch_{n+1})

term_PM_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: consecutive epochs n, n+1

term_PM_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: context(window)

term_PM_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: base_context

term_PM_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: service window

term_PM_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ψ(s₀)

term_PM_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ψ(s₀)

term_PM_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: initial state s₀

term_ER_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: advance(k, k+1)

term_ER_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: guard(k) = true

term_ER_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: stage transition k to k+1

term_ER_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: apply(effect(k))

term_ER_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: atomic(effect(k))

term_ER_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: transition effect at stage k

term_ER_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: eval(guard(k), s)

term_ER_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: s (state unchanged)

term_ER_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: guard evaluation at stage k with state s

term_ER_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: apply(e1; e2)

term_ER_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: apply(e2; e1)

term_ER_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: effects e1, e2 within same stage

term_EA_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: free(epoch(n+1))

term_EA_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: free(epoch(0))

term_EA_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: epoch boundary n to n+1

term_EA_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: grounded(epoch(n))

term_EA_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: grounded(epoch(n+1))

term_EA_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: grounded sites across epoch boundary

term_EA_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: |context(w)|

term_EA_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: windowSize(w)

term_EA_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: service window w

term_EA_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: admit(epoch(n))

term_EA_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: datum | refinement

term_EA_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: epoch admission at epoch n

term_OE_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: stage(k); stage(k+1)

term_OE_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: fused(k, k+1)

term_OE_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: adjacent stages k, k+1 with compatible guards

term_OE_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: effect(a); effect(b)

term_OE_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: effect(b); effect(a)

term_OE_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: independent effects a, b

term_OE_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: lease(A); lease(B)

term_OE_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: lease(A) || lease(B)

term_OE_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: disjoint lease sets A, B

term_OE_4a_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: sem(fused(k, k+1))

term_OE_4a_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: sem(stage(k)); sem(stage(k+1))

term_OE_4a_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: fused stages k, k+1

term_OE_4b_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: outcome(a; b)

term_OE_4b_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: outcome(b; a)

term_OE_4b_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: commuting effects a, b

term_OE_4c_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: coverage(A || B)

term_OE_4c_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: coverage(A) + coverage(B)

term_OE_4c_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: parallel leases A, B

term_CS_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: cost(stage(k))

term_CS_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: O(1)

term_CS_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: reduction step k

term_CS_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: cost(pipeline)

term_CS_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: sum(cost(stage(k)))

term_CS_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: reduction pipeline

term_CS_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: cost(rollback)

term_CS_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: cost(forward)

term_CS_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: reduction rollback operation

term_CS_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: cost(preflight)

term_CS_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: O(1)

term_CS_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: preflight check

term_CS_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: cost(reduction)

term_CS_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: n * max(cost(stage(k)))

term_CS_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: reduction with n stages

term_CS_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: thermodynamicBudget(U) < bitsWidth(unitWittLevel(U)) × ln 2

term_CS_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: reject(U) at BudgetSolvencyCheck

term_CS_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CompileUnit U

term_CS_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: unitAddress(U)

term_CS_7_rhs
type: NamedIndividual
type: LiteralExpression

term_CS_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CompileUnit U

term_FA_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: pending(q)

term_FA_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: reaches_gate(q)

term_FA_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: query q in reduction

term_FA_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: admitted(q, epoch)

term_FA_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: served(q, epoch + k)

term_FA_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: query q, bounded k

term_FA_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: lease_alloc(p1 + p2)

term_FA_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: lease_alloc(p1) + lease_alloc(p2)

term_FA_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: disjoint partitions p1, p2

term_SW_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: memory(window(w))

term_SW_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: O(w)

term_SW_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: service window of size w

term_SW_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: saturated(slide(w))

term_SW_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: saturated(w)

term_SW_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: window slide operation

term_SW_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: resources(evict(e))

term_SW_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 0

term_SW_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: evicted epoch e

term_SW_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: size(window)

term_SW_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: >= 1

term_SW_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: service window

term_LS_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: pinned(suspend(lease))

term_LS_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: pinned(lease)

term_LS_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: lease suspension

term_LS_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: resources(expire(lease))

term_LS_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 0

term_LS_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: expired lease

term_LS_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: restore(restore(checkpoint))

term_LS_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: restore(checkpoint)

term_LS_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: checkpoint restore

term_LS_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: resume(suspend(lease))

term_LS_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: lease

term_LS_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: active lease

term_TJ_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: all_or_nothing(fail)

term_TJ_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: rollback

term_TJ_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: AllOrNothing transaction with failure

term_TJ_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: best_effort(partial_fail)

term_TJ_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: commit(succeeded)

term_TJ_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: BestEffort transaction

term_TJ_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: scope(transaction)

term_TJ_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: single_epoch

term_TJ_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: reduction transaction

term_AP_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: sat(stage(k+1))

term_AP_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: >= sat(stage(k))

term_AP_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: consecutive stages k, k+1

term_AP_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: quality(epoch(n+1))

term_AP_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: >= quality(epoch(n))

term_AP_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: consecutive epochs n, n+1

term_AP_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: deferred(q)

term_AP_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: processed(q) | dropped(q)

term_AP_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: deferred query q

term_EC_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Ω⁶

term_EC_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: −1

term_EC_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: reduction phase angle Ω = e^{iπ/6}

term_EC_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: conj(conj(z))

term_EC_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: z

term_EC_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: complex z in reduction

term_EC_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: [q_A, q_B]^inf

term_EC_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 1

term_EC_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: quaternion pair q_A, q_B

term_EC_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: [q_A, q_B, q_C]^inf

term_EC_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 0

term_EC_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: octonion triple q_A, q_B, q_C

term_EC_4a_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ||[a,b,c]_{n+1}||

term_EC_4a_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: <= ||[a,b,c]_n||

term_EC_4a_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: successive associator iterates

term_EC_4b_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: steps_to_zero([a,b,c])

term_EC_4b_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: <= |three_way_sites|

term_EC_4b_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: octonion triple a, b, c

term_EC_4c_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: [a,b,c] = 0

term_EC_4c_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: associative(resolved_site_space)

term_EC_4c_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: resolved site space

term_EC_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: max_level(convergence_tower)

term_EC_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: L3_Self

term_EC_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: convergence tower

term_DA_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: CD(R, i)

term_DA_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: C

term_DA_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Cayley-Dickson doubling

term_DA_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: CD(C, j)

term_DA_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: H

term_DA_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Cayley-Dickson doubling

term_DA_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: CD(H, l)

term_DA_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: O

term_DA_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Cayley-Dickson doubling

term_DA_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: dim(normed_div_alg)

term_DA_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∈ {1, 2, 4, 8}

term_DA_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: normed division algebras over R

term_DA_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: algebra(L_k)

term_DA_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: division_algebra[k]

term_DA_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: convergence level L_k

term_DA_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: [a,b] = 0

term_DA_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: isCommutative(algebra(L_k))

term_DA_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: elements a, b in division algebra at level k

term_DA_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: [a,b,c] = 0

term_DA_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: isAssociative(algebra(L_k))

term_DA_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: elements a, b, c in division algebra at level k

term_IN_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_Δ(A,B)

term_IN_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: interaction_cost(A,B)

term_IN_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: entity pairs A, B

term_IN_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: disjoint_leases(A,B)

term_IN_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: commutator(A,B) = 0

term_IN_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: entity pairs with disjoint leases

term_IN_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: shared_sites(A,B) ≠ ∅

term_IN_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: commutator(A,B) > 0

term_IN_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: entity pairs with shared sites

term_IN_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: SR_8_session(A,B)

term_IN_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: commutator(A,B,t+1) ≤ commutator(A,B,t)

term_IN_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: entity pairs in session

term_IN_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: converged_negotiation(A,B)

term_IN_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: U(1) ⊂ SU(2)

term_IN_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: converged pairwise interactions

term_IN_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: outcome_space(pairwise_negotiation)

term_IN_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: S²

term_IN_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: pairwise negotiations

term_IN_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: converged_mutual_model(A,B,C)

term_IN_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: H ⊂ O

term_IN_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: converged triple interactions

term_IN_8_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: β_k(interaction_nerve)

term_IN_8_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: coupling_complexity(k)

term_IN_8_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: interaction nerve at dimension k

term_IN_9_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: β_2(nerve) × max_disagreement

term_IN_9_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: upper_bound(associator_norm)

term_IN_9_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: interaction nerves with β_2 > 0

term_AS_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: (δ ∘ ι) ∘ κ

term_AS_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: δ ∘ (ι ∘ κ)

term_AS_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: triple δ, ι, κ with shared registry

term_AS_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: (ι ∘ α) ∘ λ

term_AS_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ι ∘ (α ∘ λ)

term_AS_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: triple ι, α, λ with shared lease

term_AS_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: (λ ∘ κ) ∘ δ

term_AS_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: λ ∘ (κ ∘ δ)

term_AS_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: triple λ, κ, δ with shared state

term_AS_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: associator(A,B,C) ≠ 0

term_AS_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∃ mediating read-write interleaving

term_AS_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: triples with non-zero associator

term_MO_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: I ⊗ A

term_MO_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: A

term_MO_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: computations A

term_MO_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: (A⊗B)⊗C

term_MO_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: A⊗(B⊗C)

term_MO_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: computations A, B, C

term_MO_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: cert(A⊗B)

term_MO_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: cert(A) ∧ cert(B)

term_MO_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: certified computations A, B

term_MO_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: σ(A⊗B)

term_MO_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: max(σ(A), σ(B))

term_MO_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: computations A, B

term_MO_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: r(A⊗B)

term_MO_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: min(r(A), r(B))

term_MO_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: computations A, B

term_OP_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: siteCount(F(G))

term_OP_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: F.sites + Σ_i G_i.sites

term_OP_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: structural types F, G

term_OP_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: grounding(F(G(x)))

term_OP_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: F.ground(G.ground(x))

term_OP_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: structural types F, G, element x

term_OP_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_Δ(F(G))

term_OP_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: d_Δ(F) ∘ G + F ∘ d_Δ(G)

term_OP_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: structural types F, G

term_OP_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Table(Tuple(fields))

term_OP_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Sequence(Tuple(fields))

term_OP_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: tabular data

term_OP_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Tree(Symbol(leaves))

term_OP_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Graph(Symbol(leaves), acyclic)

term_OP_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: hierarchical data

term_FX_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRank(postContext(e))

term_FX_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRank(preContext(e)) − 1

term_FX_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: PinningEffect e

term_FX_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRank(postContext(e))

term_FX_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRank(preContext(e)) + 1

term_FX_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: UnbindingEffect e

term_FX_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRank(postContext(e))

term_FX_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRank(preContext(e))

term_FX_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: PhaseEffect e

term_FX_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: apply(A ; B, ctx)

term_FX_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: apply(B ; A, ctx)

term_FX_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Effects A, B with DisjointnessWitness(target(A), target(B))

term_FX_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRankDelta(E₁ ; E₂)

term_FX_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRankDelta(E₁) + freeRankDelta(E₂)

term_FX_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: CompositeEffect (E₁ ; E₂)

term_FX_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: apply(e, apply(e⁻¹, ctx))

term_FX_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ctx

term_FX_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ReversibleEffect e

term_FX_7_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRankDelta(e)

term_FX_7_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: declared freeRankDelta in EffectShape

term_FX_7_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ExternalEffect e satisfying conformance:EffectShape

term_PR_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: eval(p, x)

term_PR_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∈ {true, false}

term_PR_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Predicate p, input x

term_PR_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: state(eval(p, x, s))

term_PR_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: s

term_PR_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Predicate p, input x, state s

term_PR_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: dispatch(D, x)

term_PR_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: exactly one DispatchRule

term_PR_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: DispatchTable D with isExhaustive=true, isMutuallyExclusive=true

term_PR_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: eval(M)

term_PR_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: armResult(first matching arm)

term_PR_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: MatchExpression M with exhaustive arms

term_PR_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: advance(k, guardTarget(g))

term_PR_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: requires guardPredicate(g) = true

term_PR_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: GuardedTransition g at reduction step k

term_CG_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: advance_to(s)

term_CG_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: requires eval(g, currentState) = true

term_CG_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ReductionStep s with entryGuard g

term_CG_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: advance_from(s)

term_CG_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: requires eval(g, currentState) = true, then apply(e)

term_CG_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ReductionStep s with exitGuard g and stageEffect e

term_DIS_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: isExhaustive(D) ∧ isMutuallyExclusive(D)

term_DIS_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: true

term_DIS_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Root DispatchTable D

term_DIS_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: dispatch(D, T)

term_DIS_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: exactly one Resolver

term_DIS_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: TypeDefinition T, DispatchTable D

term_PAR_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: apply(A ⊗ B, ctx)

term_PAR_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: apply(B ⊗ A, ctx)

term_PAR_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ParallelProduct A ∥ B with DisjointnessCertificate

term_PAR_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRankDelta(A ∥ B)

term_PAR_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: freeRankDelta(A) + freeRankDelta(B)

term_PAR_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ParallelProduct A ∥ B

term_PAR_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σ |component_i|

term_PAR_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: n

term_PAR_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SitePartitioning P over n sites

term_PAR_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: finalContext(σ(A, B))

term_PAR_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: finalContext(A ⊗ B)

term_PAR_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ParallelProduct A ∥ B, any interleaving σ

term_PAR_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: cert(A ∥ B)

term_PAR_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: cert(A) ∧ cert(B) ∧ DisjointnessCertificate(A, B)

term_PAR_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: cert(A ∥ B)

term_HO_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: value(c)

term_HO_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: contentHash(referencedCertificate(c))

term_HO_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ComputationDatum c

term_HO_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: cert(output(app))

term_HO_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: cert(applicationTarget(app))

term_HO_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ApplicationMorphism app

term_HO_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: cert(f ∘ g)

term_HO_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: cert(f) ∧ cert(g) ∧ range(g) = domain(f)

term_HO_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: TransformComposition f ∘ g

term_HO_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: p

term_HO_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ApplicationMorphism(partialBase(p), boundArguments(p))

term_HO_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: PartialApplication p with remainingArity = 0

term_STR_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: reduction(e_k) converges to π

term_STR_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: true

term_STR_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Epoch e_k in ProductiveStream

term_STR_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: saturation(continuationContext(b))

term_STR_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: saturation(postContext(e_k))

term_STR_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: EpochBoundary b between e_k and e_{k+1}

term_STR_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: computationTime(P)

term_STR_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σ_{i=0}^{k−1} computationTime(epoch_i)

term_STR_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: StreamPrefix P of length k

term_STR_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: epoch_0.context

term_STR_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: seed

term_STR_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Unfold(seed, step)

term_STR_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: epoch_{k+1}.context

term_STR_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: continuationContext(boundary(e_k))

term_STR_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Unfold(seed, step), epoch e_k

term_STR_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: linearBudget(epoch_{k+1})

term_STR_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: linearBudget(epoch_k) + leaseCardinality(L)

term_STR_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: EpochBoundary b with LeaseAllocation L expiring at b

term_FLR_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: result(P)

term_FLR_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∈ {Success, Failure}

term_FLR_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: PartialComputation P

term_FLR_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: result(T)

term_FLR_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Success

term_FLR_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: TotalComputation T

term_FLR_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: result(A ⊗ B)

term_FLR_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Failure(A)

term_FLR_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: A ⊗ B where result(A) = Failure

term_FLR_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: result(A ∥ B)

term_FLR_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Failure(A) (left component)

term_FLR_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: A ∥ B where result(A) = Failure, result(B) = Success

term_FLR_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: result(apply(recoveryEffect(r), failureState(f)))

term_FLR_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ComputationResult

term_FLR_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Recovery r on Failure f

term_FLR_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: recoveryEffect(rollback(f))

term_FLR_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: PhaseEffect(conjugate)

term_FLR_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: ComplexConjugateRollback on Failure f

term_LN_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σ targetCount(site_i)

term_LN_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: n

term_LN_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: LinearTrace T over n-bit type

term_LN_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: status(f, postContext(e))

term_LN_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: pinned

term_LN_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: LinearEffect e on site f

term_LN_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: target(e′) = f

term_LN_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: forbidden

term_LN_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: LinearEffect e on site f, any subsequent effect e′

term_LN_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: remainingCount(budget after L)

term_LN_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: remainingCount(budget before L) − k

term_LN_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: LeaseAllocation L with leaseCardinality k

term_LN_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: remainingCount(budget after expiry)

term_LN_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: remainingCount(budget before expiry) + leaseCardinality(L)

term_LN_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Lease expiry on LeaseAllocation L

term_LN_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: G

term_LN_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: LinearTrace

term_LN_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: GeodesicTrace G

term_SB_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: constraints(T₁)

term_SB_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ⊇ constraints(T₂)

term_SB_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: TypeInclusion T₁ ≤ T₂

term_SB_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: resolutions(T₁)

term_SB_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ⊆ resolutions(T₂)

term_SB_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: TypeInclusion T₁ ≤ T₂, resolution R

term_SB_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: N(C(T₂))

term_SB_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: sub-complex of N(C(T₁))

term_SB_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: TypeInclusion T₁ ≤ T₂

term_SB_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: F(T₁)

term_SB_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ F(T₂)

term_SB_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Covariant position F(_), T₁ ≤ T₂

term_SB_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: F(T₂)

term_SB_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ F(T₁)

term_SB_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Contravariant position F(_), T₁ ≤ T₂

term_SB_6_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: latticeDepth

term_SB_6_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: n

term_SB_6_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: SubtypingLattice at quantum level n

term_BR_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: measureValue(stepMeasurePost(s))

term_BR_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: < measureValue(stepMeasurePre(s))

term_BR_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: RecursiveStep s

term_BR_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: depth(RecursionTrace(R))

term_BR_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ m

term_BR_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: BoundedRecursion R with initialMeasure m

term_BR_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: terminates(R)

term_BR_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: true

term_BR_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: BoundedRecursion R

term_BR_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: initialMeasure(R)

term_BR_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: structuralSize(T)

term_BR_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: StructuralRecursion R on type T

term_BR_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: eval(p, state) = true

term_BR_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: measureValue = 0

term_BR_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: BoundedRecursion R with basePredicate p

term_RG_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: workingSetRegions(W)

term_RG_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: computable from N(C(T)) and stage k site targets

term_RG_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: WorkingSet W for type T at stage k

term_RG_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∀ a, b ∈ R: d_R(a, b)

term_RG_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ diameter(R)

term_RG_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: AddressRegion R with LocalityMetric d_R

term_RG_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σ workingSetSize(stage_k)

term_RG_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ≤ totalAddressableSpace(quantumLevel)

term_RG_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: RegionAllocation A for computation C

term_RG_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: addresses accessed by resolver at stage k

term_RG_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: ⊆ addresses(W_k)

term_RG_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Reduction step k with WorkingSet W_k

term_IO_1_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: type(resultDatum(e))

term_IO_1_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: conformsTo(sourceType(s))

term_IO_1_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: IngestEffect e from Source s

term_IO_2_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: type(emittedDatum(e))

term_IO_2_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: conformsTo(sinkType(s))

term_IO_2_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: EmitEffect e to Sink s

term_IO_3_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: apply(g, ingest(s))

term_IO_3_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Datum in R_n

term_IO_3_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Source s with GroundingMap g

term_IO_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: apply(p, d)

term_IO_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: surface symbol conforming to sinkType(s)

term_IO_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: Sink s with ProjectionMap p, Datum d

term_IO_5_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: effect:effectTarget(e)

term_IO_5_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: non-empty EffectTarget

term_IO_5_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: BoundaryEffect e

term_boundarySquaredZero_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: ∂_k(∂_{k+1}(c))

term_boundarySquaredZero_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 0

term_boundarySquaredZero_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: c ∈ C_{k+1}

term_psi_4_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: β_k(K)

term_psi_4_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: rank(H_k(K))

term_psi_4_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: simplicial complex K

term_indexBridge_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: χ(K)

term_indexBridge_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Σ_k (-1)^k β_k

term_indexBridge_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: finite simplicial complex K

term_coboundarySquaredZero_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: δ^{k+1}(δ^k(f))

term_coboundarySquaredZero_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: 0

term_coboundarySquaredZero_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: f ∈ C^k

term_deRhamDuality_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: H^k(K; R)

term_deRhamDuality_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: Hom(H_k(K), R)

term_deRhamDuality_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: simplicial complex K, ring R

term_sheafCohomologyBridge_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: H^k(K; F_R)

term_sheafCohomologyBridge_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: H^k(K; R)

term_sheafCohomologyBridge_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: constant sheaf F_R over K

term_localGlobalPrinciple_lhs
type: NamedIndividual
type: LiteralExpression
literalValue: H^1(K; F) = 0

term_localGlobalPrinciple_rhs
type: NamedIndividual
type: LiteralExpression
literalValue: all local sections glue

term_localGlobalPrinciple_forAll
type: NamedIndividual
type: ForAllDeclaration
variableName: sheaf F over K

UOR Operations
type: Ontology
imports: UOR Schema
comment: Ring operations, involutions, algebraic identities, and the dihedral symmetry group D_{2^n} generated by neg and bnot.
space: kernel

Operation
type: Class
comment: An operation on the ring Z/(2^n)Z. The root class for all UOR kernel operations.
subclass of: Thing

UnaryOp
type: Class
comment: A unary operation on the ring: takes one datum and produces one datum.
subclass of: Operation

BinaryOp
type: Class
comment: A binary operation on the ring: takes two datums and produces one datum.
subclass of: Operation

Involution
type: Class
comment: A unary operation f such that f(f(x)) = x for all x in R_n. The two UOR involutions are neg (ring reflection) and bnot (hypercube reflection).
subclass of: UnaryOp

Identity
type: Class
comment: An algebraic identity: a statement that two expressions are equal for all inputs. The critical identity is neg(bnot(x)) = succ(x) for all x in R_n.
subclass of: Thing

Group
type: Class
comment: A group: a set with an associative binary operation, an identity element, and inverses for every element.
subclass of: Thing

DihedralGroup
type: Class
comment: The dihedral group D_{2^n} of order 2^(n+1), generated by the ring reflection (neg) and the hypercube reflection (bnot). This group governs the symmetry of the UOR type space.
subclass of: Group

VerificationDomain
type: Class
comment: A named mathematical discipline through which an algebraic identity is established and grounded. Every op:Identity individual references at least one VerificationDomain via op:verificationDomain. The nine canonical domain individuals are kernel-level constants defined in op/.
subclass of: Thing

GeometricCharacter
type: Class
comment: The geometric role of a ring operation in the UOR dual-geometry (ring + hypercube). Every op:Operation individual references exactly one GeometricCharacter via op:hasGeometricCharacter. The nine canonical individuals correspond to the action types of the dihedral group D_{2^n}.
subclass of: Thing

WittLevelBinding
type: Class
comment: A record linking an op:Identity individual to a specific quantum level at which it has been verified. Non-functional: one WittLevelBinding per (Identity, WittLevel) pair verified.
subclass of: Thing

QuantumThermodynamicDomain
type: Class
comment: A verification domain at the intersection of quantum superposition and classical thermodynamics. Identities in this domain require both SuperpositionDomain and Thermodynamic reasoning simultaneously.
subclass of: Thing

ValidityScopeKind
type: Class
comment: Root class for validity scope individuals. Instances are the four named scope kinds: Universal, ParametricLower, ParametricRange, and LevelSpecific.
subclass of: Thing

ComposedOperation
type: Class
comment: An operation formed by composing ring operations, witnessed by op:composedOf and morphism/CompositionLaw.
subclass of: Operation
subclass of: Composition

DispatchOperation
type: Class
comment: δ: Query × ResolverRegistry → Resolver. Non-commutative, non-associative, arity 2.
subclass of: ComposedOperation

InferenceOperation
type: Class
comment: ι = P ∘ Π ∘ G (the φ-pipeline composed). Non-commutative, non-associative, arity 2.
subclass of: ComposedOperation

AccumulationOperation
type: Class
comment: α: Binding × Context → Context. Non-commutative, associative at convergence (SR_10), arity 2.
subclass of: ComposedOperation

LeasePartitionOperation
type: Class
comment: λ: SharedContext × ℕ → ContextLease^k. Non-commutative, non-associative, arity 2.
subclass of: ComposedOperation

SessionCompositionOperation
type: Class
comment: κ: Session × Session → Session. Commutative (disjoint leases), associative (SR_8), arity 2.
subclass of: ComposedOperation

GroupPresentation
type: Class
comment: A structured group presentation: generators and relations as typed data rather than prose strings.
subclass of: Thing

arity
type: DatatypeProperty
type: FunctionalProperty
comment: The number of arguments this operation takes. 1 for unary operations, 2 for binary operations.
domain: Operation
range: nonNegativeInteger

hasGeometricCharacter
type: ObjectProperty
type: FunctionalProperty
comment: The geometric role of this operation in the UOR ring and hypercube geometry. Functional: each operation has exactly one geometric character.
domain: Operation
range: GeometricCharacter

commutative
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this binary operation satisfies op(x,y) = op(y,x) for all x, y in R_n.
domain: BinaryOp
range: boolean

associative
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this binary operation satisfies op(op(x,y),z) = op(x,op(y,z)) for all x, y, z in R_n.
domain: BinaryOp
range: boolean

identity
type: DatatypeProperty
type: FunctionalProperty
comment: The identity element of this binary operation: the value e such that op(x, e) = op(e, x) = x for all x in R_n.
domain: BinaryOp
range: integer

inverse
type: ObjectProperty
type: FunctionalProperty
comment: The inverse operation: the operation inv_op such that op(x, inv_op(x)) = e for all x, where e is the identity.
domain: Operation
range: Operation

composedOf
type: ObjectProperty
type: FunctionalProperty
comment: Ordered list of operations this operation is composed from. Uses rdf:List to preserve application order (first element applied innermost). E.g., succ = neg ∘ bnot is encoded as [op:neg, op:bnot] meaning neg applied to the result of bnot.
domain: Operation
range: List

lhs
type: ObjectProperty
type: FunctionalProperty
comment: The left-hand side of an algebraic identity as a typed AST node (schema:TermExpression).
domain: Identity
range: TermExpression

rhs
type: ObjectProperty
type: FunctionalProperty
comment: The right-hand side of an algebraic identity as a typed AST node (schema:TermExpression).
domain: Identity
range: TermExpression

forAll
type: ObjectProperty
type: FunctionalProperty
comment: The quantifier scope: a typed declaration of the variable(s) over which this identity holds.
domain: Identity
range: ForAllDeclaration

generatedBy
type: ObjectProperty
comment: An operation that generates this group. The dihedral group D_{2^n} is generated by op:neg and op:bnot.
domain: Group
range: Operation

order
type: DatatypeProperty
type: FunctionalProperty
comment: The number of elements in the group. For D_{2^n}, the order is 2^(n+1).
domain: Group
range: positiveInteger

presentation
type: AnnotationProperty
comment: The group presentation (generators and relations). Example: ⟨r, s | r^{2^n} = s² = e, srs = r⁻¹⟩
domain: Group
range: string

verificationDomain
type: ObjectProperty
comment: The mathematical discipline(s) through which this identity is established. Range is op:VerificationDomain. Non-functional: composite identities (e.g. IT_7a–IT_7d) reference multiple domain individuals.
domain: Identity
range: VerificationDomain

verifiedAtLevel
type: ObjectProperty
comment: Links an Identity individual to a WittLevelBinding attesting verification at a specific quantum level. Non-functional: one binding per (Identity, WittLevel) pair.
domain: Identity
range: WittLevelBinding

bindingLevel
type: ObjectProperty
type: FunctionalProperty
comment: The quantum level at which this WittLevelBinding was verified.
domain: WittLevelBinding
range: WittLevel

universallyValid
type: DatatypeProperty
type: FunctionalProperty
comment: True iff this identity holds for all n ≥ 1 (proved symbolically by induction on the ring axioms, not just exhaustively at Q0). Identities that reference 8-bit-specific constants receive universallyValid = false.
domain: Identity
range: boolean

enumVariant
type: ObjectProperty
type: FunctionalProperty
comment: The canonical Rust enum variant name corresponding to this VerificationDomain individual. Provides formal alignment between the OWL individual and the generated Rust enum.
domain: VerificationDomain
range: VerificationDomain

validityKind
type: ObjectProperty
type: FunctionalProperty
comment: The structured validity scope of this identity, replacing the binary universallyValid flag. Required on all new Identity individuals.
domain: Identity
range: ValidityScopeKind

validKMin
type: DatatypeProperty
type: FunctionalProperty
comment: Minimum quantum level index k for ParametricLower and ParametricRange scopes.
domain: Identity
range: nonNegativeInteger

validKMax
type: DatatypeProperty
type: FunctionalProperty
comment: Maximum quantum level index k (inclusive) for ParametricRange scope.
domain: Identity
range: nonNegativeInteger

composedOfOps
type: ObjectProperty
comment: References a constituent operation of a ComposedOperation. Non-functional: a composed operation may reference multiple constituent operations.
domain: ComposedOperation
range: Operation

operatorSignature
type: AnnotationProperty
comment: Human-readable type signature describing the domain and codomain of a composed operation (e.g., 'Query × Registry → Resolver').
domain: ComposedOperation
range: string

dispatchSource
type: ObjectProperty
type: FunctionalProperty
comment: The source selector for a dispatch operation.
domain: DispatchOperation
range: Operation

dispatchTarget
type: ObjectProperty
type: FunctionalProperty
comment: The target resolver for a dispatch operation.
domain: DispatchOperation
range: Operation

inferenceSource
type: ObjectProperty
type: FunctionalProperty
comment: The source data for an inference operation.
domain: InferenceOperation
range: Operation

inferenceTarget
type: ObjectProperty
type: FunctionalProperty
comment: The target type for an inference operation.
domain: InferenceOperation
range: Operation

inferencePipeline
type: ObjectProperty
type: FunctionalProperty
comment: The pipeline through which inference is performed.
domain: InferenceOperation
range: Operation

accumulationBase
type: ObjectProperty
type: FunctionalProperty
comment: The base value for an accumulation operation.
domain: AccumulationOperation
range: TermExpression

accumulationBinding
type: ObjectProperty
type: FunctionalProperty
comment: The binding accumulator for an accumulation operation.
domain: AccumulationOperation
range: TermExpression

leaseSource
type: ObjectProperty
type: FunctionalProperty
comment: The source context for a lease partition operation.
domain: LeasePartitionOperation
range: Operation

leaseFactor
type: ObjectProperty
type: FunctionalProperty
comment: The partition factor for a lease partition operation.
domain: LeasePartitionOperation
range: Operation

leasePartitionCount
type: DatatypeProperty
type: FunctionalProperty
comment: The number of partitions in a lease partition operation.
domain: LeasePartitionOperation
range: nonNegativeInteger

compositionLeftSession
type: ObjectProperty
type: FunctionalProperty
comment: The left session in a session composition operation.
domain: SessionCompositionOperation
range: Operation

compositionRightSession
type: ObjectProperty
type: FunctionalProperty
comment: The right session in a session composition operation.
domain: SessionCompositionOperation
range: Operation

operatorDomainType
type: ObjectProperty
type: FunctionalProperty
comment: The domain type of a composed operation.
domain: ComposedOperation
range: TypeDefinition

operatorRangeType
type: ObjectProperty
type: FunctionalProperty
comment: The range type of a composed operation.
domain: ComposedOperation
range: TypeDefinition

operatorComplexity
type: DatatypeProperty
type: FunctionalProperty
comment: The computational complexity class of a composed operation.
domain: ComposedOperation
range: string

operatorIdempotent
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this composed operation is idempotent.
domain: ComposedOperation
range: boolean

composedOperatorCount
type: DatatypeProperty
type: FunctionalProperty
comment: The number of constituent operations in a composed operation.
domain: ComposedOperation
range: nonNegativeInteger

isInvolutory
type: DatatypeProperty
type: FunctionalProperty
comment: Whether applying this operation twice yields the identity.
domain: ComposedOperation
range: boolean

convergenceGuarantee
type: ObjectProperty
type: FunctionalProperty
comment: Description of the convergence guarantee for this operation.
domain: ComposedOperation
range: TermExpression

Enumerative
type: NamedIndividual
type: VerificationDomain
comment: Established by exhaustive traversal of R_n. Valid for all identities where the ring is finite.

Algebraic
type: NamedIndividual
type: VerificationDomain
comment: Established by equational reasoning from ring or group axioms. Covers derivations via associativity, commutativity, inverse laws, and group presentations.

Geometric
type: NamedIndividual
type: VerificationDomain
comment: Established by isometry, symmetry, or GeometricCharacter arguments. Covers dihedral actions, fixed-point analysis, automorphism groups, and affine embeddings.

Analytical
type: NamedIndividual
type: VerificationDomain
comment: Established via discrete differential calculus or metric analysis. Covers ring/Hamming derivatives (DC_), metric divergence (AM_), and adiabatic scheduling (AR_).

Thermodynamic
type: NamedIndividual
type: VerificationDomain
comment: Established via entropy, Landauer bounds, or Boltzmann distributions. Covers site entropy (TH_), reversible computation (RC_), and phase transitions.

Topological
type: NamedIndividual
type: VerificationDomain
comment: Established via simplicial homology, cohomology, or constraint nerve analysis. Covers homological algebra (HA_) and ψ-pipeline identities.

Pipeline
type: NamedIndividual
type: VerificationDomain
comment: Established by the inter-algebra map structure of the resolution pipeline. Covers φ-maps (phi_1–phi_6) and ψ-maps (psi_1–psi_6).

IndexTheoretic
type: NamedIndividual
type: VerificationDomain
comment: Established by the composition of Analytical and Topological reasoning. The only domain requiring multiple op:verificationDomain assertions. Covers the UOR Index Theorem (IT_7a–IT_7d).

SuperpositionDomain
type: NamedIndividual
type: VerificationDomain
comment: Established by superposition analysis of site states. Covers identities involving superposed (non-classical) site assignments where sites carry complex amplitudes.

QuantumThermodynamic
type: NamedIndividual
type: VerificationDomain
comment: Established by the intersection of quantum superposition analysis and classical thermodynamic reasoning. Covers identities relating von Neumann entropy of superposed states to Landauer costs of projective collapse (QM_).

ArithmeticValuation
type: NamedIndividual
type: VerificationDomain
comment: Established by number-theoretic valuation arguments including p-adic absolute values, the Ostrowski product formula, and the arithmetic of global fields. Covers identities grounded in the product formula |x|_p · |x|_∞ = 1 and the Witt–Ostrowski derivation chain.

ComposedAlgebraic
type: NamedIndividual
type: VerificationDomain
comment: Verification domain for composed operation identities — algebraic properties of operator compositions including dispatch, inference, accumulation, lease, and session composition operations.

Universal
type: NamedIndividual
type: ValidityScopeKind
comment: Holds for all k in N. No minimum k constraint.

ParametricLower
type: NamedIndividual
type: ValidityScopeKind
comment: Holds for all k >= k_min, where k_min is given by validKMin.

ParametricRange
type: NamedIndividual
type: ValidityScopeKind
comment: Holds for k_min <= k <= k_max. Both validKMin and validKMax required.

LevelSpecific
type: NamedIndividual
type: ValidityScopeKind
comment: Holds only at exactly one level, given by a WittLevelBinding.

RingReflection
type: NamedIndividual
type: GeometricCharacter
comment: Reflection through the origin of the additive ring: neg(x) = -x mod 2^n. One of the two generators of D_{2^n}.

HypercubeReflection
type: NamedIndividual
type: GeometricCharacter
comment: Reflection through the centre of the hypercube: bnot(x) = (2^n-1) ⊕ x. The second generator of D_{2^n}.

Rotation
type: NamedIndividual
type: GeometricCharacter
comment: Rotation by one step: succ(x) = (x+1) mod 2^n. The composition of the two reflections.

RotationInverse
type: NamedIndividual
type: GeometricCharacter
comment: Rotation by one step in the reverse direction: pred(x) = (x-1) mod 2^n.

Translation
type: NamedIndividual
type: GeometricCharacter
comment: Translation along the ring axis: add(x,y), sub(x,y). Preserves Hamming distance locally.

Scaling
type: NamedIndividual
type: GeometricCharacter
comment: Scaling along the ring axis: mul(x,y) = (x×y) mod 2^n.

HypercubeTranslation
type: NamedIndividual
type: GeometricCharacter
comment: Translation along the hypercube axis: xor(x,y) = x ⊕ y. Preserves ring distance locally.

HypercubeProjection
type: NamedIndividual
type: GeometricCharacter
comment: Projection onto a hypercube face: and(x,y) = x ∧ y. Idempotent; collapses dimensions.

HypercubeJoin
type: NamedIndividual
type: GeometricCharacter
comment: Join on the hypercube lattice: or(x,y) = x ∨ y. Idempotent; dual to projection.

ConstraintSelection
type: NamedIndividual
type: GeometricCharacter
comment: Geometric character of dispatch: constraint-guided selection over the resolver registry lattice.

ResolutionTraversal
type: NamedIndividual
type: GeometricCharacter
comment: Geometric character of inference: traversal through the φ-pipeline resolution graph P ∘ Π ∘ G.

SiteBinding
type: NamedIndividual
type: GeometricCharacter
comment: Geometric character of accumulation: progressive pinning of site states in the context lattice.

SitePartition
type: NamedIndividual
type: GeometricCharacter
comment: Geometric character of lease partition: splitting a shared context into disjoint site-set leases.

SessionMerge
type: NamedIndividual
type: GeometricCharacter
comment: Geometric character of session composition: merging disjoint lease sessions into a unified resolution context.

neg
type: NamedIndividual
type: Involution
arity: 1
hasGeometricCharacter: RingReflection
comment: Ring reflection: neg(x) = (-x) mod 2^n. One of the two generators of the dihedral group D_{2^n}. neg(neg(x)) = x (involution property).

bnot
type: NamedIndividual
type: Involution
arity: 1
hasGeometricCharacter: HypercubeReflection
comment: Hypercube reflection: bnot(x) = (2^n - 1) ⊕ x (bitwise complement). The second generator of D_{2^n}. bnot(bnot(x)) = x.

succ
type: NamedIndividual
type: UnaryOp
arity: 1
composedOf: neg
composedOf: bnot
hasGeometricCharacter: Rotation
inverse: pred
comment: Successor: succ(x) = neg(bnot(x)) = (x + 1) mod 2^n. The critical identity: succ is the composition neg ∘ bnot.

pred
type: NamedIndividual
type: UnaryOp
arity: 1
composedOf: bnot
composedOf: neg
hasGeometricCharacter: RotationInverse
inverse: succ
comment: Predecessor: pred(x) = bnot(neg(x)) = (x - 1) mod 2^n. The inverse of succ. pred is the composition bnot ∘ neg.

add
type: NamedIndividual
type: BinaryOp
arity: 2
associative: true
commutative: true
hasGeometricCharacter: Translation
identity: 0
inverse: sub
comment: Ring addition: add(x, y) = (x + y) mod 2^n. Commutative, associative; identity element is 0.

sub
type: NamedIndividual
type: BinaryOp
arity: 2
associative: false
commutative: false
hasGeometricCharacter: Translation
comment: Ring subtraction: sub(x, y) = (x - y) mod 2^n. Not commutative, not associative.

mul
type: NamedIndividual
type: BinaryOp
arity: 2
associative: true
commutative: true
hasGeometricCharacter: Scaling
identity: 1
comment: Ring multiplication: mul(x, y) = (x × y) mod 2^n. Commutative, associative; identity element is 1.

xor
type: NamedIndividual
type: BinaryOp
arity: 2
associative: true
commutative: true
hasGeometricCharacter: HypercubeTranslation
identity: 0
comment: Bitwise exclusive or: xor(x, y) = x ⊕ y. Commutative, associative; identity element is 0.

and
type: NamedIndividual
type: BinaryOp
arity: 2
associative: true
commutative: true
hasGeometricCharacter: HypercubeProjection
comment: Bitwise and: and(x, y) = x ∧ y. Commutative, associative.

or
type: NamedIndividual
type: BinaryOp
arity: 2
associative: true
commutative: true
hasGeometricCharacter: HypercubeJoin
comment: Bitwise or: or(x, y) = x ∨ y. Commutative, associative.

dispatch
type: NamedIndividual
type: DispatchOperation
arity: 2
associative: false
commutative: false
hasGeometricCharacter: ConstraintSelection
operatorSignature: Query × ResolverRegistry → Resolver
comment: δ(q, R) = R(q): dispatches a query to the matching resolver in the registry. Non-commutative, non-associative.

infer
type: NamedIndividual
type: InferenceOperation
arity: 2
associative: false
commutative: false
hasGeometricCharacter: ResolutionTraversal
operatorSignature: Symbol × Context → ResolvedType
comment: ι(s, C) = P(Π(G(s, C))): the φ-pipeline composed into a single inference step. Non-commutative, non-associative.

accumulate
type: NamedIndividual
type: AccumulationOperation
arity: 2
associative: true
commutative: false
hasGeometricCharacter: SiteBinding
operatorSignature: Binding × Context → Context
comment: α(b, C) = C': accumulates a binding into a resolution context, pinning a site. Non-commutative, associative at convergence (SR_10).

partition_op
type: NamedIndividual
type: LeasePartitionOperation
arity: 2
associative: false
commutative: false
hasGeometricCharacter: SitePartition
operatorSignature: SharedContext × ℕ → ContextLease^k
comment: λ(S, k) = (L₁, …, Lₖ): partitions a shared context into k disjoint leases. Non-commutative, non-associative.

compose_op
type: NamedIndividual
type: SessionCompositionOperation
arity: 2
associative: true
commutative: true
hasGeometricCharacter: SessionMerge
operatorSignature: Session × Session → Session
comment: κ(S₁, S₂) = S₁ ∪ S₂: composes two sessions with disjoint leases into one. Commutative, associative (SR_8).

Critical Identity
type: NamedIndividual
type: Identity
forAll: term_criticalIdentity_forAll
lhs: succ
rhs: neg
rhs: bnot
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: The foundational theorem of the UOR kernel: neg(bnot(x)) = succ(x) for all x in R_n. This identity links the two involutions (neg and bnot) to the successor operation, making succ derivable from neg and bnot.

AD_1
type: NamedIndividual
type: Identity
forAll: term_AD_1_forAll
lhs: term_AD_1_lhs
rhs: term_AD_1_rhs
verificationDomain: Analytical
comment: Addressing bijection: addresses(glyph(d)) = d. Round-trip from datum through glyph and back is identity.

AD_2
type: NamedIndividual
type: Identity
forAll: term_AD_2_forAll
lhs: term_AD_2_lhs
rhs: term_AD_2_rhs
verificationDomain: Analytical
comment: Embedding coherence: glyph(ι(addresses(a))) = ι_addr(a). The addressing diagram commutes through embeddings.

R_A1
type: NamedIndividual
type: Identity
forAll: term_R_A1_forAll
lhs: term_R_A1_lhs
rhs: term_R_A1_rhs
verificationDomain: Algebraic
comment: Additive associativity: add(x, add(y, z)) = add(add(x, y), z).

R_A2
type: NamedIndividual
type: Identity
forAll: term_R_A2_forAll
lhs: term_R_A2_lhs
rhs: term_R_A2_rhs
verificationDomain: Algebraic
comment: Additive identity: add(x, 0) = x.

R_A3
type: NamedIndividual
type: Identity
forAll: term_R_A3_forAll
lhs: term_R_A3_lhs
rhs: term_R_A3_rhs
verificationDomain: Algebraic
comment: Additive inverse: add(x, neg(x)) = 0.

R_A4
type: NamedIndividual
type: Identity
forAll: term_R_A4_forAll
lhs: term_R_A4_lhs
rhs: term_R_A4_rhs
verificationDomain: Algebraic
comment: Additive commutativity: add(x, y) = add(y, x).

R_A5
type: NamedIndividual
type: Identity
forAll: term_R_A5_forAll
lhs: term_R_A5_lhs
rhs: term_R_A5_rhs
verificationDomain: Algebraic
comment: Subtraction definition: sub(x, y) = add(x, neg(y)).

R_A6
type: NamedIndividual
type: Identity
forAll: term_R_A6_forAll
lhs: term_R_A6_lhs
rhs: term_R_A6_rhs
verificationDomain: Algebraic
comment: Negation involution: neg(neg(x)) = x.

R_M1
type: NamedIndividual
type: Identity
forAll: term_R_M1_forAll
lhs: term_R_M1_lhs
rhs: term_R_M1_rhs
verificationDomain: Algebraic
comment: Multiplicative associativity: mul(x, mul(y, z)) = mul(mul(x, y), z).

R_M2
type: NamedIndividual
type: Identity
forAll: term_R_M2_forAll
lhs: term_R_M2_lhs
rhs: term_R_M2_rhs
verificationDomain: Algebraic
comment: Multiplicative identity: mul(x, 1) = x.

R_M3
type: NamedIndividual
type: Identity
forAll: term_R_M3_forAll
lhs: term_R_M3_lhs
rhs: term_R_M3_rhs
verificationDomain: Algebraic
comment: Multiplicative commutativity: mul(x, y) = mul(y, x).

R_M4
type: NamedIndividual
type: Identity
forAll: term_R_M4_forAll
lhs: term_R_M4_lhs
rhs: term_R_M4_rhs
verificationDomain: Algebraic
comment: Distributivity: mul(x, add(y, z)) = add(mul(x, y), mul(x, z)).

R_M5
type: NamedIndividual
type: Identity
forAll: term_R_M5_forAll
lhs: term_R_M5_lhs
rhs: term_R_M5_rhs
verificationDomain: Algebraic
comment: Annihilation: mul(x, 0) = 0.

B_1
type: NamedIndividual
type: Identity
forAll: term_B_1_forAll
lhs: term_B_1_lhs
rhs: term_B_1_rhs
verificationDomain: Algebraic
comment: XOR associativity: xor(x, xor(y, z)) = xor(xor(x, y), z).

B_2
type: NamedIndividual
type: Identity
forAll: term_B_2_forAll
lhs: term_B_2_lhs
rhs: term_B_2_rhs
verificationDomain: Algebraic
comment: XOR identity: xor(x, 0) = x.

B_3
type: NamedIndividual
type: Identity
forAll: term_B_3_forAll
lhs: term_B_3_lhs
rhs: term_B_3_rhs
verificationDomain: Algebraic
comment: XOR self-inverse: xor(x, x) = 0.

B_4
type: NamedIndividual
type: Identity
forAll: term_B_4_forAll
lhs: term_B_4_lhs
rhs: term_B_4_rhs
verificationDomain: Algebraic
comment: AND associativity: and(x, and(y, z)) = and(and(x, y), z).

B_5
type: NamedIndividual
type: Identity
forAll: term_B_5_forAll
lhs: term_B_5_lhs
rhs: term_B_5_rhs
verificationDomain: Algebraic
comment: AND identity: and(x, 2^n - 1) = x.

B_6
type: NamedIndividual
type: Identity
forAll: term_B_6_forAll
lhs: term_B_6_lhs
rhs: term_B_6_rhs
verificationDomain: Algebraic
comment: AND annihilation: and(x, 0) = 0.

B_7
type: NamedIndividual
type: Identity
forAll: term_B_7_forAll
lhs: term_B_7_lhs
rhs: term_B_7_rhs
verificationDomain: Algebraic
comment: OR associativity: or(x, or(y, z)) = or(or(x, y), z).

B_8
type: NamedIndividual
type: Identity
forAll: term_B_8_forAll
lhs: term_B_8_lhs
rhs: term_B_8_rhs
verificationDomain: Algebraic
comment: OR identity: or(x, 0) = x.

B_9
type: NamedIndividual
type: Identity
forAll: term_B_9_forAll
lhs: term_B_9_lhs
rhs: term_B_9_rhs
verificationDomain: Algebraic
comment: Absorption: and(x, or(x, y)) = x.

B_10
type: NamedIndividual
type: Identity
forAll: term_B_10_forAll
lhs: term_B_10_lhs
rhs: term_B_10_rhs
verificationDomain: Algebraic
comment: AND distributes over OR: and(x, or(y, z)) = or(and(x, y), and(x, z)).

B_11
type: NamedIndividual
type: Identity
forAll: term_B_11_forAll
lhs: term_B_11_lhs
rhs: term_B_11_rhs
verificationDomain: Algebraic
comment: De Morgan 1: bnot(and(x, y)) = or(bnot(x), bnot(y)).

B_12
type: NamedIndividual
type: Identity
forAll: term_B_12_forAll
lhs: term_B_12_lhs
rhs: term_B_12_rhs
verificationDomain: Algebraic
comment: De Morgan 2: bnot(or(x, y)) = and(bnot(x), bnot(y)).

B_13
type: NamedIndividual
type: Identity
forAll: term_B_13_forAll
lhs: term_B_13_lhs
rhs: term_B_13_rhs
verificationDomain: Algebraic
comment: Bnot involution: bnot(bnot(x)) = x.

X_1
type: NamedIndividual
type: Identity
forAll: term_X_1_forAll
lhs: term_X_1_lhs
rhs: term_X_1_rhs
verificationDomain: Algebraic
comment: Neg via subtraction: neg(x) = sub(0, x).

X_2
type: NamedIndividual
type: Identity
forAll: term_X_2_forAll
lhs: term_X_2_lhs
rhs: term_X_2_rhs
verificationDomain: Algebraic
comment: Complement via XOR: bnot(x) = xor(x, 2^n - 1).

X_3
type: NamedIndividual
type: Identity
forAll: term_X_3_forAll
lhs: term_X_3_lhs
rhs: term_X_3_rhs
verificationDomain: Algebraic
comment: Succ via addition: succ(x) = add(x, 1).

X_4
type: NamedIndividual
type: Identity
forAll: term_X_4_forAll
lhs: term_X_4_lhs
rhs: term_X_4_rhs
verificationDomain: Algebraic
comment: Pred via subtraction: pred(x) = sub(x, 1).

X_5
type: NamedIndividual
type: Identity
forAll: term_X_5_forAll
lhs: term_X_5_lhs
rhs: term_X_5_rhs
verificationDomain: Algebraic
comment: Neg-bnot bridge: neg(x) = add(bnot(x), 1).

X_6
type: NamedIndividual
type: Identity
forAll: term_X_6_forAll
lhs: term_X_6_lhs
rhs: term_X_6_rhs
verificationDomain: Algebraic
comment: Complement predecessor: bnot(x) = pred(neg(x)).

X_7
type: NamedIndividual
type: Identity
forAll: term_X_7_forAll
lhs: term_X_7_lhs
rhs: term_X_7_rhs
verificationDomain: Algebraic
comment: XOR-add bridge: xor(x, y) = add(x, y) - 2 * and(x, y) (in Z before mod).

D_1
type: NamedIndividual
type: Identity
forAll: term_D_1_forAll
lhs: term_D_1_lhs
rhs: term_D_1_rhs
verificationDomain: Algebraic
comment: Rotation order: succ^[2^n](x) = x.

D_3
type: NamedIndividual
type: Identity
forAll: term_D_3_forAll
lhs: term_D_3_lhs
rhs: term_D_3_rhs
verificationDomain: Algebraic
comment: Conjugation: neg(succ(neg(x))) = pred(x).

D_4
type: NamedIndividual
type: Identity
forAll: term_D_4_forAll
lhs: term_D_4_lhs
rhs: term_D_4_rhs
verificationDomain: Algebraic
comment: Reverse composition: bnot(neg(x)) = pred(x).

D_5
type: NamedIndividual
type: Identity
forAll: term_D_5_forAll
lhs: term_D_5_lhs
rhs: term_D_5_rhs
verificationDomain: Algebraic
comment: Group closure: D_[2^n] = [succ^k, neg ∘ succ^k : 0 ≤ k < 2^n].

U_1
type: NamedIndividual
type: Identity
forAll: term_U_1_forAll
lhs: term_U_1_lhs
rhs: term_U_1_rhs
verificationDomain: Algebraic
comment: Unit group decomposition: R_n× ≅ Z/2 × Z/2^[n-2] for n ≥ 3.

U_2
type: NamedIndividual
type: Identity
forAll: term_U_2_forAll
lhs: term_U_2_lhs
rhs: term_U_2_rhs
verificationDomain: Algebraic
comment: Unit group special cases: R_1× ≅ [1]; R_2× ≅ Z/2.

U_3
type: NamedIndividual
type: Identity
forAll: term_U_3_forAll
lhs: term_U_3_lhs
rhs: term_U_3_rhs
verificationDomain: Algebraic
comment: Multiplicative order: ord(u) = lcm(ord((-1)^a), ord(3^b)).

U_4
type: NamedIndividual
type: Identity
forAll: term_U_4_forAll
lhs: term_U_4_lhs
rhs: term_U_4_rhs
verificationDomain: Algebraic
comment: Resonance period: ord_g(2) divides φ(g).

U_5
type: NamedIndividual
type: Identity
forAll: term_U_5_forAll
lhs: term_U_5_lhs
rhs: term_U_5_rhs
verificationDomain: Algebraic
comment: Step formula derivation: step_g = 2 * ((g - (2^n mod g)) mod g) + 1.

AG_1
type: NamedIndividual
type: Identity
forAll: term_AG_1_forAll
lhs: term_AG_1_lhs
rhs: term_AG_1_rhs
verificationDomain: Algebraic
comment: Scaling not dihedral: μ_u ∉ D_[2^n] for u ≠ ±1.

AG_2
type: NamedIndividual
type: Identity
forAll: term_AG_2_forAll
lhs: term_AG_2_lhs
rhs: term_AG_2_rhs
verificationDomain: Algebraic
comment: Affine group: Aff(R_n) = R_n× ⋉ R_n.

AG_3
type: NamedIndividual
type: Identity
forAll: term_AG_3_forAll
lhs: term_AG_3_lhs
rhs: term_AG_3_rhs
verificationDomain: Algebraic
comment: Affine group order: |Aff(R_n)| = 2^[2n-1].

AG_4
type: NamedIndividual
type: Identity
forAll: term_AG_4_forAll
lhs: term_AG_4_lhs
rhs: term_AG_4_rhs
verificationDomain: Algebraic
comment: Subgroup inclusion: D_[2^n] ⊂ Aff(R_n) with u ∈ [±1].

CA_1
type: NamedIndividual
type: Identity
forAll: term_CA_1_forAll
lhs: term_CA_1_lhs
rhs: term_CA_1_rhs
verificationDomain: Algebraic
comment: Addition decomposition: add(x,y)_k = xor(x_k, xor(y_k, c_k(x,y))).

CA_2
type: NamedIndividual
type: Identity
forAll: term_CA_2_forAll
lhs: term_CA_2_lhs
rhs: term_CA_2_rhs
verificationDomain: Algebraic
comment: Carry recurrence: c_[k+1](x,y) = or(and(x_k,y_k), and(xor(x_k,y_k), c_k)).

CA_3
type: NamedIndividual
type: Identity
forAll: term_CA_3_forAll
lhs: term_CA_3_lhs
rhs: term_CA_3_rhs
verificationDomain: Algebraic
comment: Carry commutativity: c(x, y) = c(y, x).

CA_4
type: NamedIndividual
type: Identity
forAll: term_CA_4_forAll
lhs: term_CA_4_lhs
rhs: term_CA_4_rhs
verificationDomain: Algebraic
comment: Zero carry: c(x, 0) = 0 at all positions.

CA_5
type: NamedIndividual
type: Identity
forAll: term_CA_5_forAll
lhs: term_CA_5_lhs
rhs: term_CA_5_rhs
verificationDomain: Algebraic
comment: Negation carry: c(x, neg(x))_k = 1 for k > v_2(x).

CA_6
type: NamedIndividual
type: Identity
forAll: term_CA_6_forAll
lhs: term_CA_6_lhs
rhs: term_CA_6_rhs
verificationDomain: Algebraic
comment: Carry-incompatibility link: d_Δ(x, y) > 0 iff ∃ k : c_k(x,y) = 1.

C_1
type: NamedIndividual
type: Identity
forAll: term_C_1_forAll
lhs: term_C_1_lhs
rhs: term_C_1_rhs
verificationDomain: Algebraic
comment: Constraint pin union: pins of a composite constraint equal the union of component pins.

C_2
type: NamedIndividual
type: Identity
forAll: term_C_2_forAll
lhs: term_C_2_lhs
rhs: term_C_2_rhs
verificationDomain: Algebraic
comment: Constraint composition commutativity.

C_3
type: NamedIndividual
type: Identity
forAll: term_C_3_forAll
lhs: term_C_3_lhs
rhs: term_C_3_rhs
verificationDomain: Algebraic
comment: Constraint composition associativity.

C_4
type: NamedIndividual
type: Identity
forAll: term_C_4_forAll
lhs: term_C_4_lhs
rhs: term_C_4_rhs
verificationDomain: Algebraic
comment: Constraint composition idempotence.

C_5
type: NamedIndividual
type: Identity
forAll: term_C_5_forAll
lhs: term_C_5_lhs
rhs: term_C_5_rhs
verificationDomain: Algebraic
comment: Constraint composition identity element.

C_6
type: NamedIndividual
type: Identity
forAll: term_C_6_forAll
lhs: term_C_6_lhs
rhs: term_C_6_rhs
verificationDomain: Algebraic
comment: Constraint composition annihilator.

CDI
type: NamedIndividual
type: Identity
forAll: term_CDI_forAll
lhs: term_CDI_lhs
rhs: term_CDI_rhs
verificationDomain: Algebraic
comment: Constraint-depth invariant: carry complexity of the residue representation equals the type depth.

CL_1
type: NamedIndividual
type: Identity
forAll: term_CL_1_forAll
lhs: term_CL_1_lhs
rhs: term_CL_1_rhs
verificationDomain: Algebraic
comment: Constraint quotient lattice isomorphism to power set.

CL_2
type: NamedIndividual
type: Identity
forAll: term_CL_2_forAll
lhs: term_CL_2_lhs
rhs: term_CL_2_rhs
verificationDomain: Algebraic
comment: Lattice join equals constraint composition.

CL_3
type: NamedIndividual
type: Identity
forAll: term_CL_3_forAll
lhs: term_CL_3_lhs
rhs: term_CL_3_rhs
verificationDomain: Algebraic
comment: Lattice meet pins the intersection of component pins.

CL_4
type: NamedIndividual
type: Identity
forAll: term_CL_4_forAll
lhs: term_CL_4_lhs
rhs: term_CL_4_rhs
verificationDomain: Algebraic
comment: Constraint lattice distributivity.

CL_5
type: NamedIndividual
type: Identity
forAll: term_CL_5_forAll
lhs: term_CL_5_lhs
rhs: term_CL_5_rhs
verificationDomain: Algebraic
comment: Constraint lattice complement existence.

CM_1
type: NamedIndividual
type: Identity
forAll: term_CM_1_forAll
lhs: term_CM_1_lhs
rhs: term_CM_1_rhs
verificationDomain: Algebraic
comment: Constraint redundancy criterion.

CM_2
type: NamedIndividual
type: Identity
forAll: term_CM_2_forAll
lhs: term_CM_2_lhs
rhs: term_CM_2_rhs
verificationDomain: Algebraic
comment: Minimal cover via greedy irredundant removal.

CM_3
type: NamedIndividual
type: Identity
forAll: term_CM_3_forAll
lhs: term_CM_3_lhs
rhs: term_CM_3_rhs
verificationDomain: Algebraic
comment: Minimum constraint count to cover n sites.

CR_1
type: NamedIndividual
type: Identity
forAll: term_CR_1_forAll
lhs: term_CR_1_lhs
rhs: term_CR_1_rhs
verificationDomain: Algebraic
comment: Residue constraint cost is the step formula.

CR_2
type: NamedIndividual
type: Identity
forAll: term_CR_2_forAll
lhs: term_CR_2_lhs
rhs: term_CR_2_rhs
verificationDomain: Algebraic
comment: Carry constraint cost is the popcount of the pattern.

CR_3
type: NamedIndividual
type: Identity
forAll: term_CR_3_forAll
lhs: term_CR_3_lhs
rhs: term_CR_3_rhs
verificationDomain: Algebraic
comment: Depth constraint cost is sum of residue and carry costs.

CR_4
type: NamedIndividual
type: Identity
forAll: term_CR_4_forAll
lhs: term_CR_4_lhs
rhs: term_CR_4_rhs
verificationDomain: Algebraic
comment: Composite constraint cost is subadditive.

CR_5
type: NamedIndividual
type: Identity
forAll: term_CR_5_forAll
lhs: term_CR_5_lhs
rhs: term_CR_5_rhs
verificationDomain: Algebraic
comment: Optimal resolution order is increasing cost.

F_1
type: NamedIndividual
type: Identity
forAll: term_F_1_forAll
lhs: term_F_1_lhs
rhs: term_F_1_rhs
verificationDomain: Algebraic
comment: Site monotonicity: a pinned site cannot be unpinned.

F_2
type: NamedIndividual
type: Identity
forAll: term_F_2_forAll
lhs: term_F_2_lhs
rhs: term_F_2_rhs
verificationDomain: Algebraic
comment: Site budget upper bound: at most n pin operations to close.

F_3
type: NamedIndividual
type: Identity
forAll: term_F_3_forAll
lhs: term_F_3_lhs
rhs: term_F_3_rhs
verificationDomain: Algebraic
comment: Site budget conservation: pinned + free = total sites.

F_4
type: NamedIndividual
type: Identity
forAll: term_F_4_forAll
lhs: term_F_4_lhs
rhs: term_F_4_rhs
verificationDomain: Algebraic
comment: Site budget closure: closed iff all sites pinned.

FL_1
type: NamedIndividual
type: Identity
forAll: term_FL_1_forAll
lhs: term_FL_1_lhs
rhs: term_FL_1_rhs
verificationDomain: Algebraic
comment: Site lattice bottom: all sites free.

FL_2
type: NamedIndividual
type: Identity
forAll: term_FL_2_forAll
lhs: term_FL_2_lhs
rhs: term_FL_2_rhs
verificationDomain: Algebraic
comment: Site lattice top: all sites pinned.

FL_3
type: NamedIndividual
type: Identity
forAll: term_FL_3_forAll
lhs: term_FL_3_lhs
rhs: term_FL_3_rhs
verificationDomain: Algebraic
comment: Site lattice join is union of pinnings.

FL_4
type: NamedIndividual
type: Identity
forAll: term_FL_4_forAll
lhs: term_FL_4_lhs
rhs: term_FL_4_rhs
verificationDomain: Algebraic
comment: Site lattice height equals n.

FPM_1
type: NamedIndividual
type: Identity
forAll: term_FPM_1_forAll
lhs: term_FPM_1_lhs
rhs: term_FPM_1_rhs
verificationDomain: Algebraic
comment: Unit partition membership: x is a unit iff site_0(x) = 1 (x is odd).

FPM_2
type: NamedIndividual
type: Identity
forAll: term_FPM_2_forAll
lhs: term_FPM_2_lhs
rhs: term_FPM_2_rhs
verificationDomain: Algebraic
comment: Exterior partition membership: x is exterior iff x is not in the carrier of T.

FPM_3
type: NamedIndividual
type: Identity
forAll: term_FPM_3_forAll
lhs: term_FPM_3_lhs
rhs: term_FPM_3_rhs
verificationDomain: Algebraic
comment: Irreducible partition membership: x is irreducible iff x is not a unit, exterior, or reducible.

FPM_4
type: NamedIndividual
type: Identity
forAll: term_FPM_4_forAll
lhs: term_FPM_4_lhs
rhs: term_FPM_4_rhs
verificationDomain: Algebraic
comment: Reducible partition membership: x is reducible iff x is not a unit, exterior, or irreducible.

FPM_5
type: NamedIndividual
type: Identity
forAll: term_FPM_5_forAll
lhs: term_FPM_5_lhs
rhs: term_FPM_5_rhs
verificationDomain: Algebraic
comment: 2-adic decomposition: every element factors as 2^{v(x)} times an odd unit.

FPM_6
type: NamedIndividual
type: Identity
forAll: term_FPM_6_forAll
lhs: term_FPM_6_lhs
rhs: term_FPM_6_rhs
verificationDomain: Algebraic
comment: Stratum size formula.

FPM_7
type: NamedIndividual
type: Identity
forAll: term_FPM_7_forAll
lhs: term_FPM_7_lhs
rhs: term_FPM_7_rhs
verificationDomain: Algebraic
comment: Base type partition cardinalities.

FS_1
type: NamedIndividual
type: Identity
forAll: term_FS_1_forAll
lhs: term_FS_1_lhs
rhs: term_FS_1_rhs
verificationDomain: Algebraic
comment: Site extraction: site_k(x) is the k-th bit of x.

FS_2
type: NamedIndividual
type: Identity
forAll: term_FS_2_forAll
lhs: term_FS_2_lhs
rhs: term_FS_2_rhs
verificationDomain: Algebraic
comment: Site 0 is parity.

FS_3
type: NamedIndividual
type: Identity
forAll: term_FS_3_forAll
lhs: term_FS_3_lhs
rhs: term_FS_3_rhs
verificationDomain: Algebraic
comment: Progressive site determination: site_k given lower sites determines x mod 2^{k+1}.

FS_4
type: NamedIndividual
type: Identity
forAll: term_FS_4_forAll
lhs: term_FS_4_lhs
rhs: term_FS_4_rhs
verificationDomain: Algebraic
comment: Cumulative site determination: sites 0..k together determine x mod 2^{k+1}.

FS_5
type: NamedIndividual
type: Identity
forAll: term_FS_5_forAll
lhs: term_FS_5_lhs
rhs: term_FS_5_rhs
verificationDomain: Algebraic
comment: Complete site determination: all n sites determine x uniquely.

FS_6
type: NamedIndividual
type: Identity
forAll: term_FS_6_forAll
lhs: term_FS_6_lhs
rhs: term_FS_6_rhs
verificationDomain: Algebraic
comment: Stratum from sites: v_2(x) is the minimum k where site_k(x) = 1.

FS_7
type: NamedIndividual
type: Identity
forAll: term_FS_7_forAll
lhs: term_FS_7_lhs
rhs: term_FS_7_rhs
verificationDomain: Algebraic
comment: Depth bound: depth(x) ≤ v_2(x) for irreducible elements.

RE_1
type: NamedIndividual
type: Identity
forAll: term_RE_1_forAll
lhs: term_RE_1_lhs
rhs: term_RE_1_rhs
verificationDomain: Pipeline
comment: Resolution strategy equivalence: dihedral, canonical-form, and evaluation resolvers all compute the same partition.

IR_1
type: NamedIndividual
type: Identity
forAll: term_IR_1_forAll
lhs: term_IR_1_lhs
rhs: term_IR_1_rhs
verificationDomain: Pipeline
comment: Resolution monotonicity: pinned count never decreases.

IR_2
type: NamedIndividual
type: Identity
forAll: term_IR_2_forAll
lhs: term_IR_2_lhs
rhs: term_IR_2_rhs
verificationDomain: Pipeline
comment: Resolution convergence bound: at most n iterations.

IR_3
type: NamedIndividual
type: Identity
forAll: term_IR_3_forAll
lhs: term_IR_3_lhs
rhs: term_IR_3_rhs
verificationDomain: Pipeline
comment: Convergence rate definition.

IR_4
type: NamedIndividual
type: Identity
forAll: term_IR_4_forAll
lhs: term_IR_4_lhs
rhs: term_IR_4_rhs
verificationDomain: Pipeline
comment: Resolution termination: loop terminates if constraint set spans all sites.

SF_1
type: NamedIndividual
type: Identity
forAll: term_SF_1_forAll
lhs: term_SF_1_lhs
rhs: term_SF_1_rhs
verificationDomain: Pipeline
comment: Optimal resolution level for a factor: n ≡ 0 (mod ord_g(2)).

SF_2
type: NamedIndividual
type: Identity
forAll: term_SF_2_forAll
lhs: term_SF_2_lhs
rhs: term_SF_2_rhs
verificationDomain: Pipeline
comment: Constraint ordering by step cost: smaller step_g first.

RD_1
type: NamedIndividual
type: Identity
forAll: term_RD_1_forAll
lhs: term_RD_1_lhs
rhs: term_RD_1_rhs
verificationDomain: Pipeline
comment: Resolution determinism: same type and constraint sequence yield unique partition.

RD_2
type: NamedIndividual
type: Identity
forAll: term_RD_2_forAll
lhs: term_RD_2_lhs
rhs: term_RD_2_rhs
verificationDomain: Pipeline
comment: Order independence: complete constraint sets yield the same partition regardless of order.

SE_1
type: NamedIndividual
type: Identity
forAll: term_SE_1_forAll
lhs: term_SE_1_lhs
rhs: term_SE_1_rhs
verificationDomain: Pipeline
comment: Evaluation resolver directly computes the set-theoretic partition.

SE_2
type: NamedIndividual
type: Identity
forAll: term_SE_2_forAll
lhs: term_SE_2_lhs
rhs: term_SE_2_rhs
verificationDomain: Pipeline
comment: Dihedral factorization resolver yields the same partition via orbit decomposition.

SE_3
type: NamedIndividual
type: Identity
forAll: term_SE_3_forAll
lhs: term_SE_3_lhs
rhs: term_SE_3_rhs
verificationDomain: Pipeline
comment: Canonical form resolver yields the same partition via confluent rewrite.

SE_4
type: NamedIndividual
type: Identity
forAll: term_SE_4_forAll
lhs: term_SE_4_lhs
rhs: term_SE_4_rhs
verificationDomain: Pipeline
comment: All three strategies compute the same set-theoretic partition.

OO_1
type: NamedIndividual
type: Identity
forAll: term_OO_1_forAll
lhs: term_OO_1_lhs
rhs: term_OO_1_rhs
verificationDomain: Pipeline
comment: Benefit of a constraint is the number of new pins it provides.

OO_2
type: NamedIndividual
type: Identity
forAll: term_OO_2_forAll
lhs: term_OO_2_lhs
rhs: term_OO_2_rhs
verificationDomain: Pipeline
comment: Constraint cost is step or popcount depending on type.

OO_3
type: NamedIndividual
type: Identity
forAll: term_OO_3_forAll
lhs: term_OO_3_lhs
rhs: term_OO_3_rhs
verificationDomain: Pipeline
comment: Greedy selection maximizes benefit-to-cost ratio.

OO_4
type: NamedIndividual
type: Identity
forAll: term_OO_4_forAll
lhs: term_OO_4_lhs
rhs: term_OO_4_rhs
verificationDomain: Pipeline
comment: Greedy approximation achieves (1 − 1/e) ≈ 63% of optimal.

OO_5
type: NamedIndividual
type: Identity
forAll: term_OO_5_forAll
lhs: term_OO_5_lhs
rhs: term_OO_5_rhs
verificationDomain: Pipeline
comment: Tiebreaker: prefer vertical (residue) before horizontal (carry).

CB_1
type: NamedIndividual
type: Identity
forAll: term_CB_1_forAll
lhs: term_CB_1_lhs
rhs: term_CB_1_rhs
verificationDomain: Pipeline
comment: Minimum convergence rate: 1 site per iteration (worst case).

CB_2
type: NamedIndividual
type: Identity
forAll: term_CB_2_forAll
lhs: term_CB_2_lhs
rhs: term_CB_2_rhs
verificationDomain: Pipeline
comment: Maximum convergence rate: n sites in 1 iteration (best case).

CB_3
type: NamedIndividual
type: Identity
forAll: term_CB_3_forAll
lhs: term_CB_3_lhs
rhs: term_CB_3_rhs
verificationDomain: Pipeline
comment: Expected residue constraint rate: ⌊log_2(m)⌋ sites per constraint.

CB_4
type: NamedIndividual
type: Identity
forAll: term_CB_4_forAll
lhs: term_CB_4_lhs
rhs: term_CB_4_rhs
verificationDomain: Pipeline
comment: Stall detection: convergenceRate < 1 for 2 iterations suggests insufficiency.

CB_5
type: NamedIndividual
type: Identity
forAll: term_CB_5_forAll
lhs: term_CB_5_lhs
rhs: term_CB_5_rhs
verificationDomain: Pipeline
comment: Sufficiency criterion: pin union covers all site positions.

CB_6
type: NamedIndividual
type: Identity
forAll: term_CB_6_forAll
lhs: term_CB_6_lhs
rhs: term_CB_6_rhs
verificationDomain: Pipeline
comment: Iteration bound for k constraints: at most min(k, n) iterations.

OB_M1
type: NamedIndividual
type: Identity
forAll: term_OB_M1_forAll
lhs: term_OB_M1_lhs
rhs: term_OB_M1_rhs
verificationDomain: Analytical
comment: Ring metric triangle inequality.

OB_M2
type: NamedIndividual
type: Identity
forAll: term_OB_M2_forAll
lhs: term_OB_M2_lhs
rhs: term_OB_M2_rhs
verificationDomain: Analytical
comment: Hamming metric triangle inequality.

OB_M3
type: NamedIndividual
type: Identity
forAll: term_OB_M3_forAll
lhs: term_OB_M3_lhs
rhs: term_OB_M3_rhs
verificationDomain: Analytical
comment: Incompatibility metric is the absolute difference of ring and Hamming metrics.

OB_M4
type: NamedIndividual
type: Identity
forAll: term_OB_M4_forAll
lhs: term_OB_M4_lhs
rhs: term_OB_M4_rhs
verificationDomain: Analytical
comment: Negation preserves ring metric.

OB_M5
type: NamedIndividual
type: Identity
forAll: term_OB_M5_forAll
lhs: term_OB_M5_lhs
rhs: term_OB_M5_rhs
verificationDomain: Analytical
comment: Bitwise complement preserves Hamming metric.

OB_M6
type: NamedIndividual
type: Identity
forAll: term_OB_M6_forAll
lhs: term_OB_M6_lhs
rhs: term_OB_M6_rhs
verificationDomain: Analytical
comment: Successor preserves ring metric but may change Hamming metric.

OB_C1
type: NamedIndividual
type: Identity
forAll: term_OB_C1_forAll
lhs: term_OB_C1_lhs
rhs: term_OB_C1_rhs
verificationDomain: Analytical
comment: Negation-complement commutator is constant 2.

OB_C2
type: NamedIndividual
type: Identity
forAll: term_OB_C2_forAll
lhs: term_OB_C2_lhs
rhs: term_OB_C2_rhs
verificationDomain: Analytical
comment: Negation-translation commutator.

OB_C3
type: NamedIndividual
type: Identity
forAll: term_OB_C3_forAll
lhs: term_OB_C3_lhs
rhs: term_OB_C3_rhs
verificationDomain: Analytical
comment: Complement-XOR commutator is trivial.

OB_H1
type: NamedIndividual
type: Identity
forAll: term_OB_H1_forAll
lhs: term_OB_H1_lhs
rhs: term_OB_H1_rhs
verificationDomain: Analytical
comment: Additive paths have trivial monodromy (abelian ⇒ path-independent).

OB_H2
type: NamedIndividual
type: Identity
forAll: term_OB_H2_forAll
lhs: term_OB_H2_lhs
rhs: term_OB_H2_rhs
verificationDomain: Analytical
comment: Dihedral generator paths have monodromy in D_{2^n}.

OB_H3
type: NamedIndividual
type: Identity
forAll: term_OB_H3_forAll
lhs: term_OB_H3_lhs
rhs: term_OB_H3_rhs
verificationDomain: Analytical
comment: Successor-only closed path winding number.

OB_P1
type: NamedIndividual
type: Identity
forAll: term_OB_P1_forAll
lhs: term_OB_P1_lhs
rhs: term_OB_P1_rhs
verificationDomain: Analytical
comment: Path length is additive under concatenation.

OB_P2
type: NamedIndividual
type: Identity
forAll: term_OB_P2_forAll
lhs: term_OB_P2_lhs
rhs: term_OB_P2_rhs
verificationDomain: Analytical
comment: Total variation is subadditive under concatenation.

OB_P3
type: NamedIndividual
type: Identity
forAll: term_OB_P3_forAll
lhs: term_OB_P3_lhs
rhs: term_OB_P3_rhs
verificationDomain: Analytical
comment: Reduction length is additive under sequential composition.

CT_1
type: NamedIndividual
type: Identity
forAll: term_CT_1_forAll
lhs: term_CT_1_lhs
rhs: term_CT_1_rhs
verificationDomain: Analytical
comment: Catastrophe boundaries are powers of 2.

CT_2
type: NamedIndividual
type: Identity
forAll: term_CT_2_forAll
lhs: term_CT_2_lhs
rhs: term_CT_2_rhs
verificationDomain: Analytical
comment: Odd prime catastrophe transitions visibility via residue constraints.

CT_3
type: NamedIndividual
type: Identity
forAll: term_CT_3_forAll
lhs: term_CT_3_lhs
rhs: term_CT_3_rhs
verificationDomain: Analytical
comment: Catastrophe threshold is normalized step cost.

CT_4
type: NamedIndividual
type: Identity
forAll: term_CT_4_forAll
lhs: term_CT_4_lhs
rhs: term_CT_4_rhs
verificationDomain: Analytical
comment: Composite catastrophe threshold is max of component thresholds.

CF_1
type: NamedIndividual
type: Identity
forAll: term_CF_1_forAll
lhs: term_CF_1_lhs
rhs: term_CF_1_rhs
verificationDomain: Analytical
comment: Curvature flux is the sum of incompatibility along a path.

CF_2
type: NamedIndividual
type: Identity
forAll: term_CF_2_forAll
lhs: term_CF_2_lhs
rhs: term_CF_2_rhs
verificationDomain: Analytical
comment: Resolution cost is bounded below by curvature flux of optimal path.

CF_3
type: NamedIndividual
type: Identity
forAll: term_CF_3_forAll
lhs: term_CF_3_lhs
rhs: term_CF_3_rhs
verificationDomain: Analytical
comment: Successor curvature flux: trailing_ones(x) for most x, n−1 at maximum.

CF_4
type: NamedIndividual
type: Identity
forAll: term_CF_4_forAll
lhs: term_CF_4_lhs
rhs: term_CF_4_rhs
verificationDomain: Analytical
comment: Total successor curvature flux over R_n equals 2^n − 2.

HG_1
type: NamedIndividual
type: Identity
forAll: term_HG_1_forAll
lhs: term_HG_1_lhs
rhs: term_HG_1_rhs
verificationDomain: Analytical
comment: Additive holonomy is trivial (abelian group).

HG_2
type: NamedIndividual
type: Identity
forAll: term_HG_2_forAll
lhs: term_HG_2_lhs
rhs: term_HG_2_rhs
verificationDomain: Analytical
comment: Dihedral generator holonomy group is D_{2^n}.

HG_3
type: NamedIndividual
type: Identity
forAll: term_HG_3_forAll
lhs: term_HG_3_lhs
rhs: term_HG_3_rhs
verificationDomain: Analytical
comment: Unit multiplication holonomy equals the unit group.

HG_4
type: NamedIndividual
type: Identity
forAll: term_HG_4_forAll
lhs: term_HG_4_lhs
rhs: term_HG_4_rhs
verificationDomain: Analytical
comment: Full holonomy group is the affine group.

HG_5
type: NamedIndividual
type: Identity
forAll: term_HG_5_forAll
lhs: term_HG_5_lhs
rhs: term_HG_5_rhs
verificationDomain: Analytical
comment: Holonomy group decomposition into dihedral and non-trivial units.

T_C1
type: NamedIndividual
type: Identity
forAll: term_T_C1_forAll
lhs: term_T_C1_lhs
rhs: term_T_C1_rhs
verificationDomain: Geometric
comment: Category left identity: compose(id, f) = f.

T_C2
type: NamedIndividual
type: Identity
forAll: term_T_C2_forAll
lhs: term_T_C2_lhs
rhs: term_T_C2_rhs
verificationDomain: Geometric
comment: Category right identity: compose(f, id) = f.

T_C3
type: NamedIndividual
type: Identity
forAll: term_T_C3_forAll
lhs: term_T_C3_lhs
rhs: term_T_C3_rhs
verificationDomain: Geometric
comment: Category associativity of transform composition.

T_C4
type: NamedIndividual
type: Identity
forAll: term_T_C4_forAll
lhs: term_T_C4_lhs
rhs: term_T_C4_rhs
verificationDomain: Geometric
comment: Composability condition: f composesWith g iff target(f) = source(g).

T_I1
type: NamedIndividual
type: Identity
forAll: term_T_I1_forAll
lhs: term_T_I1_lhs
rhs: term_T_I1_rhs
verificationDomain: Geometric
comment: Negation is a ring-metric isometry.

T_I2
type: NamedIndividual
type: Identity
forAll: term_T_I2_forAll
lhs: term_T_I2_lhs
rhs: term_T_I2_rhs
verificationDomain: Geometric
comment: Bitwise complement is a Hamming-metric isometry.

T_I3
type: NamedIndividual
type: Identity
forAll: term_T_I3_forAll
lhs: term_T_I3_lhs
rhs: term_T_I3_rhs
verificationDomain: Geometric
comment: Successor as composed isometries: succ = neg ∘ bnot preserves d_R but not d_H.

T_I4
type: NamedIndividual
type: Identity
forAll: term_T_I4_forAll
lhs: term_T_I4_lhs
rhs: term_T_I4_rhs
verificationDomain: Geometric
comment: Ring isometries form a group under composition.

T_I5
type: NamedIndividual
type: Identity
forAll: term_T_I5_forAll
lhs: term_T_I5_lhs
rhs: term_T_I5_rhs
verificationDomain: Geometric
comment: CurvatureObservable measures failure of ring isometry to be Hamming isometry.

T_E1
type: NamedIndividual
type: Identity
forAll: term_T_E1_forAll
lhs: term_T_E1_lhs
rhs: term_T_E1_rhs
verificationDomain: Geometric
comment: Embedding injectivity.

T_E2
type: NamedIndividual
type: Identity
forAll: term_T_E2_forAll
lhs: term_T_E2_lhs
rhs: term_T_E2_rhs
verificationDomain: Geometric
comment: Embedding is a ring homomorphism.

T_E3
type: NamedIndividual
type: Identity
forAll: term_T_E3_forAll
lhs: term_T_E3_lhs
rhs: term_T_E3_rhs
verificationDomain: Geometric
comment: Embedding transitivity: composition of embeddings is an embedding.

T_E4
type: NamedIndividual
type: Identity
forAll: term_T_E4_forAll
lhs: term_T_E4_lhs
rhs: term_T_E4_rhs
verificationDomain: Geometric
comment: Embedding address coherence: glyph ∘ ι ∘ addresses is well-defined.

T_A1
type: NamedIndividual
type: Identity
forAll: term_T_A1_forAll
lhs: term_T_A1_lhs
rhs: term_T_A1_rhs
verificationDomain: Geometric
comment: Dihedral group acts on constraints by transforming them.

T_A2
type: NamedIndividual
type: Identity
forAll: term_T_A2_forAll
lhs: term_T_A2_lhs
rhs: term_T_A2_rhs
verificationDomain: Geometric
comment: Dihedral group action on partitions permutes components.

T_A3
type: NamedIndividual
type: Identity
forAll: term_T_A3_forAll
lhs: term_T_A3_lhs
rhs: term_T_A3_rhs
verificationDomain: Geometric
comment: Dihedral orbits determine irreducibility boundaries.

T_A4
type: NamedIndividual
type: Identity
forAll: term_T_A4_forAll
lhs: term_T_A4_lhs
rhs: term_T_A4_rhs
verificationDomain: Geometric
comment: Fixed points of negation are {0, 2^{n−1}}; bnot has no fixed points.

AU_1
type: NamedIndividual
type: Identity
forAll: term_AU_1_forAll
lhs: term_AU_1_lhs
rhs: term_AU_1_rhs
verificationDomain: Geometric
comment: Automorphism group consists of unit multiplications.

AU_2
type: NamedIndividual
type: Identity
forAll: term_AU_2_forAll
lhs: term_AU_2_lhs
rhs: term_AU_2_rhs
verificationDomain: Geometric
comment: Automorphism group is isomorphic to the unit group.

AU_3
type: NamedIndividual
type: Identity
forAll: term_AU_3_forAll
lhs: term_AU_3_lhs
rhs: term_AU_3_rhs
verificationDomain: Geometric
comment: Automorphism group order is 2^{n−1}.

AU_4
type: NamedIndividual
type: Identity
forAll: term_AU_4_forAll
lhs: term_AU_4_lhs
rhs: term_AU_4_rhs
verificationDomain: Geometric
comment: Intersection of automorphism group with dihedral group is {id, neg}.

AU_5
type: NamedIndividual
type: Identity
forAll: term_AU_5_forAll
lhs: term_AU_5_lhs
rhs: term_AU_5_rhs
verificationDomain: Geometric
comment: Affine group is generated by D_{2^n} and μ_3.

EF_1
type: NamedIndividual
type: Identity
forAll: term_EF_1_forAll
lhs: term_EF_1_lhs
rhs: term_EF_1_rhs
verificationDomain: Geometric
comment: Embedding functor action on morphisms.

EF_2
type: NamedIndividual
type: Identity
forAll: term_EF_2_forAll
lhs: term_EF_2_lhs
rhs: term_EF_2_rhs
verificationDomain: Geometric
comment: Embedding functor preserves composition.

EF_3
type: NamedIndividual
type: Identity
forAll: term_EF_3_forAll
lhs: term_EF_3_lhs
rhs: term_EF_3_rhs
verificationDomain: Geometric
comment: Embedding functor preserves identities.

EF_4
type: NamedIndividual
type: Identity
forAll: term_EF_4_forAll
lhs: term_EF_4_lhs
rhs: term_EF_4_rhs
verificationDomain: Geometric
comment: Embedding functor composition is functorial.

EF_5
type: NamedIndividual
type: Identity
forAll: term_EF_5_forAll
lhs: term_EF_5_lhs
rhs: term_EF_5_rhs
verificationDomain: Geometric
comment: Embedding functor preserves ring isometries.

EF_6
type: NamedIndividual
type: Identity
forAll: term_EF_6_forAll
lhs: term_EF_6_lhs
rhs: term_EF_6_rhs
verificationDomain: Geometric
comment: Embedding functor maps dihedral subgroup into target dihedral group.

EF_7
type: NamedIndividual
type: Identity
forAll: term_EF_7_forAll
lhs: term_EF_7_lhs
rhs: term_EF_7_rhs
verificationDomain: Geometric
comment: Embedding functor maps unit group into target unit group.

AA_1
type: NamedIndividual
type: Identity
forAll: term_AA_1_forAll
lhs: term_AA_1_lhs
rhs: term_AA_1_rhs
verificationDomain: Analytical
comment: Braille glyph encoding: 6-bit blocks to Braille characters.

AA_2
type: NamedIndividual
type: Identity
forAll: term_AA_2_forAll
lhs: term_AA_2_lhs
rhs: term_AA_2_rhs
verificationDomain: Analytical
comment: Braille XOR homomorphism: Braille encoding commutes with XOR.

AA_3
type: NamedIndividual
type: Identity
forAll: term_AA_3_forAll
lhs: term_AA_3_lhs
rhs: term_AA_3_rhs
verificationDomain: Analytical
comment: Braille complement: glyph of bnot(x) is character-wise complement of glyph(x).

AA_4
type: NamedIndividual
type: Identity
forAll: term_AA_4_forAll
lhs: term_AA_4_lhs
rhs: term_AA_4_rhs
verificationDomain: Analytical
comment: Addition does not lift to address space: no glyph homomorphism for add.

AA_5
type: NamedIndividual
type: Identity
forAll: term_AA_5_forAll
lhs: term_AA_5_lhs
rhs: term_AA_5_rhs
verificationDomain: Analytical
comment: Liftable operations are exactly the Boolean operations.

AA_6
type: NamedIndividual
type: Identity
forAll: term_AA_6_forAll
lhs: term_AA_6_lhs
rhs: term_AA_6_rhs
verificationDomain: Analytical
comment: Negation decomposes into liftable bnot and non-liftable succ.

AM_1
type: NamedIndividual
type: Identity
forAll: term_AM_1_forAll
lhs: term_AM_1_lhs
rhs: term_AM_1_rhs
verificationDomain: Analytical
comment: Address metric is sum of per-character Hamming distances.

AM_2
type: NamedIndividual
type: Identity
forAll: term_AM_2_forAll
lhs: term_AM_2_lhs
rhs: term_AM_2_rhs
verificationDomain: Analytical
comment: Address metric equals Hamming metric on ring elements.

AM_3
type: NamedIndividual
type: Identity
forAll: term_AM_3_forAll
lhs: term_AM_3_lhs
rhs: term_AM_3_rhs
verificationDomain: Analytical
comment: Address metric does not preserve ring metric in general.

AM_4
type: NamedIndividual
type: Identity
forAll: term_AM_4_forAll
lhs: term_AM_4_lhs
rhs: term_AM_4_rhs
verificationDomain: Analytical
comment: Address incompatibility metric.

TH_1
type: NamedIndividual
type: Identity
forAll: term_TH_1_forAll
lhs: term_TH_1_lhs
rhs: term_TH_1_rhs
verificationDomain: Thermodynamic
comment: Entropy of a site budget state.

TH_2
type: NamedIndividual
type: Identity
forAll: term_TH_2_forAll
lhs: term_TH_2_lhs
rhs: term_TH_2_rhs
verificationDomain: Thermodynamic
comment: Maximum entropy: unconstrained state has S = n × ln 2.

TH_3
type: NamedIndividual
type: Identity
forAll: term_TH_3_forAll
lhs: term_TH_3_lhs
rhs: term_TH_3_rhs
verificationDomain: Thermodynamic
comment: Zero entropy: fully resolved state has S = 0.

TH_4
type: NamedIndividual
type: Identity
forAll: term_TH_4_forAll
lhs: term_TH_4_lhs
rhs: term_TH_4_rhs
verificationDomain: Thermodynamic
comment: Landauer bound on total resolution cost.

TH_5
type: NamedIndividual
type: Identity
forAll: term_TH_5_forAll
lhs: term_TH_5_lhs
rhs: term_TH_5_rhs
verificationDomain: Thermodynamic
comment: Critical inverse temperature for UOR site system.

TH_6
type: NamedIndividual
type: Identity
forAll: term_TH_6_forAll
lhs: term_TH_6_lhs
rhs: term_TH_6_rhs
verificationDomain: Thermodynamic
comment: Constraint application removes entropy; convergence rate equals cooling rate.

TH_7
type: NamedIndividual
type: Identity
forAll: term_TH_7_forAll
lhs: term_TH_7_lhs
rhs: term_TH_7_rhs
verificationDomain: Thermodynamic
comment: CatastropheThreshold is the temperature of a partition phase transition.

TH_8
type: NamedIndividual
type: Identity
forAll: term_TH_8_forAll
lhs: term_TH_8_lhs
rhs: term_TH_8_rhs
verificationDomain: Thermodynamic
comment: Step formula as free-energy cost of a constraint boundary.

TH_9
type: NamedIndividual
type: Identity
forAll: term_TH_9_forAll
lhs: term_TH_9_lhs
rhs: term_TH_9_rhs
verificationDomain: Thermodynamic
comment: Computational hardness maps to type incompleteness (high temperature).

TH_10
type: NamedIndividual
type: Identity
forAll: term_TH_10_forAll
lhs: term_TH_10_lhs
rhs: term_TH_10_rhs
verificationDomain: Thermodynamic
comment: Type resolution is measurement; cost ≥ entropy removed.

AR_1
type: NamedIndividual
type: Identity
forAll: term_AR_1_forAll
lhs: term_AR_1_lhs
rhs: term_AR_1_rhs
verificationDomain: Analytical
comment: Adiabatic schedule: decreasing freeRank, cost-per-site ordering.

AR_2
type: NamedIndividual
type: Identity
forAll: term_AR_2_forAll
lhs: term_AR_2_lhs
rhs: term_AR_2_rhs
verificationDomain: Analytical
comment: Adiabatic cost is sum of constraint costs in optimal order.

AR_3
type: NamedIndividual
type: Identity
forAll: term_AR_3_forAll
lhs: term_AR_3_lhs
rhs: term_AR_3_rhs
verificationDomain: Analytical
comment: Adiabatic cost satisfies Landauer bound.

AR_4
type: NamedIndividual
type: Identity
forAll: term_AR_4_forAll
lhs: term_AR_4_lhs
rhs: term_AR_4_rhs
verificationDomain: Analytical
comment: Adiabatic efficiency η ≤ 1.

AR_5
type: NamedIndividual
type: Identity
forAll: term_AR_5_forAll
lhs: term_AR_5_lhs
rhs: term_AR_5_rhs
verificationDomain: Analytical
comment: Greedy vs adiabatic cost difference: ≤ 5% for n ≤ 16.

PD_1
type: NamedIndividual
type: Identity
forAll: term_PD_1_forAll
lhs: term_PD_1_lhs
rhs: term_PD_1_rhs
verificationDomain: Thermodynamic
comment: Phase space definition.

PD_2
type: NamedIndividual
type: Identity
forAll: term_PD_2_forAll
lhs: term_PD_2_lhs
rhs: term_PD_2_rhs
verificationDomain: Thermodynamic
comment: Phase boundaries occur where g divides 2^n − 1 or g is a power of 2.

PD_3
type: NamedIndividual
type: Identity
forAll: term_PD_3_forAll
lhs: term_PD_3_lhs
rhs: term_PD_3_rhs
verificationDomain: Thermodynamic
comment: Parity boundary divides R_n into equal halves.

PD_4
type: NamedIndividual
type: Identity
forAll: term_PD_4_forAll
lhs: term_PD_4_lhs
rhs: term_PD_4_rhs
verificationDomain: Thermodynamic
comment: Resonance lines in the phase diagram.

PD_5
type: NamedIndividual
type: Identity
forAll: term_PD_5_forAll
lhs: term_PD_5_lhs
rhs: term_PD_5_rhs
verificationDomain: Thermodynamic
comment: Phase count at level n is at most 2^n (typically O(n)).

RC_1
type: NamedIndividual
type: Identity
forAll: term_RC_1_forAll
lhs: term_RC_1_lhs
rhs: term_RC_1_rhs
verificationDomain: Thermodynamic
comment: Reversible pinning stores prior state in ancilla site.

RC_2
type: NamedIndividual
type: Identity
forAll: term_RC_2_forAll
lhs: term_RC_2_lhs
rhs: term_RC_2_rhs
verificationDomain: Thermodynamic
comment: Reversible pinning has zero total entropy change.

RC_3
type: NamedIndividual
type: Identity
forAll: term_RC_3_forAll
lhs: term_RC_3_lhs
rhs: term_RC_3_rhs
verificationDomain: Thermodynamic
comment: Deferred Landauer cost at ancilla erasure.

RC_4
type: NamedIndividual
type: Identity
forAll: term_RC_4_forAll
lhs: term_RC_4_lhs
rhs: term_RC_4_rhs
verificationDomain: Thermodynamic
comment: Reversible total cost equals irreversible total cost (redistributed).

RC_5
type: NamedIndividual
type: Identity
forAll: term_RC_5_forAll
lhs: term_RC_5_lhs
rhs: term_RC_5_rhs
verificationDomain: Thermodynamic
comment: Quantum UOR: superposed sites, cost proportional to winning path.

DC_1
type: NamedIndividual
type: Identity
forAll: term_DC_1_forAll
lhs: term_DC_1_lhs
rhs: term_DC_1_rhs
verificationDomain: Analytical
comment: Ring derivative: ∂_R f(x) = f(succ(x)) - f(x).

DC_2
type: NamedIndividual
type: Identity
forAll: term_DC_2_forAll
lhs: term_DC_2_lhs
rhs: term_DC_2_rhs
verificationDomain: Analytical
comment: Hamming derivative: ∂_H f(x) = f(bnot(x)) - f(x).

DC_3
type: NamedIndividual
type: Identity
forAll: term_DC_3_forAll
lhs: term_DC_3_lhs
rhs: term_DC_3_rhs
verificationDomain: Analytical
comment: Hamming derivative of identity: ∂_H id(x) = -(2x + 1) mod 2^n.

DC_4
type: NamedIndividual
type: Identity
forAll: term_DC_4_forAll
lhs: term_DC_4_lhs
rhs: term_DC_4_rhs
verificationDomain: Analytical
comment: Commutator from derivatives: [neg, bnot](x) = ∂_R neg(x) - ∂_H neg(x).

DC_5
type: NamedIndividual
type: Identity
forAll: term_DC_5_forAll
lhs: term_DC_5_lhs
rhs: term_DC_5_rhs
verificationDomain: Analytical
comment: Carry dependence: the difference ∂_R f - ∂_H f decomposes into carry contributions.

DC_6
type: NamedIndividual
type: Identity
forAll: term_DC_6_forAll
lhs: term_DC_6_lhs
rhs: term_DC_6_rhs
verificationDomain: Analytical
comment: Jacobian definition: J_k(x) = ∂_R f_k(x) where f_k = site_k.

DC_7
type: NamedIndividual
type: Identity
forAll: term_DC_7_forAll
lhs: term_DC_7_lhs
rhs: term_DC_7_rhs
verificationDomain: Analytical
comment: Top-site anomaly: J_{n-1}(x) may differ from lower sites.

DC_8
type: NamedIndividual
type: Identity
forAll: term_DC_8_forAll
lhs: term_DC_8_lhs
rhs: term_DC_8_rhs
verificationDomain: Analytical
comment: Rank-curvature identity: rank(J(x)) = d_H(x, succ(x)) - 1 for generic x.

DC_9
type: NamedIndividual
type: Identity
forAll: term_DC_9_forAll
lhs: term_DC_9_lhs
rhs: term_DC_9_rhs
verificationDomain: Analytical
comment: Total curvature from Jacobian: κ(x) = Σ_k J_k(x).

DC_10
type: NamedIndividual
type: Identity
forAll: term_DC_10_forAll
lhs: term_DC_10_lhs
rhs: term_DC_10_rhs
verificationDomain: Analytical
comment: Curvature-weighted ordering: optimal next constraint maximizes J_k over free sites.

DC_11
type: NamedIndividual
type: Identity
forAll: term_DC_11_forAll
lhs: term_DC_11_lhs
rhs: term_DC_11_rhs
verificationDomain: Analytical
comment: Curvature equipartition: Σ_{x} J_k(x) ≈ (2^n - 2)/n for each k.

HA_1
type: NamedIndividual
type: Identity
forAll: term_HA_1_forAll
lhs: term_HA_1_lhs
rhs: term_HA_1_rhs
verificationDomain: Topological
comment: Constraint nerve: N(C) is the simplicial complex on constraints.

HA_2
type: NamedIndividual
type: Identity
forAll: term_HA_2_forAll
lhs: term_HA_2_lhs
rhs: term_HA_2_rhs
verificationDomain: Topological
comment: Stall iff non-trivial homology: resolution stalls ⟺ H_k(N(C)) ≠ 0 for some k > 0.

HA_3
type: NamedIndividual
type: Identity
forAll: term_HA_3_forAll
lhs: term_HA_3_lhs
rhs: term_HA_3_rhs
verificationDomain: Topological
comment: Betti-entropy theorem: S_residual ≥ Σ_k β_k × ln 2.

IT_2
type: NamedIndividual
type: Identity
forAll: term_IT_2_forAll
lhs: term_IT_2_lhs
rhs: term_IT_2_rhs
verificationDomain: Topological
comment: Euler-Poincaré formula for constraint nerve.

IT_3
type: NamedIndividual
type: Identity
forAll: term_IT_3_forAll
lhs: term_IT_3_lhs
rhs: term_IT_3_rhs
verificationDomain: Topological
comment: Spectral Euler characteristic.

IT_6
type: NamedIndividual
type: Identity
forAll: term_IT_6_forAll
lhs: term_IT_6_lhs
rhs: term_IT_6_rhs
verificationDomain: Topological
comment: Spectral gap bounds convergence rate from below.

IT_7a
type: NamedIndividual
type: Identity
forAll: term_IT_7a_forAll
lhs: term_IT_7a_lhs
rhs: term_IT_7a_rhs
verificationDomain: IndexTheoretic
verificationDomain: Analytical
verificationDomain: Topological
comment: UOR index theorem (topological form): total curvature minus Euler characteristic equals residual entropy in bits.

IT_7b
type: NamedIndividual
type: Identity
forAll: term_IT_7b_forAll
lhs: term_IT_7b_lhs
rhs: term_IT_7b_rhs
verificationDomain: IndexTheoretic
verificationDomain: Analytical
verificationDomain: Topological
comment: UOR index theorem (entropy-topology duality).

IT_7c
type: NamedIndividual
type: Identity
forAll: term_IT_7c_forAll
lhs: term_IT_7c_lhs
rhs: term_IT_7c_rhs
verificationDomain: IndexTheoretic
verificationDomain: Analytical
verificationDomain: Topological
comment: UOR index theorem (spectral cost bound): resolution cost ≥ n - χ(N(C)).

IT_7d
type: NamedIndividual
type: Identity
forAll: term_IT_7d_forAll
lhs: term_IT_7d_lhs
rhs: term_IT_7d_rhs
verificationDomain: IndexTheoretic
verificationDomain: Analytical
verificationDomain: Topological
comment: UOR index theorem (completeness criterion): resolution is complete iff χ(N(C)) = n and all Betti numbers vanish.

phi_1
type: NamedIndividual
type: Identity
forAll: term_phi_1_forAll
lhs: term_phi_1_lhs
rhs: term_phi_1_rhs
verificationDomain: Pipeline
comment: Ring → Constraints map: negation transforms a residue constraint.

phi_2
type: NamedIndividual
type: Identity
forAll: term_phi_2_forAll
lhs: term_phi_2_lhs
rhs: term_phi_2_rhs
verificationDomain: Pipeline
comment: Constraints → Sites map: composition maps to union of site pinnings.

phi_3
type: NamedIndividual
type: Identity
forAll: term_phi_3_forAll
lhs: term_phi_3_lhs
rhs: term_phi_3_rhs
verificationDomain: Pipeline
comment: Sites → Partition map: a closed site state determines a unique 4-component partition.

phi_4
type: NamedIndividual
type: Identity
forAll: term_phi_4_forAll
lhs: term_phi_4_lhs
rhs: term_phi_4_rhs
verificationDomain: Pipeline
comment: Resolution pipeline: φ₄ = φ₃ ∘ φ₂ ∘ φ₁ is the composite resolution map.

phi_5
type: NamedIndividual
type: Identity
forAll: term_phi_5_forAll
lhs: term_phi_5_lhs
rhs: term_phi_5_rhs
verificationDomain: Pipeline
comment: Operations → Observables map: negation preserves d_R, may change d_H.

phi_6
type: NamedIndividual
type: Identity
forAll: term_phi_6_forAll
lhs: term_phi_6_lhs
rhs: term_phi_6_rhs
verificationDomain: Pipeline
comment: Observables → Refinement map: observables on a state yield a refinement suggestion.

psi_1
type: NamedIndividual
type: Identity
forAll: term_psi_1_forAll
lhs: term_psi_1_lhs
rhs: term_psi_1_rhs
verificationDomain: Topological
comment: ψ_1: Constraints → SimplicialComplex (nerve construction). Maps a set of constraints to its nerve simplicial complex.

psi_2
type: NamedIndividual
type: Identity
forAll: term_psi_2_forAll
lhs: term_psi_2_lhs
rhs: term_psi_2_rhs
verificationDomain: Topological
comment: ψ_2: SimplicialComplex → ChainComplex (chain functor). Maps a simplicial complex to its chain complex.

psi_3
type: NamedIndividual
type: Identity
forAll: term_psi_3_forAll
lhs: term_psi_3_lhs
rhs: term_psi_3_rhs
verificationDomain: Topological
comment: ψ_3: ChainComplex → HomologyGroups (homology functor). Computes homology groups from a chain complex.

psi_5
type: NamedIndividual
type: Identity
forAll: term_psi_5_forAll
lhs: term_psi_5_lhs
rhs: term_psi_5_rhs
verificationDomain: Topological
comment: ψ_5: ChainComplex → CochainComplex (dualization functor). Dualizes a chain complex to obtain a cochain complex.

psi_6
type: NamedIndividual
type: Identity
forAll: term_psi_6_forAll
lhs: term_psi_6_lhs
rhs: term_psi_6_rhs
verificationDomain: Topological
comment: ψ_6: CochainComplex → CohomologyGroups (cohomology functor). Computes cohomology groups from a cochain complex.

D_{2^n}
type: NamedIndividual
type: DihedralGroup
generatedBy: neg
generatedBy: bnot
presentation: ⟨r, s | r^{2^n} = s² = e, srs = r⁻¹⟩
comment: The dihedral group of order 2^(n+1), generated by neg (ring reflection) and bnot (hypercube reflection). Every element of this group acts as an isometry on the type space 𝒯_n.

Surface Symmetry
type: NamedIndividual
type: Identity
forAll: term_surfaceSymmetry_forAll
lhs: term_surfaceSymmetry_lhs
rhs: term_surfaceSymmetry_rhs
verificationDomain: Pipeline
comment: The Surface Symmetry Theorem: the composite P∘Π∘G is a well-typed morphism iff G and P share the same state:Frame F. When the shared-frame condition holds, the output lands in the type-equivalent neighbourhood of the source symbol.

CC_1
type: NamedIndividual
type: Identity
forAll: term_CC_1_forAll
lhs: term_CC_1_lhs
rhs: term_CC_1_rhs
verificationDomain: Pipeline
comment: Completeness implies O(1) resolution: a CompleteType T satisfies ∀ x ∈ R_n, resolution(x, T) terminates in O(1).

CC_2
type: NamedIndividual
type: Identity
forAll: term_CC_2_forAll
lhs: term_CC_2_lhs
rhs: term_CC_2_rhs
verificationDomain: Algebraic
comment: Completeness is monotone: if T ⊆ T' (T' has more constraints), then completeness(T) implies completeness(T').

CC_3
type: NamedIndividual
type: Identity
forAll: term_CC_3_forAll
lhs: term_CC_3_lhs
rhs: term_CC_3_rhs
verificationDomain: Algebraic
comment: Witness composition: concat(W₁, W₂) is a valid audit trail iff W₁.sitesClosed + W₂.sitesClosed = n.

CC_4
type: NamedIndividual
type: Identity
forAll: term_CC_4_forAll
lhs: term_CC_4_lhs
rhs: term_CC_4_rhs
verificationDomain: Pipeline
comment: The CompletenessResolver is the unique fixed point of the ψ-pipeline applied to a CompletenessCandidate.

CC_5
type: NamedIndividual
type: Identity
forAll: term_CC_5_forAll
lhs: term_CC_5_lhs
rhs: term_CC_5_rhs
verificationDomain: Topological
comment: CompletenessCertificate soundness: cert.verified = true implies χ(N(C)) = n and for all k: β_k = 0.

QL_1
type: NamedIndividual
type: Identity
forAll: term_QL_1_forAll
lhs: term_QL_1_lhs
rhs: term_QL_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: neg(bnot(x)) = succ(x) holds in Z/(2ⁿ)Z for all n ≥ 1. Universal form of the Critical Identity across all quantum levels.

QL_2
type: NamedIndividual
type: Identity
forAll: term_QL_2_forAll
lhs: term_QL_2_lhs
rhs: term_QL_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: The dihedral group D_{2ⁿ} generated by neg and bnot has order 2ⁿ⁺¹ for all n ≥ 1.

QL_3
type: NamedIndividual
type: Identity
forAll: term_QL_3_forAll
lhs: term_QL_3_lhs
rhs: term_QL_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Thermodynamic
comment: The reduction distribution P(j) = 2^{-j} is the Boltzmann distribution at β* = ln 2 for all n ≥ 1.

QL_4
type: NamedIndividual
type: Identity
forAll: term_QL_4_forAll
lhs: term_QL_4_lhs
rhs: term_QL_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: The site budget of a PrimitiveType at quantum level n is exactly n (one site per bit).

QL_5
type: NamedIndividual
type: Identity
forAll: term_QL_5_forAll
lhs: term_QL_5_lhs
rhs: term_QL_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Resolution complexity for a CompleteType is O(1) for all n ≥ 1.

QL_6
type: NamedIndividual
type: Identity
forAll: term_QL_6_forAll
lhs: term_QL_6_lhs
rhs: term_QL_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Content addressing is a bijection for all n ≥ 1 (AD_1/AD_2 universality).

QL_7
type: NamedIndividual
type: Identity
forAll: term_QL_7_forAll
lhs: term_QL_7_lhs
rhs: term_QL_7_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: The ψ-pipeline produces a valid ChainComplex for any ConstrainedType at any quantum level n ≥ 1.

GR_1
type: NamedIndividual
type: Identity
forAll: term_GR_1_forAll
lhs: term_GR_1_lhs
rhs: term_GR_1_rhs
verificationDomain: Algebraic
comment: Binding monotonicity: freeRank(B_{i+1}) ≤ freeRank(B_i) for all i in a Session.

GR_2
type: NamedIndividual
type: Identity
forAll: term_GR_2_forAll
lhs: term_GR_2_lhs
rhs: term_GR_2_rhs
verificationDomain: Pipeline
comment: Binding soundness: a Binding b is sound iff b.datum resolves under b.constraint in O(1).

GR_3
type: NamedIndividual
type: Identity
forAll: term_GR_3_forAll
lhs: term_GR_3_lhs
rhs: term_GR_3_rhs
verificationDomain: Algebraic
comment: Session convergence: a Session S converges iff there exists i such that freeRank(B_i) = 0.

GR_4
type: NamedIndividual
type: Identity
forAll: term_GR_4_forAll
lhs: term_GR_4_lhs
rhs: term_GR_4_rhs
verificationDomain: Algebraic
comment: Context reset isolation: bindings in C_fresh are independent of bindings in C_prior after a SessionBoundary.

GR_5
type: NamedIndividual
type: Identity
forAll: term_GR_5_forAll
lhs: term_GR_5_lhs
rhs: term_GR_5_rhs
verificationDomain: Algebraic
comment: Contradiction detection: ContradictionBoundary fires iff there exist bindings b, b' with the same address, different datum, under the same constraint.

TS_1
type: NamedIndividual
type: Identity
forAll: term_TS_1_forAll
lhs: term_TS_1_lhs
rhs: term_TS_1_rhs
verificationDomain: Pipeline
comment: Nerve realisability: for any target (χ*, β₀* = 1, β_k* = 0 for k ≥ 1) with χ* ≤ n, there exists a ConstrainedType T over R_n whose constraint nerve realises the target.

TS_2
type: NamedIndividual
type: Identity
forAll: term_TS_2_forAll
lhs: term_TS_2_lhs
rhs: term_TS_2_rhs
verificationDomain: Pipeline
comment: Minimal basis bound: for the IT_7d target (χ* = n, all β* = 0), the MinimalConstraintBasis has size exactly n (one constraint per site position). No redundant constraints exist in the minimal basis.

TS_3
type: NamedIndividual
type: Identity
forAll: term_TS_3_forAll
lhs: term_TS_3_lhs
rhs: term_TS_3_rhs
verificationDomain: Pipeline
comment: Synthesis monotonicity: adding a constraint to a synthesis candidate never decreases the Euler characteristic of the resulting nerve (χ is monotone non-decreasing under constraint addition).

TS_4
type: NamedIndividual
type: Identity
forAll: term_TS_4_forAll
lhs: term_TS_4_lhs
rhs: term_TS_4_rhs
verificationDomain: Pipeline
comment: Synthesis convergence: the TypeSynthesisResolver terminates for any realisable target in at most n constraint additions (for n-site types).

TS_5
type: NamedIndividual
type: Identity
forAll: term_TS_5_forAll
lhs: term_TS_5_lhs
rhs: term_TS_5_rhs
verificationDomain: Pipeline
comment: Synthesis–certification duality: a SynthesizedType T achieves the IT_7d target if and only if the CompletenessResolver certifies T as a CompleteType. The forward ψ-pipeline and the inverse TypeSynthesisResolver are dual computations.

TS_6
type: NamedIndividual
type: Identity
forAll: term_TS_6_forAll
lhs: term_TS_6_lhs
rhs: term_TS_6_rhs
verificationDomain: Pipeline
comment: Jacobian-guided synthesis efficiency: using the Jacobian (DC_10) to select the next constraint reduces the expected number of synthesis steps from O(n²) (uninformed) to O(n log n) (Jacobian-guided).

TS_7
type: NamedIndividual
type: Identity
forAll: term_TS_7_forAll
lhs: term_TS_7_lhs
rhs: term_TS_7_rhs
verificationDomain: Pipeline
comment: Unreachable signatures: a Betti profile with β₀ = 0 is unreachable by any non-empty ConstrainedType — the nerve of a non-empty constraint set is always connected (β₀ ≥ 1).

WLS_1
type: NamedIndividual
type: Identity
forAll: term_WLS_1_forAll
lhs: term_WLS_1_lhs
rhs: term_WLS_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: Lift unobstructedness criterion: WittLift T' is a CompleteType iff the spectral sequence E_r^{p,q} collapses at E_2 (d_2 = 0 and all higher differentials zero).

WLS_2
type: NamedIndividual
type: Identity
forAll: term_WLS_2_forAll
lhs: term_WLS_2_lhs
rhs: term_WLS_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: Obstruction localisation: a non-trivial LiftObstruction is localised to a specific site at bit position n+1. The obstruction class lives in H²(N(C(T))) and is killed by adding one constraint involving the new site.

WLS_3
type: NamedIndividual
type: Identity
forAll: term_WLS_3_forAll
lhs: term_WLS_3_lhs
rhs: term_WLS_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: Monotone lifting for trivially obstructed types: if T is a CompleteType at Q_n and its constraint set is closed under the Q_{n+1} extension map, then T' is a CompleteType at Q_{n+1} with basisSize(T') = basisSize(T) + 1.

WLS_4
type: NamedIndividual
type: Identity
forAll: term_WLS_4_forAll
lhs: term_WLS_4_lhs
rhs: term_WLS_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: Spectral sequence convergence bound: for constraint configurations of homological depth d (H_k(N(C(T))) = 0 for k > d), the spectral sequence converges by page E_{d+2}.

WLS_5
type: NamedIndividual
type: Identity
forAll: term_WLS_5_forAll
lhs: term_WLS_5_lhs
rhs: term_WLS_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: Universal identity preservation: every op:universallyValid identity holds in ℤ/(2^{n+1})ℤ with the lifted constraint set. The lift does not invalidate any certified universal identity.

WLS_6
type: NamedIndividual
type: Identity
forAll: term_WLS_6_forAll
lhs: term_WLS_6_lhs
rhs: term_WLS_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: ψ-pipeline universality for quantum lifts: the ψ-pipeline produces a valid ChainComplex for any WittLift T' — the chain complex of T' restricts to the chain complex of T on the base nerve, and the extension is well-formed by construction.

MN_1
type: NamedIndividual
type: Identity
forAll: term_MN_1_forAll
lhs: term_MN_1_lhs
rhs: term_MN_1_rhs
verificationDomain: Topological
comment: Holonomy group containment: HolonomyGroup(T) ≤ D_{2^n} for all ConstrainedTypes T over R_n. The holonomy group is always a subgroup of the full dihedral group.

MN_2
type: NamedIndividual
type: Identity
forAll: term_MN_2_forAll
lhs: term_MN_2_lhs
rhs: term_MN_2_rhs
verificationDomain: Topological
comment: Additive flatness (extends OB_H1): if all constraints in T are additive (ResidueConstraint or DepthConstraint type), then HolonomyGroup(T) = {id} — T is a FlatType.

MN_3
type: NamedIndividual
type: Identity
forAll: term_MN_3_forAll
lhs: term_MN_3_lhs
rhs: term_MN_3_rhs
verificationDomain: Topological
comment: Dihedral generation: if T contains both a neg-related and a bnot-related constraint in a common closed path, then HolonomyGroup(T) = D_{2^n} — T has full dihedral holonomy.

MN_4
type: NamedIndividual
type: Identity
forAll: term_MN_4_forAll
lhs: term_MN_4_lhs
rhs: term_MN_4_rhs
verificationDomain: Topological
comment: Holonomy-Betti implication: HolonomyGroup(T) ≠ {id} ⟹ β₁(N(C(T))) ≥ 1. Non-trivial monodromy requires a topological loop. (Converse is false: β₁ ≥ 1 does not imply non-trivial holonomy.)

MN_5
type: NamedIndividual
type: Identity
forAll: term_MN_5_forAll
lhs: term_MN_5_lhs
rhs: term_MN_5_rhs
verificationDomain: Topological
comment: CompleteType holonomy: a CompleteType (IT_7d: χ = n, all β = 0) has trivial holonomy. IT_7d implies FlatType because IT_7d requires β₁ = 0, which by MN_4 implies trivial monodromy.

MN_6
type: NamedIndividual
type: Identity
forAll: term_MN_6_forAll
lhs: term_MN_6_lhs
rhs: term_MN_6_rhs
verificationDomain: Topological
comment: Monodromy composition: if p₁ and p₂ are closed constraint paths, then monodromy(p₁ · p₂) = monodromy(p₁) · monodromy(p₂) in D_{2^n} (group homomorphism from loops to dihedral elements).

MN_7
type: NamedIndividual
type: Identity
forAll: term_MN_7_forAll
lhs: term_MN_7_lhs
rhs: term_MN_7_rhs
verificationDomain: Topological
comment: TwistedType obstruction class: the monodromy of a TwistedType contributes a non-zero class to H²(N(C(T')); ℤ/2ℤ) where T' is any WittLift of T. TwistedTypes always have non-trivial lift obstructions.

PT_1
type: NamedIndividual
type: Identity
forAll: term_PT_1_forAll
lhs: term_PT_1_lhs
rhs: term_PT_1_rhs
verificationDomain: Algebraic
comment: Product type site additivity: siteBudget(A × B) = siteBudget(A) + siteBudget(B).

PT_2
type: NamedIndividual
type: Identity
forAll: term_PT_2_forAll
lhs: term_PT_2_lhs
rhs: term_PT_2_rhs
verificationDomain: Topological
comment: Product type partition product: partition(A × B) = partition(A) ⊗ partition(B).

PT_3
type: NamedIndividual
type: Identity
forAll: term_PT_3_forAll
lhs: term_PT_3_lhs
rhs: term_PT_3_rhs
verificationDomain: IndexTheoretic
comment: Product type Euler additivity: χ(N(C(A × B))) = χ(N(C(A))) + χ(N(C(B))).

PT_4
type: NamedIndividual
type: Identity
forAll: term_PT_4_forAll
lhs: term_PT_4_lhs
rhs: term_PT_4_rhs
verificationDomain: Thermodynamic
comment: Product type entropy additivity: S(A × B) = S(A) + S(B).

ST_1
type: NamedIndividual
type: Identity
forAll: term_ST_1_forAll
lhs: term_ST_1_lhs
rhs: term_ST_1_rhs
verificationDomain: Algebraic
comment: Sum type site budget: siteBudget(A + B) = max(siteBudget(A), siteBudget(B)).

ST_2
type: NamedIndividual
type: Identity
forAll: term_ST_2_forAll
lhs: term_ST_2_lhs
rhs: term_ST_2_rhs
verificationDomain: Thermodynamic
comment: Sum type entropy: S(A + B) = ln 2 + max(S(A), S(B)).

GS_1
type: NamedIndividual
type: Identity
forAll: term_GS_1_forAll
lhs: term_GS_1_lhs
rhs: term_GS_1_rhs
verificationDomain: Thermodynamic
comment: Context temperature: T_ctx(C) = freeRank(C) × ln 2 / n.

GS_2
type: NamedIndividual
type: Identity
forAll: term_GS_2_forAll
lhs: term_GS_2_lhs
rhs: term_GS_2_rhs
verificationDomain: Algebraic
comment: Grounding degree: σ(C) = (n − freeRank(C)) / n.

GS_3
type: NamedIndividual
type: Identity
forAll: term_GS_3_forAll
lhs: term_GS_3_lhs
rhs: term_GS_3_rhs
verificationDomain: Algebraic
comment: Grounding monotonicity: σ(B_{i+1}) ≥ σ(B_i) for all i in a Session.

GS_4
type: NamedIndividual
type: Identity
forAll: term_GS_4_forAll
lhs: term_GS_4_lhs
rhs: term_GS_4_rhs
verificationDomain: Thermodynamic
comment: Ground state equivalence: σ(C) = 1 ↔ freeRank(C) = 0 ↔ S(C) = 0 ↔ T_ctx(C) = 0.

GS_5
type: NamedIndividual
type: Identity
forAll: term_GS_5_forAll
lhs: term_GS_5_lhs
rhs: term_GS_5_rhs
verificationDomain: Pipeline
comment: O(1) resolution guarantee: freeRank(C) = 0 ∧ q.address ∈ bindings(C) → stepCount(q, C) = 0.

GS_6
type: NamedIndividual
type: Identity
forAll: term_GS_6_forAll
lhs: term_GS_6_lhs
rhs: term_GS_6_rhs
verificationDomain: Algebraic
comment: Pre-reduction of effective budget: effectiveBudget(q, C) = max(0, siteBudget(q.type) − |pinnedSites(C) ∩ q.siteSet|).

GS_7
type: NamedIndividual
type: Identity
forAll: term_GS_7_forAll
lhs: term_GS_7_lhs
rhs: term_GS_7_rhs
verificationDomain: Thermodynamic
comment: Thermodynamic cooling cost: Cost_saturation(C) = n × k_B T × ln 2.

MS_1
type: NamedIndividual
type: Identity
forAll: term_MS_1_forAll
lhs: term_MS_1_lhs
rhs: term_MS_1_rhs
verificationDomain: Pipeline
comment: Connectivity lower bound: β₀(N(C)) ≥ 1 for all non-empty C.

MS_2
type: NamedIndividual
type: Identity
forAll: term_MS_2_forAll
lhs: term_MS_2_lhs
rhs: term_MS_2_rhs
verificationDomain: Algebraic
comment: Euler capacity ceiling: χ(N(C)) ≤ n for all C at quantum level n.

MS_3
type: NamedIndividual
type: Identity
forAll: term_MS_3_forAll
lhs: term_MS_3_lhs
rhs: term_MS_3_rhs
verificationDomain: Topological
comment: Betti monotonicity under addition: χ(N(C + c)) ≥ χ(N(C)) for any constraint c added to C.

MS_4
type: NamedIndividual
type: Identity
forAll: term_MS_4_forAll
lhs: term_MS_4_lhs
rhs: term_MS_4_rhs
verificationDomain: Pipeline
comment: Level-relative achievability: a signature achievable at quantum level n remains achievable at level n+1.

MS_5
type: NamedIndividual
type: Identity
forAll: term_MS_5_forAll
lhs: term_MS_5_lhs
rhs: term_MS_5_rhs
verificationDomain: Pipeline
comment: Empirical completeness convergence: verified SynthesisSignature individuals converge to the exact morphospace boundary.

GD_1
type: NamedIndividual
type: Identity
forAll: term_GD_1_forAll
lhs: term_GD_1_lhs
rhs: term_GD_1_rhs
verificationDomain: Analytical
comment: Geodesic condition: a ComputationTrace is a geodesic iff its steps are AR_1-ordered and each step selects the constraint maximising J_k over free sites (DC_10).

GD_2
type: NamedIndividual
type: Identity
forAll: term_GD_2_forAll
lhs: term_GD_2_lhs
rhs: term_GD_2_rhs
verificationDomain: Thermodynamic
comment: Geodesic entropy bound: ΔS_step(i) = ln 2 for every step i of a geodesic trace.

GD_3
type: NamedIndividual
type: Identity
forAll: term_GD_3_forAll
lhs: term_GD_3_lhs
rhs: term_GD_3_rhs
verificationDomain: Thermodynamic
comment: Total geodesic cost: Cost_geodesic(T) = freeRank_initial × k_B T ln 2 = TH_4.

GD_4
type: NamedIndividual
type: Identity
forAll: term_GD_4_forAll
lhs: term_GD_4_lhs
rhs: term_GD_4_rhs
verificationDomain: Analytical
comment: Geodesic uniqueness up to step-order equivalence: all geodesics for the same ConstrainedType share stepCount and constraint set.

GD_5
type: NamedIndividual
type: Identity
forAll: term_GD_5_forAll
lhs: term_GD_5_lhs
rhs: term_GD_5_rhs
verificationDomain: Pipeline
comment: Subgeodesic detectability: a trace is sub-geodesic iff ∃ step i where J_k(step_i) < max_{free sites} J_k(state_i).

QM_1
type: NamedIndividual
type: Identity
forAll: term_QM_1_forAll
lhs: term_QM_1_lhs
rhs: term_QM_1_rhs
verificationDomain: QuantumThermodynamic
comment: Von Neumann–Landauer bridge: S_vonNeumann(ψ) = Cost_Landauer(collapse(ψ)).

QM_2
type: NamedIndividual
type: Identity
forAll: term_QM_2_forAll
lhs: term_QM_2_lhs
rhs: term_QM_2_rhs
verificationDomain: Topological
comment: Measurement as site topology change: projective collapse on a SuperposedSiteState is topologically equivalent to applying a ResidueConstraint that pins the collapsed site.

QM_3
type: NamedIndividual
type: Identity
forAll: term_QM_3_forAll
lhs: term_QM_3_lhs
rhs: term_QM_3_rhs
verificationDomain: QuantumThermodynamic
comment: Superposition entropy bound: 0 ≤ S_vN(ψ) ≤ ln 2 for any single-site SuperposedSiteState.

QM_4
type: NamedIndividual
type: Identity
forAll: term_QM_4_forAll
lhs: term_QM_4_lhs
rhs: term_QM_4_rhs
verificationDomain: Algebraic
comment: Collapse idempotence: collapse(collapse(ψ)) = collapse(ψ). Measurement on an already-collapsed state is a no-op.

QM_5
type: NamedIndividual
type: Identity
forAll: term_QM_5_forAll
lhs: term_QM_5_lhs
rhs: term_QM_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: QuantumThermodynamic
comment: Amplitude normalization (Born rule): the sum of squared amplitudes equals 1 for any well-formed SuperposedSiteState.

RC_6
type: NamedIndividual
type: Identity
forAll: term_RC_6_forAll
lhs: term_RC_6_lhs
rhs: term_RC_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: SuperpositionDomain
comment: Amplitude renormalization: a SuperposedSiteState can always be normalized to satisfy QM_5.

FPM_8
type: NamedIndividual
type: Identity
forAll: term_FPM_8_forAll
lhs: term_FPM_8_lhs
rhs: term_FPM_8_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Enumerative
comment: Partition exhaustiveness: the four component cardinalities sum to the ring size.

FPM_9
type: NamedIndividual
type: Identity
forAll: term_FPM_9_forAll
lhs: term_FPM_9_lhs
rhs: term_FPM_9_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Exterior membership criterion: x is exterior iff x is not in the carrier of T.

MN_8
type: NamedIndividual
type: Identity
forAll: term_MN_8_forAll
lhs: term_MN_8_lhs
rhs: term_MN_8_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: Holonomy classification covering: every ConstrainedType with a computed holonomy group is either flat or twisted, not both.

QL_8
type: NamedIndividual
type: Identity
forAll: term_QL_8_forAll
lhs: term_QL_8_lhs
rhs: term_QL_8_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Witt level chain inverse: wittLevelPredecessor is the left inverse of nextWittLevel.

D_7
type: NamedIndividual
type: Identity
forAll: term_D_7_forAll
lhs: term_D_7_lhs
rhs: term_D_7_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Geometric
comment: Dihedral composition rule: (rᵃ sᵖ)(rᵇ sᵠ) = r^(a + (-1)ᵖ b mod 2ⁿ) s^(p xor q).

SP_1
type: NamedIndividual
type: Identity
forAll: term_SP_1_forAll
lhs: term_SP_1_lhs
rhs: term_SP_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: SuperpositionDomain
comment: Classical embedding: superposition resolution of a classical (non-superposed) datum reduces to classical resolution.

SP_2
type: NamedIndividual
type: Identity
forAll: term_SP_2_forAll
lhs: term_SP_2_lhs
rhs: term_SP_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: QuantumThermodynamic
comment: Collapse–resolve commutativity: collapsing then resolving classically equals resolving in superposition then collapsing.

SP_3
type: NamedIndividual
type: Identity
forAll: term_SP_3_forAll
lhs: term_SP_3_lhs
rhs: term_SP_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: SuperpositionDomain
comment: Amplitude preservation: the SuperpositionResolver preserves the normalized amplitude vector during ψ-pipeline traversal.

SP_4
type: NamedIndividual
type: Identity
forAll: term_SP_4_forAll
lhs: term_SP_4_lhs
rhs: term_SP_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: QuantumThermodynamic
comment: Born rule outcome probability: the probability of collapsing to site k equals the squared amplitude of that site.

PT_2a
type: NamedIndividual
type: Identity
forAll: term_PT_2a_forAll
lhs: term_PT_2a_lhs
rhs: term_PT_2a_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Product type partition tensor: the partition of a product type is the tensor product of the component partitions.

PT_2b
type: NamedIndividual
type: Identity
forAll: term_PT_2b_forAll
lhs: term_PT_2b_lhs
rhs: term_PT_2b_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Sum type partition coproduct: the partition of a sum type is the coproduct of the variant partitions.

GD_6
type: NamedIndividual
type: Identity
forAll: term_GD_6_forAll
lhs: term_GD_6_lhs
rhs: term_GD_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Analytical
comment: Geodesic predicate decomposition: a trace is geodesic iff it is both AR_1-ordered and DC_10-selected.

WT_1
type: NamedIndividual
type: Identity
forAll: term_WT_1_forAll
lhs: term_WT_1_lhs
rhs: term_WT_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: LiftChain(Q_j, Q_k) is valid CompleteType tower iff every chainStep WittLift has trivial or resolved LiftObstruction.

WT_2
type: NamedIndividual
type: Identity
forAll: term_WT_2_forAll
lhs: term_WT_2_lhs
rhs: term_WT_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: Obstruction count bound: the number of non-trivial LiftObstructions in a chain is at most the chain length.

WT_3
type: NamedIndividual
type: Identity
forAll: term_WT_3_forAll
lhs: term_WT_3_lhs
rhs: term_WT_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: Resolved basis size formula: the basis size at Q_k equals basisSize(Q_j) + chainLength + obstructionResolutionCost.

WT_4
type: NamedIndividual
type: Identity
forAll: term_WT_4_forAll
lhs: term_WT_4_lhs
rhs: term_WT_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: Flat tower characterization: isFlat(chain) iff obstructionCount = 0 iff HolonomyGroup trivial at every step.

WT_5
type: NamedIndividual
type: Identity
forAll: term_WT_5_forAll
lhs: term_WT_5_lhs
rhs: term_WT_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: LiftChainCertificate existence: a CompleteType at Q_k satisfies IT_7d with a witness chain.

WT_6
type: NamedIndividual
type: Identity
forAll: term_WT_6_forAll
lhs: term_WT_6_lhs
rhs: term_WT_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: Single-step reduction: QT_3 with chainLength=1 and cost=0 reduces to QLS_3.

WT_7
type: NamedIndividual
type: Identity
forAll: term_WT_7_forAll
lhs: term_WT_7_lhs
rhs: term_WT_7_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: Flat chain basis size: for flat chains, resolvedBasisSize(Q_k) = basisSize(Q_j) + (k - j).

CC_PINS
type: NamedIndividual
type: Identity
forAll: term_CC_PINS_forAll
lhs: term_CC_PINS_lhs
rhs: term_CC_PINS_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Carry-constraint site-pinning map: pinsSites(CarryConstraint(p)) equals the set of bit positions where p has a 1 plus the first-zero stopping position.

CC_COST_SITE
type: NamedIndividual
type: Identity
forAll: term_CC_COST_SITE_forAll
lhs: term_CC_COST_SITE_lhs
rhs: term_CC_COST_SITE_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Enumerative
comment: Carry-constraint cost-to-site count: the number of sites pinned by a CarryConstraint equals popcount plus one for the stopping position.

jsat_RR
type: NamedIndividual
type: Identity
forAll: term_jsat_RR_forAll
lhs: term_jsat_RR_lhs
rhs: term_jsat_RR_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: CRT joint satisfiability: two ResidueConstraints are jointly satisfiable iff the gcd of their moduli divides the difference of their residues.

jsat_CR
type: NamedIndividual
type: Identity
forAll: term_jsat_CR_forAll
lhs: term_jsat_CR_lhs
rhs: term_jsat_CR_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Carry-residue joint satisfiability: a CarryConstraint and ResidueConstraint are jointly satisfiable iff the carry-pinned sites are compatible with the residue class.

jsat_CC
type: NamedIndividual
type: Identity
forAll: term_jsat_CC_forAll
lhs: term_jsat_CC_lhs
rhs: term_jsat_CC_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Enumerative
comment: Carry-carry joint satisfiability: two CarryConstraints are jointly satisfiable iff their bit-patterns have no conflicting positions.

D_8
type: NamedIndividual
type: Identity
forAll: term_D_8_forAll
lhs: term_D_8_lhs
rhs: term_D_8_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Dihedral inverse formula: the inverse of r^a s^p in D_(2^n) is r^(-(−1)^p a mod 2^n) s^p.

D_9
type: NamedIndividual
type: Identity
forAll: term_D_9_forAll
lhs: term_D_9_lhs
rhs: term_D_9_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Dihedral reflection order: every reflection element r^k s^1 in D_(2^n) has order 2.

EXP_1
type: NamedIndividual
type: Identity
forAll: term_EXP_1_forAll
lhs: term_EXP_1_lhs
rhs: term_EXP_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Monotone carrier characterization: a ConstrainedType has an upward-closed carrier iff every ResidueConstraint has residue = modulus - 1 and no CarryConstraint or DepthConstraint appears.

EXP_2
type: NamedIndividual
type: Identity
forAll: term_EXP_2_forAll
lhs: term_EXP_2_lhs
rhs: term_EXP_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Enumerative
comment: Monotone constraint count: the number of expressible monotone ConstrainedTypes at quantum level Q_n is 2^n, corresponding to the principal filter count.

EXP_3
type: NamedIndividual
type: Identity
forAll: term_EXP_3_forAll
lhs: term_EXP_3_lhs
rhs: term_EXP_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: SumType carrier semantics: the carrier of a SumType is the coproduct (disjoint union) of component carriers, not the set-theoretic union.

ST_3
type: NamedIndividual
type: Identity
forAll: term_ST_3_forAll
lhs: term_ST_3_lhs
rhs: term_ST_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: SumType Euler characteristic additivity: for a SumType with topologically disjoint component nerves, the Euler characteristic is additive.

ST_4
type: NamedIndividual
type: Identity
forAll: term_ST_4_forAll
lhs: term_ST_4_lhs
rhs: term_ST_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: SumType Betti number additivity: for disjoint component nerves, all Betti numbers are additive.

ST_5
type: NamedIndividual
type: Identity
forAll: term_ST_5_forAll
lhs: term_ST_5_lhs
rhs: term_ST_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: SumType completeness transfer: a SumType A+B is CompleteType iff both A and B are CompleteType and they have equal quantum levels.

TS_8
type: NamedIndividual
type: Identity
forAll: term_TS_8_forAll
lhs: term_TS_8_lhs
rhs: term_TS_8_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Betti-1 minimum constraint count: the minimum number of constraints needed to achieve first Betti number beta_1 = k in the constraint nerve is 2k + 1.

TS_9
type: NamedIndividual
type: Identity
forAll: term_TS_9_forAll
lhs: term_TS_9_lhs
rhs: term_TS_9_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: TypeSynthesisResolver termination: the resolver terminates in at most 2^n steps for any target signature at quantum level Q_n, returning either a ConstrainedType or a ForbiddenSignature certificate.

TS_10
type: NamedIndividual
type: Identity
forAll: term_TS_10_forAll
lhs: term_TS_10_lhs
rhs: term_TS_10_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: ForbiddenSignature membership criterion: a topological signature is a ForbiddenSignature iff no ConstrainedType with at most n constraints realises it at quantum level Q_n.

WT_8
type: NamedIndividual
type: Identity
forAll: term_WT_8_forAll
lhs: term_WT_8_lhs
rhs: term_WT_8_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: ObstructionChain length bound: the length of the ObstructionChain from Q_j to Q_k is at most (k-j) times C(basisSize(Q_j), 3).

WT_9
type: NamedIndividual
type: Identity
forAll: term_WT_9_forAll
lhs: term_WT_9_lhs
rhs: term_WT_9_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: TowerCompletenessResolver termination: the resolver terminates for any finite LiftChain within the QT_8 bound, producing a CompleteType certificate or a bounded ObstructionChain.

COEFF_1
type: NamedIndividual
type: Identity
forAll: term_COEFF_1_forAll
lhs: term_COEFF_1_lhs
rhs: term_COEFF_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Standard coefficient ring: the coefficient ring for all psi-pipeline cohomology computations in uor.foundation is Z/2Z, consistent with MN_7.

GO_1
type: NamedIndividual
type: Identity
forAll: term_GO_1_forAll
lhs: term_GO_1_lhs
rhs: term_GO_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: GluingObstruction feedback: given a GluingObstruction class in H^1(N(C)), the killing RefinementSuggestion adds a constraint whose pinned sites contain the intersection of the cycle-generating pair.

GR_6
type: NamedIndividual
type: Identity
forAll: term_GR_6_forAll
lhs: term_GR_6_lhs
rhs: term_GR_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Grounding re-entry free rank: for a session at full grounding, a new query q has freeRank equal to the number of q's site coordinates not already bound.

GR_7
type: NamedIndividual
type: Identity
forAll: term_GR_7_forAll
lhs: term_GR_7_lhs
rhs: term_GR_7_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Grounding degree degradation: after re-entry with query q, the grounding degree becomes min(current sigma, 1 - freeRank(q)/n).

QM_6
type: NamedIndividual
type: Identity
forAll: term_QM_6_forAll
lhs: term_QM_6_lhs
rhs: term_QM_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: SuperpositionDomain
comment: Amplitude index set characterization: the amplitude index set of a SuperposedSiteState over ConstrainedType T at Q_n is the set of monotone pinning trajectories consistent with T's constraints.

CIC_1
type: NamedIndividual
type: Identity
forAll: term_CIC_1_forAll
lhs: term_CIC_1_lhs
rhs: term_CIC_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Certificate issuance: every valid Transform admits a TransformCertificate attesting correct source-to-target mapping.

CIC_2
type: NamedIndividual
type: Identity
forAll: term_CIC_2_forAll
lhs: term_CIC_2_lhs
rhs: term_CIC_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Geometric
comment: Certificate issuance: every metric-preserving Transform admits an IsometryCertificate attesting distance preservation.

CIC_3
type: NamedIndividual
type: Identity
forAll: term_CIC_3_forAll
lhs: term_CIC_3_lhs
rhs: term_CIC_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Certificate issuance: every involutive operation f where f(f(x)) = x admits an InvolutionCertificate.

CIC_4
type: NamedIndividual
type: Identity
forAll: term_CIC_4_forAll
lhs: term_CIC_4_lhs
rhs: term_CIC_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Thermodynamic
comment: Certificate issuance: full saturation (σ = 1, freeRank = 0) admits a GroundingCertificate.

CIC_5
type: NamedIndividual
type: Identity
forAll: term_CIC_5_forAll
lhs: term_CIC_5_lhs
rhs: term_CIC_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Certificate issuance: an AR_1-ordered and DC_10-selected trace admits a GeodesicCertificate.

CIC_6
type: NamedIndividual
type: Identity
forAll: term_CIC_6_forAll
lhs: term_CIC_6_lhs
rhs: term_CIC_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: QuantumThermodynamic
comment: Certificate issuance: a MeasurementEvent verifying the von Neumann–Landauer bridge at β* = ln 2 admits a MeasurementCertificate.

CIC_7
type: NamedIndividual
type: Identity
forAll: term_CIC_7_forAll
lhs: term_CIC_7_lhs
rhs: term_CIC_7_rhs
universallyValid: true
validityKind: Universal
verificationDomain: QuantumThermodynamic
comment: Certificate issuance: a MeasurementEvent verifying P(outcome k) = |α_k|² admits a BornRuleVerification certificate.

GC_1
type: NamedIndividual
type: Identity
forAll: term_GC_1_forAll
lhs: term_GC_1_lhs
rhs: term_GC_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Certificate issuance: shared-frame grounding that lands in the type-equivalent neighbourhood admits a GroundingCertificate.

GR_8
type: NamedIndividual
type: Identity
forAll: term_GR_8_forAll
lhs: term_GR_8_lhs
rhs: term_GR_8_rhs
universallyValid: false
validKMin: 0
validityKind: ParametricLower
verificationDomain: Algebraic
comment: Session composition validity: compose(S_A, S_B) is valid at Q_k iff all pinned-site intersections agree at every tower level Q_0 through Q_k.

GR_9
type: NamedIndividual
type: Identity
forAll: term_GR_9_forAll
lhs: term_GR_9_lhs
rhs: term_GR_9_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: ContextLease disjointness: two distinct leases on the same SharedContext have non-overlapping site sets.

GR_10
type: NamedIndividual
type: Identity
forAll: term_GR_10_forAll
lhs: term_GR_10_lhs
rhs: term_GR_10_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: ExecutionPolicy confluence: different execution policies on the same pending query set produce the same final resolved state (Church-Rosser for session resolution).

MC_1
type: NamedIndividual
type: Identity
forAll: term_MC_1_forAll
lhs: term_MC_1_lhs
rhs: term_MC_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Lease partition conserves total budget: the sum of freeRank over all leases equals the SharedContext freeRank.

MC_2
type: NamedIndividual
type: Identity
forAll: term_MC_2_forAll
lhs: term_MC_2_lhs
rhs: term_MC_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Per-lease binding monotonicity: within a leased sub-domain, freeRank decreases monotonically (SR_1 restricted to lease).

MC_3
type: NamedIndividual
type: Identity
forAll: term_MC_3_forAll
lhs: term_MC_3_lhs
rhs: term_MC_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: General composition freeRank via inclusion-exclusion.

MC_4
type: NamedIndividual
type: Identity
forAll: term_MC_4_forAll
lhs: term_MC_4_lhs
rhs: term_MC_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Disjoint-lease composition is additive: the intersection term vanishes when leases are site-disjoint (SR_9).

MC_5
type: NamedIndividual
type: Identity
forAll: term_MC_5_forAll
lhs: term_MC_5_lhs
rhs: term_MC_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Policy-invariant final binding set: different execution policies produce identical SiteBinding records.

MC_6
type: NamedIndividual
type: Identity
forAll: term_MC_6_forAll
lhs: term_MC_6_lhs
rhs: term_MC_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Full lease coverage implies composed saturation: k sessions on disjoint covering leases, each locally converged, produce a GroundedContext via composition.

MC_7
type: NamedIndividual
type: Identity
forAll: term_MC_7_forAll
lhs: term_MC_7_lhs
rhs: term_MC_7_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Distributed O(1) resolution: a query against a composed GroundedContext resolves in zero steps.

MC_8
type: NamedIndividual
type: Identity
forAll: term_MC_8_forAll
lhs: term_MC_8_lhs
rhs: term_MC_8_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Parallelism bound: per-session resolution work is bounded by lease size, not by total site count n.

WC_1
type: NamedIndividual
type: Identity
forAll: term_WC_1_forAll
lhs: term_WC_1_lhs
rhs: term_WC_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Witt coordinate identification: the bit coordinates (x_0, …, x_[n−1]) of x ∈ Z/(2ⁿ)Z are exactly its Witt coordinates under the canonical isomorphism W_n(F_2) ≅ Z/(2ⁿ)Z.

WC_2
type: NamedIndividual
type: Identity
forAll: term_WC_2_forAll
lhs: term_WC_2_lhs
rhs: term_WC_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Witt sum correction equals carry: the k-th Witt addition polynomial correction term S_k − x_k − y_k (mod 2) is exactly the carry c_k(x,y).

WC_3
type: NamedIndividual
type: Identity
forAll: term_WC_3_forAll
lhs: term_WC_3_lhs
rhs: term_WC_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Carry recurrence is the Witt polynomial recurrence: CA_2 implements the ghost equation for S_[k+1] at p=2.

WC_4
type: NamedIndividual
type: Identity
forAll: term_WC_4_forAll
lhs: term_WC_4_lhs
rhs: term_WC_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: The δ-correction at level k equals the single-level carry c_[k+1](x,y). Each application of δ divides by 2, consuming one unit of 2-adic valuation.

WC_5
type: NamedIndividual
type: Identity
forAll: term_WC_5_forAll
lhs: term_WC_5_lhs
rhs: term_WC_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: LiftObstruction is equivalent to δ-nonvanishing: a nontrivial LiftObstruction at Q_[k+1] means δ_k ≠ 0 for some element pair.

WC_6
type: NamedIndividual
type: Identity
forAll: term_WC_6_forAll
lhs: term_WC_6_lhs
rhs: term_WC_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Analytical
comment: Metric discrepancy equals Witt defect: d_Δ(x,y) > 0 iff the ghost map correction (carry) is nonzero.

WC_7
type: NamedIndividual
type: Identity
forAll: term_WC_7_forAll
lhs: term_WC_7_lhs
rhs: term_WC_7_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: D_1 is the Witt truncation order relation: succ^[2ⁿ](x) = x is the group relation r^[2ⁿ] = 1 in the Witt-Burnside ring of D_[2∞].

WC_8
type: NamedIndividual
type: Identity
forAll: term_WC_8_forAll
lhs: term_WC_8_lhs
rhs: term_WC_8_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: D_3 is the Witt-Burnside conjugation relation: neg(succ(neg(x))) = pred(x) is srs = r⁻¹ in the pro-dihedral group.

WC_9
type: NamedIndividual
type: Identity
forAll: term_WC_9_forAll
lhs: term_WC_9_lhs
rhs: term_WC_9_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: D_4 is a Witt-Burnside reflection composition: bnot(neg(x)) = pred(x) is the product of two reflections yielding inverse rotation.

WC_10
type: NamedIndividual
type: Identity
forAll: term_WC_10_forAll
lhs: term_WC_10_lhs
rhs: term_WC_10_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: The δ-ring Frobenius lift on W_n(F_2) is the identity map because F_2 is a perfect field of characteristic 2 (a² = a for a ∈ F_2).

WC_11
type: NamedIndividual
type: Identity
forAll: term_WC_11_forAll
lhs: term_WC_11_lhs
rhs: term_WC_11_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: The Verschiebung on W_n(F_2) is multiplication by 2: V(x) = 2x = add(x,x). This is a coordinate shift with zero Witt defect.

WC_12
type: NamedIndividual
type: Identity
forAll: term_WC_12_forAll
lhs: term_WC_12_lhs
rhs: term_WC_12_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: The δ-operator on W_n(F_2) is the squaring defect divided by 2: δ(x) = (x − mul(x,x)) / 2. Expressible entirely in existing op/ primitives (sub, mul, arithmetic right shift).

OA_1
type: NamedIndividual
type: Identity
forAll: term_OA_1_forAll
lhs: term_OA_1_lhs
rhs: term_OA_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ArithmeticValuation
comment: Ostrowski product formula at p=2: |2|_2 · |2|_∞ = 1. The 2-adic and Archimedean absolute values of 2 are multiplicative inverses.

OA_2
type: NamedIndividual
type: Identity
forAll: term_OA_2_forAll
lhs: term_OA_2_lhs
rhs: term_OA_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ArithmeticValuation
comment: Crossing cost equals ln 2: the Archimedean image of one unit of 2-adic valuation, under the product formula, is ln 2 nats.

OA_3
type: NamedIndividual
type: Identity
forAll: term_OA_3_forAll
lhs: term_OA_3_lhs
rhs: term_OA_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ArithmeticValuation
comment: QM_1 grounding: the Landauer cost β* = ln 2 is the crossing cost from OA_2, derived from the prime p=2 that structures the Witt tower.

OA_4
type: NamedIndividual
type: Identity
forAll: term_OA_4_forAll
lhs: term_OA_4_lhs
rhs: term_OA_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ArithmeticValuation
comment: Born rule bridge (conditional on amplitude rationality): P(outcome k) = |α_k|_∞², where |·|_∞ is the Archimedean image of the 2-adic amplitude via the product formula.

OA_5
type: NamedIndividual
type: Identity
forAll: term_OA_5_forAll
lhs: term_OA_5_lhs
rhs: term_OA_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ArithmeticValuation
comment: Entropy per δ-level equals the crossing cost: each application of the δ-operator (division by 2) costs ln 2 nats in the Archimedean completion, which is the per-bit Landauer cost.

HT_1
type: NamedIndividual
type: Identity
forAll: term_HT_1_forAll
lhs: term_HT_1_lhs
rhs: term_HT_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: KanComplex(N(C)) — the constraint nerve satisfies the Kan extension condition for all horns of dimension ≤ d where d is the maximum simplex dimension of N(C).

HT_2
type: NamedIndividual
type: Identity
forAll: term_HT_2_forAll
lhs: term_HT_2_lhs
rhs: term_HT_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: Path components of nerve recover β₀: π₀(N(C)) ≅ Z^{β₀} counts the connected components of the constraint configuration.

HT_3
type: NamedIndividual
type: Identity
forAll: term_HT_3_forAll
lhs: term_HT_3_lhs
rhs: term_HT_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: MN_6 monodromy is abelianisation of full π₁: the fundamental group π₁(N(C)) surjects onto the HolonomyGroup D_{2^n} via abelianisation.

HT_4
type: NamedIndividual
type: Identity
forAll: term_HT_4_forAll
lhs: term_HT_4_lhs
rhs: term_HT_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: Higher homotopy groups vanish above nerve dimension: π_k(N(C)) = 0 for all k > dim(N(C)), because the nerve is a finite CW-complex.

HT_5
type: NamedIndividual
type: Identity
forAll: term_HT_5_forAll
lhs: term_HT_5_lhs
rhs: term_HT_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: 1-truncation determines flat/twisted classification: τ_{≤1}(N(C)) captures the holonomy action that distinguishes FlatType from TwistedType.

HT_6
type: NamedIndividual
type: Identity
forAll: term_HT_6_forAll
lhs: term_HT_6_lhs
rhs: term_HT_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: Trivial k-invariants beyond depth d imply spectral collapse: if κ_k is trivial for all k > d then the spectral sequence collapses at E_{d+2}.

HT_7
type: NamedIndividual
type: Identity
forAll: term_HT_7_forAll
lhs: term_HT_7_lhs
rhs: term_HT_7_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: Non-trivial Whitehead product implies lift obstruction: [α, β] ≠ 0 in π_{p+q−1} implies a non-trivial LiftObstruction that Betti numbers alone cannot detect.

HT_8
type: NamedIndividual
type: Identity
forAll: term_HT_8_forAll
lhs: term_HT_8_lhs
rhs: term_HT_8_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: Hurewicz isomorphism for first non-vanishing group: π_k(N(C)) ⊗ Z ≅ H_k(N(C); Z) for the smallest k with π_k ≠ 0, linking homotopy invariants to homology.

psi_7
type: NamedIndividual
type: Identity
forAll: term_psi_7_forAll
lhs: term_psi_7_lhs
rhs: term_psi_7_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: ψ_7: KanComplex → PostnikovTower — compute the Postnikov truncations τ_{≤k} for k = 0, 1, …, dim(N(C)).

psi_8
type: NamedIndividual
type: Identity
forAll: term_psi_8_forAll
lhs: term_psi_8_lhs
rhs: term_psi_8_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: ψ_8: PostnikovTower → HomotopyGroups — extract the homotopy groups π_k from each truncation stage.

psi_9
type: NamedIndividual
type: Identity
forAll: term_psi_9_forAll
lhs: term_psi_9_lhs
rhs: term_psi_9_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: ψ_9: HomotopyGroups → KInvariants — compute the k-invariants κ_k classifying the Postnikov tower.

HP_1
type: NamedIndividual
type: Identity
forAll: term_HP_1_forAll
lhs: term_HP_1_lhs
rhs: term_HP_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Pipeline composition: nerve construction + Kan promotion = ψ_7 ∘ ψ_1.

HP_2
type: NamedIndividual
type: Identity
forAll: term_HP_2_forAll
lhs: term_HP_2_lhs
rhs: term_HP_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Homotopy extraction agrees with homology on k-skeleton.

HP_3
type: NamedIndividual
type: Identity
forAll: term_HP_3_forAll
lhs: term_HP_3_lhs
rhs: term_HP_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: k-invariant computation detects QLS_4 convergence.

HP_4
type: NamedIndividual
type: Identity
forAll: term_HP_4_forAll
lhs: term_HP_4_lhs
rhs: term_HP_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Analytical
comment: Complexity bound for homotopy type computation.

MD_1
type: NamedIndividual
type: Identity
forAll: term_MD_1_forAll
lhs: term_MD_1_lhs
rhs: term_MD_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Moduli space dimension equals basis size of any contained type.

MD_2
type: NamedIndividual
type: Identity
forAll: term_MD_2_forAll
lhs: term_MD_2_lhs
rhs: term_MD_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Zeroth deformation cohomology = automorphism group intersected with dihedral group.

MD_3
type: NamedIndividual
type: Identity
forAll: term_MD_3_forAll
lhs: term_MD_3_lhs
rhs: term_MD_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: First deformation cohomology = tangent space to the moduli space at T.

MD_4
type: NamedIndividual
type: Identity
forAll: term_MD_4_forAll
lhs: term_MD_4_lhs
rhs: term_MD_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: Second deformation cohomology = LiftObstruction space.

MD_5
type: NamedIndividual
type: Identity
forAll: term_MD_5_forAll
lhs: term_MD_5_lhs
rhs: term_MD_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: FlatType stratum has codimension zero in the moduli space.

MD_6
type: NamedIndividual
type: Identity
forAll: term_MD_6_forAll
lhs: term_MD_6_lhs
rhs: term_MD_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: TwistedType stratum has codimension at least 1.

MD_7
type: NamedIndividual
type: Identity
forAll: term_MD_7_forAll
lhs: term_MD_7_lhs
rhs: term_MD_7_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: VersalDeformation existence is guaranteed when the obstruction space H² vanishes.

MD_8
type: NamedIndividual
type: Identity
forAll: term_MD_8_forAll
lhs: term_MD_8_lhs
rhs: term_MD_8_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: A deformation family preserves completeness iff H²(Def(T_t)) = 0 along the entire path.

MD_9
type: NamedIndividual
type: Identity
forAll: term_MD_9_forAll
lhs: term_MD_9_lhs
rhs: term_MD_9_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: The site of a ModuliTowerMap at T has dimension 1 when the obstruction is trivial.

MD_10
type: NamedIndividual
type: Identity
forAll: term_MD_10_forAll
lhs: term_MD_10_lhs
rhs: term_MD_10_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: The site of a ModuliTowerMap at T is empty iff T is a TwistedType at every level.

MR_1
type: NamedIndividual
type: Identity
forAll: term_MR_1_forAll
lhs: term_MR_1_lhs
rhs: term_MR_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: ModuliResolver boundary agrees with MorphospaceBoundary.

MR_2
type: NamedIndividual
type: Identity
forAll: term_MR_2_forAll
lhs: term_MR_2_lhs
rhs: term_MR_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: StratificationRecord covers every CompleteType in exactly one stratum.

MR_3
type: NamedIndividual
type: Identity
forAll: term_MR_3_forAll
lhs: term_MR_3_lhs
rhs: term_MR_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Analytical
comment: ModuliResolver complexity bound.

MR_4
type: NamedIndividual
type: Identity
forAll: term_MR_4_forAll
lhs: term_MR_4_lhs
rhs: term_MR_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Achievable signatures correspond to membership in some HolonomyStratum.

CY_1
type: NamedIndividual
type: Identity
forAll: term_CY_1_forAll
lhs: term_CY_1_lhs
rhs: term_CY_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Carry generates at site k iff and(x_k, y_k) = 1. Extends CA_1 (addition decomposition) and WC_2 (Witt sum correction).

CY_2
type: NamedIndividual
type: Identity
forAll: term_CY_2_forAll
lhs: term_CY_2_lhs
rhs: term_CY_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Carry propagates at site k iff xor(x_k, y_k) = 1 and c_k = 1. Extends CA_2 (carry recurrence) and WC_3 (Witt polynomial recurrence).

CY_3
type: NamedIndividual
type: Identity
forAll: term_CY_3_forAll
lhs: term_CY_3_lhs
rhs: term_CY_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Carry kills at site k iff and(x_k, y_k) = 0 and xor(x_k, y_k) = 0. Complement of CY_1 and CY_2.

CY_4
type: NamedIndividual
type: Identity
forAll: term_CY_4_forAll
lhs: term_CY_4_lhs
rhs: term_CY_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: d_Δ(x,y) = |carryCount(x+y) − hammingDistance(x,y)|. The metric incompatibility IS the discrepancy between carry count and Hamming distance. Strengthens WC_6.

CY_5
type: NamedIndividual
type: Identity
forAll: term_CY_5_forAll
lhs: term_CY_5_lhs
rhs: term_CY_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Optimal encoding theorem: the encoding that minimizes Σ d_Δ over observed pairs is the one where the carry chain’s significance hierarchy matches the domain’s dependency structure.

CY_6
type: NamedIndividual
type: Identity
forAll: term_CY_6_forAll
lhs: term_CY_6_lhs
rhs: term_CY_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Site ordering theorem: d_Δ is minimized when high-significance sites (upstream in the carry chain) encode the most structurally informative observables.

CY_7
type: NamedIndividual
type: Identity
forAll: term_CY_7_forAll
lhs: term_CY_7_lhs
rhs: term_CY_7_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Carry lookahead: the carry chain for n sites is computable in O(log n) using prefix computation on generate/propagate pairs.

BM_1
type: NamedIndividual
type: Identity
forAll: term_BM_1_forAll
lhs: term_BM_1_lhs
rhs: term_BM_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: σ(C) = (n − freeRank(C)) / n. The saturation metric is the complement of free site ratio. Derives from SC_2.

BM_2
type: NamedIndividual
type: Identity
forAll: term_BM_2_forAll
lhs: term_BM_2_lhs
rhs: term_BM_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: χ = Σ(−1)^k β_k. The Euler characteristic of the constraint nerve. Derives from IT_2.

BM_3
type: NamedIndividual
type: Identity
forAll: term_BM_3_forAll
lhs: term_BM_3_lhs
rhs: term_BM_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: Index theorem: Σκ_k − χ = S_residual / ln 2. Links all six metrics. Derives from IT_7a.

BM_4
type: NamedIndividual
type: Identity
forAll: term_BM_4_forAll
lhs: term_BM_4_lhs
rhs: term_BM_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: J_k = 0 for pinned sites. The Jacobian vanishes on resolved sites.

BM_5
type: NamedIndividual
type: Identity
forAll: term_BM_5_forAll
lhs: term_BM_5_lhs
rhs: term_BM_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: d_Δ > 0 iff carry ≠ 0. The metric discrepancy equals the Witt defect. Derives from WC_6.

BM_6
type: NamedIndividual
type: Identity
forAll: term_BM_6_forAll
lhs: term_BM_6_lhs
rhs: term_BM_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: IndexTheoretic
comment: Metric composition tower: d_Δ → {σ, J_k} → β_k → χ → r. Each metric derives from previous ones.

GL_1
type: NamedIndividual
type: Identity
forAll: term_GL_1_forAll
lhs: term_GL_1_lhs
rhs: term_GL_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: σ = lower adjoint evaluated at current type. The saturation metric is the lower adjoint of the Galois connection. Derives from SC_2.

GL_2
type: NamedIndividual
type: Identity
forAll: term_GL_2_forAll
lhs: term_GL_2_lhs
rhs: term_GL_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: r = complement of upper adjoint image. The residual freedom is what the type closure does not reach.

GL_3
type: NamedIndividual
type: Identity
forAll: term_GL_3_forAll
lhs: term_GL_3_lhs
rhs: term_GL_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: CompleteType = fixpoint of Galois connection, σ=1, r=0. Derives from IT_7d.

GL_4
type: NamedIndividual
type: Identity
forAll: term_GL_4_forAll
lhs: term_GL_4_lhs
rhs: term_GL_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: Type refinement = ascending in type lattice = descending in site freedom. The Galois connection reverses order.

NV_1
type: NamedIndividual
type: Identity
forAll: term_NV_1_forAll
lhs: term_NV_1_lhs
rhs: term_NV_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: nerve(C₁ ∪ C₂) = nerve(C₁) ∪ nerve(C₂) for disjoint constraint domains.

NV_2
type: NamedIndividual
type: Identity
forAll: term_NV_2_forAll
lhs: term_NV_2_lhs
rhs: term_NV_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: Mayer–Vietoris: β_k(C₁ ∪ C₂) computable from parts and intersection.

NV_3
type: NamedIndividual
type: Identity
forAll: term_NV_3_forAll
lhs: term_NV_3_lhs
rhs: term_NV_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: Single constraint addition: Δβ_k ∈ {−1, 0, +1} per dimension.

NV_4
type: NamedIndividual
type: Identity
forAll: term_NV_4_forAll
lhs: term_NV_4_lhs
rhs: term_NV_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: Constraint accumulation monotonicity: β_k non-increasing under SR_1. Derives from SR_1.

SD_1
type: NamedIndividual
type: Identity
forAll: term_SD_1_forAll
lhs: term_SD_1_lhs
rhs: term_SD_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: ScalarType grounding: quantize(value, range, bits) produces ring element where d_R reflects value proximity.

SD_2
type: NamedIndividual
type: Identity
forAll: term_SD_2_forAll
lhs: term_SD_2_lhs
rhs: term_SD_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: SymbolType grounding: argmin_{encoding} Σ d_Δ over observed pairs (CY_5).

SD_3
type: NamedIndividual
type: Identity
forAll: term_SD_3_forAll
lhs: term_SD_3_lhs
rhs: term_SD_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: SequenceType = free monoid on element type with backbone constraint.

SD_4
type: NamedIndividual
type: Identity
forAll: term_SD_4_forAll
lhs: term_SD_4_lhs
rhs: term_SD_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: TupleType site count = Σ field site counts, site ordering follows CY_6.

SD_5
type: NamedIndividual
type: Identity
forAll: term_SD_5_forAll
lhs: term_SD_5_lhs
rhs: term_SD_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: GraphType constraint nerve = graph nerve, β_k equality.

SD_6
type: NamedIndividual
type: Identity
forAll: term_SD_6_forAll
lhs: term_SD_6_lhs
rhs: term_SD_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: SetType d_Δ invariant under element permutation, D_{2n} symmetry.

SD_7
type: NamedIndividual
type: Identity
forAll: term_SD_7_forAll
lhs: term_SD_7_lhs
rhs: term_SD_7_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: TreeType β_1=0, β_0=1 topological constraints.

SD_8
type: NamedIndividual
type: Identity
forAll: term_SD_8_forAll
lhs: term_SD_8_lhs
rhs: term_SD_8_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: TableType = SequenceType(TupleType(S)), functorial decomposition.

DD_1
type: NamedIndividual
type: Identity
forAll: term_DD_1_forAll
lhs: term_DD_1_lhs
rhs: term_DD_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Dispatch determinism: same query and same registry always yield the same resolver.

DD_2
type: NamedIndividual
type: Identity
forAll: term_DD_2_forAll
lhs: term_DD_2_lhs
rhs: term_DD_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Dispatch coverage: every query in the registry domain has a matching resolver.

PI_1
type: NamedIndividual
type: Identity
forAll: term_PI_1_forAll
lhs: term_PI_1_lhs
rhs: term_PI_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Inference idempotence: ι(ι(s,C),C) = ι(s,C) on GroundedContext.

PI_2
type: NamedIndividual
type: Identity
forAll: term_PI_2_forAll
lhs: term_PI_2_lhs
rhs: term_PI_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Inference soundness: ι(s,C) resolves to a type consistent with C.

PI_3
type: NamedIndividual
type: Identity
forAll: term_PI_3_forAll
lhs: term_PI_3_lhs
rhs: term_PI_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Inference composition: ι = P ∘ Π ∘ G (references phi_4).

PI_4
type: NamedIndividual
type: Identity
forAll: term_PI_4_forAll
lhs: term_PI_4_lhs
rhs: term_PI_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Inference complexity: O(n) worst case, O(1) on CompleteType.

PI_5
type: NamedIndividual
type: Identity
forAll: term_PI_5_forAll
lhs: term_PI_5_lhs
rhs: term_PI_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Inference coherence: roundTrip(P(Π(G(s)))) = s.

PA_1
type: NamedIndividual
type: Identity
forAll: term_PA_1_forAll
lhs: term_PA_1_lhs
rhs: term_PA_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Accumulation permutation invariance: accumulating bindings in any order yields the same saturated context (derives from SR_10).

PA_2
type: NamedIndividual
type: Identity
forAll: term_PA_2_forAll
lhs: term_PA_2_lhs
rhs: term_PA_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Accumulation monotonicity: α(b,C) ⊇ C (the context only grows, never loses bindings). Derives from SR_1.

PA_3
type: NamedIndividual
type: Identity
forAll: term_PA_3_forAll
lhs: term_PA_3_lhs
rhs: term_PA_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Accumulation soundness: α(b,C) preserves all previously satisfied constraints. Derives from SR_2.

PA_4
type: NamedIndividual
type: Identity
forAll: term_PA_4_forAll
lhs: term_PA_4_lhs
rhs: term_PA_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Accumulation base preservation: α does not modify previously pinned sites.

PA_5
type: NamedIndividual
type: Identity
forAll: term_PA_5_forAll
lhs: term_PA_5_lhs
rhs: term_PA_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Accumulation identity: α(∅, C) = C.

PL_1
type: NamedIndividual
type: Identity
forAll: term_PL_1_forAll
lhs: term_PL_1_lhs
rhs: term_PL_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Lease disjointness: partitioned leases have pairwise disjoint site sets (derives from SR_9).

PL_2
type: NamedIndividual
type: Identity
forAll: term_PL_2_forAll
lhs: term_PL_2_lhs
rhs: term_PL_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Lease conservation: union of all leases equals the original shared context (derives from MC_1).

PL_3
type: NamedIndividual
type: Identity
forAll: term_PL_3_forAll
lhs: term_PL_3_lhs
rhs: term_PL_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Lease coverage: every site in the shared context appears in exactly one lease (derives from MC_6).

PK_1
type: NamedIndividual
type: Identity
forAll: term_PK_1_forAll
lhs: term_PK_1_lhs
rhs: term_PK_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Composition validity: κ(S₁,S₂) is a valid session if S₁,S₂ have disjoint leases (derives from SR_8).

PK_2
type: NamedIndividual
type: Identity
forAll: term_PK_2_forAll
lhs: term_PK_2_lhs
rhs: term_PK_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Distributed resolution: resolving in κ(S₁,S₂) equals resolving in S₁ or S₂ independently (derives from MC_7).

PP_1
type: NamedIndividual
type: Identity
forAll: term_PP_1_forAll
lhs: term_PP_1_lhs
rhs: term_PP_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Pipeline unification: κ(λₖ(α*(ι(s,·))),C) = resolve(s,C). The full composed pipeline equals the top-level resolution function.

PE_1
type: NamedIndividual
type: Identity
forAll: term_PE_1_forAll
lhs: term_PE_1_lhs
rhs: term_PE_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Stage 0 initializes state vector to 1.

PE_2
type: NamedIndividual
type: Identity
forAll: term_PE_2_forAll
lhs: term_PE_2_lhs
rhs: term_PE_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Stage 1 dispatches resolver (δ selects).

PE_3
type: NamedIndividual
type: Identity
forAll: term_PE_3_forAll
lhs: term_PE_3_lhs
rhs: term_PE_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Stage 2 produces valid ring address (G grounds).

PE_4
type: NamedIndividual
type: Identity
forAll: term_PE_4_forAll
lhs: term_PE_4_lhs
rhs: term_PE_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Stage 3 resolves constraints (Π terminates).

PE_5
type: NamedIndividual
type: Identity
forAll: term_PE_5_forAll
lhs: term_PE_5_lhs
rhs: term_PE_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Stage 4 accumulates without contradiction (α consistent).

PE_6
type: NamedIndividual
type: Identity
forAll: term_PE_6_forAll
lhs: term_PE_6_lhs
rhs: term_PE_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Stage 5 extracts coherent output (P projects).

PE_7
type: NamedIndividual
type: Identity
forAll: term_PE_7_forAll
lhs: term_PE_7_lhs
rhs: term_PE_7_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Full pipeline is the composition PE_6 ∘ … ∘ PE_1.

PM_1
type: NamedIndividual
type: Identity
forAll: term_PM_1_forAll
lhs: term_PM_1_lhs
rhs: term_PM_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Phase rotation Ω^k at stage k.

PM_2
type: NamedIndividual
type: Identity
forAll: term_PM_2_forAll
lhs: term_PM_2_lhs
rhs: term_PM_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Phase gate checks Ω^k at boundary.

PM_3
type: NamedIndividual
type: Identity
forAll: term_PM_3_forAll
lhs: term_PM_3_lhs
rhs: term_PM_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Gate failure triggers complex conjugate rollback.

PM_4
type: NamedIndividual
type: Identity
forAll: term_PM_4_forAll
lhs: term_PM_4_lhs
rhs: term_PM_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Rollback is involutory: (z̄)̄ = z.

PM_5
type: NamedIndividual
type: Identity
forAll: term_PM_5_forAll
lhs: term_PM_5_lhs
rhs: term_PM_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Epoch boundary preserves saturation.

PM_6
type: NamedIndividual
type: Identity
forAll: term_PM_6_forAll
lhs: term_PM_6_lhs
rhs: term_PM_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Service window provides base context.

PM_7
type: NamedIndividual
type: Identity
forAll: term_PM_7_forAll
lhs: term_PM_7_lhs
rhs: term_PM_7_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Machine is deterministic given initial state.

ER_1
type: NamedIndividual
type: Identity
forAll: term_ER_1_forAll
lhs: term_ER_1_lhs
rhs: term_ER_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Stage transition requires guard satisfaction.

ER_2
type: NamedIndividual
type: Identity
forAll: term_ER_2_forAll
lhs: term_ER_2_lhs
rhs: term_ER_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Effect application is atomic.

ER_3
type: NamedIndividual
type: Identity
forAll: term_ER_3_forAll
lhs: term_ER_3_lhs
rhs: term_ER_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Guard evaluation is side-effect-free.

ER_4
type: NamedIndividual
type: Identity
forAll: term_ER_4_forAll
lhs: term_ER_4_lhs
rhs: term_ER_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Effect composition is order-independent within a stage.

EA_1
type: NamedIndividual
type: Identity
forAll: term_EA_1_forAll
lhs: term_EA_1_lhs
rhs: term_EA_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Epoch boundary resets free sites.

EA_2
type: NamedIndividual
type: Identity
forAll: term_EA_2_forAll
lhs: term_EA_2_lhs
rhs: term_EA_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Grounding carries across epochs.

EA_3
type: NamedIndividual
type: Identity
forAll: term_EA_3_forAll
lhs: term_EA_3_lhs
rhs: term_EA_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Service window bounds context size.

EA_4
type: NamedIndividual
type: Identity
forAll: term_EA_4_forAll
lhs: term_EA_4_lhs
rhs: term_EA_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Epoch admits one datum or one refinement pass.

OE_1
type: NamedIndividual
type: Identity
forAll: term_OE_1_forAll
lhs: term_OE_1_lhs
rhs: term_OE_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Adjacent stages with compatible guards may fuse.

OE_2
type: NamedIndividual
type: Identity
forAll: term_OE_2_forAll
lhs: term_OE_2_lhs
rhs: term_OE_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Independent effects commute.

OE_3
type: NamedIndividual
type: Identity
forAll: term_OE_3_forAll
lhs: term_OE_3_lhs
rhs: term_OE_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Disjoint leases parallelize.

OE_4a
type: NamedIndividual
type: Identity
forAll: term_OE_4a_forAll
lhs: term_OE_4a_lhs
rhs: term_OE_4a_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Stage fusion preserves semantics.

OE_4b
type: NamedIndividual
type: Identity
forAll: term_OE_4b_forAll
lhs: term_OE_4b_lhs
rhs: term_OE_4b_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Effect commutation preserves outcome.

OE_4c
type: NamedIndividual
type: Identity
forAll: term_OE_4c_forAll
lhs: term_OE_4c_lhs
rhs: term_OE_4c_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Lease parallelism preserves coverage.

CS_1
type: NamedIndividual
type: Identity
forAll: term_CS_1_forAll
lhs: term_CS_1_lhs
rhs: term_CS_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Each stage has bounded cost.

CS_2
type: NamedIndividual
type: Identity
forAll: term_CS_2_forAll
lhs: term_CS_2_lhs
rhs: term_CS_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Pipeline cost = sum of stage costs.

CS_3
type: NamedIndividual
type: Identity
forAll: term_CS_3_forAll
lhs: term_CS_3_lhs
rhs: term_CS_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Rollback cost is at most forward cost.

CS_4
type: NamedIndividual
type: Identity
forAll: term_CS_4_forAll
lhs: term_CS_4_lhs
rhs: term_CS_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Preflight cost is O(1).

CS_5
type: NamedIndividual
type: Identity
forAll: term_CS_5_forAll
lhs: term_CS_5_lhs
rhs: term_CS_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Total reduction cost bounded by n × stage_max_cost.

CS_6
type: NamedIndividual
type: Identity
forAll: term_CS_6_forAll
lhs: term_CS_6_lhs
rhs: term_CS_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Budget solvency rejection: a CompileUnit whose declared thermodynamicBudget is strictly less than the Landauer minimum (bitsWidth(Q_k) × ln 2) is rejected at the BudgetSolvencyCheck preflight.

CS_7
type: NamedIndividual
type: Identity
forAll: term_CS_7_forAll
lhs: term_CS_7_lhs
rhs: term_CS_7_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Unit address identity: the unitAddress of a CompileUnit is the u:Element computed by hashing the canonical byte serialization of the root term’s transitive closure.

FA_1
type: NamedIndividual
type: Identity
forAll: term_FA_1_forAll
lhs: term_FA_1_lhs
rhs: term_FA_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Every pending query eventually reaches a stage gate.

FA_2
type: NamedIndividual
type: Identity
forAll: term_FA_2_forAll
lhs: term_FA_2_lhs
rhs: term_FA_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: No starvation under bounded epoch admission.

FA_3
type: NamedIndividual
type: Identity
forAll: term_FA_3_forAll
lhs: term_FA_3_lhs
rhs: term_FA_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Fair lease allocation under disjoint composition.

SW_1
type: NamedIndividual
type: Identity
forAll: term_SW_1_forAll
lhs: term_SW_1_lhs
rhs: term_SW_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Service window bounds context memory.

SW_2
type: NamedIndividual
type: Identity
forAll: term_SW_2_forAll
lhs: term_SW_2_lhs
rhs: term_SW_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Window slide preserves saturation invariant.

SW_3
type: NamedIndividual
type: Identity
forAll: term_SW_3_forAll
lhs: term_SW_3_lhs
rhs: term_SW_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Evicted epochs release lease resources.

SW_4
type: NamedIndividual
type: Identity
forAll: term_SW_4_forAll
lhs: term_SW_4_lhs
rhs: term_SW_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Window size ≥ 1 (non-empty).

LS_1
type: NamedIndividual
type: Identity
forAll: term_LS_1_forAll
lhs: term_LS_1_lhs
rhs: term_LS_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Suspended lease preserves pinned state.

LS_2
type: NamedIndividual
type: Identity
forAll: term_LS_2_forAll
lhs: term_LS_2_lhs
rhs: term_LS_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Lease expiry triggers resource release.

LS_3
type: NamedIndividual
type: Identity
forAll: term_LS_3_forAll
lhs: term_LS_3_lhs
rhs: term_LS_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Checkpoint restore is idempotent.

LS_4
type: NamedIndividual
type: Identity
forAll: term_LS_4_forAll
lhs: term_LS_4_lhs
rhs: term_LS_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Active → Suspended → Active round-trip preserves state.

TJ_1
type: NamedIndividual
type: Identity
forAll: term_TJ_1_forAll
lhs: term_TJ_1_lhs
rhs: term_TJ_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: AllOrNothing transaction rolls back on any failure.

TJ_2
type: NamedIndividual
type: Identity
forAll: term_TJ_2_forAll
lhs: term_TJ_2_lhs
rhs: term_TJ_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: BestEffort transaction commits partial results.

TJ_3
type: NamedIndividual
type: Identity
forAll: term_TJ_3_forAll
lhs: term_TJ_3_lhs
rhs: term_TJ_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Transaction atomicity within a single epoch.

AP_1
type: NamedIndividual
type: Identity
forAll: term_AP_1_forAll
lhs: term_AP_1_lhs
rhs: term_AP_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Partial saturation is monotonically non-decreasing across stages.

AP_2
type: NamedIndividual
type: Identity
forAll: term_AP_2_forAll
lhs: term_AP_2_lhs
rhs: term_AP_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Approximation quality improves with additional epochs.

AP_3
type: NamedIndividual
type: Identity
forAll: term_AP_3_forAll
lhs: term_AP_3_lhs
rhs: term_AP_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Deferred queries are eventually processed or explicitly dropped.

EC_1
type: NamedIndividual
type: Identity
forAll: term_EC_1_forAll
lhs: term_EC_1_lhs
rhs: term_EC_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: Ω⁶ = −1: reduction converges in 6 stages (phase half-turn).

EC_2
type: NamedIndividual
type: Identity
forAll: term_EC_2_forAll
lhs: term_EC_2_lhs
rhs: term_EC_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: Complex conjugate rollback involutory: (z̄)̄ = z.

EC_3
type: NamedIndividual
type: Identity
forAll: term_EC_3_forAll
lhs: term_EC_3_lhs
rhs: term_EC_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: Pairwise convergence: commutator converges to identity.

EC_4
type: NamedIndividual
type: Identity
forAll: term_EC_4_forAll
lhs: term_EC_4_lhs
rhs: term_EC_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: Triple convergence: associator converges to zero.

EC_4a
type: NamedIndividual
type: Identity
forAll: term_EC_4a_forAll
lhs: term_EC_4a_lhs
rhs: term_EC_4a_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: Associator monotonicity: associator norm non-increasing.

EC_4b
type: NamedIndividual
type: Identity
forAll: term_EC_4b_forAll
lhs: term_EC_4b_lhs
rhs: term_EC_4b_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: Associator finiteness: reaches 0 in bounded steps.

EC_4c
type: NamedIndividual
type: Identity
forAll: term_EC_4c_forAll
lhs: term_EC_4c_lhs
rhs: term_EC_4c_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: Associator vanishing implies associativity on resolved site space.

EC_5
type: NamedIndividual
type: Identity
forAll: term_EC_5_forAll
lhs: term_EC_5_lhs
rhs: term_EC_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: Adams termination: no convergence level beyond L3_Self.

DA_1
type: NamedIndividual
type: Identity
forAll: term_DA_1_forAll
lhs: term_DA_1_lhs
rhs: term_DA_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Cayley-Dickson R→C: adjoin i with i²=−1, conjugation (a+bi)* = a−bi.

DA_2
type: NamedIndividual
type: Identity
forAll: term_DA_2_forAll
lhs: term_DA_2_lhs
rhs: term_DA_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Cayley-Dickson C→H: adjoin j with j²=−1, ij=k, ji=−k, k²=−1.

DA_3
type: NamedIndividual
type: Identity
forAll: term_DA_3_forAll
lhs: term_DA_3_lhs
rhs: term_DA_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Cayley-Dickson H→O: adjoin l, Fano plane products, associativity fails.

DA_4
type: NamedIndividual
type: Identity
forAll: term_DA_4_forAll
lhs: term_DA_4_lhs
rhs: term_DA_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Adams theorem: no normed division algebra of dimension 16 exists.

DA_5
type: NamedIndividual
type: Identity
forAll: term_DA_5_forAll
lhs: term_DA_5_lhs
rhs: term_DA_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Convergence level k lives in k-th division algebra: L0 in R, L1 in C, L2 in H, L3 in O.

DA_6
type: NamedIndividual
type: Identity
forAll: term_DA_6_forAll
lhs: term_DA_6_lhs
rhs: term_DA_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Commutator vanishes iff algebra at that level is commutative.

DA_7
type: NamedIndividual
type: Identity
forAll: term_DA_7_forAll
lhs: term_DA_7_lhs
rhs: term_DA_7_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Associator vanishes iff algebra at that level is associative.

IN_1
type: NamedIndividual
type: Identity
forAll: term_IN_1_forAll
lhs: term_IN_1_lhs
rhs: term_IN_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: d_Δ as interaction cost between entities.

IN_2
type: NamedIndividual
type: Identity
forAll: term_IN_2_forAll
lhs: term_IN_2_lhs
rhs: term_IN_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Disjoint leases imply commutator = 0.

IN_3
type: NamedIndividual
type: Identity
forAll: term_IN_3_forAll
lhs: term_IN_3_lhs
rhs: term_IN_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Shared sites imply commutator > 0.

IN_4
type: NamedIndividual
type: Identity
forAll: term_IN_4_forAll
lhs: term_IN_4_lhs
rhs: term_IN_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: SR_8 implies negotiation converges (commutator decreases monotonically).

IN_5
type: NamedIndividual
type: Identity
forAll: term_IN_5_forAll
lhs: term_IN_5_lhs
rhs: term_IN_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Convergent negotiation selects U(1) ⊂ SU(2).

IN_6
type: NamedIndividual
type: Identity
forAll: term_IN_6_forAll
lhs: term_IN_6_lhs
rhs: term_IN_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Outcome space of pairwise negotiation is S².

IN_7
type: NamedIndividual
type: Identity
forAll: term_IN_7_forAll
lhs: term_IN_7_lhs
rhs: term_IN_7_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Mutual modeling selects H ⊂ O.

IN_8
type: NamedIndividual
type: Identity
forAll: term_IN_8_forAll
lhs: term_IN_8_lhs
rhs: term_IN_8_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Interaction nerve β_k bounds coupling complexity at dimension k.

IN_9
type: NamedIndividual
type: Identity
forAll: term_IN_9_forAll
lhs: term_IN_9_lhs
rhs: term_IN_9_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: β_2(nerve) × max_disagreement bounds associator norm.

AS_1
type: NamedIndividual
type: Identity
forAll: term_AS_1_forAll
lhs: term_AS_1_lhs
rhs: term_AS_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: δ-ι-κ non-associativity: δ reads registry written by κ.

AS_2
type: NamedIndividual
type: Identity
forAll: term_AS_2_forAll
lhs: term_AS_2_lhs
rhs: term_AS_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: ι-α-λ non-associativity: λ reads lease state written by α.

AS_3
type: NamedIndividual
type: Identity
forAll: term_AS_3_forAll
lhs: term_AS_3_lhs
rhs: term_AS_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: λ-κ-δ non-associativity: δ reads state written by κ.

AS_4
type: NamedIndividual
type: Identity
forAll: term_AS_4_forAll
lhs: term_AS_4_lhs
rhs: term_AS_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: ComposedAlgebraic
comment: Root cause: non-associativity from read-write interleaving through mediating entity.

MO_1
type: NamedIndividual
type: Identity
forAll: term_MO_1_forAll
lhs: term_MO_1_lhs
rhs: term_MO_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Unit law: I ⊗ A ≅ A ≅ A ⊗ I.

MO_2
type: NamedIndividual
type: Identity
forAll: term_MO_2_forAll
lhs: term_MO_2_lhs
rhs: term_MO_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Associativity: (A⊗B)⊗C ≅ A⊗(B⊗C).

MO_3
type: NamedIndividual
type: Identity
forAll: term_MO_3_forAll
lhs: term_MO_3_lhs
rhs: term_MO_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Certificate composition: cert(A⊗B) contains cert(A) and cert(B).

MO_4
type: NamedIndividual
type: Identity
forAll: term_MO_4_forAll
lhs: term_MO_4_lhs
rhs: term_MO_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: σ(A⊗B) ≥ max(σ(A), σ(B)): sequential composition does not lose saturation.

MO_5
type: NamedIndividual
type: Identity
forAll: term_MO_5_forAll
lhs: term_MO_5_lhs
rhs: term_MO_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: r(A⊗B) ≤ min(r(A), r(B)): residual can only shrink under sequential composition.

OP_1
type: NamedIndividual
type: Identity
forAll: term_OP_1_forAll
lhs: term_OP_1_lhs
rhs: term_OP_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Site additivity: siteCount(F(G)) = F.sites + Σ_i G_i.sites.

OP_2
type: NamedIndividual
type: Identity
forAll: term_OP_2_forAll
lhs: term_OP_2_lhs
rhs: term_OP_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Grounding distributivity: grounding(F(G(x))) = F.ground(G.ground(x)).

OP_3
type: NamedIndividual
type: Identity
forAll: term_OP_3_forAll
lhs: term_OP_3_lhs
rhs: term_OP_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: d_Δ decomposition: d_Δ(F(G)) decomposes into outer + inner d_Δ.

OP_4
type: NamedIndividual
type: Identity
forAll: term_OP_4_forAll
lhs: term_OP_4_lhs
rhs: term_OP_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Table(Tuple(fields)): standard tabular data structure decomposition.

OP_5
type: NamedIndividual
type: Identity
forAll: term_OP_5_forAll
lhs: term_OP_5_lhs
rhs: term_OP_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Tree(leaves): standard hierarchical data structure (AST, XML, JSON).

FX_1
type: NamedIndividual
type: Identity
forAll: term_FX_1_forAll
lhs: term_FX_1_lhs
rhs: term_FX_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Pinning decrements free count by exactly 1.

FX_2
type: NamedIndividual
type: Identity
forAll: term_FX_2_forAll
lhs: term_FX_2_lhs
rhs: term_FX_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Unbinding increments free count by exactly 1.

FX_3
type: NamedIndividual
type: Identity
forAll: term_FX_3_forAll
lhs: term_FX_3_lhs
rhs: term_FX_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Phase effects preserve site budget.

FX_4
type: NamedIndividual
type: Identity
forAll: term_FX_4_forAll
lhs: term_FX_4_lhs
rhs: term_FX_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Disjoint effects commute.

FX_5
type: NamedIndividual
type: Identity
forAll: term_FX_5_forAll
lhs: term_FX_5_lhs
rhs: term_FX_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Composite free-count delta is additive.

FX_6
type: NamedIndividual
type: Identity
forAll: term_FX_6_forAll
lhs: term_FX_6_lhs
rhs: term_FX_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Every ReversibleEffect has an inverse (PinningEffect⁻¹ = UnbindingEffect on same site, PhaseEffect⁻¹ = conjugate phase).

FX_7
type: NamedIndividual
type: Identity
forAll: term_FX_7_forAll
lhs: term_FX_7_lhs
rhs: term_FX_7_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: External effects must match their declared shape.

PR_1
type: NamedIndividual
type: Identity
forAll: term_PR_1_forAll
lhs: term_PR_1_lhs
rhs: term_PR_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Every predicate is total: evaluation terminates for all inputs.

PR_2
type: NamedIndividual
type: Identity
forAll: term_PR_2_forAll
lhs: term_PR_2_lhs
rhs: term_PR_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Every predicate is pure: evaluation does not modify state.

PR_3
type: NamedIndividual
type: Identity
forAll: term_PR_3_forAll
lhs: term_PR_3_lhs
rhs: term_PR_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Exhaustive + mutually exclusive dispatch is deterministic.

PR_4
type: NamedIndividual
type: Identity
forAll: term_PR_4_forAll
lhs: term_PR_4_lhs
rhs: term_PR_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Match evaluation is deterministic given exhaustive, ordered arms.

PR_5
type: NamedIndividual
type: Identity
forAll: term_PR_5_forAll
lhs: term_PR_5_lhs
rhs: term_PR_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Stage transition requires typed guard satisfaction.

CG_1
type: NamedIndividual
type: Identity
forAll: term_CG_1_forAll
lhs: term_CG_1_lhs
rhs: term_CG_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Entry guard must be satisfied to enter a stage.

CG_2
type: NamedIndividual
type: Identity
forAll: term_CG_2_forAll
lhs: term_CG_2_lhs
rhs: term_CG_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Exit guard must be satisfied, then the stage effect is applied.

DIS_1
type: NamedIndividual
type: Identity
forAll: term_DIS_1_forAll
lhs: term_DIS_1_lhs
rhs: term_DIS_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: The root resolver dispatch table is exhaustive and mutually exclusive over all TypeDefinitions.

DIS_2
type: NamedIndividual
type: Identity
forAll: term_DIS_2_forAll
lhs: term_DIS_2_lhs
rhs: term_DIS_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Resolver dispatch is deterministic for every type.

PAR_1
type: NamedIndividual
type: Identity
forAll: term_PAR_1_forAll
lhs: term_PAR_1_lhs
rhs: term_PAR_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Disjoint parallel computations commute: A ⊗ B = B ⊗ A when site targets are disjoint.

PAR_2
type: NamedIndividual
type: Identity
forAll: term_PAR_2_forAll
lhs: term_PAR_2_lhs
rhs: term_PAR_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Parallel free-count deltas are additive.

PAR_3
type: NamedIndividual
type: Identity
forAll: term_PAR_3_forAll
lhs: term_PAR_3_lhs
rhs: term_PAR_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Partitioning is exhaustive: component cardinalities sum to total site budget.

PAR_4
type: NamedIndividual
type: Identity
forAll: term_PAR_4_forAll
lhs: term_PAR_4_lhs
rhs: term_PAR_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: All interleavings of disjoint parallel computations yield the same final context.

PAR_5
type: NamedIndividual
type: Identity
forAll: term_PAR_5_forAll
lhs: term_PAR_5_lhs
rhs: term_PAR_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Parallel certificate is the conjunction of component certificates plus disjointness.

HO_1
type: NamedIndividual
type: Identity
forAll: term_HO_1_forAll
lhs: term_HO_1_lhs
rhs: term_HO_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: A ComputationDatum’s ring value is the content hash of its certificate.

HO_2
type: NamedIndividual
type: Identity
forAll: term_HO_2_forAll
lhs: term_HO_2_lhs
rhs: term_HO_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Application preserves certification.

HO_3
type: NamedIndividual
type: Identity
forAll: term_HO_3_forAll
lhs: term_HO_3_lhs
rhs: term_HO_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Composition certification requires both components certified and type-compatible.

HO_4
type: NamedIndividual
type: Identity
forAll: term_HO_4_forAll
lhs: term_HO_4_lhs
rhs: term_HO_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Fully saturated partial application equals direct application.

STR_1
type: NamedIndividual
type: Identity
forAll: term_STR_1_forAll
lhs: term_STR_1_lhs
rhs: term_STR_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Every epoch terminates: the reduction within each epoch reaches convergence angle π.

STR_2
type: NamedIndividual
type: Identity
forAll: term_STR_2_forAll
lhs: term_STR_2_lhs
rhs: term_STR_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Grounding preservation across epoch boundaries.

STR_3
type: NamedIndividual
type: Identity
forAll: term_STR_3_forAll
lhs: term_STR_3_lhs
rhs: term_STR_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Every finite prefix computes in finite time.

STR_4
type: NamedIndividual
type: Identity
forAll: term_STR_4_forAll
lhs: term_STR_4_lhs
rhs: term_STR_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: The first epoch starts from the unfold seed context.

STR_5
type: NamedIndividual
type: Identity
forAll: term_STR_5_forAll
lhs: term_STR_5_lhs
rhs: term_STR_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Each subsequent epoch starts from the previous boundary’s continuation context.

STR_6
type: NamedIndividual
type: Identity
forAll: term_STR_6_forAll
lhs: term_STR_6_lhs
rhs: term_STR_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Lease expiry at an epoch boundary returns claimed sites to the next epoch’s linear budget.

FLR_1
type: NamedIndividual
type: Identity
forAll: term_FLR_1_forAll
lhs: term_FLR_1_lhs
rhs: term_FLR_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Every partial computation produces exactly one of Success or Failure.

FLR_2
type: NamedIndividual
type: Identity
forAll: term_FLR_2_forAll
lhs: term_FLR_2_lhs
rhs: term_FLR_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: A total computation always succeeds.

FLR_3
type: NamedIndividual
type: Identity
forAll: term_FLR_3_forAll
lhs: term_FLR_3_lhs
rhs: term_FLR_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Sequential failure propagation: if A fails, B is not evaluated.

FLR_4
type: NamedIndividual
type: Identity
forAll: term_FLR_4_forAll
lhs: term_FLR_4_lhs
rhs: term_FLR_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Parallel failure independence: one component’s failure does not prevent the other’s success.

FLR_5
type: NamedIndividual
type: Identity
forAll: term_FLR_5_forAll
lhs: term_FLR_5_lhs
rhs: term_FLR_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Recovery produces a new ComputationResult.

FLR_6
type: NamedIndividual
type: Identity
forAll: term_FLR_6_forAll
lhs: term_FLR_6_lhs
rhs: term_FLR_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: The reduction’s existing rollback mechanism is a Recovery whose effect is the conjugate phase rotation.

LN_1
type: NamedIndividual
type: Identity
forAll: term_LN_1_forAll
lhs: term_LN_1_lhs
rhs: term_LN_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: In a linear trace, every site is targeted exactly once. Total effect count equals site budget.

LN_2
type: NamedIndividual
type: Identity
forAll: term_LN_2_forAll
lhs: term_LN_2_lhs
rhs: term_LN_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: After a LinearEffect, the target site is pinned.

LN_3
type: NamedIndividual
type: Identity
forAll: term_LN_3_forAll
lhs: term_LN_3_lhs
rhs: term_LN_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: A consumed LinearSite cannot be targeted again.

LN_4
type: NamedIndividual
type: Identity
forAll: term_LN_4_forAll
lhs: term_LN_4_lhs
rhs: term_LN_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Lease allocation decrements the linear budget by the lease cardinality.

LN_5
type: NamedIndividual
type: Identity
forAll: term_LN_5_forAll
lhs: term_LN_5_lhs
rhs: term_LN_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Lease expiry returns claimed sites to the budget.

LN_6
type: NamedIndividual
type: Identity
forAll: term_LN_6_forAll
lhs: term_LN_6_lhs
rhs: term_LN_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: Every geodesic trace is a linear trace.

SB_1
type: NamedIndividual
type: Identity
forAll: term_SB_1_forAll
lhs: term_SB_1_lhs
rhs: term_SB_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Subtyping is constraint superset: more constraints = more specific.

SB_2
type: NamedIndividual
type: Identity
forAll: term_SB_2_forAll
lhs: term_SB_2_lhs
rhs: term_SB_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Subtype has fewer valid resolutions.

SB_3
type: NamedIndividual
type: Identity
forAll: term_SB_3_forAll
lhs: term_SB_3_lhs
rhs: term_SB_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: The constraint nerve of the supertype is a sub-complex of the subtype’s nerve.

SB_4
type: NamedIndividual
type: Identity
forAll: term_SB_4_forAll
lhs: term_SB_4_lhs
rhs: term_SB_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Covariance preserves inclusion.

SB_5
type: NamedIndividual
type: Identity
forAll: term_SB_5_forAll
lhs: term_SB_5_lhs
rhs: term_SB_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Contravariance reverses inclusion.

SB_6
type: NamedIndividual
type: Identity
forAll: term_SB_6_forAll
lhs: term_SB_6_lhs
rhs: term_SB_6_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Lattice depth equals site budget.

BR_1
type: NamedIndividual
type: Identity
forAll: term_BR_1_forAll
lhs: term_BR_1_lhs
rhs: term_BR_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Every recursive step strictly decreases the descent measure.

BR_2
type: NamedIndividual
type: Identity
forAll: term_BR_2_forAll
lhs: term_BR_2_lhs
rhs: term_BR_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Recursion depth is bounded by the initial measure value.

BR_3
type: NamedIndividual
type: Identity
forAll: term_BR_3_forAll
lhs: term_BR_3_lhs
rhs: term_BR_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Every bounded recursion terminates.

BR_4
type: NamedIndividual
type: Identity
forAll: term_BR_4_forAll
lhs: term_BR_4_lhs
rhs: term_BR_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Structural recursion’s measure is the input type’s structural size.

BR_5
type: NamedIndividual
type: Identity
forAll: term_BR_5_forAll
lhs: term_BR_5_lhs
rhs: term_BR_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: The base predicate is satisfied exactly when the measure reaches zero.

RG_1
type: NamedIndividual
type: Identity
forAll: term_RG_1_forAll
lhs: term_RG_1_lhs
rhs: term_RG_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Topological
comment: The working set is determined by the constraint nerve and the stage’s site targets.

RG_2
type: NamedIndividual
type: Identity
forAll: term_RG_2_forAll
lhs: term_RG_2_lhs
rhs: term_RG_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Analytical
comment: All addresses within a region are within the region’s diameter under the chosen metric.

RG_3
type: NamedIndividual
type: Identity
forAll: term_RG_3_forAll
lhs: term_RG_3_lhs
rhs: term_RG_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Total working set size is bounded by the addressable space at the quantum level.

RG_4
type: NamedIndividual
type: Identity
forAll: term_RG_4_forAll
lhs: term_RG_4_lhs
rhs: term_RG_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: The resolver at stage k accesses only addresses within its working set.

IO_1
type: NamedIndividual
type: Identity
forAll: term_IO_1_forAll
lhs: term_IO_1_lhs
rhs: term_IO_1_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Ingested data conforms to the source’s declared type.

IO_2
type: NamedIndividual
type: Identity
forAll: term_IO_2_forAll
lhs: term_IO_2_lhs
rhs: term_IO_2_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Emitted data conforms to the sink’s declared type.

IO_3
type: NamedIndividual
type: Identity
forAll: term_IO_3_forAll
lhs: term_IO_3_lhs
rhs: term_IO_3_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Every ingestion through a source produces a valid ring datum via grounding.

IO_4
type: NamedIndividual
type: Identity
forAll: term_IO_4_forAll
lhs: term_IO_4_lhs
rhs: term_IO_4_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Pipeline
comment: Every emission through a sink produces a valid surface symbol via projection.

IO_5
type: NamedIndividual
type: Identity
forAll: term_IO_5_forAll
lhs: term_IO_5_lhs
rhs: term_IO_5_rhs
universallyValid: true
validityKind: Universal
verificationDomain: Algebraic
comment: Every boundary effect touches at least one site.

UOR Queries
type: Ontology
imports: UOR Schema
imports: UOR Content Addressing
imports: UOR Type System
imports: UOR Partitions
imports: UOR Transforms and Morphisms
imports: UOR State
comment: Information extraction queries. Users initiate queries; the kernel resolves them against the ring substrate.
space: bridge

Query
type: Class
comment: A request for information from the UOR kernel. The root abstraction for all query types.
subclass of: Thing

CoordinateQuery
type: Class
comment: A query for the ring-coordinate position of a datum: its stratum, spectrum, and address within the ring geometry.
subclass of: Query

MetricQuery
type: Class
comment: A query for a metric value between two datums: ring distance, Hamming distance, or their divergence (curvature).
subclass of: Query

RepresentationQuery
type: Class
comment: A query for the canonical representation of a datum or term: its normal form under the active resolver strategy.
subclass of: Query

RelationQuery
type: Class
comment: A Query with a known source address, a typed relation constraint, and an open target partition:FreeRank. Represents any question of the form: given this source symbol and relation type, what is the target? The relation type is read from the observable coordinate triple (d_R, d_H, d_I) — not externally supplied. Applies across NLP, ARC-AGI grids, music, images, sensor streams, and logical formulae.
subclass of: Query

TriadProjection
type: Class
comment: A classification of coordinate types that a CoordinateQuery can extract. Each TriadProjection individual names a specific coordinate system (stratum, spectrum, address) replacing the string-valued query:coordinate property.
subclass of: Thing

SessionQuery
type: Class
comment: A RelationQuery that explicitly declares its session membership, allowing the conformance suite to validate session-scoped site reduction across multi-turn Prism deployments.
subclass of: RelationQuery

inputType
type: ObjectProperty
type: FunctionalProperty
comment: The type of input the query accepts.
domain: Query
range: Class

outputType
type: ObjectProperty
type: FunctionalProperty
comment: The type of output the query produces.
domain: Query
range: Class

hasTriadProjection
type: ObjectProperty
type: FunctionalProperty
comment: The typed coordinate kind this query extracts. Replaces the string-valued query:coordinate property with a typed reference to a TriadProjection individual.
domain: CoordinateQuery
range: TriadProjection

sourceAddress
type: ObjectProperty
type: FunctionalProperty
comment: The ring address of the grounded source symbol.
domain: RelationQuery
range: Element

relationType
type: ObjectProperty
type: FunctionalProperty
comment: The transformation type, expressed as a type:CompositeConstraint composed from the primitive basis. At inference time this is the output of an observable coordinate read on example pairs — computed from (d_R, d_H, d_I). Not an externally supplied input; read from the representation space.
domain: RelationQuery
range: Constraint

targetSite
type: ObjectProperty
type: FunctionalProperty
comment: The open site budget for the unknown target. Begins fully free; closes to isClosed = true upon resolution.
domain: RelationQuery
range: FreeRank

groundingMap
type: ObjectProperty
type: FunctionalProperty
comment: The GroundingMap that resolved the source symbol to its ring address.
domain: RelationQuery
range: GroundingMap

projectionMap
type: ObjectProperty
type: FunctionalProperty
comment: The ProjectionMap that will render the resolved address neighbourhood back to surface symbols.
domain: RelationQuery
range: ProjectionMap

sessionContext
type: ObjectProperty
type: FunctionalProperty
comment: Accumulated session state carrying all state:Binding instances produced by prior queries. Each binding records a resolved address, its datum, and the type under which it was resolved. Prior bindings monotonically reduce the free site space for subsequent queries.
domain: RelationQuery
range: Context

sessionMembership
type: ObjectProperty
type: FunctionalProperty
comment: The session this query belongs to.
domain: SessionQuery
range: Session

TwoAdicValuation
type: NamedIndividual
type: TriadProjection
comment: The stratum coordinate: the layer position of a datum within the ring's stratification.

WalshHadamardImage
type: NamedIndividual
type: TriadProjection
comment: The spectrum coordinate: the spectral decomposition of a datum under the ring's Fourier analysis.

RingElement
type: NamedIndividual
type: TriadProjection
comment: The address coordinate: the content-addressable position of a datum in the Braille glyph encoding.

UOR Resolvers
type: Ontology
imports: UOR Schema
imports: UOR Queries
imports: UOR Partitions
imports: UOR Type System
imports: UOR Predicates and Dispatch
comment: Type resolution strategies implementing the partition map Π : T_n → Part(R_n). Resolvers transform type declarations into ring partitions.
space: bridge

Resolver
type: Class
disjoint with: ResolutionState
disjoint with: RefinementSuggestion
comment: A strategy for resolving a type declaration into a partition of the ring. The kernel dispatches to a specific resolver based on the type's structure.
subclass of: Thing

DihedralFactorizationResolver
type: Class
comment: Resolves types by factoring the ring under dihedral group action. Identifies orbits under D_{2^n} to determine irreducibility boundaries.
subclass of: Resolver

CanonicalFormResolver
type: Class
comment: Resolves types by computing canonical forms via term rewriting. Applies the critical identity and normalization rules to reduce terms to unique canonical representatives.
subclass of: Resolver

EvaluationResolver
type: Class
comment: Resolves types by direct evaluation: applies operations to enumerate ring elements and classify them as irreducible, reducible, unit, or exterior.
subclass of: Resolver

ResolutionState
type: Class
disjoint with: Resolver
disjoint with: RefinementSuggestion
comment: The current state of an iterative resolution: tracks how many iterations have been performed, whether the resolution is complete, and the current site deficit.
subclass of: Thing

RefinementSuggestion
type: Class
disjoint with: Resolver
disjoint with: ResolutionState
comment: A suggestion from the resolver for how to refine an incomplete resolution: which metric axis to explore, which class to narrow to, and which sites to target.
subclass of: Thing

ComplexityClass
type: Class
comment: A computational complexity classification for resolvers. Each resolver's asymptotic runtime is typed as a named ComplexityClass individual rather than a free string.
subclass of: Thing

CechNerve
type: Class
comment: The simplicial complex whose vertices are constraints and where a k-simplex exists iff the corresponding k+1 constraints have nonempty intersection. The nerve's topology governs resolution convergence: trivial homology ↔ smooth convergence, non-trivial homology ↔ potential stalls.
subclass of: Thing
subclass of: SimplicialComplex
subclass of: KanComplex

CompletenessResolver
type: Class
comment: A specialisation of Resolver driving the completeness certification loop. Accepts a CompletenessCandidate, runs the ψ-pipeline (reading nerveEulerCharacteristic from ResolutionState), and either issues a CompletenessCertificate or produces a RefinementSuggestion.
subclass of: Resolver

WittLevelResolver
type: Class
comment: A Resolver parameterised by quantum level. The same resolver strategy runs at any quantum level n ≥ 1 by substituting the appropriate R_n ring.
subclass of: Resolver

SessionResolver
type: Class
comment: A Resolver that maintains a BindingAccumulator across multiple RelationQuery evaluations. The top-level resolver for multi-turn Prism deployments.
subclass of: Resolver

TypeSynthesisResolver
type: Class
comment: A Resolver that runs the ψ-pipeline in inverse mode. Accepts a TypeSynthesisGoal and returns a TypeSynthesisResult. Internally maintains a ConstraintSearchState tracking which constraint combinations have been explored and which Betti profiles they realise.
subclass of: Resolver

ConstraintSearchState
type: Class
comment: Internal resolver state tracking the boundary of explored constraint combinations during synthesis. Carries exploredCount, currentCandidate, and a link to the best SynthesisSignature achieved so far.
subclass of: Thing

IncrementalCompletenessResolver
type: Class
comment: A Resolver that determines whether a CompleteType T at Q_n lifts to a CompleteType at Q_{n+1} without re-running the full ψ-pipeline from scratch. It computes the SpectralSequencePage sequence, reads the LiftObstruction, and either confirms the lift or returns a LiftRefinementSuggestion.
subclass of: Resolver

LiftRefinementSuggestion
type: Class
comment: A RefinementSuggestion produced when a WittLift has a non-trivial LiftObstruction. Specialises RefinementSuggestion with liftSitePosition (the new bit position n+1) and obstructionClass.
subclass of: RefinementSuggestion

MonodromyResolver
type: Class
comment: A Resolver that computes the HolonomyGroup of a ConstrainedType by enumerating closed paths in the constraint nerve and accumulating DihedralElement values. Outputs a MonodromyClass and classifies the type as FlatType or TwistedType.
subclass of: Resolver

JacobianGuidedResolver
type: Class
comment: A resolver that uses the Jacobian matrix to guide constraint selection, implementing DC_10: select the constraint that maximises total curvature reduction.
subclass of: Resolver

SuperpositionResolver
type: Class
comment: A resolver that handles superposed site states, computing amplitudes and determining when superposition collapses to a classical site assignment (Amendment 32).
subclass of: Resolver

GroundingAwareResolver
type: Class
comment: A resolver that exploits accumulated session bindings at full saturation (σ = 1) to provide O(1) resolution via direct coordinate reads (SC_5).
subclass of: Resolver

GeodesicValidator
type: Class
comment: A resolver that validates whether a ComputationTrace satisfies the dual geodesic condition (AR_1-ordered and DC_10-selected). Produces GeodesicViolation individuals on failure.
subclass of: Resolver

MeasurementResolver
type: Class
comment: A resolver that handles projective collapse of SuperposedSiteState components. Issues MeasurementCertificate upon successful collapse with QM_1 verification.
subclass of: Resolver

TowerCompletenessResolver
type: Class
comment: A Resolver that constructs a LiftChain from liftSourceLevel to an arbitrary liftTargetLevel Q_k by iterating IncrementalCompletenessResolver step by step.
subclass of: Resolver

ExecutionPolicy
type: Class
disjoint with: Resolver
disjoint with: ResolutionState
disjoint with: RefinementSuggestion
comment: A strategy class that defines how a SessionResolver orders pending RelationQuery instances for dispatch. The policy reads the targetSite.freeRank of each pending query and applies an ordering function.
subclass of: Thing

ExecutionPolicyKind
type: Class
comment: A typed controlled vocabulary for ExecutionPolicy individuals. Follows the SessionBoundaryType pattern: a single class with named individuals rather than a subclass hierarchy.
subclass of: Thing

HomotopyResolver
type: Class
comment: A resolver that runs the extended ψ-pipeline (ψ_7–ψ_9) to compute the full homotopy type of a CechNerve. Returns HomotopyGroup observables and PostnikovTruncation records.
subclass of: Resolver

ModuliResolver
type: Class
comment: A resolver that computes the local structure of the moduli space at a given CompleteType: constructs the DeformationComplex, determines the HolonomyStratum, and computes tangent/obstruction dimensions.
subclass of: Resolver

inputType
type: ObjectProperty
type: FunctionalProperty
comment: The type of input this resolver accepts.
domain: Resolver
range: TypeDefinition

outputType
type: ObjectProperty
type: FunctionalProperty
comment: The type of output this resolver produces. For all UOR resolvers, the output is a partition:Partition.
domain: Resolver
range: Partition

strategy
type: ObjectProperty
type: FunctionalProperty
comment: A human-readable description of the resolution strategy this resolver implements.
domain: Resolver
range: TermExpression

resolutionState
type: ObjectProperty
type: FunctionalProperty
comment: The current resolution state of this resolver.
domain: Resolver
range: ResolutionState

isComplete
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this resolution is complete: all sites are pinned and the partition is fully determined.
domain: ResolutionState
range: boolean

iterationCount
type: DatatypeProperty
type: FunctionalProperty
comment: The number of refinement iterations performed so far.
domain: ResolutionState
range: nonNegativeInteger

siteDeficit
type: ObjectProperty
type: FunctionalProperty
comment: The site budget showing the remaining unpinned sites. When all sites are pinned, the deficit is zero and resolution is complete.
domain: ResolutionState
range: FreeRank

suggestion
type: ObjectProperty
comment: A refinement suggestion for advancing this resolution.
domain: ResolutionState
range: RefinementSuggestion

suggestedAxis
type: ObjectProperty
type: FunctionalProperty
comment: The metric axis this suggestion recommends exploring.
domain: RefinementSuggestion
range: MetricAxis

suggestedClass
type: ObjectProperty
type: FunctionalProperty
comment: The constraint class this suggestion recommends applying.
domain: RefinementSuggestion
range: Class

targetSites
type: ObjectProperty
comment: The site coordinates this suggestion targets for pinning.
domain: RefinementSuggestion
range: SiteIndex

convergenceRate
type: DatatypeProperty
type: FunctionalProperty
comment: The rate at which sites are being pinned per iteration. A higher rate indicates faster convergence toward a complete resolution.
domain: ResolutionState
range: decimal

hasComplexityClass
type: ObjectProperty
type: FunctionalProperty
comment: The computational complexity class of this resolver. Replaces the string-valued resolver:complexity property.
domain: Resolver
range: ComplexityClass

cechNerve
type: ObjectProperty
type: FunctionalProperty
comment: The constraint nerve associated with this resolution state.
domain: ResolutionState
range: CechNerve

residualEntropy
type: DatatypeProperty
type: FunctionalProperty
comment: The residual Shannon entropy of the resolution state: S = freeRank × ln 2. Measures remaining uncertainty.
domain: ResolutionState
range: decimal

topologicallyComplete
type: DatatypeProperty
type: FunctionalProperty
comment: Whether all Betti numbers of the constraint nerve are zero, indicating no topological obstructions to resolution.
domain: ResolutionState
range: boolean

nerveEulerCharacteristic
type: DatatypeProperty
type: FunctionalProperty
comment: The Euler characteristic χ(N(C)) of the active constraint nerve at this resolution state. IT_7d requires this value to equal n (the quantum level) for resolution to be complete. Cached here to avoid recomputing the full ψ pipeline on each iteration check.
domain: ResolutionState
range: integer

completenessTarget
type: ObjectProperty
type: FunctionalProperty
comment: The CompletenessCandidate this resolver is certifying.
domain: CompletenessResolver
range: CompletenessCandidate

quantumLevel
type: ObjectProperty
type: FunctionalProperty
comment: The quantum level this resolver instance is configured for.
domain: WittLevelResolver
range: WittLevel

sessionAccumulator
type: ObjectProperty
type: FunctionalProperty
comment: The BindingAccumulator this session resolver maintains across multiple RelationQuery evaluations.
domain: SessionResolver
range: BindingAccumulator

synthesisGoal
type: ObjectProperty
type: FunctionalProperty
comment: The goal this type synthesis resolver is working to achieve.
domain: TypeSynthesisResolver
range: TypeSynthesisGoal

exploredCount
type: DatatypeProperty
type: FunctionalProperty
comment: Number of constraint combinations evaluated so far during synthesis.
domain: ConstraintSearchState
range: nonNegativeInteger

currentCandidate
type: ObjectProperty
type: FunctionalProperty
comment: The type candidate currently being evaluated during synthesis.
domain: ConstraintSearchState
range: ConstrainedType

liftTarget
type: ObjectProperty
type: FunctionalProperty
comment: The WittLift this incremental completeness resolver is evaluating.
domain: IncrementalCompletenessResolver
range: WittLift

liftSitePosition
type: ObjectProperty
type: FunctionalProperty
comment: The new site position at Q_{n+1} that the lift refinement suggestion targets.
domain: LiftRefinementSuggestion
range: SiteIndex

obstructionClass
type: ObjectProperty
type: FunctionalProperty
comment: The obstruction class this lift refinement suggestion is designed to kill.
domain: LiftRefinementSuggestion
range: LiftObstructionClass

monodromyTarget
type: ObjectProperty
type: FunctionalProperty
comment: The type whose holonomy this monodromy resolver is computing.
domain: MonodromyResolver
range: ConstrainedType

holonomyResult
type: ObjectProperty
type: FunctionalProperty
comment: The HolonomyGroup produced by this monodromy resolver run.
domain: MonodromyResolver
range: HolonomyGroup

guidingJacobian
type: ObjectProperty
type: FunctionalProperty
comment: The Jacobian observable guiding constraint selection at this resolution state (DC_10).
domain: ResolutionState
range: Jacobian

usedGrounding
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this resolver used the saturation shortcut (SC_5) to bypass the ψ-pipeline and return a direct coordinate read.
domain: GroundingAwareResolver
range: boolean

validateGeodesic
type: ObjectProperty
type: FunctionalProperty
comment: The GeodesicTrace being validated by this GeodesicValidator.
domain: GeodesicValidator
range: GeodesicTrace

collapseAmplitude
type: DatatypeProperty
type: FunctionalProperty
comment: The amplitude of the SuperposedSiteState prior to projective collapse by this MeasurementResolver.
domain: MeasurementResolver
range: decimal

collapsedSite
type: DatatypeProperty
type: FunctionalProperty
comment: The site index that was collapsed (pinned to a classical value) by the projective measurement.
domain: MeasurementResolver
range: nonNegativeInteger

measurementOutcome
type: DatatypeProperty
type: FunctionalProperty
comment: The classical value obtained from the projective collapse. Either 0 or 1 for a single-site measurement.
domain: MeasurementResolver
range: nonNegativeInteger

amplitudeVector
type: DatatypeProperty
type: FunctionalProperty
comment: The amplitude vector of all branches maintained by this SuperpositionResolver during ψ-pipeline traversal. Encoded as a comma-separated list of decimal amplitudes. Must satisfy Σ|αᵢ|² = 1 (QM_5) after normalization.
domain: SuperpositionResolver
range: decimal

priorAmplitudeVector
type: DatatypeProperty
type: FunctionalProperty
comment: The full vector of all branch amplitudes before projective collapse. Recorded by the MeasurementResolver to enable Born rule verification (QM_5): P(outcome k) = |α_k|².
domain: MeasurementResolver
range: decimal

towerSourceLevel
type: ObjectProperty
type: FunctionalProperty
comment: The level at which the tower starts.
domain: TowerCompletenessResolver
range: WittLevel

towerTargetLevel
type: ObjectProperty
type: FunctionalProperty
comment: The level to which the tower is being built.
domain: TowerCompletenessResolver
range: WittLevel

currentChain
type: ObjectProperty
type: FunctionalProperty
comment: The LiftChain under construction.
domain: TowerCompletenessResolver
range: LiftChain

towerStepResolver
type: ObjectProperty
type: FunctionalProperty
comment: The IncrementalCompletenessResolver used for each single-step lift.
domain: TowerCompletenessResolver
range: IncrementalCompletenessResolver

executionPolicy
type: ObjectProperty
type: FunctionalProperty
comment: The ordering strategy this resolver applies to pending queries. Defaults to FifoPolicy if unset.
domain: SessionResolver
range: ExecutionPolicy

homotopyTarget
type: ObjectProperty
type: FunctionalProperty
comment: The CechNerve whose homotopy type this resolver computes.
domain: HomotopyResolver
range: CechNerve

homotopyResult
type: ObjectProperty
comment: A HomotopyGroup observable produced by this resolver.
domain: HomotopyResolver
range: HomotopyGroup

moduliTarget
type: ObjectProperty
type: FunctionalProperty
comment: The CompleteType whose local moduli structure this resolver computes.
domain: ModuliResolver
range: CompleteType

moduliDeformation
type: ObjectProperty
type: FunctionalProperty
comment: The DeformationComplex constructed by this resolver.
domain: ModuliResolver
range: DeformationComplex

dispatchTable
type: ObjectProperty
type: FunctionalProperty
comment: The dispatch table governing resolver selection for this resolver class.
domain: Resolver
range: DispatchTable

resolverPredicate
type: ObjectProperty
type: FunctionalProperty
comment: The predicate that selects this specific resolver. When the predicate evaluates to true on an input type, this resolver is chosen.
domain: Resolver
range: TypePredicate

ConstantTime
type: NamedIndividual
type: ComplexityClass
comment: O(1) complexity — the resolver runs in constant time regardless of ring size.

LogarithmicTime
type: NamedIndividual
type: ComplexityClass
comment: O(log n) complexity — the resolver runs in logarithmic time in the quantum level.

LinearTime
type: NamedIndividual
type: ComplexityClass
comment: O(n) complexity — the resolver runs in time linear in the quantum level.

ExponentialTime
type: NamedIndividual
type: ComplexityClass
comment: O(2^n) complexity — the resolver runs in time exponential in the quantum level.

FifoPolicy
type: NamedIndividual
type: ExecutionPolicyKind
comment: Process queries in arrival order. The implicit pre-Amendment 48 behavior.

MinFreeCountFirst
type: NamedIndividual
type: ExecutionPolicyKind
comment: Process the query with the smallest targetSite.freeRank first. Favors cheapest resolutions, accelerating early grounding gain.

MaxFreeCountFirst
type: NamedIndividual
type: ExecutionPolicyKind
comment: Process the query with the largest targetSite.freeRank first. Favors hardest resolutions, maximizing information gain per step.

DisjointFirst
type: NamedIndividual
type: ExecutionPolicyKind
comment: Process queries whose targetSite is disjoint from all other pending queries' site sets first. Minimizes contention when operating against a SharedContext.

UOR Type System
type: Ontology
imports: UOR Schema
imports: UOR Content Addressing
imports: UOR Operations
comment: Runtime type declarations that parameterize the resolution pipeline. Types are declared by Prism applications and resolved to partitions of the ring.
space: user

TypeDefinition
type: Class
disjoint with: Constraint
disjoint with: MetricAxis
comment: A runtime type declaration. The root class for all UOR types. Each TypeDefinition, when resolved, produces a partition of the ring at the specified quantum level.
subclass of: Thing

PrimitiveType
type: Class
comment: A primitive type defined by a fixed bit width. The carrier is the entire ring Z/(2^n)Z at the specified quantum level.
subclass of: TypeDefinition

ProductType
type: Class
comment: A product (Cartesian) type formed from multiple component types. The carrier is the product of the component carriers.
subclass of: TypeDefinition

SumType
type: Class
comment: A sum (disjoint union) type formed from multiple variant types. The carrier is the disjoint union of the variant carriers.
subclass of: TypeDefinition

ConstrainedType
type: Class
comment: A type formed by constraining a base type with a predicate. The carrier is the subset of the base carrier satisfying the constraint.
subclass of: TypeDefinition

Constraint
type: Class
disjoint with: TypeDefinition
disjoint with: MetricAxis
comment: A composable predicate that refines a type by pinning one or more site coordinates. Constraints are the parameterization mechanism for ConstrainedType.
subclass of: Thing

ResidueConstraint
type: Class
disjoint with: CarryConstraint
disjoint with: DepthConstraint
disjoint with: CompositeConstraint
disjoint with: HammingConstraint
disjoint with: SiteConstraint
disjoint with: AffineConstraint
comment: A constraint based on residue class membership: x ≡ r (mod m). Pins sites corresponding to the residue pattern.
subclass of: Constraint

CarryConstraint
type: Class
disjoint with: ResidueConstraint
disjoint with: DepthConstraint
disjoint with: CompositeConstraint
disjoint with: HammingConstraint
disjoint with: SiteConstraint
disjoint with: AffineConstraint
comment: A constraint based on carry propagation patterns in ring arithmetic. Pins sites corresponding to carry positions.
subclass of: Constraint

DepthConstraint
type: Class
disjoint with: ResidueConstraint
disjoint with: CarryConstraint
disjoint with: CompositeConstraint
disjoint with: HammingConstraint
disjoint with: SiteConstraint
disjoint with: AffineConstraint
comment: A constraint on factorization depth: the minimum and maximum number of irreducible factors. Pins sites by bounding the factorization tree depth.
subclass of: Constraint

CompositeConstraint
type: Class
disjoint with: ResidueConstraint
disjoint with: CarryConstraint
disjoint with: DepthConstraint
disjoint with: HammingConstraint
disjoint with: SiteConstraint
disjoint with: AffineConstraint
comment: A constraint formed by composing two or more simpler constraints. The composite pins the union of sites pinned by its components.
subclass of: Constraint

HammingConstraint
type: Class
disjoint with: ResidueConstraint
disjoint with: CarryConstraint
disjoint with: DepthConstraint
disjoint with: CompositeConstraint
disjoint with: SiteConstraint
disjoint with: AffineConstraint
comment: Pins the Hamming weight of the Datum to at most the bound. The horizontal axis of the tri-metric.
subclass of: Constraint

SiteConstraint
type: Class
disjoint with: ResidueConstraint
disjoint with: CarryConstraint
disjoint with: DepthConstraint
disjoint with: CompositeConstraint
disjoint with: HammingConstraint
disjoint with: AffineConstraint
comment: Pins a single site coordinate to 0 or 1. The atomic unit of the site budget.
subclass of: Constraint

AffineConstraint
type: Class
disjoint with: ResidueConstraint
disjoint with: CarryConstraint
disjoint with: DepthConstraint
disjoint with: CompositeConstraint
disjoint with: HammingConstraint
disjoint with: SiteConstraint
comment: Pins the Datum to an affine subspace specified by an offset and a set of generators.
subclass of: Constraint

MetricAxis
type: Class
disjoint with: TypeDefinition
disjoint with: Constraint
comment: A classification axis for constraints by their geometric effect. The three axes — vertical (ring/additive), horizontal (Hamming/bitwise), diagonal (incompatibility) — form the tri-metric coordinate system of UOR.
subclass of: Thing

CompleteType
type: Class
comment: A TypeDefinition certified to satisfy the UOR completeness criterion (IT_7d): its constraint nerve N(C) has Euler characteristic χ = n and all Betti numbers β_k = 0. A CompleteType guarantees that resolution closes the site budget in O(1) — no iterative refinement is required. Completeness is attested by a cert:CompletenessCertificate linked via cert:certifiedType. This class is not addressable from a type-expr position in the term language; references from term-language positions are rejected by the resolver.
subclass of: TypeDefinition

CompletenessCandidate
type: Class
disjoint with: CompleteType
comment: A ConstrainedType actively undergoing the completeness certification pipeline. Links to the resolver:ResolutionState tracking the current iteration and to the resolver:CechNerve being computed. Disjoint from CompleteType (which is already certified). This class is not addressable from a type-expr position in the term language; references from term-language positions are rejected by the resolver.
subclass of: ConstrainedType

CompletenessWitness
type: Class
comment: A record of a single site-closing event: one constraint application that reduced the FreeRank deficit. Carries the applied constraint and the sitesClosed count. Forms the ordered audit trail between ConstrainedType and CompleteType.
subclass of: Thing

TypeSynthesisGoal
type: Class
comment: A specification of the desired topological properties of a type to be synthesised. Carries a target Euler characteristic (targetEulerCharacteristic) and a target Betti profile (zero or more targetBettiNumber assertions). The minimal goal for O(1) resolution is: targetEulerCharacteristic = n and all targetBettiNumber = 0 — the IT_7d profile.
subclass of: Thing

TypeSynthesisResult
type: Class
comment: The output of a TypeSynthesisResolver run. Contains the SynthesizedType, the realised topological signature (as a SynthesisSignature), and the SynthesisTrace recording the construction steps.
subclass of: Thing

SynthesizedType
type: Class
comment: A ConstrainedType produced by the TypeSynthesisResolver. Distinguished from a hand-authored ConstrainedType by the presence of a type:synthesisResult link. May or may not be a CompleteType, depending on the synthesis goal. This class is not addressable from a type-expr position in the term language; references from term-language positions are rejected by the resolver.
subclass of: ConstrainedType

MinimalConstraintBasis
type: Class
comment: The minimal set of constraints in the SynthesizedType's constraint set that is sufficient to realise the target topological signature. The minimality criterion is that removing any single member changes the realised signature.
subclass of: Thing

WittLift
type: Class
comment: A ConstrainedType T' over R_{n+1} obtained by extending a ConstrainedType T over R_n. Carries a link to the base type (liftBase), the quantum level it lifts to (liftTargetLevel), and the LiftObstruction (if the lift fails to transfer completeness). A WittLift is a CompleteType iff its LiftObstruction is trivial.
subclass of: Thing

LiftObstruction
type: Class
comment: The algebraic obstruction to a WittLift inheriting the completeness of its base type. Computed as the image of the spectral sequence differential d_2. If trivial (zero), the base type's completeness lifts. If non-trivial, at least one additional constraint is needed at the new quantum level.
subclass of: Thing

TwistedType
type: Class
disjoint with: FlatType
comment: A ConstrainedType whose HolonomyGroup is non-trivial — at least one closed constraint path produces a non-identity dihedral element. A TwistedType may still be a CompleteType (IT_7d is a homological, not holonomic, criterion), but its resolution paths require tracking dihedral accumulation. This class is not addressable from a type-expr position in the term language; references from term-language positions are rejected by the resolver.
subclass of: ConstrainedType

FlatType
type: Class
disjoint with: TwistedType
comment: A ConstrainedType whose HolonomyGroup is trivial — all closed constraint paths have identity monodromy. The constraint configuration is topologically flat: resolution is path-independent. This class is not addressable from a type-expr position in the term language; references from term-language positions are rejected by the resolver.
subclass of: ConstrainedType

SuperposedSiteState
type: Class
comment: A type representing a superposition of site states where sites carry complex amplitudes rather than binary pinned/free assignments. Ontological realisation of RC_5 (Amendment 32). This class is not addressable from a type-expr position in the term language; references from term-language positions are rejected by the resolver.
subclass of: TypeDefinition

ForbiddenSignature
type: Class
comment: A topological signature (χ, β_k) that is formally impossible to achieve for any ConstrainedType. Witnessed by an ImpossibilityWitness in proof/.
subclass of: Thing

CollapsedSiteState
type: Class
comment: A site state that has undergone projective collapse from a SuperposedSiteState to a definitive classical value. Topologically equivalent to a classically pinned site (QM_2).
subclass of: Thing

LiftChain
type: Class
comment: An ordered composition of WittLift steps from liftSourceLevel (Q_j) to liftTargetLevel (Q_k) for any j < k. The canonical object certifying type completeness at arbitrary Q_k.
subclass of: Thing

ObstructionChain
type: Class
comment: The complete ordered sequence of LiftObstruction records encountered and resolved along a LiftChain. An empty ObstructionChain means the tower is flat.
subclass of: Thing

ModuliSpace
type: Class
comment: The space of all CompleteTypes over R_n at a given quantum level.
subclass of: Thing

HolonomyStratum
type: Class
comment: A stratum indexed by a conjugacy class of subgroups of D_{2^n}.
subclass of: Thing

DeformationFamily
type: Class
comment: A one-parameter family of constraint configurations parameterizing a path.
subclass of: Thing

VersalDeformation
type: Class
comment: The versal deformation of a CompleteType T.
subclass of: Thing

ModuliTowerMap
type: Class
comment: The map induced by WittLift from one moduli space to the next.
subclass of: Thing

GaloisConnection
type: Class
comment: The adjunction between the type lattice and the site lattice. The upper adjoint is type closure; the lower adjoint is site interior. σ = lower adjoint evaluation; r = complement of upper adjoint image.
subclass of: Thing

TypeInclusion
type: Class
comment: A morphism T₁ → T₂ witnessing that T₁ is a subtype of T₂: the constraint set of T₁ is a superset of the constraint set of T₂.
subclass of: Transform

VarianceAnnotation
type: Class
comment: The variance of a structural type position under operad composition. One of Covariant, Contravariant, Invariant, or Bivariant.
subclass of: Thing

SubtypingLattice
type: Class
comment: The partial order on types induced by TypeInclusion. The top element is PrimitiveType (no constraints); the bottom elements are CompleteTypes (all sites pinned).
subclass of: Thing

bitWidth
type: DatatypeProperty
type: FunctionalProperty
comment: The bit width of a primitive type (the quantum level n). The carrier is Z/(2^n)Z.
domain: PrimitiveType
range: positiveInteger

component
type: ObjectProperty
comment: A component type in a product type.
domain: ProductType
range: TypeDefinition

baseType
type: ObjectProperty
type: FunctionalProperty
comment: The base type that a constrained type restricts.
domain: ConstrainedType
range: TypeDefinition

contentAddress
type: ObjectProperty
type: FunctionalProperty
comment: The content-derived address of this type definition, uniquely identifying the type in the UOR address space.
domain: TypeDefinition
range: Element

hasConstraint
type: ObjectProperty
comment: A typed constraint object applied to this constrained type. Replaces the deprecated string-based type:constraint property.
domain: ConstrainedType
range: Constraint

modulus
type: DatatypeProperty
type: FunctionalProperty
comment: The modulus m of a residue constraint: x ≡ r (mod m).
domain: ResidueConstraint
range: positiveInteger

residue
type: DatatypeProperty
type: FunctionalProperty
comment: The residue value r of a residue constraint: x ≡ r (mod m).
domain: ResidueConstraint
range: nonNegativeInteger

carryPattern
type: ObjectProperty
type: FunctionalProperty
comment: The carry propagation pattern of a carry constraint, expressed as a Datum at the appropriate quantum level.
domain: CarryConstraint
range: Datum

minDepth
type: DatatypeProperty
type: FunctionalProperty
comment: The minimum factorization depth required by a depth constraint.
domain: DepthConstraint
range: nonNegativeInteger

maxDepth
type: DatatypeProperty
type: FunctionalProperty
comment: The maximum factorization depth allowed by a depth constraint.
domain: DepthConstraint
range: nonNegativeInteger

composedFrom
type: ObjectProperty
comment: A component constraint of this composite constraint.
domain: CompositeConstraint
range: Constraint

hammingBound
type: DatatypeProperty
type: FunctionalProperty
comment: Upper bound on the Hamming weight of the Datum.
domain: HammingConstraint
range: nonNegativeInteger

siteIndex
type: DatatypeProperty
type: FunctionalProperty
comment: Zero-based index of the pinned site coordinate.
domain: SiteConstraint
range: nonNegativeInteger

siteValue
type: ObjectProperty
type: FunctionalProperty
comment: The value the pinned site coordinate must equal (a Datum in the set {0, 1}).
domain: SiteConstraint
range: Datum

affineOffset
type: ObjectProperty
type: FunctionalProperty
comment: Constant offset defining the affine subspace.
domain: AffineConstraint
range: Datum

affineGenerator
type: ObjectProperty
comment: A generator of the affine subspace. Non-functional: multiple generators span the subspace.
domain: AffineConstraint
range: Datum

metricAxis
type: ObjectProperty
type: FunctionalProperty
comment: The metric axis along which this constraint operates: vertical (ring), horizontal (Hamming), or diagonal (incompatibility).
domain: Constraint
range: MetricAxis

pinsSites
type: ObjectProperty
comment: A site coordinate that this constraint pins when applied.
domain: Constraint
range: SiteIndex

affectsAxis
type: ObjectProperty
comment: The metric axis that this operation affects. Replaces the string-valued type:axisSignature property with a typed reference to MetricAxis individuals. Non-functional: an operation may affect multiple axes.
domain: Operation
range: MetricAxis

crossingCost
type: DatatypeProperty
type: FunctionalProperty
comment: The cost of applying this constraint in terms of axis crossings: the number of metric boundaries that must be traversed.
domain: Constraint
range: nonNegativeInteger

completenessCandidate
type: ObjectProperty
comment: The ConstrainedType being evaluated for completeness by this CompletenessCandidate.
domain: CompletenessCandidate
range: ConstrainedType

candidateNerve
type: ObjectProperty
type: FunctionalProperty
comment: The constraint nerve being computed for this candidate. The CompletenessResolver reads χ(N(C)) from this nerve at each iteration via resolver:nerveEulerCharacteristic.
domain: CompletenessCandidate
range: CechNerve

witnessConstraint
type: ObjectProperty
type: FunctionalProperty
comment: The constraint applied in this witness step.
domain: CompletenessWitness
range: Constraint

sitesClosed
type: DatatypeProperty
type: FunctionalProperty
comment: Number of sites closed by this witness step.
domain: CompletenessWitness
range: nonNegativeInteger

targetEulerCharacteristic
type: DatatypeProperty
type: FunctionalProperty
comment: The target χ(N(C)) value. For O(1) resolution: set equal to n (the quantum level).
domain: TypeSynthesisGoal
range: integer

targetBettiNumber
type: DatatypeProperty
comment: Non-functional. Each assertion specifies a target Betti number value for a given homological degree. Multiple assertions permitted, one per degree.
domain: TypeSynthesisGoal
range: nonNegativeInteger

synthesisResult
type: ObjectProperty
type: FunctionalProperty
comment: Links a SynthesizedType back to the synthesis run that produced it.
domain: SynthesizedType
range: TypeSynthesisResult

basisConstraint
type: ObjectProperty
comment: Non-functional. One assertion per constraint in the minimal basis.
domain: MinimalConstraintBasis
range: Constraint

basisSize
type: DatatypeProperty
type: FunctionalProperty
comment: The cardinality of the minimal basis. The theoretical lower bound is n (one constraint per site).
domain: MinimalConstraintBasis
range: nonNegativeInteger

liftBase
type: ObjectProperty
type: FunctionalProperty
comment: The base type being lifted to the next quantum level.
domain: WittLift
range: ConstrainedType

liftTargetLevel
type: ObjectProperty
type: FunctionalProperty
comment: The quantum level this lift targets.
domain: WittLift
range: WittLevel

liftObstruction
type: ObjectProperty
type: FunctionalProperty
comment: The LiftObstruction for this lift. Trivial (zero class) iff the lift inherits completeness.
domain: WittLift
range: LiftObstruction

obstructionTrivial
type: DatatypeProperty
type: FunctionalProperty
comment: True iff the obstruction class is zero — the base type's completeness transfers to the lifted quantum level without additional constraints.
domain: LiftObstruction
range: boolean

obstructionSite
type: ObjectProperty
type: FunctionalProperty
comment: The site at the new quantum level where the obstruction is located. Ranges over the new bit position introduced at Q_{n+1}.
domain: LiftObstruction
range: SiteIndex

holonomyGroup
type: ObjectProperty
type: FunctionalProperty
comment: The HolonomyGroup of this type. Computed by the MonodromyResolver.
domain: ConstrainedType
range: HolonomyGroup

monodromyClass
type: ObjectProperty
type: FunctionalProperty
comment: The MonodromyClass classifying this type as flat or twisted.
domain: ConstrainedType
range: MonodromyClass

amplitude
type: DatatypeProperty
type: FunctionalProperty
comment: The amplitude coefficient for this superposed site state.
domain: SuperposedSiteState
range: decimal

targetForbidden
type: DatatypeProperty
type: FunctionalProperty
comment: Whether the target signature of this TypeSynthesisGoal is a ForbiddenSignature. If true, synthesis is provably impossible.
domain: TypeSynthesisGoal
range: boolean

collapsedFrom
type: ObjectProperty
type: FunctionalProperty
comment: The SuperposedSiteState from which this CollapsedSiteState was produced by projective measurement.
domain: CollapsedSiteState
range: SuperposedSiteState

survivingAmplitude
type: DatatypeProperty
type: FunctionalProperty
comment: The amplitude of the surviving branch after projective collapse. |α|² is the probability of this outcome under the Born rule.
domain: CollapsedSiteState
range: decimal

normalizationVerified
type: DatatypeProperty
type: FunctionalProperty
comment: Whether the amplitude vector of this SuperposedSiteState satisfies the normalization condition Σ|αᵢ|² = 1 (QM_5). Set by the SuperpositionResolver after verification.
domain: SuperposedSiteState
range: boolean

holonomyClassified
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this ConstrainedType has been classified as FlatType or TwistedType by the MonodromyResolver. The MN_8 identity guarantees this is a bivalent classification: holonomyClassified(T) iff isFlatType(T) xor isTwistedType(T).
domain: ConstrainedType
range: boolean

liftSourceLevel
type: ObjectProperty
type: FunctionalProperty
comment: The quantum level at the base of the chain.
domain: LiftChain
range: WittLevel

chainLength
type: DatatypeProperty
type: FunctionalProperty
comment: The number of WittLift steps in the chain (k - j).
domain: LiftChain
range: nonNegativeInteger

chainStep
type: ObjectProperty
comment: A WittLift step in this chain. Non-functional: one per step.
domain: LiftChain
range: WittLift

chainObstructionProfile
type: ObjectProperty
type: FunctionalProperty
comment: The full obstruction history of this chain.
domain: LiftChain
range: ObstructionChain

resolvedBasisSize
type: DatatypeProperty
type: FunctionalProperty
comment: The basis size of the CompleteType at the chain target level.
domain: LiftChain
range: nonNegativeInteger

obstructionAt
type: ObjectProperty
comment: A non-trivial LiftObstruction in this chain. Non-functional.
domain: ObstructionChain
range: LiftObstruction

obstructionCount
type: DatatypeProperty
type: FunctionalProperty
comment: Total number of non-trivial LiftObstruction records.
domain: ObstructionChain
range: nonNegativeInteger

isFlat
type: DatatypeProperty
type: FunctionalProperty
comment: True iff obstructionCount = 0 (flat tower).
domain: ObstructionChain
range: boolean

moduliWittLevel
type: ObjectProperty
type: FunctionalProperty
comment: The quantum level at which this moduli space is defined.
domain: ModuliSpace
range: WittLevel

moduliPoint
type: ObjectProperty
comment: A CompleteType that is a point of this moduli space.
domain: ModuliSpace
range: CompleteType

moduliDimension
type: DatatypeProperty
type: FunctionalProperty
comment: The dimension of this moduli space.
domain: ModuliSpace
range: nonNegativeInteger

stratumHolonomyClass
type: ObjectProperty
type: FunctionalProperty
comment: The MonodromyClass indexing this stratum.
domain: HolonomyStratum
range: MonodromyClass

stratumCodimension
type: DatatypeProperty
type: FunctionalProperty
comment: The codimension of this stratum within the moduli space.
domain: HolonomyStratum
range: nonNegativeInteger

stratumModuli
type: ObjectProperty
type: FunctionalProperty
comment: The moduli space containing this stratum.
domain: HolonomyStratum
range: ModuliSpace

familyParameter
type: ObjectProperty
comment: A CompleteType along the deformation path.
domain: DeformationFamily
range: CompleteType

familyPreservesCompleteness
type: DatatypeProperty
type: FunctionalProperty
comment: Whether every member of this deformation family remains a CompleteType.
domain: DeformationFamily
range: boolean

versalBase
type: ObjectProperty
type: FunctionalProperty
comment: The CompleteType at the base of this versal deformation.
domain: VersalDeformation
range: CompleteType

versalDimension
type: DatatypeProperty
type: FunctionalProperty
comment: The dimension of the versal deformation space.
domain: VersalDeformation
range: nonNegativeInteger

towerMapSource
type: ObjectProperty
type: FunctionalProperty
comment: The source moduli space of this tower map.
domain: ModuliTowerMap
range: ModuliSpace

upperAdjoint
type: ObjectProperty
type: FunctionalProperty
comment: The upper adjoint (type closure) of the Galois connection, expressed as a symbolic formula string.
domain: GaloisConnection
range: TermExpression

lowerAdjoint
type: ObjectProperty
type: FunctionalProperty
comment: The lower adjoint (site interior) of the Galois connection, expressed as a symbolic formula string.
domain: GaloisConnection
range: TermExpression

fixpointCondition
type: ObjectProperty
type: FunctionalProperty
comment: The fixpoint condition for the Galois connection: when upper(lower(T)) = T, the type is complete.
domain: GaloisConnection
range: TermExpression

refinementDirection
type: DatatypeProperty
type: FunctionalProperty
comment: Description of the refinement direction: ascending in type lattice corresponds to descending in site freedom.
domain: GaloisConnection
range: nonNegativeInteger

structuralSiteCount
type: AnnotationProperty
comment: Formula or description of the site count for a structural type individual.
range: string

structuralGrounding
type: AnnotationProperty
comment: Description of the grounding rule that maps values of this structural type onto the UOR ring.
range: string

structuralConstraint
type: AnnotationProperty
comment: Description of the constraint characterizing this structural type, if applicable.
range: string

galoisClosureProperty
type: ObjectProperty
type: FunctionalProperty
comment: The closure property of a Galois connection.
domain: GaloisConnection
range: TermExpression

galoisInteriorProperty
type: ObjectProperty
type: FunctionalProperty
comment: The interior property of a Galois connection.
domain: GaloisConnection
range: TermExpression

groundingMapRef
type: ObjectProperty
type: FunctionalProperty
comment: Reference to the grounding map used for this type.
range: GroundingMap

compositionLaw
type: ObjectProperty
type: FunctionalProperty
comment: The algebraic composition law governing this type.
range: TermExpression

siteOrderingConstraint
type: ObjectProperty
type: FunctionalProperty
comment: Constraint on the ordering of sites for this type.
range: TermExpression

backboneThreshold
type: DatatypeProperty
type: FunctionalProperty
comment: The threshold for backbone inclusion in the type lattice.
range: nonNegativeInteger

permutationGroup
type: ObjectProperty
type: FunctionalProperty
comment: The permutation group acting on sites of this type.
range: Group

acyclicityWitness
type: DatatypeProperty
type: FunctionalProperty
comment: Witness certifying acyclicity of the type dependency graph.
range: boolean

connectivityWitness
type: DatatypeProperty
type: FunctionalProperty
comment: Witness certifying connectivity of the type graph.
range: boolean

schemaConstraint
type: ObjectProperty
type: FunctionalProperty
comment: Constraint imposed by the schema on this type.
range: TermExpression

alphabetSize
type: AnnotationProperty
comment: The size of the alphabet for symbol-based types.
range: nonNegativeInteger

quantizationRange
type: ObjectProperty
type: FunctionalProperty
comment: The range specification for quantized types.
range: TermExpression

elementTypeRef
type: ObjectProperty
type: FunctionalProperty
comment: Reference to the element type within a composite type.
range: TypeDefinition

inclusionSource
type: ObjectProperty
type: FunctionalProperty
comment: The subtype (more constraints).
domain: TypeInclusion
range: ConstrainedType

inclusionTarget
type: ObjectProperty
type: FunctionalProperty
comment: The supertype (fewer constraints).
domain: TypeInclusion
range: ConstrainedType

positionVariance
type: ObjectProperty
type: FunctionalProperty
comment: The variance of each operand position.
domain: StructuralOperad
range: VarianceAnnotation

inclusionWitness
type: DatatypeProperty
type: FunctionalProperty
comment: True iff constraints(source) ⊇ constraints(target). Computed, not asserted.
domain: TypeInclusion
range: boolean

latticeDepth
type: DatatypeProperty
type: FunctionalProperty
comment: Maximum chain length from PrimitiveType (top) to any CompleteType (bottom). Equals the site budget n at quantum level Q_k.
domain: SubtypingLattice
range: nonNegativeInteger

verticalAxis
type: NamedIndividual
type: MetricAxis
comment: The vertical (ring/additive) metric axis. Constraints on this axis operate through ring arithmetic: residue classes, divisibility, and additive structure.

horizontalAxis
type: NamedIndividual
type: MetricAxis
comment: The horizontal (Hamming/bitwise) metric axis. Constraints on this axis operate through bitwise structure: carry patterns, bit positions, and Hamming distance.

diagonalAxis
type: NamedIndividual
type: MetricAxis
comment: The diagonal (incompatibility) metric axis. Constraints on this axis measure the gap between ring and Hamming metrics — the curvature of UOR geometry.

ScalarType
type: NamedIndividual
type: PrimitiveType
comment: A single value from an ordered domain. siteCount = n (quantization bits).
structuralGrounding: quantize(value, range, bits) produces ring element where d_R reflects value proximity
structuralSiteCount: n (quantization bits)

SymbolType
type: NamedIndividual
type: PrimitiveType
comment: A value from a finite unordered set. siteCount = ceil(log2(|alphabet|)).
structuralGrounding: argmin_{encoding} sum d_delta over observed pairs (CY_5)
structuralSiteCount: ceil(log2(|alphabet|))

SequenceType
type: NamedIndividual
type: ProductType
comment: An ordered list of elements with backbone constraint. The free monoid on the element type.
structuralConstraint: backbone ordering: elements indexed by position
structuralGrounding: free monoid on element type, backbone constraint
structuralSiteCount: sum of element site counts

TupleType
type: NamedIndividual
type: ProductType
comment: A fixed collection of named fields.
structuralGrounding: site ordering follows CY_6 (optimal site ordering)
structuralSiteCount: sum of field site counts

GraphType
type: NamedIndividual
type: ConstrainedType
comment: Nodes with edge constraints. Constraint nerve = graph topology.
structuralConstraint: edge constraints: adjacency preserved under grounding
structuralGrounding: constraint nerve = graph nerve, beta_k equality
structuralSiteCount: sum of node site counts + edge overhead

SetType
type: NamedIndividual
type: ConstrainedType
comment: Unordered collection. d_delta is permutation-invariant.
structuralConstraint: permutation invariance: encoding is order-independent
structuralGrounding: d_delta invariant under element permutation, D_{2n} symmetry
structuralSiteCount: sum of element site counts

TreeType
type: NamedIndividual
type: ConstrainedType
comment: Hierarchical structure. beta_1=0 (acyclic), beta_0=1 (connected).
structuralConstraint: beta_1=0 (acyclic), beta_0=1 (connected)
structuralGrounding: parent-child encoding with acyclicity constraint
structuralSiteCount: sum of node site counts

TableType
type: NamedIndividual
type: ProductType
comment: Collection of tuples sharing a schema = Sequence(Tuple(S)). Functorial decomposition.
structuralConstraint: shared schema: all rows conform to tuple type S
structuralGrounding: Sequence(Tuple(S)), functorial decomposition
structuralSiteCount: row_count * tuple_site_count

Covariant
type: NamedIndividual
type: VarianceAnnotation
comment: The structural position preserves TypeInclusion: if T₁ ≤ T₂, then F(T₁) ≤ F(T₂).

Contravariant
type: NamedIndividual
type: VarianceAnnotation
comment: The structural position reverses TypeInclusion: if T₁ ≤ T₂, then F(T₂) ≤ F(T₁).

Invariant
type: NamedIndividual
type: VarianceAnnotation
comment: The structural position requires exact type equality: F(T₁) ≤ F(T₂) only if T₁ = T₂.

Bivariant
type: NamedIndividual
type: VarianceAnnotation
comment: The structural position ignores the type parameter: F(T₁) ≤ F(T₂) for all T₁, T₂.

EitherType
type: NamedIndividual
type: SumType
comment: The canonical binary disjoint union type whose carrier is L + R.

OptionType
type: NamedIndividual
type: SumType
comment: The canonical A + Unit idiom for optional values.

UOR Partitions
type: Ontology
imports: UOR Schema
imports: UOR Type System
comment: Irreducibility partitions produced by type resolution. A partition divides the ring into four disjoint components: Irreducible, Reducible, Units, and Exterior.
space: bridge

Partition
type: Class
comment: A four-component partition of R_n produced by resolving a type declaration. The four components — Irreducible, Reducible, Units, Exterior — are mutually disjoint and exhaustive over the carrier.
subclass of: Thing

Component
type: Class
disjoint with: SiteIndex
disjoint with: FreeRank
comment: A single component of a partition: a set of datum values belonging to one of the four categories.
subclass of: Thing

IrreducibleSet
type: Class
disjoint with: ReducibleSet
disjoint with: UnitGroup
disjoint with: Complement
comment: The set of irreducible elements under the active type: elements whose only factorizations involve units or themselves. Analogous to prime elements in a ring.
subclass of: Component

ReducibleSet
type: Class
disjoint with: IrreducibleSet
disjoint with: UnitGroup
disjoint with: Complement
comment: The set of reducible non-unit elements: elements that can be expressed as a product of two or more non-unit elements.
subclass of: Component

UnitGroup
type: Class
disjoint with: IrreducibleSet
disjoint with: ReducibleSet
disjoint with: Complement
comment: The set of invertible elements (units) in the carrier: elements with a multiplicative inverse. In Z/(2^n)Z, the units are the odd integers.
subclass of: Component

Complement
type: Class
disjoint with: IrreducibleSet
disjoint with: ReducibleSet
disjoint with: UnitGroup
comment: Elements of R_n that fall outside the active carrier — i.e., outside the type's domain. These are ring elements that do not participate in the current type resolution.
subclass of: Component

SiteIndex
type: Class
disjoint with: FreeRank
disjoint with: Component
comment: A single site coordinate in the iterated Z/2Z fibration. Each site represents one binary degree of freedom in the ring's structure. The total number of sites equals the quantum level n.
subclass of: Thing

FreeRank
type: Class
disjoint with: SiteIndex
disjoint with: Component
comment: The site budget for a partition: an accounting of how many sites are pinned (determined by constraints) versus free (still available for further refinement). A closed budget means all sites are pinned and the type is fully resolved.
subclass of: Thing

SiteBinding
type: Class
comment: A record of a single site being pinned by a constraint. Links a specific site coordinate to the constraint that determined its value.
subclass of: Thing

PartitionProduct
type: Class
disjoint with: PartitionCoproduct
comment: The tensor product of two partitions: partition(A × B) = partition(A) ⊗ partition(B). The four-component structure combines component-wise under the product type construction (PT_2a). Carries leftFactor and rightFactor links to the operand partitions.
subclass of: Thing

PartitionCoproduct
type: Class
disjoint with: PartitionProduct
comment: The coproduct (disjoint union) of two partitions: partition(A + B) = partition(A) ⊕ partition(B). The four-component structure combines via disjoint union under the sum type construction (PT_2b). Carries leftSummand and rightSummand links to the operand partitions.
subclass of: Thing

irreducibles
type: ObjectProperty
type: FunctionalProperty
comment: The irreducible component of this partition.
domain: Partition
range: IrreducibleSet

reducibles
type: ObjectProperty
type: FunctionalProperty
comment: The reducible component of this partition.
domain: Partition
range: ReducibleSet

units
type: ObjectProperty
type: FunctionalProperty
comment: The units component of this partition.
domain: Partition
range: UnitGroup

exterior
type: ObjectProperty
type: FunctionalProperty
comment: The exterior component of this partition.
domain: Partition
range: Complement

member
type: ObjectProperty
comment: A datum value belonging to this partition component.
domain: Component
range: Datum

cardinality
type: DatatypeProperty
type: FunctionalProperty
comment: The number of elements in this partition component. The cardinalities of the four components must sum to 2^n.
domain: Component
range: nonNegativeInteger

density
type: DatatypeProperty
type: FunctionalProperty
comment: The irreducible density of this partition: |Irr| / |A|, where A is the active carrier.
domain: Partition
range: decimal

sourceType
type: ObjectProperty
type: FunctionalProperty
comment: The type declaration that was resolved to produce this partition.
domain: Partition
range: TypeDefinition

wittLength
type: DatatypeProperty
type: FunctionalProperty
comment: The Witt level n at which this partition was computed. The ring has 2^n elements at this level.
domain: Partition
range: positiveInteger

sitePosition
type: DatatypeProperty
type: FunctionalProperty
comment: The zero-based position of this site coordinate within the iterated fibration. Position 0 is the least significant bit; position n-1 is the most significant.
domain: SiteIndex
range: nonNegativeInteger

siteState
type: DatatypeProperty
type: FunctionalProperty
comment: The current state of this site coordinate: 'pinned' if determined by a constraint, 'free' if still available for refinement.
domain: SiteIndex
range: nonNegativeInteger

siteBudget
type: ObjectProperty
type: FunctionalProperty
comment: The site budget associated with this partition.
domain: Partition
range: FreeRank

totalSites
type: DatatypeProperty
type: FunctionalProperty
comment: The total number of site coordinates in this budget, equal to the quantum level n.
domain: FreeRank
range: nonNegativeInteger

pinnedCount
type: DatatypeProperty
type: FunctionalProperty
comment: The number of site coordinates currently pinned by constraints.
domain: FreeRank
range: nonNegativeInteger

freeRank
type: DatatypeProperty
type: FunctionalProperty
comment: The number of site coordinates still free (not yet pinned). Equals totalSites - pinnedCount.
domain: FreeRank
range: nonNegativeInteger

isClosed
type: DatatypeProperty
type: FunctionalProperty
comment: Whether all sites in this budget are pinned. A closed budget means the type is fully resolved and the partition is complete.
domain: FreeRank
range: boolean

hasSite
type: ObjectProperty
comment: A site coordinate belonging to this budget.
domain: FreeRank
range: SiteIndex

pinnedBy
type: ObjectProperty
type: FunctionalProperty
comment: The constraint that pins this site coordinate.
domain: SiteBinding
range: Constraint

pinsCoordinate
type: ObjectProperty
type: FunctionalProperty
comment: The site coordinate that this pinning determines.
domain: SiteBinding
range: SiteIndex

hasBinding
type: ObjectProperty
comment: A site pinning record in this budget.
domain: FreeRank
range: SiteBinding

ancillaSite
type: ObjectProperty
type: FunctionalProperty
comment: An ancilla site coordinate paired with this site for reversible computation (RC_1–RC_4 ancilla model).
domain: SiteIndex
range: SiteIndex

reversibleStrategy
type: DatatypeProperty
type: FunctionalProperty
comment: True when this site budget uses a reversible computation strategy preserving information through ancilla sites.
domain: FreeRank
range: boolean

exteriorCriteria
type: ObjectProperty
type: FunctionalProperty
comment: The formal membership criterion for this Complement: x ∈ Ext(T) iff x ∉ carrier(T). The Complement is context-dependent on the active type T (FPM_9).
domain: Complement
range: TermExpression

isExhaustive
type: DatatypeProperty
type: FunctionalProperty
comment: Whether the four components of this partition are exhaustive over R_n: |Irr| + |Red| + |Unit| + |Ext| = 2^n (FPM_8). Set by the kernel after verification.
domain: Partition
range: boolean

leftFactor
type: ObjectProperty
type: FunctionalProperty
comment: The left operand partition of this tensor product.
domain: PartitionProduct
range: Partition

rightFactor
type: ObjectProperty
type: FunctionalProperty
comment: The right operand partition of this tensor product.
domain: PartitionProduct
range: Partition

leftSummand
type: ObjectProperty
type: FunctionalProperty
comment: The left operand partition of this coproduct.
domain: PartitionCoproduct
range: Partition

rightSummand
type: ObjectProperty
type: FunctionalProperty
comment: The right operand partition of this coproduct.
domain: PartitionCoproduct
range: Partition

UOR Observables
type: Ontology
imports: UOR Operations
imports: UOR Schema
imports: UOR Partitions
imports: UOR Type System
comment: Observable quantities and metrics computed by the UOR kernel. Includes ring-metric, Hamming-metric, curvature, holonomy, and catastrophe-theoretic observables.
space: bridge

Observable
type: Class
comment: A measurable quantity in the UOR Framework. All observables are kernel-computed and user-consumed.
subclass of: Thing

StratumObservable
type: Class
comment: An observable measuring stratum-level properties: position within the ring's layer structure.
subclass of: Observable

MetricObservable
type: Class
comment: An observable measuring geometric distance between ring elements under a specific metric.
subclass of: Observable

PathObservable
type: Class
comment: An observable measuring properties of paths through the ring: path length, total variation, winding number.
subclass of: Observable

ReductionObservable
type: Class
comment: An observable measuring reduction properties: the length and count of operation sequences.
subclass of: Observable

CatastropheObservable
type: Class
comment: An observable measuring catastrophe-theoretic properties: thresholds at which qualitative changes occur in the partition.
subclass of: Observable

CurvatureObservable
type: Class
comment: An observable measuring the curvature of the UOR geometry: the gap between ring-isometry and Hamming-isometry for a given transform.
subclass of: Observable

HolonomyObservable
type: Class
comment: An observable measuring holonomy: the accumulated transformation when traversing a closed path in the ring.
subclass of: Observable

RingMetric
type: Class
comment: Distance between two ring elements under the ring metric: d_R(x, y) = |x - y| mod 2^n.
subclass of: MetricObservable

HammingMetric
type: Class
comment: Distance between two ring elements under the Hamming metric: the number of bit positions where they differ.
subclass of: MetricObservable

IncompatibilityMetric
type: Class
comment: The metric incompatibility between two ring elements: the divergence between their ring-metric and Hamming-metric distances, measuring geometric curvature.
subclass of: MetricObservable

StratumValue
type: Class
comment: The stratum index of a ring element.
subclass of: StratumObservable

StratumDelta
type: Class
comment: The difference in stratum between two ring elements.
subclass of: StratumObservable

StratumTrajectory
type: Class
comment: The sequence of strata traversed by a path through the ring.
subclass of: StratumObservable

PathLength
type: Class
comment: The length of a path through the ring, measured in operation steps.
subclass of: PathObservable

TotalVariation
type: Class
comment: The total variation of a path: the sum of metric distances between consecutive elements.
subclass of: PathObservable

WindingNumber
type: Class
comment: The winding number of a closed path: the number of times the path wraps around the ring.
subclass of: PathObservable

ReductionLength
type: Class
comment: The number of operation applications in a reduction sequence.
subclass of: ReductionObservable

ReductionCount
type: Class
comment: The number of distinct reduction sequences in a computation.
subclass of: ReductionObservable

CatastropheThreshold
type: Class
comment: A critical value at which a qualitative change occurs in the partition structure.
subclass of: CatastropheObservable

CatastropheCount
type: Class
comment: The number of catastrophe events (qualitative partition changes) in a computation.
subclass of: CatastropheObservable

Commutator
type: Class
comment: The commutator [f, g](x) = f(g(x)) - g(f(x)) of two operations, measuring their non-commutativity.
subclass of: CurvatureObservable

CurvatureFlux
type: Class
comment: The integrated curvature over a region of type space: the total metric incompatibility accumulated.
subclass of: CurvatureObservable

Monodromy
type: Class
comment: The monodromy of a closed path: the net transformation accumulated when traversing a loop in the type space.
subclass of: HolonomyObservable

ParallelTransport
type: Class
comment: The parallel transport of a vector along a path: the canonical lift of the path to the tangent bundle of the ring.
subclass of: HolonomyObservable

DihedralElement
type: Class
comment: An element of the dihedral group D_{2^n} acting on the type space. Each dihedral element induces an isometry of 𝒯_n.
subclass of: HolonomyObservable

MeasurementUnit
type: Class
comment: A unit of measurement for observable quantities. Each MeasurementUnit individual names a specific unit (bits, ring steps, dimensionless) replacing the string-valued observable:unit property.
subclass of: Thing

Jacobian
type: Class
comment: Site-by-site curvature decomposition. J_k measures the discrete derivative of the incompatibility metric at site position k: J_k = |d_R(x, succ(x)) - d_H(x, succ(x))| restricted to position k.
subclass of: CurvatureObservable

TopologicalObservable
type: Class
comment: An observable measuring a topological invariant of the resolution space. Topological observables are invariant under continuous deformations of the constraint configuration.
subclass of: Observable

BettiNumber
type: Class
comment: The rank of a homology group of the constraint nerve. β_k = rank(H_k(N(C))) counts the k-dimensional holes in the constraint configuration.
subclass of: TopologicalObservable

SpectralGap
type: Class
comment: The smallest positive eigenvalue of the constraint nerve Laplacian. Controls the convergence rate of iterative resolution: larger gap = faster convergence.
subclass of: TopologicalObservable

ThermoObservable
type: Class
comment: An observable measuring thermodynamic properties of the resolution process: residual entropy, Landauer cost, and reduction distribution statistics.
subclass of: Observable

ResidualEntropy
type: Class
comment: S_residual: the residual Shannon entropy of the site distribution after partial resolution. Computed as S = (Σ κ_k − χ(N(C))) × ln 2 (IT_7b). Unit: Nats.
subclass of: ThermoObservable

LandauerCost
type: Class
comment: The minimum thermodynamic cost (in units of k_B T ln 2) of erasing one bit of site uncertainty. The UOR ring operates at β* = ln 2 — the Landauer temperature.
subclass of: ThermoObservable

ReductionEntropy
type: Class
comment: The Shannon entropy of the reduction distribution P(j) = 2^{−j}. At the Landauer temperature, this equals ln 2 per reduction step — each step erases exactly one bit of site uncertainty.
subclass of: ThermoObservable

SynthesisSignature
type: Class
comment: A named topological signature: a pair (realised Euler characteristic, realised Betti profile). Linked from TypeSynthesisResult. Allows comparison between the goal signature and the actually achieved signature.
subclass of: Thing

SpectralSequencePage
type: Class
comment: A single page E_r of the quantum level spectral sequence. Carries the page index r and the differential d_r. The sequence converges when all differentials vanish — typically by E_3 for simple constraint configurations.
subclass of: Thing

LiftObstructionClass
type: Class
comment: The cohomology class in H^2(N(C(T))) representing the LiftObstruction for a specific WittLift. The class is zero iff the obstruction is trivial. When non-zero, it indexes the specific site pair at Q_{n+1} that cannot be closed by the lifted constraint set alone.
subclass of: Thing

MonodromyClass
type: Class
comment: A classification of a type's holonomy: the subgroup of D_{2^n} generated by all Monodromy observables computed over closed paths in the type's constraint nerve. Trivial iff every closed constraint path returns to its starting site assignment without net dihedral transformation.
subclass of: Thing

HolonomyGroup
type: Class
comment: The holonomy group of a ConstrainedType: the group of all Monodromy elements achievable by closed paths in the constraint nerve. Always a subgroup of D_{2^n}. Trivial iff the type has trivial monodromy everywhere; equals D_{2^n} iff paths involving both neg and bnot involutions are present.
subclass of: Thing

ClosedConstraintPath
type: Class
comment: A sequence of constraint applications forming a closed loop in the constraint nerve — beginning and ending at the same site assignment. The Monodromy of the loop is the net DihedralElement accumulated when traversing it.
subclass of: Thing

PhaseBoundaryType
type: Class
comment: A classification of phase boundary in the catastrophe diagram: period boundary (g divides 2^n − 1) or power-of-two boundary (g = 2^k).
subclass of: Thing

AchievabilityStatus
type: Class
comment: The achievability classification of a topological signature in the morphospace. Either Achievable or Forbidden (witnessed by ImpossibilityWitness).
subclass of: Thing

HomotopyGroup
type: Class
comment: The k-th homotopy group πk(N(C), v) of the constraint nerve based at vertex v.
subclass of: Thing

HigherMonodromy
type: Class
comment: The image of πk(N(C)) → Aut(sitek) for k > 1. Generalises the MN_6 monodromy homomorphism.
subclass of: Thing

WhiteheadProduct
type: Class
comment: The Whitehead product [α, β] ∈ πp+q−1 for α ∈ πp, β ∈ πq.
subclass of: Thing

StratificationRecord
type: Class
comment: A record of the holonomy stratification of the moduli space at a given quantum level: the list of HolonomyStrata, their codimensions, and their relationship to the MorphospaceBoundary.
subclass of: Thing

BaseMetric
type: Class
comment: Superclass for the six universal measurements. Every computation on the ring produces these six quantities: d_Δ, σ, J_k, β_k, χ, r.
subclass of: Observable

GroundingObservable
type: Class
comment: The grounding metric σ = pinned sites / total sites. Ranges from 0 (no sites pinned) to 1 (fully grounded).
subclass of: Observable

EulerCharacteristicObservable
type: Class
comment: The Euler characteristic χ = Σ(−1)^k β_k of the constraint nerve. An integer-valued topological invariant.
subclass of: Observable

value
type: DatatypeProperty
type: FunctionalProperty
comment: The numeric value of an observable measurement.
domain: Observable
range: decimal

source
type: ObjectProperty
type: FunctionalProperty
comment: The source object of this measurement (datum, partition, or path start point).
domain: Observable
range: Thing

target
type: ObjectProperty
type: FunctionalProperty
comment: The target object of this measurement (for metrics and path-end measurements).
domain: Observable
range: Thing

hasUnit
type: ObjectProperty
type: FunctionalProperty
comment: The measurement unit of this observable. Replaces the string-valued observable:unit property with a typed reference to a MeasurementUnit individual.
domain: Observable
range: MeasurementUnit

sitePosition
type: DatatypeProperty
type: FunctionalProperty
comment: The site position k at which this Jacobian entry is measured.
domain: Jacobian
range: nonNegativeInteger

derivativeValue
type: DatatypeProperty
type: FunctionalProperty
comment: The discrete derivative value at this site position.
domain: Jacobian
range: decimal

dimension
type: DatatypeProperty
type: FunctionalProperty
comment: The dimension k of the topological observable (e.g., the degree of the Betti number or the dimension of the spectral gap).
domain: TopologicalObservable
range: nonNegativeInteger

realisedEuler
type: DatatypeProperty
type: FunctionalProperty
comment: The Euler characteristic actually achieved by this synthesis signature.
domain: SynthesisSignature
range: integer

realisedBetti
type: DatatypeProperty
comment: Non-functional. Realised Betti number values, one assertion per homological degree.
domain: SynthesisSignature
range: nonNegativeInteger

pageIndex
type: DatatypeProperty
type: FunctionalProperty
comment: The page r of this spectral sequence page. r=1 is the initial page; convergence is declared when all d_r are zero.
domain: SpectralSequencePage
range: nonNegativeInteger

differentialIsZero
type: DatatypeProperty
type: FunctionalProperty
comment: True iff d_r = 0 on this page — no further corrections to the lifted homology.
domain: SpectralSequencePage
range: boolean

convergedAt
type: DatatypeProperty
type: FunctionalProperty
comment: The page index r at which the spectral sequence converged (all subsequent differentials zero).
domain: SpectralSequencePage
range: nonNegativeInteger

obstructionClass
type: ObjectProperty
type: FunctionalProperty
comment: The cohomology class in H^2(N(C(T))) representing this obstruction.
domain: LiftObstructionClass
range: CohomologyGroup

monodromyLoop
type: ObjectProperty
type: FunctionalProperty
comment: The closed path that generates this monodromy value.
domain: Monodromy
range: ClosedConstraintPath

monodromyElement
type: ObjectProperty
type: FunctionalProperty
comment: The dihedral element g in D_{2^n} accumulated when traversing the loop. The monodromy is trivial iff this element is the group identity.
domain: Monodromy
range: DihedralElement

isTrivialMonodromy
type: DatatypeProperty
type: FunctionalProperty
comment: True iff the monodromyElement is the identity in D_{2^n}.
domain: Monodromy
range: boolean

holonomyGroup
type: ObjectProperty
comment: Non-functional. The generators of the holonomy group: one DihedralElement per generating monodromy.
domain: HolonomyGroup
range: DihedralElement

holonomyGroupOrder
type: DatatypeProperty
type: FunctionalProperty
comment: The order of the holonomy group as a subgroup of D_{2^n}. For a FlatType: 1. For full dihedral holonomy: 2^{n+1}.
domain: HolonomyGroup
range: positiveInteger

pathLength
type: DatatypeProperty
type: FunctionalProperty
comment: The number of constraint application steps in this closed path.
domain: ClosedConstraintPath
range: nonNegativeInteger

pathConstraints
type: ObjectProperty
comment: Non-functional. The ordered sequence of constraints traversed. One assertion per step.
domain: ClosedConstraintPath
range: Constraint

dihedralElementValue
type: ObjectProperty
comment: Non-functional. One assertion per generator in the normal form of the element — the sequence of neg and/or bnot operations that realises this dihedral element when composed.
domain: DihedralElement
range: Operation

isIdentityElement
type: DatatypeProperty
type: FunctionalProperty
comment: True iff this element is the group identity (the trivial monodromy value).
domain: DihedralElement
range: boolean

elementOrder
type: DatatypeProperty
type: FunctionalProperty
comment: The order of this element in D_{2^n}: the smallest k >= 1 such that g^k = id. For neg and bnot: order 2.
domain: DihedralElement
range: positiveInteger

hardnessEstimate
type: DatatypeProperty
type: FunctionalProperty
comment: An estimated computational hardness for a ThermoObservable, connecting thermodynamic cost to complexity (TH_9 realisation).
domain: ThermoObservable
range: decimal

phaseN
type: DatatypeProperty
type: FunctionalProperty
comment: The ring dimension coordinate n in the (n, g) catastrophe phase diagram (PD_1 n-coordinate).
domain: CatastropheObservable
range: positiveInteger

phaseG
type: DatatypeProperty
type: FunctionalProperty
comment: The group-order coordinate g in the (n, g) catastrophe phase diagram (PD_1 g-coordinate).
domain: CatastropheObservable
range: positiveInteger

onResonanceLine
type: DatatypeProperty
type: FunctionalProperty
comment: True when g divides 2^n − 1, placing this observable on a resonance line in the phase diagram (PD_4).
domain: CatastropheObservable
range: boolean

phaseBoundaryType
type: ObjectProperty
type: FunctionalProperty
comment: The type of phase boundary at this point in the catastrophe diagram: PeriodBoundary or PowerOfTwoBoundary (PD_2).
domain: CatastropheObservable
range: PhaseBoundaryType

achievabilityStatus
type: ObjectProperty
type: FunctionalProperty
comment: The achievability classification of this observable's topological signature in the morphospace.
domain: SynthesisSignature
range: AchievabilityStatus

isAchievable
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this signature has been empirically verified as achievable at some quantum level.
domain: SynthesisSignature
range: boolean

isForbidden
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this signature has been formally proven impossible by an ImpossibilityWitness.
domain: SynthesisSignature
range: boolean

achievabilityWitness
type: ObjectProperty
type: FunctionalProperty
comment: The proof individual (ImpossibilityWitness or AxiomaticDerivation) that grounds this signature's achievability classification.
domain: SynthesisSignature
range: Proof

rotationExponent
type: DatatypeProperty
type: FunctionalProperty
comment: The rotation exponent k ∈ \[0, 2^n) of this dihedral element in the standard representation r^k s^s. Together with reflectionBit, uniquely identifies the element within D_\{2^n\}.
domain: DihedralElement
range: nonNegativeInteger

reflectionBit
type: DatatypeProperty
type: FunctionalProperty
comment: The reflection flag s ∈ \{0,1\} of this dihedral element. False = pure rotation (r^k), true = reflection (r^k·s). D_7 defines composition: (r^a s^p)(r^b s^q) = r^(a + (-1)^p b) s^(p XOR q).
domain: DihedralElement
range: boolean

homotopyDimension
type: DatatypeProperty
type: FunctionalProperty
comment: The dimension k of this homotopy group πk.
domain: HomotopyGroup
range: nonNegativeInteger

homotopyRank
type: DatatypeProperty
type: FunctionalProperty
comment: The rank of this homotopy group (number of free generators).
domain: HomotopyGroup
range: nonNegativeInteger

homotopyBasepoint
type: ObjectProperty
type: FunctionalProperty
comment: The basepoint vertex v at which this homotopy group is computed.
domain: HomotopyGroup
range: Constraint

higherMonodromyDimension
type: DatatypeProperty
type: FunctionalProperty
comment: The dimension k > 1 at which this higher monodromy acts.
domain: HigherMonodromy
range: nonNegativeInteger

whiteheadTrivial
type: DatatypeProperty
type: FunctionalProperty
comment: True iff this Whitehead product is trivial (zero in πp+q−1).
domain: WhiteheadProduct
range: boolean

postnikovTruncation
type: ObjectProperty
type: FunctionalProperty
comment: The Postnikov truncation associated with this spectral sequence page.
domain: SpectralSequencePage
range: PostnikovTruncation

stratificationLevel
type: ObjectProperty
type: FunctionalProperty
comment: The quantum level at which this stratification is computed.
domain: StratificationRecord
range: WittLevel

stratificationStratum
type: ObjectProperty
comment: A HolonomyStratum in this stratification record.
domain: StratificationRecord
range: HolonomyStratum

metricDomain
type: DatatypeProperty
type: FunctionalProperty
comment: The mathematical domain of this base metric.
domain: BaseMetric
range: string

metricRange
type: DatatypeProperty
type: FunctionalProperty
comment: The mathematical range (codomain) of this base metric.
domain: BaseMetric
range: string

metricComposition
type: ObjectProperty
type: FunctionalProperty
comment: How this metric composes with others in the measurement tower.
domain: BaseMetric
range: TermExpression

referencesClass
type: ObjectProperty
type: FunctionalProperty
comment: The existing observable class that this base metric references.
domain: BaseMetric
range: Observable

referencesIdentity
type: ObjectProperty
type: FunctionalProperty
comment: The existing identity that defines this base metric.
domain: BaseMetric
range: Identity

saturationNumerator
type: DatatypeProperty
type: FunctionalProperty
comment: The count of pinned sites (numerator of σ).
domain: GroundingObservable
range: nonNegativeInteger

saturationDenominator
type: DatatypeProperty
type: FunctionalProperty
comment: The total site count (denominator of σ).
domain: GroundingObservable
range: positiveInteger

alternatingSum
type: ObjectProperty
type: FunctionalProperty
comment: The alternating sum formula for Euler characteristic.
domain: EulerCharacteristicObservable
range: TermExpression

metricUnit
type: ObjectProperty
type: FunctionalProperty
comment: The unit of measurement for this base metric.
domain: BaseMetric
range: MeasurementUnit

metricPrecision
type: DatatypeProperty
type: FunctionalProperty
comment: The precision or resolution of this base metric.
domain: BaseMetric
range: nonNegativeInteger

metricMonotonicity
type: ObjectProperty
type: FunctionalProperty
comment: Monotonicity property of this metric (e.g., non-decreasing).
domain: BaseMetric
range: TermExpression

metricDecomposition
type: ObjectProperty
type: FunctionalProperty
comment: The decomposition rule for this metric into sub-metrics.
domain: BaseMetric
range: TermExpression

metricTowerPosition
type: DatatypeProperty
type: FunctionalProperty
comment: The position of this metric in the metric tower.
domain: BaseMetric
range: nonNegativeInteger

metricComputationCost
type: ObjectProperty
type: FunctionalProperty
comment: The computational cost of evaluating this metric.
domain: BaseMetric
range: TermExpression

metricBound
type: ObjectProperty
type: FunctionalProperty
comment: Upper or lower bound on the metric value.
domain: BaseMetric
range: TermExpression

Bits
type: NamedIndividual
type: MeasurementUnit
comment: Information-theoretic unit: the measurement is in bits (e.g., Hamming weight, entropy).

RingSteps
type: NamedIndividual
type: MeasurementUnit
comment: Ring-arithmetic unit: the measurement is in ring distance steps (|x - y| mod 2^n).

Dimensionless
type: NamedIndividual
type: MeasurementUnit
comment: Dimensionless unit: the measurement is a pure number (e.g., winding number, Betti number, spectral gap).

Nats
type: NamedIndividual
type: MeasurementUnit
comment: Natural information unit: entropy measured in nats (using natural logarithm). S_residual is expressed in nats when computed as (Σ κ_k − χ) × ln 2.

PeriodBoundary
type: NamedIndividual
type: PhaseBoundaryType
comment: A phase boundary where g divides 2^n − 1, meaning g is a period of the multiplicative structure of R_n.

PowerOfTwoBoundary
type: NamedIndividual
type: PhaseBoundaryType
comment: A phase boundary where g = 2^k, meaning g aligns with the binary stratification of R_n.

Achievable
type: NamedIndividual
type: AchievabilityStatus
comment: The signature has been verified as achievable at some quantum level by an AxiomaticDerivation proof.

Forbidden
type: NamedIndividual
type: AchievabilityStatus
comment: The signature has been formally proven impossible by an ImpossibilityWitness deriving from MS_1, MS_2, or other impossibility theorems.

d_delta_metric
type: NamedIndividual
type: BaseMetric
metricDomain: pair of ring elements
metricRange: non-negative integer
referencesClass: IncompatibilityMetric
comment: d_Δ: the incompatibility metric |d_R − d_H| per site pair.

sigma_metric
type: NamedIndividual
type: BaseMetric
metricDomain: computation state
metricRange: decimal in [0, 1]
referencesIdentity: GS_2
comment: σ: the grounding metric, pinned sites / total sites.

jacobian_metric
type: NamedIndividual
type: BaseMetric
metricDomain: computation state × site index
metricRange: decimal
referencesClass: Jacobian
referencesIdentity: DC_6
comment: J_k: per-site curvature, ∂_R f_k.

betti_metric
type: NamedIndividual
type: BaseMetric
metricDomain: simplicial complex × dimension
metricRange: non-negative integer
referencesClass: BettiNumber
comment: β_k: per-dimension Betti number of the constraint nerve.

euler_metric
type: NamedIndividual
type: BaseMetric
metricDomain: simplicial complex
metricRange: integer
referencesIdentity: IT_2
comment: χ: Euler characteristic, Σ(−1)^k β_k.

residual_metric
type: NamedIndividual
type: BaseMetric
metricDomain: computation state
metricRange: non-negative integer
referencesClass: ResidualEntropy
comment: r: count of free (unpinned) sites, the residual entropy.

UOR Carry Algebra
type: Ontology
imports: UOR Operations
imports: UOR Observables
imports: UOR Partitions
comment: Carry chain algebra: generate/propagate/kill event classification, carry profiles, encoding configurations, and encoding quality metrics for d_Δ optimization.
space: kernel

CarryChain
type: Class
comment: The Boolean function chain c_{k+1} = or(and(x_k, y_k), and(xor(x_k, y_k), c_k)). The carry chain is the algebraic mechanism behind the incompatibility metric d_Δ.
subclass of: Thing

CarryEvent
type: Class
comment: A single carry event at site k. Three kinds: Generate (and(x_k, y_k) = 1), Propagate (xor(x_k, y_k) = 1 and c_k = 1), Kill (neither generate nor propagate).
subclass of: Thing

CarryProfile
type: Class
comment: The complete carry pattern for an addition x + y. Aggregates carry events across all sites into counts and position masks.
subclass of: Thing

EncodingConfiguration
type: Class
comment: A mapping from a finite symbol set S to Z/(2^k)Z where 2^k ≥ |S|. Determines how domain values are represented as ring elements.
subclass of: Thing

EncodingQuality
type: Class
comment: The d_Δ quality metric for an encoding over observed data. Measures how well an encoding minimizes carry-induced metric incompatibility.
subclass of: Thing

chainLength
type: DatatypeProperty
type: FunctionalProperty
comment: The number of sites in this carry chain.
domain: CarryChain
range: nonNegativeInteger

generateMask
type: DatatypeProperty
type: FunctionalProperty
comment: Bit mask of site positions where carry is generated: and(x_k, y_k) = 1.
domain: CarryChain
range: nonNegativeInteger

propagateMask
type: DatatypeProperty
type: FunctionalProperty
comment: Bit mask of site positions where carry propagates: xor(x_k, y_k) = 1.
domain: CarryChain
range: nonNegativeInteger

killMask
type: DatatypeProperty
type: FunctionalProperty
comment: Bit mask of site positions where carry is killed: neither generated nor propagated.
domain: CarryChain
range: nonNegativeInteger

eventKind
type: DatatypeProperty
type: FunctionalProperty
comment: The kind of carry event: Generate, Propagate, or Kill.
domain: CarryEvent
range: nonNegativeInteger

sitePosition
type: DatatypeProperty
type: FunctionalProperty
comment: The site index k at which this carry event occurs.
domain: CarryEvent
range: nonNegativeInteger

carryCount
type: DatatypeProperty
type: FunctionalProperty
comment: The total number of carry events in this profile.
domain: CarryProfile
range: nonNegativeInteger

maxPropagationLength
type: DatatypeProperty
type: FunctionalProperty
comment: The longest consecutive propagation run in this profile.
domain: CarryProfile
range: nonNegativeInteger

profileChain
type: ObjectProperty
type: FunctionalProperty
comment: The carry chain that this profile summarizes.
domain: CarryProfile
range: CarryChain

symbolSetSize
type: DatatypeProperty
type: FunctionalProperty
comment: The cardinality of the symbol set S being encoded.
domain: EncodingConfiguration
range: positiveInteger

quantizationBits
type: DatatypeProperty
type: FunctionalProperty
comment: The number of bits k used for encoding (2^k ≥ |S|).
domain: EncodingConfiguration
range: positiveInteger

encodingMap
type: ObjectProperty
type: FunctionalProperty
comment: String representation of the mapping from symbols to ring elements.
domain: EncodingConfiguration
range: TermExpression

meanDelta
type: DatatypeProperty
type: FunctionalProperty
comment: The mean d_Δ over observed pairs for this encoding.
domain: EncodingQuality
range: decimal

discriminationRatio
type: DatatypeProperty
type: FunctionalProperty
comment: The ratio of distinguishable pairs to total pairs under this encoding.
domain: EncodingQuality
range: decimal

isOptimalEncoding
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this encoding minimizes Σ d_Δ over observed pairs.
domain: EncodingQuality
range: boolean

UOR Homology
type: Ontology
imports: UOR Schema
imports: UOR Type System
imports: UOR Resolvers
imports: UOR Partitions
imports: UOR Observables
imports: UOR Operations
comment: Simplicial complexes, chain complexes, boundary operators, and homology groups for structural reasoning.
space: bridge

Simplex
type: Class
comment: A k-simplex: a finite set of k+1 vertices drawn from constraint objects.
subclass of: Thing

SimplicialComplex
type: Class
comment: A simplicial complex: a set of simplices closed under taking faces.
subclass of: Thing

FaceMap
type: Class
comment: A face map d_i: removes vertex i from a simplex, producing a face.
subclass of: Thing

ChainGroup
type: Class
comment: A free abelian group generated by k-simplices (the k-th chain group C_k).
subclass of: Thing

BoundaryOperator
type: Class
comment: The boundary operator ∂_k: C_k → C_{k-1}. Satisfies ∂² = 0.
subclass of: Thing

ChainComplex
type: Class
comment: A chain complex: a sequence of chain groups connected by boundary operators.
subclass of: Thing

HomologyGroup
type: Class
comment: The k-th homology group H_k = ker(∂_k) / im(∂_{k+1}). Measures k-dimensional holes.
subclass of: Thing

NerveFunctor
type: Class
comment: The nerve functor N: maps a set of constraints to a simplicial complex.
subclass of: Thing

ChainFunctor
type: Class
comment: The chain functor C: maps a simplicial complex to a chain complex.
subclass of: Thing

KanComplex
type: Class
comment: A simplicial set satisfying the Kan extension condition. The constraint nerve, when promoted from a SimplicialComplex to a KanComplex, carries a full homotopy type — not just its homology groups.
subclass of: SimplicialComplex

HornFiller
type: Class
comment: A witness that a given horn (an incomplete simplex boundary) can be filled, certifying the Kan condition at a specific dimension and position.
subclass of: Thing

PostnikovTruncation
type: Class
comment: The k-th Postnikov truncation τ≤k of the constraint nerve: a KanComplex whose homotopy groups πj vanish for j > k.
subclass of: Thing

KInvariant
type: Class
comment: The k-invariant κk that classifies the extension from the (k−1)-truncation to the k-truncation of the Postnikov tower. Trivial κk means the truncation splits as a product.
subclass of: Thing

DeformationComplex
type: Class
comment: The deformation complex of a CompleteType T: a chain complex whose H⁰ = automorphisms, H¹ = first-order deformations, H² = obstructions to extending deformations.
subclass of: ChainComplex

dimension
type: DatatypeProperty
type: FunctionalProperty
comment: The dimension k of a simplex (number of vertices minus one).
domain: Simplex
range: integer

vertex
type: ObjectProperty
comment: A vertex of this simplex, drawn from the set of constraint objects.
domain: Simplex
range: Constraint

vertexCount
type: DatatypeProperty
type: FunctionalProperty
comment: The number of vertices in this simplex (dimension + 1).
domain: Simplex
range: positiveInteger

isFaceOf
type: ObjectProperty
comment: Indicates that this simplex is a face of another simplex.
domain: Simplex
range: Simplex

pinIntersection
type: ObjectProperty
comment: A site coordinate in the partition whose intersection pins this simplex.
domain: Simplex
range: SiteIndex

hasSimplex
type: ObjectProperty
comment: A simplex belonging to this simplicial complex.
domain: SimplicialComplex
range: Simplex

maxDimension
type: DatatypeProperty
type: FunctionalProperty
comment: The maximum dimension of any simplex in this simplicial complex.
domain: SimplicialComplex
range: integer

eulerCharacteristic
type: DatatypeProperty
type: FunctionalProperty
comment: The Euler characteristic of this simplicial complex: the alternating sum of simplex counts by dimension.
domain: SimplicialComplex
range: integer

simplicialVertexCount
type: DatatypeProperty
type: FunctionalProperty
comment: The total number of vertices (0-simplices) in this simplicial complex.
domain: SimplicialComplex
range: nonNegativeInteger

removesVertex
type: DatatypeProperty
type: FunctionalProperty
comment: The index i of the vertex removed by this face map d_i.
domain: FaceMap
range: nonNegativeInteger

sourceSimplex
type: ObjectProperty
type: FunctionalProperty
comment: The source simplex of this face map.
domain: FaceMap
range: Simplex

targetFace
type: ObjectProperty
type: FunctionalProperty
comment: The target face (result simplex) of this face map.
domain: FaceMap
range: Simplex

degree
type: DatatypeProperty
type: FunctionalProperty
comment: The degree k of this chain group (the dimension of its generating simplices).
domain: ChainGroup
range: integer

rank
type: DatatypeProperty
type: FunctionalProperty
comment: The rank of a free abelian group (number of generators).
range: nonNegativeInteger

generatedBy
type: ObjectProperty
comment: A simplex that generates this chain group.
domain: ChainGroup
range: Simplex

sourceGroup
type: ObjectProperty
type: FunctionalProperty
comment: The source chain group C_k of this boundary operator.
domain: BoundaryOperator
range: ChainGroup

targetGroup
type: ObjectProperty
type: FunctionalProperty
comment: The target chain group C_{k-1} of this boundary operator.
domain: BoundaryOperator
range: ChainGroup

satisfiesBoundarySquaredZero
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this boundary operator satisfies the fundamental property ∂² = 0.
domain: BoundaryOperator
range: boolean

hasChainGroup
type: ObjectProperty
comment: A chain group belonging to this chain complex.
domain: ChainComplex
range: ChainGroup

hasBoundary
type: ObjectProperty
comment: A boundary operator belonging to this chain complex.
domain: ChainComplex
range: BoundaryOperator

homologyDegree
type: DatatypeProperty
type: FunctionalProperty
comment: The degree k of this homology group H_k.
domain: HomologyGroup
range: integer

bettiNumber
type: DatatypeProperty
type: FunctionalProperty
comment: The Betti number β_k = rank(H_k): the rank of this homology group.
domain: HomologyGroup
range: nonNegativeInteger

kanWitness
type: ObjectProperty
comment: A horn filler witnessing the Kan condition for this complex.
domain: KanComplex
range: HornFiller

hornDimension
type: DatatypeProperty
type: FunctionalProperty
comment: The dimension of the horn that this filler completes.
domain: HornFiller
range: nonNegativeInteger

hornPosition
type: DatatypeProperty
type: FunctionalProperty
comment: The position (missing face index) of the horn that this filler completes.
domain: HornFiller
range: nonNegativeInteger

truncationLevel
type: DatatypeProperty
type: FunctionalProperty
comment: The truncation level k of this Postnikov truncation τ≤k.
domain: PostnikovTruncation
range: nonNegativeInteger

truncationSource
type: ObjectProperty
type: FunctionalProperty
comment: The KanComplex from which this Postnikov truncation is derived.
domain: PostnikovTruncation
range: KanComplex

kInvariant
type: ObjectProperty
type: FunctionalProperty
comment: The k-invariant classifying the extension at this truncation level.
domain: PostnikovTruncation
range: KInvariant

kInvariantTrivial
type: DatatypeProperty
type: FunctionalProperty
comment: True iff this k-invariant is trivial, meaning the Postnikov truncation splits as a product.
domain: KInvariant
range: boolean

deformationBase
type: ObjectProperty
type: FunctionalProperty
comment: The CompleteType whose deformation complex this is.
domain: DeformationComplex
range: CompleteType

tangentDimension
type: DatatypeProperty
type: FunctionalProperty
comment: The dimension of the tangent space H¹(Def(T)): the number of first-order deformations.
domain: DeformationComplex
range: nonNegativeInteger

obstructionDimension
type: DatatypeProperty
type: FunctionalProperty
comment: The dimension of the obstruction space H²(Def(T)): the number of independent obstructions to extending deformations.
domain: DeformationComplex
range: nonNegativeInteger

groundedIn
type: ObjectProperty
type: FunctionalProperty
comment: The homology group that grounds this Betti number observable.
domain: BettiNumber
range: HomologyGroup

laplacianOf
type: ObjectProperty
type: FunctionalProperty
comment: The chain complex whose Laplacian determines this spectral gap.
domain: SpectralGap
range: ChainComplex

homologicalAnalysis
type: ObjectProperty
type: FunctionalProperty
comment: The chain complex used for homological analysis of a resolution state.
domain: ResolutionState
range: ChainComplex

boundarySquaredZero
type: NamedIndividual
type: Identity
forAll: term_boundarySquaredZero_forAll
lhs: term_boundarySquaredZero_lhs
rhs: term_boundarySquaredZero_rhs
verificationDomain: Topological
comment: ∂² = 0: the boundary of a boundary is zero.

nerveFunctorN
type: NamedIndividual
type: NerveFunctor
comment: The nerve functor N: constraints → simplicial complex.

chainFunctorC
type: NamedIndividual
type: ChainFunctor
comment: The chain functor C: simplicial complex → chain complex.

psi_4
type: NamedIndividual
type: Identity
forAll: term_psi_4_forAll
lhs: term_psi_4_lhs
rhs: term_psi_4_rhs
verificationDomain: Topological
comment: ψ_4: HomologyGroups → BettiNumbers (extraction functor).

indexBridge
type: NamedIndividual
type: Identity
forAll: term_indexBridge_forAll
lhs: term_indexBridge_lhs
rhs: term_indexBridge_rhs
verificationDomain: Topological
comment: Index bridge: connects Euler characteristic to alternating Betti sum.

UOR Cohomology
type: Ontology
imports: UOR Schema
imports: UOR Homology
imports: UOR Resolvers
imports: UOR Observables
imports: UOR Partitions
comment: Cochain complexes, sheaf cohomology, and local-to-global obstruction detection.
space: bridge

CochainGroup
type: Class
comment: A cochain group: the dual of a chain group, maps chains to coefficients.
subclass of: Thing

CoboundaryOperator
type: Class
comment: The coboundary operator δ^k: C^k → C^{k+1}. Satisfies δ² = 0.
subclass of: Thing

CochainComplex
type: Class
comment: A cochain complex: a sequence of cochain groups connected by coboundary operators.
subclass of: Thing

CohomologyGroup
type: Class
comment: The k-th cohomology group H^k = ker(δ^k) / im(δ^{k-1}). Measures k-dimensional obstructions.
subclass of: Thing

Sheaf
type: Class
comment: A sheaf F over a simplicial complex: assigns data to each simplex with restriction maps.
subclass of: Thing

Stalk
type: Class
comment: A stalk F_σ: the local data of a sheaf at a simplex σ.
subclass of: Thing

Section
type: Class
comment: A global section of a sheaf: a consistent choice of local data across all simplices.
subclass of: Thing

LocalSection
type: Class
comment: A local section: a consistent choice of data over a subcomplex.
subclass of: Section

RestrictionMap
type: Class
comment: A restriction map ρ_{σ,τ}: maps data from a simplex to a face.
subclass of: Thing

GluingObstruction
type: Class
comment: A gluing obstruction: a cohomology class that detects when local sections fail to glue.
subclass of: Thing

cochainDegree
type: DatatypeProperty
type: FunctionalProperty
comment: The degree k of this cochain group C^k.
domain: CochainGroup
range: integer

cochainRank
type: DatatypeProperty
type: FunctionalProperty
comment: The rank (dimension) of this cochain group.
domain: CochainGroup
range: nonNegativeInteger

dualOf
type: ObjectProperty
type: FunctionalProperty
comment: The chain group that this cochain group is dual to.
domain: CochainGroup
range: ChainGroup

coboundarySource
type: ObjectProperty
type: FunctionalProperty
comment: The source cochain group of this coboundary operator.
domain: CoboundaryOperator
range: CochainGroup

coboundaryTarget
type: ObjectProperty
type: FunctionalProperty
comment: The target cochain group of this coboundary operator.
domain: CoboundaryOperator
range: CochainGroup

satisfiesCoboundarySquaredZero
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this coboundary operator satisfies δ² = 0.
domain: CoboundaryOperator
range: boolean

hasCochainGroup
type: ObjectProperty
comment: A cochain group belonging to this cochain complex.
domain: CochainComplex
range: CochainGroup

hasCoboundary
type: ObjectProperty
comment: A coboundary operator belonging to this cochain complex.
domain: CochainComplex
range: CoboundaryOperator

cohomologyDegree
type: DatatypeProperty
type: FunctionalProperty
comment: The degree k of this cohomology group H^k.
domain: CohomologyGroup
range: integer

cohomologyRank
type: DatatypeProperty
type: FunctionalProperty
comment: The rank (dimension) of this cohomology group.
domain: CohomologyGroup
range: nonNegativeInteger

sheafOver
type: ObjectProperty
type: FunctionalProperty
comment: The simplicial complex that this sheaf is defined over.
domain: Sheaf
range: SimplicialComplex

coefficientIn
type: ObjectProperty
type: FunctionalProperty
comment: The coefficient ring of this sheaf.
domain: Sheaf
range: Ring

hasStalks
type: ObjectProperty
comment: A stalk belonging to this sheaf.
domain: Sheaf
range: Stalk

stalkAt
type: ObjectProperty
type: FunctionalProperty
comment: The simplex at which this stalk is located.
domain: Stalk
range: Simplex

restrictsFrom
type: ObjectProperty
type: FunctionalProperty
comment: The source simplex of this restriction map.
domain: RestrictionMap
range: Simplex

restrictsTo
type: ObjectProperty
type: FunctionalProperty
comment: The target simplex (face) of this restriction map.
domain: RestrictionMap
range: Simplex

hasGlobalSection
type: ObjectProperty
comment: A global section of this sheaf.
domain: Sheaf
range: Section

obstructionClass
type: ObjectProperty
type: FunctionalProperty
comment: The cohomology class that this gluing obstruction represents.
domain: GluingObstruction
range: CohomologyGroup

addressesSuggestion
type: ObjectProperty
comment: The refinement suggestion that, if applied, would resolve this gluing obstruction. Computed by the kernel when ψ₆ detects H^1 ≠ 0: the obstruction class indexes the site pair that is incompatible, and the suggestion targets that pair with a new bridging constraint.
domain: GluingObstruction
range: RefinementSuggestion

sheafAnalysis
type: ObjectProperty
type: FunctionalProperty
comment: The sheaf analysis associated with a resolution state.
domain: ResolutionState
range: Sheaf

coboundarySquaredZero
type: NamedIndividual
type: Identity
forAll: term_coboundarySquaredZero_forAll
lhs: term_coboundarySquaredZero_lhs
rhs: term_coboundarySquaredZero_rhs
verificationDomain: Topological
comment: δ² = 0: the coboundary of a coboundary is zero.

deRhamDuality
type: NamedIndividual
type: Identity
forAll: term_deRhamDuality_forAll
lhs: term_deRhamDuality_lhs
rhs: term_deRhamDuality_rhs
verificationDomain: Topological
comment: Discrete de Rham duality: H^k ≅ Hom(H_k, R).

sheafCohomologyBridge
type: NamedIndividual
type: Identity
forAll: term_sheafCohomologyBridge_forAll
lhs: term_sheafCohomologyBridge_lhs
rhs: term_sheafCohomologyBridge_rhs
verificationDomain: Topological
comment: Sheaf cohomology equals simplicial cohomology for constant sheaves.

localGlobalPrinciple
type: NamedIndividual
type: Identity
forAll: term_localGlobalPrinciple_forAll
lhs: term_localGlobalPrinciple_lhs
rhs: term_localGlobalPrinciple_rhs
verificationDomain: Topological
comment: Local-global principle: H^1(K; F) = 0 implies all local sections glue to global sections.

UOR Proofs
type: Ontology
imports: UOR Schema
imports: UOR Operations
imports: UOR Derivations
imports: UOR Observables
comment: Kernel-produced verification proofs attesting to algebraic properties of UOR objects and operations.
space: bridge

Proof
type: Class
comment: A kernel-produced attestation that a given algebraic property holds. The root class for all proof types.
subclass of: Thing

CoherenceProof
type: Class
comment: A proof of coherence: the type system and ring structure are mutually consistent at a given quantum level.
subclass of: Proof

ComputationCertificate
type: Class
comment: A proof confirmed by exhaustive execution over R_n at a specific quantum level. The kernel ran the identity against all 2^n inputs and observed that it holds. The proof:atWittLevel property records the level; proof:witness links to the WitnessData. CriticalIdentityProof is a subclass of ComputationCertificate.
subclass of: Proof

AxiomaticDerivation
type: Class
comment: A proof that follows from previously established axioms or definitions by equational, structural, or topological reasoning. The proof:derivationWitness property links to a derivation:Derivation individual recording the rewrite chain. All pipeline, constraint, observable, and topological identities are AxiomaticDerivations.
subclass of: Proof

CriticalIdentityProof
type: Class
comment: A proof of the critical identity: neg(bnot(x)) = succ(x) for all x in R_n. This is the foundational theorem of the UOR kernel.
subclass of: ComputationCertificate

WitnessData
type: Class
comment: Supporting data for a proof: specific examples, counter-examples checked, or intermediate computation results.
subclass of: Thing

ImpossibilityWitness
type: Class
comment: A formal witness that a topological signature (χ, β_k) is impossible to achieve for any ConstrainedType. Carries the algebraic reason and the verification domain grounding the impossibility.
subclass of: Proof

MorphospaceRecord
type: Class
comment: A formal record of a morphospace boundary point — either an achievable or forbidden topological signature. Aggregated by MorphospaceBoundary to form the queryable morphospace map.
subclass of: Thing

MorphospaceBoundary
type: Class
comment: An aggregate of ImpossibilityWitness instances forming the queryable morphospace map. SPARQL over this structure answers achievability queries in O(1).
subclass of: Thing

InductiveProof
type: Class
comment: A proof by structural induction on the quantum level index k. Carries a base case proof, an inductive step proof, and the minimum k for which the induction holds.
subclass of: Proof

ProofStrategy
type: Class
comment: A controlled vocabulary of proof methods. Each proof individual carries exactly one strategy from this vocabulary, enabling compilation to verified theorem provers.
subclass of: Thing

DerivationTerm
type: Class
comment: Root AST node for proof construction terms. Distinct from schema:TermExpression which represents mathematical terms; DerivationTerm represents proof constructions (tactic applications, lemma invocations, induction scaffolding).
subclass of: Thing

TacticApplication
type: Class
comment: A proof step applying a named tactic (from ProofStrategy) with arguments. Maps to a Lean4 tactic invocation.
subclass of: DerivationTerm

LemmaInvocation
type: Class
comment: A proof step invoking a previously proved identity as a lemma. References the identity via proof:dependsOn.
subclass of: DerivationTerm

InductionStep
type: Class
comment: A proof step performing structural induction: base case derivation, inductive hypothesis, and step derivation.
subclass of: DerivationTerm

ComputationStep
type: Class
comment: A proof step performing exhaustive computation at a specific quantum level as verification witness.
subclass of: DerivationTerm

verified
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this proof has been verified by the kernel.
domain: Proof
range: boolean

timestamp
type: DatatypeProperty
type: FunctionalProperty
comment: The time at which this proof was produced.
domain: Proof
range: dateTime

witness
type: ObjectProperty
comment: Supporting witness data for this proof.
domain: Proof
range: WitnessData

criticalIdentity
type: AnnotationProperty
comment: Human-readable statement of the critical identity proven. E.g., 'neg(bnot(x)) = succ(x) for all x in R_n'. Annotation only — proof:provesIdentity is the typed reference.
domain: CriticalIdentityProof
range: string

x
type: DatatypeProperty
comment: A specific input value used as a witness for the critical identity check.
domain: WitnessData
range: integer

bnot_x
type: DatatypeProperty
comment: The value bnot(x) for a witness x.
domain: WitnessData
range: integer

neg_bnot_x
type: DatatypeProperty
comment: The value neg(bnot(x)) for a witness x.
domain: WitnessData
range: integer

succ_x
type: DatatypeProperty
comment: The value succ(x) for a witness x.
domain: WitnessData
range: integer

holds
type: DatatypeProperty
comment: Whether the identity neg(bnot(x)) = succ(x) holds for this specific witness.
domain: WitnessData
range: boolean

provesIdentity
type: ObjectProperty
type: FunctionalProperty
comment: The algebraic identity this proof establishes. Provides a canonical object reference alongside the existing proof:criticalIdentity string property, which remains for human readability.
domain: Proof
range: Identity

atWittLevel
type: ObjectProperty
type: FunctionalProperty
comment: The quantum level at which this computation certificate was produced. A ComputationCertificate at schema:Q0 confirms the identity holds for all 256 inputs of R_8. A certificate at schema:Q1 confirms it for all 65,536 inputs of R_16.
domain: ComputationCertificate
range: WittLevel

wittNote
type: AnnotationProperty
comment: Human-readable quantum level note, e.g. 'n=8, 256 inputs'. Annotation only — proof:atWittLevel is the typed assertion.
domain: ComputationCertificate
range: string

universalScope
type: DatatypeProperty
type: FunctionalProperty
comment: True when this axiomatic derivation holds for all quantum levels by the definition of Z/(2^n)Z. False when the derivation depends on a property specific to a particular ring size. All current AxiomaticDerivation individuals in the spec carry universalScope true.
domain: AxiomaticDerivation
range: boolean

derivationWitness
type: ObjectProperty
comment: The derivation chain that witnesses this axiomatic derivation. Links a proof:AxiomaticDerivation to the derivation:Derivation individual recording the rewrite sequence. Optional at the spec level — the conformance suite requires only that the proof individual exists; full derivation chains live in generated artifacts.
domain: AxiomaticDerivation
range: Derivation

forbidsSignature
type: AnnotationProperty
comment: The topological signature (χ, β_k) that this ImpossibilityWitness formally forbids, expressed as a symbolic notation string (e.g., 'β₀ = 0').
domain: ImpossibilityWitness
range: string

impossibilityReason
type: DatatypeProperty
type: FunctionalProperty
comment: Human-readable statement of the algebraic reason the signature is impossible (e.g., 'β₀ = 0 violates MS_1').
domain: ImpossibilityWitness
range: string

impossibilityDomain
type: ObjectProperty
type: FunctionalProperty
comment: The verification domain grounding the impossibility (e.g., Pipeline for β₀ = 0, Algebraic for χ > n).
domain: ImpossibilityWitness
range: VerificationDomain

achievabilityStatus
type: ObjectProperty
type: FunctionalProperty
comment: The achievability classification of a proof-linked observable signature: Achievable or Forbidden.
domain: ImpossibilityWitness
range: AchievabilityStatus

verifiedAtLevel
type: ObjectProperty
comment: The specific quantum level at which an empirical verification or impossibility witness was established.
domain: Proof
range: WittLevel

morphospaceRecord
type: ObjectProperty
comment: Links a MorphospaceBoundary to one of its constituent MorphospaceRecord individuals.
domain: MorphospaceBoundary
range: MorphospaceRecord

boundaryType
type: ObjectProperty
type: FunctionalProperty
comment: Whether this MorphospaceRecord represents an impossibility boundary (from below) or an achievability boundary (from above).
domain: MorphospaceRecord
range: AchievabilityStatus

baseCase
type: ObjectProperty
type: FunctionalProperty
comment: The proof that the claim holds at the base level k_0.
domain: InductiveProof
range: Proof

inductiveStep
type: ObjectProperty
type: FunctionalProperty
comment: The proof that if the claim holds at Q_k, it holds at Q_{k+1}.
domain: InductiveProof
range: Proof

validForKAtLeast
type: DatatypeProperty
type: FunctionalProperty
comment: The minimum k for which the induction is valid.
domain: InductiveProof
range: nonNegativeInteger

strategy
type: ObjectProperty
type: FunctionalProperty
comment: The proof method from the ProofStrategy controlled vocabulary. Determines the compilation target (e.g., `by ring`, `by simp`, `by induction`).
domain: Proof
range: ProofStrategy

dependsOn
type: ObjectProperty
comment: An identity that this proof depends on as a lemma. Forms the proof dependency DAG. Leaf proofs (provable from definitions alone) have no dependsOn assertions.
domain: Proof
range: Identity

formalDerivation
type: ObjectProperty
type: FunctionalProperty
comment: The formal proof construction term: a DerivationTerm AST node encoding the tactic script, lemma chain, or induction scaffold that constitutes the proof.
domain: Proof
range: DerivationTerm

RingAxiom
type: NamedIndividual
type: ProofStrategy
comment: Follows from ZMod ring axioms. Lean4 tactic: `by ring`.

DecideQ0
type: NamedIndividual
type: ProofStrategy
comment: Decidable at Q0 by exhaustive evaluation. Lean4: `by native_decide`.

BitwiseInduction
type: NamedIndividual
type: ProofStrategy
comment: Induction on bit width n. Lean4: `by induction n`.

GroupPresentation
type: NamedIndividual
type: ProofStrategy
comment: From dihedral group presentation. Lean4: `by group`.

Simplification
type: NamedIndividual
type: ProofStrategy
comment: By simplification with cited lemmas. Lean4: `by simp [lemmalist]`.

ChineseRemainder
type: NamedIndividual
type: ProofStrategy
comment: By Chinese Remainder Theorem. Lean4: `by exact ZMod.chineseRemainder ...`.

EulerPoincare
type: NamedIndividual
type: ProofStrategy
comment: By Euler-Poincare formula applied to the constraint nerve.

ProductFormula
type: NamedIndividual
type: ProofStrategy
comment: By Ostrowski product formula or derived valuation arguments.

Composition
type: NamedIndividual
type: ProofStrategy
comment: By composing proofs of sub-identities. Lean4: `by exact ...`.

Contradiction
type: NamedIndividual
type: ProofStrategy
comment: By deriving contradiction for impossibility witnesses. Lean4: `by contradiction`.

Computation
type: NamedIndividual
type: ProofStrategy
comment: By computation at a specified quantum level. Lean4: `by native_decide`.

prf_criticalIdentity
type: NamedIndividual
type: CriticalIdentityProof
atWittLevel: W8
provesIdentity: Critical Identity
strategy: Computation
verified: true
comment: Computation certificate for the critical identity neg(bnot(x)) = succ(x) at Q0.

prf_criticalIdentity_axiomatic
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: Critical Identity
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of the critical identity neg(bnot(x)) = succ(x). Holds at all quantum levels.

prf_phi_1
type: NamedIndividual
type: ComputationCertificate
atWittLevel: W8
provesIdentity: phi_1
strategy: Computation
verified: true
comment: Computation certificate for phi_1 at Q0.

prf_phi_2
type: NamedIndividual
type: ComputationCertificate
atWittLevel: W8
provesIdentity: phi_2
strategy: Computation
verified: true
comment: Computation certificate for phi_2 at Q0.

prf_phi_3
type: NamedIndividual
type: ComputationCertificate
atWittLevel: W8
provesIdentity: phi_3
strategy: Computation
verified: true
comment: Computation certificate for phi_3 at Q0.

prf_phi_4
type: NamedIndividual
type: ComputationCertificate
atWittLevel: W8
provesIdentity: phi_4
strategy: Computation
verified: true
comment: Computation certificate for phi_4 at Q0.

prf_phi_5
type: NamedIndividual
type: ComputationCertificate
atWittLevel: W8
provesIdentity: phi_5
strategy: Computation
verified: true
comment: Computation certificate for phi_5 at Q0.

prf_phi_6
type: NamedIndividual
type: ComputationCertificate
atWittLevel: W8
provesIdentity: phi_6
strategy: Computation
verified: true
comment: Computation certificate for phi_6 at Q0.

prf_AD_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AD_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AD_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AD_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AD_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AD_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_R_A1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: R_A1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of R_A1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_R_A2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: R_A2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of R_A2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_R_A3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: R_A3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of R_A3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_R_A4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: R_A4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of R_A4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_R_A5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: R_A5
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of R_A5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_R_A6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: R_A6
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of R_A6. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_R_M1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: R_M1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of R_M1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_R_M2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: R_M2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of R_M2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_R_M3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: R_M3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of R_M3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_R_M4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: R_M4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of R_M4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_R_M5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: R_M5
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of R_M5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_B_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: B_1
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of B_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_B_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: B_2
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of B_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_B_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: B_3
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of B_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_B_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: B_4
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of B_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_B_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: B_5
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of B_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_B_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: B_6
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of B_6. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_B_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: B_7
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of B_7. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_B_8
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: B_8
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of B_8. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_B_9
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: B_9
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of B_9. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_B_10
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: B_10
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of B_10. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_B_11
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: B_11
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of B_11. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_B_12
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: B_12
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of B_12. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_B_13
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: B_13
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of B_13. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_X_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: X_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of X_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_X_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: X_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of X_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_X_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: X_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of X_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_X_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: X_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of X_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_X_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: X_5
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of X_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_X_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: X_6
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of X_6. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_X_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: X_7
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of X_7. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_D_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: D_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of D_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_D_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: D_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of D_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_D_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: D_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of D_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_D_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: D_5
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of D_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_U_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: U_1
strategy: ChineseRemainder
universalScope: true
verified: true
comment: Axiomatic derivation of U_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_U_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: U_2
strategy: ChineseRemainder
universalScope: true
verified: true
comment: Axiomatic derivation of U_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_U_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: U_3
strategy: ChineseRemainder
universalScope: true
verified: true
comment: Axiomatic derivation of U_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_U_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: U_4
strategy: ChineseRemainder
universalScope: true
verified: true
comment: Axiomatic derivation of U_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_U_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: U_5
strategy: ChineseRemainder
universalScope: true
verified: true
comment: Axiomatic derivation of U_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AG_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AG_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of AG_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AG_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AG_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of AG_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AG_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AG_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of AG_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AG_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AG_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of AG_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CA_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CA_1
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of CA_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CA_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CA_2
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of CA_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CA_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CA_3
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of CA_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CA_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CA_4
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of CA_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CA_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CA_5
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of CA_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CA_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CA_6
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of CA_6. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_C_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: C_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of C_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_C_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: C_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of C_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_C_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: C_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of C_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_C_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: C_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of C_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_C_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: C_5
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of C_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_C_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: C_6
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of C_6. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CDI
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CDI
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CDI. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CL_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CL_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CL_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CL_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CL_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CL_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CL_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CL_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CL_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CL_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CL_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CL_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CL_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CL_5
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CL_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CM_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CM_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CM_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CM_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CM_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CM_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CM_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CM_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CM_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CR_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CR_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CR_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CR_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CR_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CR_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CR_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CR_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CR_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CR_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CR_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CR_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CR_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CR_5
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CR_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_F_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: F_1
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of F_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_F_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: F_2
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of F_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_F_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: F_3
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of F_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_F_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: F_4
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of F_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_FL_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FL_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FL_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_FL_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FL_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FL_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_FL_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FL_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FL_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_FL_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FL_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FL_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_FPM_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FPM_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FPM_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_FPM_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FPM_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FPM_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_FPM_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FPM_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FPM_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_FPM_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FPM_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FPM_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_FPM_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FPM_5
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FPM_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_FPM_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FPM_6
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FPM_6. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_FPM_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FPM_7
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FPM_7. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_FS_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FS_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FS_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_FS_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FS_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FS_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_FS_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FS_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FS_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_FS_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FS_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FS_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_FS_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FS_5
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FS_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_FS_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FS_6
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FS_6. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_FS_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FS_7
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FS_7. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_RE_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: RE_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of RE_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_IR_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IR_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of IR_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_IR_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IR_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of IR_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_IR_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IR_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of IR_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_IR_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IR_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of IR_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_SF_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SF_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of SF_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_SF_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SF_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of SF_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_RD_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: RD_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of RD_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_RD_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: RD_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of RD_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_SE_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SE_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of SE_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_SE_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SE_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of SE_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_SE_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SE_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of SE_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_SE_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SE_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of SE_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OO_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OO_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OO_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OO_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OO_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OO_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OO_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OO_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OO_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OO_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OO_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OO_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OO_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OO_5
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OO_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CB_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CB_1
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of CB_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CB_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CB_2
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of CB_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CB_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CB_3
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of CB_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CB_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CB_4
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of CB_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CB_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CB_5
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of CB_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CB_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CB_6
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of CB_6. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OB_M1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OB_M1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OB_M1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OB_M2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OB_M2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OB_M2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OB_M3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OB_M3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OB_M3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OB_M4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OB_M4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OB_M4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OB_M5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OB_M5
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OB_M5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OB_M6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OB_M6
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OB_M6. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OB_C1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OB_C1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OB_C1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OB_C2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OB_C2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OB_C2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OB_C3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OB_C3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OB_C3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OB_H1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OB_H1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OB_H1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OB_H2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OB_H2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OB_H2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OB_H3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OB_H3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OB_H3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OB_P1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OB_P1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OB_P1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OB_P2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OB_P2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OB_P2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_OB_P3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OB_P3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OB_P3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CT_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CT_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of CT_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CT_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CT_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of CT_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CT_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CT_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of CT_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CT_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CT_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of CT_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CF_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CF_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of CF_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CF_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CF_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of CF_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CF_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CF_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of CF_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_CF_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CF_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of CF_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_HG_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HG_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of HG_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_HG_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HG_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of HG_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_HG_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HG_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of HG_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_HG_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HG_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of HG_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_HG_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HG_5
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of HG_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_T_C1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: T_C1
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of T_C1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_T_C2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: T_C2
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of T_C2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_T_C3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: T_C3
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of T_C3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_T_C4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: T_C4
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of T_C4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_T_I1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: T_I1
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of T_I1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_T_I2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: T_I2
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of T_I2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_T_I3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: T_I3
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of T_I3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_T_I4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: T_I4
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of T_I4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_T_I5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: T_I5
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of T_I5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_T_E1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: T_E1
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of T_E1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_T_E2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: T_E2
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of T_E2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_T_E3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: T_E3
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of T_E3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_T_E4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: T_E4
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of T_E4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_T_A1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: T_A1
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of T_A1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_T_A2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: T_A2
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of T_A2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_T_A3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: T_A3
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of T_A3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_T_A4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: T_A4
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of T_A4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AU_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AU_1
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of AU_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AU_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AU_2
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of AU_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AU_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AU_3
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of AU_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AU_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AU_4
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of AU_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AU_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AU_5
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of AU_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_EF_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EF_1
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of EF_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_EF_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EF_2
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of EF_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_EF_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EF_3
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of EF_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_EF_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EF_4
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of EF_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_EF_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EF_5
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of EF_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_EF_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EF_6
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of EF_6. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_EF_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EF_7
strategy: GroupPresentation
universalScope: true
verified: true
comment: Axiomatic derivation of EF_7. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AA_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AA_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AA_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AA_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AA_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AA_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AA_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AA_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AA_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AA_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AA_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AA_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AA_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AA_5
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AA_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AA_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AA_6
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AA_6. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AM_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AM_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AM_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AM_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AM_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AM_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AM_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AM_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AM_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AM_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AM_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AM_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_TH_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TH_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of TH_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_TH_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TH_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of TH_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_TH_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TH_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of TH_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_TH_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TH_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of TH_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_TH_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TH_5
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of TH_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_TH_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TH_6
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of TH_6. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_TH_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TH_7
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of TH_7. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_TH_8
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TH_8
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of TH_8. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_TH_9
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TH_9
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of TH_9. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_TH_10
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TH_10
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of TH_10. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AR_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AR_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AR_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AR_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AR_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AR_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AR_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AR_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AR_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AR_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AR_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AR_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_AR_5
type: NamedIndividual
type: InductiveProof
baseCase: prf_AR_5_base
inductiveStep: prf_AR_5_step
provesIdentity: AR_5
strategy: BitwiseInduction
universalScope: true
validForKAtLeast: 0
verified: true
comment: Inductive proof of AR_5: greedy vs adiabatic cost difference is at most 5%. Base case at Q0 by exhaustive evaluation; inductive step by QLS_5 (identity preservation under lift).

prf_PD_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PD_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PD_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_PD_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PD_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PD_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_PD_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PD_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PD_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_PD_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PD_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PD_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_PD_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PD_5
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PD_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_RC_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: RC_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of RC_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_RC_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: RC_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of RC_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_RC_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: RC_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of RC_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_RC_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: RC_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of RC_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_RC_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: RC_5
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of RC_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_DC_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DC_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of DC_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_DC_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DC_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of DC_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_DC_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DC_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of DC_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_DC_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DC_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of DC_4. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_DC_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DC_5
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of DC_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_DC_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DC_6
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of DC_6. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_DC_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DC_7
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of DC_7. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_DC_8
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DC_8
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of DC_8. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_DC_9
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DC_9
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of DC_9. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_DC_10
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DC_10
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of DC_10. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_DC_11
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DC_11
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of DC_11. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_HA_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HA_1
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of HA_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_HA_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HA_2
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of HA_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_HA_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HA_3
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of HA_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_IT_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IT_2
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of IT_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_IT_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IT_3
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of IT_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_IT_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IT_6
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of IT_6. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_IT_7a
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IT_7a
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of IT_7a. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_IT_7b
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IT_7b
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of IT_7b. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_IT_7c
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IT_7c
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of IT_7c. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_IT_7d
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IT_7d
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of IT_7d. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_psi_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: psi_1
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of psi_1. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_psi_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: psi_2
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of psi_2. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_psi_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: psi_3
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of psi_3. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_psi_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: psi_5
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of psi_5. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_psi_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: psi_6
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of psi_6. Holds at all quantum levels by definition of Z/(2^n)Z.

prf_boundarySquaredZero
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: boundarySquaredZero
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of homology:boundarySquaredZero. Holds at all quantum levels by topological reasoning.

prf_psi_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: psi_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of homology:psi_4. Holds at all quantum levels by topological reasoning.

prf_indexBridge
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: indexBridge
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of homology:indexBridge. Holds at all quantum levels by topological reasoning.

prf_coboundarySquaredZero
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: coboundarySquaredZero
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of cohomology:coboundarySquaredZero. Holds at all quantum levels by topological reasoning.

prf_deRhamDuality
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: deRhamDuality
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of cohomology:deRhamDuality. Holds at all quantum levels by topological reasoning.

prf_sheafCohomologyBridge
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: sheafCohomologyBridge
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of cohomology:sheafCohomologyBridge. Holds at all quantum levels by topological reasoning.

prf_localGlobalPrinciple
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: localGlobalPrinciple
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of cohomology:localGlobalPrinciple. Holds at all quantum levels by topological reasoning.

prf_surfaceSymmetry
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: Surface Symmetry
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of the Surface Symmetry Theorem. Holds at all quantum levels: the composite P∘Π∘G is a well-typed morphism whenever G and P share the same state:Frame. Follows from the definition of the shared-frame condition and the type-equivalence algebra.

prf_CC_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CC_1
strategy: Simplification
universalScope: true
verified: true
comment: Proof that a CompleteType T satisfies: resolution(x, T) terminates in O(1) for all x ∈ R_n. Follows from IT_7d: when χ(N(C)) = n and all β_k = 0, the resolver has no topological obstructions.

prf_CC_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CC_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof that the ψ pipeline is monotone: each constraint application cannot increase the site deficit. Derived from the definition of the partition refinement order.

prf_CC_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CC_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof that a CompletenessCertificate implies CompleteType: the certificate attestation is only issued when IT_7d holds, by construction of the CompletenessResolver.

prf_CC_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CC_4
strategy: Simplification
universalScope: true
verified: true
comment: Proof that the CompletenessAuditTrail witnessCount equals the number of CompletenessWitness records in the trail. Structural invariant of the audit accumulation protocol.

prf_CC_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CC_5
strategy: EulerPoincare
universalScope: true
verified: true
comment: Proof that the CechNerve nerve computation is deterministic: the same constraint set always produces the same nerve topology. Follows from the nerve functor being a functor (functoriality).

prf_QL_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: QL_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Universal proof that neg(bnot(x)) = succ(x) in Z/(2^n)Z for all n ≥ 1. Derived symbolically from ring axioms: bnot is bitwise complement, neg is two's complement negation, succ is modular increment. The critical identity in universal form.

prf_QL_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: QL_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Universal proof that the ring carrier set size is exactly 2^n for all n ≥ 1. Follows from the definition of Z/(2^n)Z.

prf_QL_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: QL_3
strategy: Simplification
universalScope: true
verified: true
comment: Universal proof that Landauer erasure cost scales as n × k_B T ln 2 at quantum level n. Follows from the thermodynamic interpretation: each bit erased from an n-bit ring costs k_B T ln 2.

prf_QL_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: QL_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Universal proof that the dihedral group D_{2^n} action on Z/(2^n)Z is faithful for all n ≥ 1. The stabilizer of any element is trivial under the full dihedral action.

prf_QL_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: QL_5
strategy: Simplification
universalScope: true
verified: true
comment: Universal proof that canonical form rewriting terminates at all quantum levels. The rewriting system is terminating by lexicographic descent on the term complexity measure.

prf_QL_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: QL_6
strategy: RingAxiom
universalScope: true
verified: true
comment: Universal proof that the completeness criterion χ(N(C)) = n generalizes to all quantum levels. Derived from the definition of the nerve construction and the Euler characteristic formula for simplicial complexes.

prf_QL_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: QL_7
strategy: EulerPoincare
universalScope: true
verified: true
comment: Universal proof of the ring topology Euler characteristic identity: χ = 1 − n at quantum level n. Derived from the CW decomposition of the n-dimensional torus formed by the ring's cyclic group action.

prf_GR_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GR_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of binding monotonicity: freeRank(B_{i+1}) ≤ freeRank(B_i) for all i in a Session. Follows from the definition of the BindingAccumulator: each appended binding either pins sites or is a no-op; it never frees them.

prf_GR_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GR_2
strategy: Simplification
universalScope: true
verified: true
comment: Proof that the empty session is the identity element of the session algebra: freeRank(B_0) = total site space. The empty accumulator has no pinned sites by definition.

prf_GR_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GR_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of session convergence: a session terminates iff freeRank reaches its minimum (the maximum pinned by the given constraint set). Follows from the compactness of the site space and monotonicity.

prf_GR_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GR_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof that disjoint bindings compose without site conflict: if two bindings address disjoint site sets, their composition is well-defined and their union is also a valid binding.

prf_GR_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GR_5
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of contradiction detection correctness: ContradictionBoundary fires iff there exist bindings b, b' in the same Context with the same address, different datum, and same constraint. This is the minimal condition for type contradiction.

prf_TS_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TS_1
strategy: Simplification
universalScope: true
verified: true
comment: Proof of nerve realisability: for any target profile with χ* ≤ n and β₀* = 1, there exists a ConstrainedType whose constraint nerve realises the target. Follows from the constructive synthesis algorithm.

prf_TS_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TS_2
strategy: Simplification
universalScope: true
verified: true
comment: Proof of minimal basis bound: the MinimalConstraintBasis for the IT_7d target has size exactly n. Follows from the site-by-site construction and the minimality criterion.

prf_TS_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TS_3
strategy: Simplification
universalScope: true
verified: true
comment: Proof of synthesis monotonicity: adding a constraint never decreases the Euler characteristic of the constraint nerve. Follows from the nerve inclusion principle.

prf_TS_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TS_4
strategy: Simplification
universalScope: true
verified: true
comment: Proof of synthesis convergence: the TypeSynthesisResolver terminates in at most n steps. Follows from monotonicity (TS_3) and the finite site budget bound.

prf_TS_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TS_5
strategy: Simplification
universalScope: true
verified: true
comment: Proof of synthesis-certification duality: a SynthesizedType achieves IT_7d iff the CompletenessResolver certifies it as CompleteType. The duality follows from the shared topological criterion.

prf_TS_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TS_6
strategy: Simplification
universalScope: true
verified: true
comment: Proof of Jacobian-guided synthesis efficiency: the Jacobian oracle reduces expected steps from O(n²) to O(n log n). Follows from the information content of the Jacobian at each synthesis step.

prf_TS_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TS_7
strategy: Simplification
universalScope: true
verified: true
comment: Proof of unreachable signatures: β₀ = 0 is unreachable by any non-empty ConstrainedType. Follows from the nerve connectedness of non-empty constraint sets.

prf_WLS_1
type: NamedIndividual
type: InductiveProof
baseCase: prf_WLS_1_base
inductiveStep: prf_WLS_6
provesIdentity: WLS_1
strategy: BitwiseInduction
universalScope: true
validForKAtLeast: 0
verified: true
comment: Proof of lift unobstructedness criterion: WittLift T' is CompleteType iff the spectral sequence collapses at E_2. Follows from the Leray spectral sequence of the quantum level extension.

prf_WLS_2
type: NamedIndividual
type: InductiveProof
baseCase: prf_WLS_2_base
inductiveStep: prf_WLS_2_step
provesIdentity: WLS_2
strategy: BitwiseInduction
universalScope: true
validForKAtLeast: 0
verified: true
comment: Proof of obstruction localisation: a non-trivial LiftObstruction is localised to a specific site at bit position n+1. Follows from the local-to-global structure of the constraint nerve.

prf_WLS_3
type: NamedIndividual
type: InductiveProof
baseCase: prf_WLS_3_base
inductiveStep: prf_WLS_3_step
provesIdentity: WLS_3
strategy: BitwiseInduction
universalScope: true
validForKAtLeast: 0
verified: true
comment: Proof of monotone lifting: basisSize(T') = basisSize(T) + 1 for trivially obstructed lifts. Follows from the minimal basis construction at Q_{n+1}.

prf_WLS_4
type: NamedIndividual
type: InductiveProof
baseCase: prf_WLS_4_base
inductiveStep: prf_WLS_4_step
provesIdentity: WLS_4
strategy: BitwiseInduction
universalScope: true
validForKAtLeast: 0
verified: true
comment: Proof of spectral sequence convergence bound: the spectral sequence converges by page E_{d+2} for depth-d configurations. Follows from the filtration length of the constraint nerve chain complex.

prf_WLS_5
type: NamedIndividual
type: InductiveProof
baseCase: prf_WLS_5_base
inductiveStep: prf_WLS_6
provesIdentity: WLS_5
strategy: BitwiseInduction
universalScope: true
validForKAtLeast: 0
verified: true
comment: Proof of universal identity preservation under quantum lifts: every universallyValid identity holds in the lifted ring. Follows from the universal validity definition and ring extension properties.

prf_WLS_6
type: NamedIndividual
type: InductiveProof
baseCase: prf_WLS_6_base
inductiveStep: prf_WLS_6_step
provesIdentity: WLS_6
strategy: BitwiseInduction
universalScope: true
validForKAtLeast: 0
verified: true
comment: Proof of ψ-pipeline universality for quantum lifts: the ψ-pipeline produces a valid ChainComplex for any WittLift. Follows from the functorial construction of the chain complex.

prf_WLS_1_base
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WLS_1
strategy: Composition
universalScope: false
verified: true
comment: Base case for QLS_1 at Q0: lift unobstructedness holds trivially for 8-bit rings where the constraint nerve is contractible.

prf_WLS_2_base
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WLS_2
strategy: Composition
universalScope: false
verified: true
comment: Base case for QLS_2 at Q0: obstruction localisation holds at the 8-bit level where sites are directly inspectable.

prf_WLS_2_step
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WLS_2
strategy: Composition
universalScope: true
verified: true
comment: Inductive step for QLS_2: if obstruction is localised at Q_k, the local-to-global structure of the constraint nerve preserves localisation at Q_{k+1}.

prf_WLS_3_base
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WLS_3
strategy: Composition
universalScope: false
verified: true
comment: Base case for QLS_3 at Q0: monotone lifting basis size increment holds trivially for 8-bit to 16-bit extension.

prf_WLS_3_step
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WLS_3
strategy: Composition
universalScope: true
verified: true
comment: Inductive step for QLS_3: the minimal basis construction at Q_{k+1} adds exactly one element from the trivially obstructed site.

prf_WLS_4_base
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WLS_4
strategy: Composition
universalScope: false
verified: true
comment: Base case for QLS_4 at Q0: spectral sequence convergence at E_{d+2} holds for 8-bit filtrations by direct computation.

prf_WLS_4_step
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WLS_4
strategy: Composition
universalScope: true
verified: true
comment: Inductive step for QLS_4: filtration length at Q_{k+1} extends by at most one page from Q_k, preserving the E_{d+2} bound.

prf_WLS_5_base
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WLS_5
strategy: Composition
universalScope: false
verified: true
comment: Base case for QLS_5 at Q0: universallyValid identities hold in the 8-bit ring by definition of universal validity.

prf_WLS_6_base
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WLS_6
strategy: Composition
universalScope: false
verified: true
comment: Base case for QLS_6 at Q0: the psi-pipeline produces a valid ChainComplex for 8-bit WittLifts by direct construction.

prf_WLS_6_step
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WLS_6
strategy: Composition
universalScope: true
verified: true
comment: Inductive step for QLS_6: the functorial construction of the chain complex commutes with quantum level extension.

prf_WT_3_base
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WT_3
strategy: Composition
universalScope: false
verified: true
comment: Base case for QT_3: resolved basis size formula holds for chain length 1 by direct construction.

prf_WT_5_base
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WT_5
strategy: Composition
universalScope: false
verified: true
comment: Base case for QT_5: LiftChainCertificate existence for tower height 1 follows from single-step certificate issuance.

prf_AR_5_base
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AR_5
strategy: Simplification
universalScope: false
verified: true
comment: Base case for AR_5 at Q0: greedy vs adiabatic cost difference verified by exhaustive enumeration over Z/256Z.

prf_AR_5_step
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AR_5
strategy: Simplification
universalScope: true
verified: true
comment: Inductive step for AR_5: if greedy vs adiabatic bound holds at Q_k, it holds at Q_{k+1} by QLS_5 (universal identity preservation under quantum lift).

prf_QM_6_base
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: QM_6
strategy: Simplification
universalScope: false
verified: true
comment: Base case for QM_6 at Q0: amplitude index set equals monotone pinning trajectories by exhaustive trajectory enumeration over the 8-bit site lattice.

prf_QM_6_step
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: QM_6
strategy: Simplification
universalScope: true
verified: true
comment: Inductive step for QM_6: monotone pinning trajectories at Q_{k+1} extend those at Q_k by the site lattice ordering (monotone extension property).

prf_MN_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MN_1
strategy: EulerPoincare
universalScope: true
verified: true
comment: Proof of holonomy group containment: HolonomyGroup(T) ≤ D_{2^n}. Follows from the fact that all constraint applications are dihedral group elements.

prf_MN_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MN_2
strategy: EulerPoincare
universalScope: true
verified: true
comment: Proof of additive flatness: additive constraints (ResidueConstraint, DepthConstraint) generate only the identity in D_{2^n}. Follows from the additive structure of the dihedral action.

prf_MN_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MN_3
strategy: EulerPoincare
universalScope: true
verified: true
comment: Proof of dihedral generation: neg and bnot together generate D_{2^n}. Follows from the standard presentation of the dihedral group by involutions.

prf_MN_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MN_4
strategy: EulerPoincare
universalScope: true
verified: true
comment: Proof of holonomy-Betti implication: non-trivial holonomy implies β₁ ≥ 1. Follows from the fact that a non-trivial monodromy requires a topological loop in the constraint nerve.

prf_MN_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MN_5
strategy: EulerPoincare
universalScope: true
verified: true
comment: Proof of CompleteType holonomy: IT_7d (β₁ = 0) implies trivial holonomy (FlatType). Follows from MN_4 contrapositive: trivial holonomy ← β₁ = 0.

prf_MN_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MN_6
strategy: EulerPoincare
universalScope: true
verified: true
comment: Proof of monodromy composition: the monodromy map is a group homomorphism from the loop space to D_{2^n}. Follows from the composition of dihedral group elements.

prf_MN_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MN_7
strategy: EulerPoincare
universalScope: true
verified: true
comment: Proof of TwistedType obstruction class: a TwistedType always contributes a non-zero class to H²(N(C(T')); ℤ/2ℤ) for any WittLift T'. Follows from MN_4 and the obstruction theory of dihedral torsors.

prf_PT_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PT_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of PT_1: product type site additivity. siteBudget(A × B) = siteBudget(A) + siteBudget(B). Follows from the definition of ProductType as an independent concatenation of site spaces.

prf_PT_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PT_2
strategy: EulerPoincare
universalScope: true
verified: true
comment: Proof of PT_2: product type partition factorisation. partition(A × B) = partition(A) ⊗ partition(B). Follows from the tensor product structure of constraint nerves over independent site spaces.

prf_PT_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PT_3
strategy: Composition
universalScope: true
verified: true
comment: Proof of PT_3: product type Euler characteristic additivity. χ(N(C(A × B))) = χ(N(C(A))) + χ(N(C(B))). Follows from the Künneth formula applied to the join of constraint nerves.

prf_PT_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PT_4
strategy: Simplification
universalScope: true
verified: true
comment: Proof of PT_4: product type entropy additivity. S(A × B) = S(A) + S(B). Follows from PT_1 (site additivity) and TH_1 (S = freeRank × ln 2).

prf_ST_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: ST_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of ST_1: sum type site budget maximum. siteBudget(A + B) = max(siteBudget(A), siteBudget(B)). Follows from SumType requiring capacity for the larger variant.

prf_ST_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: ST_2
strategy: Simplification
universalScope: true
verified: true
comment: Proof of ST_2: sum type entropy. S(A + B) = ln 2 + max(S(A), S(B)). The ln 2 term accounts for the variant discriminant bit; the max reflects that only one variant is active at a time.

prf_GS_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GS_1
strategy: Simplification
universalScope: true
verified: true
comment: Proof of SC_1: context temperature. T_ctx(C) = freeRank(C) × ln 2 / n. Derived from TH_1 normalized per site.

prf_GS_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GS_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of SC_2: saturation degree. σ(C) = (n − freeRank(C)) / n. Definitional identity.

prf_GS_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GS_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of SC_3: saturation monotonicity. Corollary of SR_1 through order-reversing SC_2.

prf_GS_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GS_4
strategy: Simplification
universalScope: true
verified: true
comment: Proof of SC_4: ground state equivalence. Four equivalent conditions for full saturation derived from SC_2, TH_1, SC_1.

prf_GS_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GS_5
strategy: Simplification
universalScope: true
verified: true
comment: Proof of SC_5: O(1) resolution guarantee at saturation. Derived from SR_2 and FreeRank.isClosed at freeRank = 0.

prf_GS_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GS_6
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of SC_6: pre-reduction of effective budget. Derived from session-scoped site reduction at partial saturation.

prf_GS_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GS_7
strategy: Simplification
universalScope: true
verified: true
comment: Proof of SC_7: thermodynamic cooling cost. n site-closures at Landauer cost each via SR_1 + TH_4.

prf_MS_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MS_1
strategy: Simplification
universalScope: true
verified: true
comment: Proof of MS_1: connectivity lower bound β₀ ≥ 1. Formalisation of TS_7.

prf_MS_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MS_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of MS_2: Euler capacity ceiling χ ≤ n. Derived from TS_1 constraint nerve dimension bound.

prf_MS_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MS_3
strategy: EulerPoincare
universalScope: true
verified: true
comment: Proof of MS_3: Betti monotonicity under constraint addition. Formalisation of TS_3.

prf_MS_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MS_4
strategy: Simplification
universalScope: true
verified: true
comment: Proof of MS_4: level-relative achievability. Derived from WittLift construction (Amendment 29).

prf_MS_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MS_5
strategy: Simplification
universalScope: true
verified: true
comment: Proof of MS_5: empirical completeness convergence. Convergence statement for verification accumulation.

prf_GD_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GD_1
strategy: Simplification
universalScope: true
verified: true
comment: Proof of GD_1: geodesic condition. Dual condition from AR_1 ordering + DC_10 selection.

prf_GD_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GD_2
strategy: Simplification
universalScope: true
verified: true
comment: Proof of GD_2: geodesic entropy bound. Each step on a geodesic erases exactly ln 2 nats.

prf_GD_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GD_3
strategy: Simplification
universalScope: true
verified: true
comment: Proof of GD_3: total geodesic cost equals Landauer bound TH_4 with equality.

prf_GD_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GD_4
strategy: Simplification
universalScope: true
verified: true
comment: Proof of GD_4: geodesic uniqueness up to step-order equivalence. Equal-J_k permutations are interchangeable.

prf_GD_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GD_5
strategy: Simplification
universalScope: true
verified: true
comment: Proof of GD_5: subgeodesic detectability via step-by-step J_k check.

prf_QM_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: QM_1
strategy: ProductFormula
universalScope: true
verified: true
comment: Proof of QM_1: von Neumann–Landauer bridge. S_vN equals Landauer erasure cost at the Landauer temperature β* = ln 2.

prf_QM_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: QM_2
strategy: EulerPoincare
universalScope: true
verified: true
comment: Proof of QM_2: measurement as site topology change. Projective collapse ≅ classical ResidueConstraint pinning.

prf_QM_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: QM_3
strategy: ProductFormula
universalScope: true
verified: true
comment: Proof of QM_3: superposition entropy bound. 0 ≤ S_vN ≤ ln 2 for single-site superpositions.

prf_QM_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: QM_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of QM_4: collapse idempotence. Re-measurement of a collapsed state is a no-op.

prf_QM_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: QM_5
strategy: ProductFormula
universalScope: true
verified: true
comment: Proof of QM_5: amplitude normalization (Born rule). Σ|αᵢ|² = 1 for well-formed SuperposedSiteState.

prf_RC_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: RC_6
strategy: Simplification
universalScope: true
verified: true
comment: Proof of RC_6: amplitude renormalization. Division by norm yields a normalized SuperposedSiteState.

prf_FPM_8
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FPM_8
strategy: DecideQ0
universalScope: true
verified: true
comment: Proof of FPM_8: partition exhaustiveness. The four component cardinalities sum to 2ⁿ.

prf_FPM_9
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FPM_9
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of FPM_9: exterior membership criterion. x ∈ Ext(T) iff x ∉ carrier(T).

prf_MN_8
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MN_8
strategy: EulerPoincare
universalScope: true
verified: true
comment: Proof of MN_8: holonomy classification covering. Every ConstrainedType is flat xor twisted.

prf_QL_8
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: QL_8
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of QL_8: quantum level chain inverse. wittLevelPredecessor is the left inverse of nextLevel.

prf_D_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: D_7
strategy: GroupPresentation
universalScope: true
verified: true
comment: Proof of D_7: dihedral composition rule from the semidirect product presentation.

prf_SP_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SP_1
strategy: Simplification
universalScope: true
verified: true
comment: Proof of SP_1: classical embedding. Superposition resolution of a classical datum reduces to classical resolution.

prf_SP_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SP_2
strategy: ProductFormula
universalScope: true
verified: true
comment: Proof of SP_2: collapse–resolve commutativity. The collapse and resolve operations commute.

prf_SP_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SP_3
strategy: Simplification
universalScope: true
verified: true
comment: Proof of SP_3: amplitude preservation. The SuperpositionResolver preserves the normalized amplitude vector.

prf_SP_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SP_4
strategy: ProductFormula
universalScope: true
verified: true
comment: Proof of SP_4: Born rule outcome probability. P(collapse to site k) = |α_k|².

prf_PT_2a
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PT_2a
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of PT_2a: product type partition tensor. Π(A × B) = PartitionProduct(Π(A), Π(B)).

prf_PT_2b
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PT_2b
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of PT_2b: sum type partition coproduct. Π(A + B) = PartitionCoproduct(Π(A), Π(B)).

prf_GD_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GD_6
strategy: Simplification
universalScope: true
verified: true
comment: Proof of GD_6: geodesic predicate decomposition. isGeodesic = isAR1Ordered ∧ isDC10Selected.

iw_beta0_bound
type: NamedIndividual
type: ImpossibilityWitness
forbidsSignature: β₀ = 0
impossibilityDomain: Pipeline
impossibilityReason: β₀ = 0 violates MS_1: constraint nerve of non-empty set is connected
strategy: Contradiction
verified: true
comment: Impossibility witness for MS_1: β₀ = 0 is forbidden for any non-empty ConstrainedType because the constraint nerve is always connected.

iw_chi_ceiling
type: NamedIndividual
type: ImpossibilityWitness
forbidsSignature: χ > n
impossibilityDomain: Algebraic
impossibilityReason: χ > n violates MS_2: Euler characteristic bounded by quantum level
strategy: Contradiction
verified: true
comment: Impossibility witness for MS_2: χ > n is forbidden at quantum level n. The Euler characteristic cannot exceed the quantum level.

mr_completeness_target
type: NamedIndividual
type: MorphospaceRecord
boundaryType: Achievable
verified: true
comment: Morphospace record: the IT_7d completeness target χ = n is achievable (sits at the ceiling of MS_2).

mr_connectivity_lower
type: NamedIndividual
type: MorphospaceRecord
boundaryType: Forbidden
verified: true
comment: Morphospace record: the connectivity lower bound β₀ ≥ 1 marks the forbidden region from below.

prf_WT_1
type: NamedIndividual
type: InductiveProof
baseCase: prf_WLS_1
inductiveStep: prf_WLS_6
provesIdentity: WT_1
strategy: BitwiseInduction
universalScope: true
validForKAtLeast: 0
verified: true
comment: Proof of tower chain validity by induction on chain length.

prf_WT_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WT_2
strategy: Composition
universalScope: true
verified: true
comment: Proof of obstruction count bound: direct from QLS_2 localization.

prf_WT_3
type: NamedIndividual
type: InductiveProof
baseCase: prf_WT_3_base
inductiveStep: prf_WLS_3
provesIdentity: WT_3
strategy: BitwiseInduction
universalScope: true
validForKAtLeast: 0
verified: true
comment: Proof of resolved basis size formula by induction on chain length.

prf_WT_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WT_4
strategy: EulerPoincare
universalScope: true
verified: true
comment: Proof of flat tower characterization: isFlat iff trivial holonomy at every step.

prf_WT_5
type: NamedIndividual
type: InductiveProof
baseCase: prf_WT_5_base
inductiveStep: prf_WT_1
provesIdentity: WT_5
strategy: BitwiseInduction
universalScope: true
validForKAtLeast: 0
verified: true
comment: Proof of LiftChainCertificate existence by induction on tower height.

prf_WT_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WT_6
strategy: Composition
universalScope: true
verified: true
comment: Proof of single-step reduction: QT_3 with chainLength=1 reduces to QLS_3.

prf_WT_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WT_7
strategy: Composition
universalScope: true
verified: true
comment: Proof of flat chain basis size formula.

prf_CC_PINS
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CC_PINS
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of CC_PINS: carry-constraint site-pinning map follows from ring carry propagation rule.

prf_CC_COST_SITE
type: NamedIndividual
type: ComputationCertificate
atWittLevel: W8
provesIdentity: CC_COST_SITE
strategy: Computation
verified: true
comment: Computation certificate for CC_COST_SITE: exhaustive enumeration at Q0 confirms |pinsSites| = popcount + 1.

prf_jsat_RR
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: jsat_RR
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of jsat_RR: CRT joint satisfiability follows from the Chinese Remainder Theorem.

prf_jsat_CR
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: jsat_CR
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of jsat_CR: carry-residue joint satisfiability follows from the carry stopping rule and residue class intersection.

prf_jsat_CC
type: NamedIndividual
type: ComputationCertificate
atWittLevel: W8
provesIdentity: jsat_CC
strategy: Computation
verified: true
comment: Computation certificate for jsat_CC: bit-pattern exhaustive enumeration at Q0.

prf_D_8
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: D_8
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of D_8: dihedral inverse formula follows from D_5 group presentation and D_7 composition rule.

prf_D_9
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: D_9
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of D_9: reflection order 2 follows from D_7 composition: (r^k s)(r^k s) = identity.

prf_EXP_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EXP_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of EXP_1: monotone carrier characterization follows from site lattice monotonicity.

prf_EXP_2
type: NamedIndividual
type: ComputationCertificate
atWittLevel: W8
provesIdentity: EXP_2
strategy: Computation
verified: true
comment: Computation certificate for EXP_2: principal filter count verified by exhaustive enumeration at Q0.

prf_EXP_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EXP_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of EXP_3: SumType carrier is coproduct by definitional architectural decision.

prf_ST_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: ST_3
strategy: Composition
universalScope: true
verified: true
comment: Proof of ST_3: Euler characteristic additivity for disjoint simplicial complexes.

prf_ST_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: ST_4
strategy: EulerPoincare
universalScope: true
verified: true
comment: Proof of ST_4: Betti number additivity via Mayer-Vietoris for disjoint union.

prf_ST_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: ST_5
strategy: Composition
universalScope: true
verified: true
comment: Proof of ST_5: SumType completeness transfer follows from ST_3 + ST_4 + IT_7d.

prf_TS_8
type: NamedIndividual
type: InductiveProof
baseCase: prf_HA_1
inductiveStep: prf_TS_4
provesIdentity: TS_8
strategy: BitwiseInduction
universalScope: true
validForKAtLeast: 1
verified: true
comment: Inductive proof of TS_8: minimum constraint count for beta_1 = k is 2k + 1. Base case at k=1 requires 3 mutually overlapping constraints.

prf_TS_9
type: NamedIndividual
type: InductiveProof
baseCase: prf_TS_1
inductiveStep: prf_TS_4
provesIdentity: TS_9
strategy: BitwiseInduction
universalScope: true
validForKAtLeast: 1
verified: true
comment: Inductive proof of TS_9: TypeSynthesisResolver terminates in at most 2^n steps. Base case at n=1 has 2 constraint combinations.

prf_TS_10
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TS_10
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of TS_10: ForbiddenSignature membership follows from exhaustive enumeration bound.

prf_WT_8
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WT_8
strategy: Composition
universalScope: true
verified: true
comment: Proof of QT_8: ObstructionChain length bound follows from QLS_2 and spectral sequence convergence.

prf_WT_9
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WT_9
strategy: Simplification
universalScope: true
verified: true
comment: Proof of QT_9: TowerCompletenessResolver termination follows from finite chain length and QT_8 bound.

prf_COEFF_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: COEFF_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of COEFF_1: Z/2Z coefficient ring is definitional, consistent with MN_7.

prf_GO_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GO_1
strategy: EulerPoincare
universalScope: true
verified: true
comment: Proof of GO_1: cohomology killing lemma for GluingObstruction feedback.

prf_GR_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GR_6
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of SR_6: saturation re-entry free count follows from SR_1 monotone accumulation and SC_2.

prf_GR_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GR_7
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of SR_7: saturation degree degradation follows from SC_2 definition and SR_1 monotonicity.

prf_QM_6
type: NamedIndividual
type: InductiveProof
baseCase: prf_QM_6_base
inductiveStep: prf_QM_6_step
provesIdentity: QM_6
strategy: BitwiseInduction
universalScope: true
validForKAtLeast: 0
verified: true
comment: Inductive proof of QM_6: amplitude index set equals monotone pinning trajectories. Base case at Q0 by exhaustive trajectory enumeration; inductive step by site lattice ordering.

prf_CIC_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CIC_1
strategy: Simplification
universalScope: true
verified: true
comment: Proof of CIC_1: TransformCertificate issuance coverage.

prf_CIC_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CIC_2
strategy: GroupPresentation
universalScope: true
verified: true
comment: Proof of CIC_2: IsometryCertificate issuance coverage.

prf_CIC_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CIC_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of CIC_3: InvolutionCertificate issuance coverage.

prf_CIC_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CIC_4
strategy: Simplification
universalScope: true
verified: true
comment: Proof of CIC_4: GroundingCertificate issuance coverage.

prf_CIC_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CIC_5
strategy: Simplification
universalScope: true
verified: true
comment: Proof of CIC_5: GeodesicCertificate issuance coverage.

prf_CIC_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CIC_6
strategy: ProductFormula
universalScope: true
verified: true
comment: Proof of CIC_6: MeasurementCertificate issuance coverage.

prf_CIC_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CIC_7
strategy: ProductFormula
universalScope: true
verified: true
comment: Axiomatic derivation of CIC_7: BornRuleVerification issuance coverage. Follows by composition from corrected OA_4 (product formula chain) and QM_5 (amplitude normalization).

prf_GC_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GC_1
strategy: Simplification
universalScope: true
verified: true
comment: Proof of GC_1: GroundingCertificate issuance coverage.

prf_GR_8
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GR_8
strategy: RingAxiom
universalScope: false
verified: true
comment: Proof of SR_8: session composition tower consistency.

prf_GR_9
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GR_9
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of SR_9: ContextLease site disjointness.

prf_GR_10
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GR_10
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of SR_10: ExecutionPolicy confluence.

prf_MC_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MC_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of MC_1: lease partition conserves total budget.

prf_MC_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MC_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of MC_2: per-lease binding monotonicity.

prf_MC_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MC_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of MC_3: composition freeRank inclusion-exclusion.

prf_MC_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MC_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of MC_4: disjoint-lease composition additivity.

prf_MC_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MC_5
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of MC_5: policy-invariant final binding set.

prf_MC_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MC_6
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of MC_6: full lease coverage implies composed saturation.

prf_MC_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MC_7
strategy: Simplification
universalScope: true
verified: true
comment: Proof of MC_7: distributed O(1) resolution.

prf_MC_8
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MC_8
strategy: RingAxiom
universalScope: true
verified: true
comment: Proof of MC_8: parallelism bound on per-session resolution work.

prf_WC_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WC_1
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of WC_1: Witt coordinate = bit coordinate at p=2.

prf_WC_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WC_2
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of WC_2: Witt sum correction = carry.

prf_WC_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WC_3
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of WC_3: carry recurrence = Witt polynomial recurrence.

prf_WC_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WC_4
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of WC_4: δ-correction = single-level carry.

prf_WC_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WC_5
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of WC_5: LiftObstruction = δ nonvanishing.

prf_WC_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WC_6
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of WC_6: metric discrepancy = Witt defect.

prf_WC_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WC_7
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of WC_7: D_1 = Witt truncation order.

prf_WC_8
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WC_8
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of WC_8: D_3 = Witt-Burnside conjugation.

prf_WC_9
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WC_9
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of WC_9: D_4 = reflection composition.

prf_WC_10
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WC_10
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of WC_10: Frobenius = identity on W_n(F_2).

prf_WC_11
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WC_11
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of WC_11: Verschiebung = doubling.

prf_WC_12
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: WC_12
strategy: BitwiseInduction
universalScope: true
verified: true
comment: Axiomatic derivation of WC_12: δ = squaring defect / 2.

prf_OA_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OA_1
strategy: ProductFormula
universalScope: true
verified: true
comment: Axiomatic derivation of OA_1: Ostrowski product formula at p=2.

prf_OA_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OA_2
strategy: ProductFormula
universalScope: true
verified: true
comment: Axiomatic derivation of OA_2: crossing cost = ln 2.

prf_OA_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OA_3
strategy: ProductFormula
universalScope: true
verified: true
comment: Axiomatic derivation of OA_3: Landauer cost grounding via product formula.

prf_OA_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OA_4
strategy: ProductFormula
universalScope: true
verified: true
comment: Axiomatic derivation of OA_4: Born rule bridge follows from the product formula chain OA_1 (Ostrowski) -> OA_2 (crossing cost) -> OA_3 (Landauer grounding) -> OA_4.

prf_OA_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OA_5
strategy: ProductFormula
universalScope: true
verified: true
comment: Axiomatic derivation of OA_5: entropy per δ-level = crossing cost.

prf_HT_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HT_1
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of HT_1: the constraint nerve satisfies the Kan extension condition.

prf_HT_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HT_2
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of HT_2: path components recover Betti number β₀.

prf_HT_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HT_3
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of HT_3: π₁ factors through HolonomyGroup.

prf_HT_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HT_4
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of HT_4: higher homotopy groups vanish above dimension.

prf_HT_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HT_5
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of HT_5: 1-truncation determines flat/twisted classification.

prf_HT_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HT_6
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of HT_6: trivial k-invariants imply spectral collapse.

prf_HT_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HT_7
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of HT_7: Whitehead product detects lift obstructions.

prf_HT_8
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HT_8
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of HT_8: Hurewicz isomorphism for first non-vanishing group.

prf_psi_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: psi_7
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of ψ_7: KanComplex to PostnikovTower functor.

prf_psi_8
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: psi_8
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of ψ_8: PostnikovTower to HomotopyGroups extraction.

prf_psi_9
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: psi_9
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of ψ_9: HomotopyGroups to KInvariants computation.

prf_HP_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HP_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of HP_1: pipeline composition ψ_7 ∘ ψ_1.

prf_HP_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HP_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of HP_2: homotopy extraction agrees with homology on truncation.

prf_HP_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HP_3
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of HP_3: ψ_9 detects QLS_4 convergence.

prf_HP_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HP_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of HP_4: HomotopyResolver complexity bound.

prf_MD_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MD_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of MD_1: moduli dimension equals basis cardinality.

prf_MD_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MD_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of MD_2: automorphism group is subgroup of D_{2^n}.

prf_MD_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MD_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of MD_3: first-order deformations parameterize tangent space.

prf_MD_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MD_4
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of MD_4: obstruction space equals LiftObstruction.

prf_MD_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MD_5
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of MD_5: FlatType stratum is open (codimension 0).

prf_MD_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MD_6
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of MD_6: TwistedType strata have codimension at least 1.

prf_MD_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MD_7
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of MD_7: versal deformation exists when unobstructed.

prf_MD_8
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MD_8
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of MD_8: completeness-preserving iff obstruction vanishes along path.

prf_MD_9
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MD_9
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of MD_9: tower map site dimension is 1 when unobstructed.

prf_MD_10
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MD_10
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of MD_10: tower map site empty iff twisted at every level.

prf_MR_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MR_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of MR_1: moduli resolver agrees with MorphospaceBoundary.

prf_MR_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MR_2
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of MR_2: stratification record is complete.

prf_MR_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MR_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of MR_3: moduli resolver complexity bound.

prf_MR_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MR_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of MR_4: moduli-morphospace consistency.

prf_CY_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CY_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CY_1: carry generation condition.

prf_CY_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CY_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CY_2: carry propagation condition.

prf_CY_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CY_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CY_3: carry kill condition.

prf_CY_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CY_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CY_4: d_Δ as carry–Hamming discrepancy.

prf_CY_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CY_5
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CY_5: optimal encoding theorem.

prf_CY_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CY_6
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CY_6: site ordering theorem.

prf_CY_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CY_7
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CY_7: carry lookahead via prefix computation.

prf_BM_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: BM_1
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of BM_1: saturation metric definition.

prf_BM_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: BM_2
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of BM_2: Euler characteristic formula.

prf_BM_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: BM_3
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of BM_3: index theorem linking all six metrics.

prf_BM_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: BM_4
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of BM_4: Jacobian vanishes on pinned sites.

prf_BM_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: BM_5
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of BM_5: d_delta equals Witt defect.

prf_BM_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: BM_6
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of BM_6: metric composition tower.

prf_GL_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GL_1
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of GL_1: σ as lower adjoint.

prf_GL_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GL_2
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of GL_2: r as complement of upper adjoint.

prf_GL_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GL_3
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of GL_3: completeness as Galois fixpoint.

prf_GL_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: GL_4
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of GL_4: Galois order reversal.

prf_NV_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: NV_1
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of NV_1: nerve additivity for disjoint domains.

prf_NV_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: NV_2
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of NV_2: Mayer–Vietoris for constraint nerves.

prf_NV_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: NV_3
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of NV_3: Betti number bounded change.

prf_NV_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: NV_4
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of NV_4: accumulation monotonicity.

prf_SD_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SD_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of SD_1: scalar grounding identity.

prf_SD_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SD_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of SD_2: symbol grounding identity.

prf_SD_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SD_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of SD_3: sequence free monoid identity.

prf_SD_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SD_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of SD_4: tuple site additivity.

prf_SD_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SD_5
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of SD_5: graph nerve equality.

prf_SD_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SD_6
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of SD_6: set permutation invariance.

prf_SD_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SD_7
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of SD_7: tree acyclicity constraint.

prf_SD_8
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SD_8
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of SD_8: table functorial decomposition.

prf_DD_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DD_1
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of DD_1: dispatch determinism.

prf_DD_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DD_2
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of DD_2: dispatch coverage.

prf_PI_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PI_1
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of PI_1: inference idempotence.

prf_PI_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PI_2
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of PI_2: inference soundness.

prf_PI_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PI_3
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of PI_3: inference composition.

prf_PI_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PI_4
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of PI_4: inference complexity.

prf_PI_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PI_5
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of PI_5: inference coherence.

prf_PA_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PA_1
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of PA_1: accumulation permutation invariance.

prf_PA_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PA_2
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of PA_2: accumulation monotonicity.

prf_PA_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PA_3
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of PA_3: accumulation soundness.

prf_PA_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PA_4
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of PA_4: accumulation base preservation.

prf_PA_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PA_5
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of PA_5: accumulation identity element.

prf_PL_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PL_1
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of PL_1: lease disjointness.

prf_PL_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PL_2
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of PL_2: lease conservation.

prf_PL_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PL_3
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of PL_3: lease coverage.

prf_PK_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PK_1
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of PK_1: composition validity.

prf_PK_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PK_2
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of PK_2: distributed resolution.

prf_PP_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PP_1
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of PP_1: pipeline unification.

prf_PE_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PE_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PE_1: state initialization.

prf_PE_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PE_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PE_2: resolver dispatch.

prf_PE_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PE_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PE_3: ring address grounding.

prf_PE_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PE_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PE_4: constraint resolution.

prf_PE_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PE_5
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PE_5: consistent accumulation.

prf_PE_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PE_6
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PE_6: coherent extraction.

prf_PE_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PE_7
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PE_7: full pipeline composition.

prf_PM_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PM_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PM_1: phase rotation.

prf_PM_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PM_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PM_2: phase gate check.

prf_PM_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PM_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PM_3: conjugate rollback.

prf_PM_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PM_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PM_4: rollback involution.

prf_PM_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PM_5
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PM_5: epoch saturation preservation.

prf_PM_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PM_6
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PM_6: service window context.

prf_PM_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PM_7
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PM_7: machine determinism.

prf_ER_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: ER_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of ER_1: guard satisfaction.

prf_ER_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: ER_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of ER_2: effect atomicity.

prf_ER_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: ER_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of ER_3: guard purity.

prf_ER_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: ER_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of ER_4: intra-stage commutativity.

prf_EA_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EA_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of EA_1: epoch boundary reset.

prf_EA_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EA_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of EA_2: saturation monotonicity.

prf_EA_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EA_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of EA_3: service window bound.

prf_EA_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EA_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of EA_4: epoch admission exclusivity.

prf_OE_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OE_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OE_1: stage fusion.

prf_OE_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OE_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OE_2: effect commutativity.

prf_OE_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OE_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OE_3: lease parallelism.

prf_OE_4a
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OE_4a
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OE_4a: fusion semantics preservation.

prf_OE_4b
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OE_4b
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OE_4b: commutation outcome preservation.

prf_OE_4c
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OE_4c
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of OE_4c: parallelism coverage preservation.

prf_CS_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CS_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of CS_1: bounded stage cost.

prf_CS_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CS_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of CS_2: additive pipeline cost.

prf_CS_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CS_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of CS_3: bounded rollback cost.

prf_CS_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CS_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of CS_4: constant preflight cost.

prf_CS_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CS_5
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of CS_5: total reduction cost bound.

prf_FA_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FA_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of FA_1: query liveness.

prf_FA_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FA_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of FA_2: no starvation.

prf_FA_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FA_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of FA_3: fair lease allocation.

prf_SW_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SW_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of SW_1: memory boundedness.

prf_SW_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SW_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of SW_2: saturation invariance.

prf_SW_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SW_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of SW_3: eviction releases resources.

prf_SW_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SW_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of SW_4: non-empty window.

prf_LS_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: LS_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of LS_1: suspension preserves pinned state.

prf_LS_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: LS_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of LS_2: expiry releases resources.

prf_LS_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: LS_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of LS_3: checkpoint restore idempotence.

prf_LS_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: LS_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of LS_4: suspend-resume round-trip.

prf_TJ_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TJ_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of TJ_1: AllOrNothing rollback.

prf_TJ_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TJ_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of TJ_2: BestEffort partial commit.

prf_TJ_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: TJ_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of TJ_3: epoch-scoped atomicity.

prf_AP_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AP_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AP_1: saturation monotonicity.

prf_AP_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AP_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AP_2: quality improvement.

prf_AP_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AP_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of AP_3: deferred query liveness.

prf_EC_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EC_1
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of EC_1: phase half-turn convergence.

prf_EC_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EC_2
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of EC_2: conjugate involution.

prf_EC_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EC_3
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of EC_3: pairwise commutator convergence.

prf_EC_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EC_4
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of EC_4: triple associator convergence.

prf_EC_4a
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EC_4a
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of EC_4a: associator monotonicity.

prf_EC_4b
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EC_4b
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of EC_4b: associator finiteness.

prf_EC_4c
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EC_4c
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of EC_4c: vanishing implies associativity.

prf_EC_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: EC_5
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of EC_5: Adams termination.

prf_DA_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DA_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of DA_1: Cayley-Dickson R→C.

prf_DA_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DA_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of DA_2: Cayley-Dickson C→H.

prf_DA_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DA_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of DA_3: Cayley-Dickson H→O.

prf_DA_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DA_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of DA_4: Adams dimension restriction.

prf_DA_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DA_5
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of DA_5: convergence-level algebra correspondence.

prf_DA_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DA_6
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of DA_6: commutator-commutativity equivalence.

prf_DA_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DA_7
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of DA_7: associator-associativity equivalence.

prf_IN_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IN_1
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of IN_1: interaction cost.

prf_IN_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IN_2
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of IN_2: disjoint leases commute.

prf_IN_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IN_3
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of IN_3: shared sites nonzero commutator.

prf_IN_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IN_4
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of IN_4: negotiation convergence.

prf_IN_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IN_5
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of IN_5: commutative subspace selection.

prf_IN_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IN_6
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of IN_6: pairwise outcome space.

prf_IN_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IN_7
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of IN_7: associative subalgebra selection.

prf_IN_8
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IN_8
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of IN_8: nerve Betti coupling bound.

prf_IN_9
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IN_9
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of IN_9: Betti-disagreement associator bound.

prf_AS_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AS_1
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of AS_1: δ-ι-κ non-associativity.

prf_AS_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AS_2
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of AS_2: ι-α-λ non-associativity.

prf_AS_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AS_3
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of AS_3: λ-κ-δ non-associativity.

prf_AS_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: AS_4
strategy: Composition
universalScope: true
verified: true
comment: Axiomatic derivation of AS_4: non-associativity root cause.

prf_MO_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MO_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of MO_1: monoidal unit law.

prf_MO_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MO_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of MO_2: monoidal associativity.

prf_MO_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MO_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of MO_3: certificate composition.

prf_MO_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MO_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of MO_4: saturation monotonicity.

prf_MO_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: MO_5
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of MO_5: residual monotonicity.

prf_OP_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OP_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of OP_1: site additivity.

prf_OP_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OP_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of OP_2: grounding distributivity.

prf_OP_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OP_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of OP_3: d_Δ decomposition.

prf_OP_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OP_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of OP_4: tabular data decomposition.

prf_OP_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: OP_5
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of OP_5: hierarchical data structure.

prf_FX_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FX_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FX_1: pinning site budget decrement.

prf_FX_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FX_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FX_2: unbinding site budget increment.

prf_FX_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FX_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FX_3: phase budget invariance.

prf_FX_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FX_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FX_4: disjoint commutativity.

prf_FX_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FX_5
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FX_5: composite delta additivity.

prf_FX_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FX_6
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FX_6: reversible inverse.

prf_FX_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FX_7
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of FX_7: external shape compliance.

prf_PR_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PR_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PR_1: predicate totality.

prf_PR_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PR_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PR_2: predicate purity.

prf_PR_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PR_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PR_3: deterministic dispatch.

prf_PR_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PR_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PR_4: deterministic match.

prf_PR_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PR_5
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PR_5: typed guard transition.

prf_CG_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CG_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of CG_1: typed entry guard.

prf_CG_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CG_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of CG_2: typed exit guard with effect.

prf_DIS_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DIS_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of DIS_1: exhaustive exclusive table.

prf_DIS_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: DIS_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of DIS_2: deterministic resolver selection.

prf_PAR_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PAR_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of PAR_1: disjoint commutativity.

prf_PAR_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PAR_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of PAR_2: delta additivity.

prf_PAR_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PAR_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of PAR_3: exhaustive partitioning.

prf_PAR_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PAR_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of PAR_4: interleaving invariance.

prf_PAR_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: PAR_5
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of PAR_5: certificate conjunction.

prf_HO_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HO_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of HO_1: content-addressed certification.

prf_HO_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HO_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of HO_2: application certification.

prf_HO_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HO_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of HO_3: composition certification.

prf_HO_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: HO_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of HO_4: saturation equivalence.

prf_STR_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: STR_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of STR_1: epoch termination.

prf_STR_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: STR_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of STR_2: saturation preservation.

prf_STR_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: STR_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of STR_3: finite prefix computability.

prf_STR_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: STR_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of STR_4: unfold seed initialization.

prf_STR_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: STR_5
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of STR_5: continuation chaining.

prf_STR_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: STR_6
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of STR_6: lease expiry budget return.

prf_FLR_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FLR_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FLR_1: coproduct exhaustiveness.

prf_FLR_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FLR_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of FLR_2: total computation guarantee.

prf_FLR_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FLR_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FLR_3: sequential propagation.

prf_FLR_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FLR_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of FLR_4: parallel independence.

prf_FLR_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FLR_5
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of FLR_5: recovery result.

prf_FLR_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: FLR_6
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of FLR_6: conjugate rollback.

prf_LN_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: LN_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of LN_1: exact coverage.

prf_LN_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: LN_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of LN_2: pinning effect.

prf_LN_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: LN_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of LN_3: consumption linearity.

prf_LN_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: LN_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of LN_4: lease budget decrement.

prf_LN_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: LN_5
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of LN_5: lease expiry return.

prf_LN_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: LN_6
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of LN_6: geodesic linearity.

prf_SB_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SB_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of SB_1: constraint superset.

prf_SB_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SB_2
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of SB_2: resolution subset.

prf_SB_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SB_3
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of SB_3: nerve sub-complex.

prf_SB_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SB_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of SB_4: covariance.

prf_SB_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SB_5
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of SB_5: contravariance.

prf_SB_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: SB_6
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of SB_6: lattice depth.

prf_BR_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: BR_1
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of BR_1: strict descent.

prf_BR_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: BR_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of BR_2: depth bound.

prf_BR_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: BR_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of BR_3: termination guarantee.

prf_BR_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: BR_4
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of BR_4: structural measure.

prf_BR_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: BR_5
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of BR_5: base predicate zero.

prf_RG_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: RG_1
strategy: EulerPoincare
universalScope: true
verified: true
comment: Axiomatic derivation of RG_1: nerve-determined working set.

prf_RG_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: RG_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of RG_2: diameter bound.

prf_RG_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: RG_3
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of RG_3: addressable space bound.

prf_RG_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: RG_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of RG_4: working set containment.

prf_IO_1
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IO_1
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of IO_1: ingest type conformance.

prf_IO_2
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IO_2
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of IO_2: emit type conformance.

prf_IO_3
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IO_3
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of IO_3: grounding validity.

prf_IO_4
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IO_4
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of IO_4: projection validity.

prf_IO_5
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: IO_5
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of IO_5: non-vacuous crossing.

prf_CS_6
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CS_6
strategy: Simplification
universalScope: true
verified: true
comment: Axiomatic derivation of CS_6: budget solvency rejection.

prf_CS_7
type: NamedIndividual
type: AxiomaticDerivation
provesIdentity: CS_7
strategy: RingAxiom
universalScope: true
verified: true
comment: Axiomatic derivation of CS_7: unit address computation.

UOR Derivations
type: Ontology
imports: UOR Observables
imports: UOR Operations
imports: UOR Resolvers
imports: UOR Schema
imports: UOR Type System
comment: Computation witnesses recording term rewriting sequences from original terms to their canonical forms.
space: bridge

Derivation
type: Class
comment: A complete term rewriting witness: the full sequence of rewrite steps transforming an original term into its canonical form.
subclass of: Thing

DerivationStep
type: Class
comment: An abstract step in a derivation. Concrete subclasses are RewriteStep (term-level rewriting) and RefinementStep (type-level refinement).
subclass of: Thing

RewriteStep
type: Class
comment: A single rewrite step in a derivation: the application of one rewrite rule to transform a term.
subclass of: DerivationStep

RefinementStep
type: Class
comment: A type-level refinement step: the application of a constraint to narrow a type, pinning additional site coordinates. Complements RewriteStep (term-level) in the derivation hierarchy.
subclass of: DerivationStep

RewriteRule
type: Class
comment: A named rewrite rule that can be applied in a derivation step. Each RewriteRule individual represents a specific algebraic law or normalization strategy used during term rewriting.
subclass of: Thing

TermMetrics
type: Class
comment: Metrics describing the size and complexity of a term.
subclass of: Thing

SynthesisStep
type: Class
comment: A single step in the construction of a SynthesizedType: one constraint added to the synthesis candidate and the resulting change in the constraint nerve's topological signature. Ordered by derivation:stepIndex. Analogous to derivation:RewriteStep in the forward pipeline.
subclass of: Thing

SynthesisCheckpoint
type: Class
comment: A persistent snapshot of a ConstraintSearchState at a specific SynthesisStep, allowing a TypeSynthesisResolver to resume exploration after interruption. Essential at Q1+ scale where exhaustive synthesis is computationally significant.
subclass of: Thing

originalTerm
type: ObjectProperty
type: FunctionalProperty
comment: The term at the start of the derivation, before any rewriting.
domain: Derivation
range: Term

canonicalTerm
type: ObjectProperty
type: FunctionalProperty
comment: The canonical form produced at the end of the derivation.
domain: Derivation
range: Term

result
type: ObjectProperty
type: FunctionalProperty
comment: The datum value obtained by evaluating the canonical term.
domain: Derivation
range: Datum

step
type: ObjectProperty
comment: A rewrite step in this derivation.
domain: Derivation
range: RewriteStep

termMetrics
type: ObjectProperty
type: FunctionalProperty
comment: Metrics for the canonical term produced by this derivation.
domain: Derivation
range: TermMetrics

from
type: ObjectProperty
type: FunctionalProperty
comment: The term before this rewrite step.
domain: RewriteStep
range: Term

to
type: ObjectProperty
type: FunctionalProperty
comment: The term after this rewrite step.
domain: RewriteStep
range: Term

hasRewriteRule
type: ObjectProperty
type: FunctionalProperty
comment: The typed rewrite rule applied in this step. Provides a structured reference to a named RewriteRule individual, complementing the string-valued derivation:rule property.
domain: RewriteStep
range: RewriteRule

groundedIn
type: ObjectProperty
type: FunctionalProperty
comment: The algebraic identity that grounds this rewrite rule. Links a RewriteRule to the op:Identity individual that justifies its application.
domain: RewriteRule
range: Identity

stepCount
type: DatatypeProperty
type: FunctionalProperty
comment: The total number of rewrite steps in this derivation.
domain: TermMetrics
range: nonNegativeInteger

termSize
type: DatatypeProperty
type: FunctionalProperty
comment: The number of nodes in the canonical term's syntax tree.
domain: TermMetrics
range: nonNegativeInteger

previousType
type: ObjectProperty
type: FunctionalProperty
comment: The type before this refinement step was applied.
domain: RefinementStep
range: TypeDefinition

appliedConstraint
type: ObjectProperty
type: FunctionalProperty
comment: The constraint that was applied in this refinement step.
domain: RefinementStep
range: Constraint

refinedType
type: ObjectProperty
type: FunctionalProperty
comment: The type after this refinement step was applied.
domain: RefinementStep
range: TypeDefinition

sitesClosed
type: DatatypeProperty
type: FunctionalProperty
comment: The number of site coordinates pinned by this refinement step.
domain: RefinementStep
range: nonNegativeInteger

stepIndex
type: DatatypeProperty
type: FunctionalProperty
comment: Zero-based sequential index of this step within the synthesis derivation.
domain: SynthesisStep
range: nonNegativeInteger

addedConstraint
type: ObjectProperty
type: FunctionalProperty
comment: The constraint added in this synthesis step.
domain: SynthesisStep
range: Constraint

signatureBefore
type: ObjectProperty
type: FunctionalProperty
comment: The constraint nerve signature before this synthesis step.
domain: SynthesisStep
range: SynthesisSignature

signatureAfter
type: ObjectProperty
type: FunctionalProperty
comment: The constraint nerve signature after this synthesis step.
domain: SynthesisStep
range: SynthesisSignature

checkpointStep
type: ObjectProperty
type: FunctionalProperty
comment: The SynthesisStep at which this checkpoint was taken.
domain: SynthesisCheckpoint
range: SynthesisStep

checkpointState
type: ObjectProperty
type: FunctionalProperty
comment: The ConstraintSearchState snapshot captured by this checkpoint.
domain: SynthesisCheckpoint
range: ConstraintSearchState

towerCheckpoint
type: ObjectProperty
comment: Links a TowerCompletenessResolver to a SynthesisCheckpoint issued at each completed step. Cross-namespace bridge property: domain is resolver:TowerCompletenessResolver.
domain: TowerCompletenessResolver
range: SynthesisCheckpoint

CriticalIdentityRule
type: NamedIndividual
type: RewriteRule
groundedIn: Critical Identity
comment: The rewrite rule applying the critical identity: neg(bnot(x)) → succ(x). Grounded in op:criticalIdentity.

InvolutionRule
type: NamedIndividual
type: RewriteRule
comment: The rewrite rule applying involution cancellation: f(f(x)) → x for any involution f.

AssociativityRule
type: NamedIndividual
type: RewriteRule
comment: The rewrite rule applying associativity to re-bracket nested binary operations.

CommutativityRule
type: NamedIndividual
type: RewriteRule
comment: The rewrite rule applying commutativity to reorder operands of commutative operations.

IdentityElementRule
type: NamedIndividual
type: RewriteRule
comment: The rewrite rule eliminating identity elements: add(x, 0) → x, mul(x, 1) → x, xor(x, 0) → x.

NormalizationRule
type: NamedIndividual
type: RewriteRule
comment: The rewrite rule normalizing compound expressions to canonical ordering (e.g., sorting operands by address).

UOR Computation Traces
type: Ontology
imports: UOR Schema
imports: UOR Operations
imports: UOR Certificates
imports: UOR Observables
comment: Execution traces recording the sequence of kernel operations, intermediate results, and accumulated metrics for a computation.
space: bridge

ComputationTrace
type: Class
comment: A complete record of a kernel computation: the input, output, every operation step, and accumulated metrics.
subclass of: Thing

ComputationStep
type: Class
comment: A single step in a computation trace: one operation applied to produce one output from one or more inputs.
subclass of: Thing

TraceMetrics
type: Class
comment: Summary metrics for a computation trace: total steps, accumulated ring distance, and accumulated Hamming distance.
subclass of: Thing

GeodesicTrace
type: Class
comment: A computation trace that satisfies the dual geodesic condition (GD_1): AR_1-ordered and DC_10-selected. The path of least dissipation through the resolution landscape.
subclass of: ComputationTrace

GeodesicViolation
type: Class
comment: A record of a geodesic condition violation at a specific step of a computation trace. Produced by GeodesicValidator when J_k(step_i) < max_\{free\} J_k(state_i).
subclass of: Thing

MeasurementEvent
type: Class
comment: A specialized computation step recording a single projective collapse of a SuperposedSiteState. Carries pre-collapse entropy and post-collapse Landauer cost (QM_1).
subclass of: ComputationStep

MeasurementOutcome
type: Class
comment: A single outcome of a projective measurement on a SuperposedSiteState, recording the classical site index (outcomeValue) and its Born-rule probability |α_k|² (outcomeProbability). Multiple outcomes form the probability distribution of a measurement.
subclass of: Thing

input
type: ObjectProperty
type: FunctionalProperty
comment: The input datum of this computation.
domain: ComputationTrace
range: Datum

output
type: ObjectProperty
type: FunctionalProperty
comment: The output datum of this computation.
domain: ComputationTrace
range: Datum

step
type: ObjectProperty
comment: A computation step in this trace.
domain: ComputationTrace
range: ComputationStep

monodromy
type: ObjectProperty
type: FunctionalProperty
comment: The monodromy accumulated by this computation: the net dihedral group element produced by the full operation sequence.
domain: ComputationTrace
range: DihedralElement

from
type: ObjectProperty
type: FunctionalProperty
comment: The input datum of this computation step.
domain: ComputationStep
range: Datum

to
type: ObjectProperty
type: FunctionalProperty
comment: The output datum of this computation step.
domain: ComputationStep
range: Datum

operation
type: ObjectProperty
type: FunctionalProperty
comment: The operation applied in this computation step.
domain: ComputationStep
range: Operation

index
type: DatatypeProperty
type: FunctionalProperty
comment: The zero-based sequential index of this step within its trace.
domain: ComputationStep
range: nonNegativeInteger

stepCount
type: DatatypeProperty
type: FunctionalProperty
comment: Total number of computation steps in this trace.
domain: TraceMetrics
range: nonNegativeInteger

totalRingDistance
type: DatatypeProperty
type: FunctionalProperty
comment: Total ring-metric distance accumulated across all steps.
domain: TraceMetrics
range: nonNegativeInteger

totalHammingDistance
type: DatatypeProperty
type: FunctionalProperty
comment: Total Hamming-metric distance accumulated across all steps.
domain: TraceMetrics
range: nonNegativeInteger

certifiedBy
type: ObjectProperty
type: FunctionalProperty
comment: The certificate that attests to the correctness of this computation trace.
domain: ComputationTrace
range: Certificate

residualEntropy
type: ObjectProperty
type: FunctionalProperty
comment: The residual entropy observable remaining after this computation trace, linking to the ThermoObservable taxonomy (TH_9 connection).
domain: ComputationTrace
range: ResidualEntropy

isGeodesic
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this computation trace satisfies the dual geodesic condition (GD_1): AR_1-ordered and DC_10-selected.
domain: ComputationTrace
range: boolean

geodesicCertificate
type: ObjectProperty
type: FunctionalProperty
comment: The GeodesicCertificate attesting that this trace satisfied both GD_1 conditions.
domain: GeodesicTrace
range: GeodesicCertificate

geodesicViolation
type: ObjectProperty
comment: A GeodesicViolation record indicating where the trace deviated from the geodesic condition.
domain: ComputationTrace
range: GeodesicViolation

stepEntropyCost
type: DatatypeProperty
type: FunctionalProperty
comment: The entropy cost of a single computation step. On a geodesic, this equals ln 2 for every step (GD_2).
domain: ComputationStep
range: decimal

cumulativeEntropyCost
type: DatatypeProperty
type: FunctionalProperty
comment: The total entropy cost accumulated across all steps of a trace. On a geodesic, equals freeRank_initial × ln 2 (GD_3).
domain: ComputationTrace
range: decimal

jacobianAtStep
type: DatatypeProperty
type: FunctionalProperty
comment: The Jacobian value J_k at this step, used by the GeodesicValidator to check DC_10 maximality.
domain: ComputationStep
range: decimal

adiabaticallyOrdered
type: DatatypeProperty
type: FunctionalProperty
comment: Whether the step sequence of this trace follows the AR_1 adiabatic ordering (decreasing freeRank × cost-per-site).
domain: ComputationTrace
range: boolean

violationReason
type: DatatypeProperty
type: FunctionalProperty
comment: Human-readable description of why a geodesic violation occurred, citing the step index and the unused higher-J_k option.
domain: GeodesicViolation
range: string

measurementEvent
type: ObjectProperty
comment: A MeasurementEvent step within this computation trace.
domain: ComputationTrace
range: MeasurementEvent

preCollapseEntropy
type: DatatypeProperty
type: FunctionalProperty
comment: The von Neumann entropy S_vN of the SuperposedSiteState before projective collapse.
domain: MeasurementEvent
range: decimal

postCollapseLandauerCost
type: DatatypeProperty
type: FunctionalProperty
comment: The Landauer cost incurred by the projective collapse. Equals preCollapseEntropy at β* = ln 2 (QM_1).
domain: MeasurementEvent
range: decimal

collapseStep
type: DatatypeProperty
type: FunctionalProperty
comment: The step index within the enclosing ComputationTrace at which this projective collapse occurred.
domain: MeasurementEvent
range: nonNegativeInteger

amplitudeVector
type: DatatypeProperty
type: FunctionalProperty
comment: The full pre-collapse amplitude vector of all branches at the time of measurement. Enables Born rule verification (QM_5): P(outcome k) = |α_k|² / Σ|αᵢ|².
domain: MeasurementEvent
range: decimal

isAR1Ordered
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this computation trace has steps ordered by the AR_1 adiabatic metric (decreasing freeRank × cost-per-site). One of the two sub-predicates of isGeodesic (GD_6).
domain: ComputationTrace
range: boolean

isDC10Selected
type: DatatypeProperty
type: FunctionalProperty
comment: Whether each step of this computation trace was selected by the DC_10 Jacobian criterion (maximal J_k among free sites). One of the two sub-predicates of isGeodesic (GD_6).
domain: ComputationTrace
range: boolean

outcomeValue
type: DatatypeProperty
type: FunctionalProperty
comment: The classical site index selected by projective collapse in this measurement outcome.
domain: MeasurementOutcome
range: nonNegativeInteger

outcomeProbability
type: DatatypeProperty
type: FunctionalProperty
comment: The Born-rule probability of this measurement outcome: |α_k|² where α_k is the amplitude of the collapsed site.
domain: MeasurementOutcome
range: decimal

geodesic_Q0
type: NamedIndividual
type: GeodesicTrace
comment: Canonical geodesic trace at quantum level Q0 (n=8). Demonstrates GD_1 through GD_3 at the base level.
adiabaticallyOrdered: true
isGeodesic: true

geodesic_Q1
type: NamedIndividual
type: GeodesicTrace
comment: Canonical geodesic trace at quantum level Q1 (n=16). Demonstrates geodesic scaling from Q0 to Q1.
adiabaticallyOrdered: true
isGeodesic: true

geodesic_Q2
type: NamedIndividual
type: GeodesicTrace
comment: Canonical geodesic trace at quantum level Q2 (n=32). Demonstrates geodesic scaling from Q1 to Q2.
adiabaticallyOrdered: true
isGeodesic: true

geodesic_Q3
type: NamedIndividual
type: GeodesicTrace
comment: Canonical geodesic trace at quantum level Q3 (n=64). Demonstrates geodesic scaling from Q2 to Q3.
adiabaticallyOrdered: true
isGeodesic: true

collapse_equal_superposition
type: NamedIndividual
type: MeasurementEvent
comment: Canonical measurement event: collapse of an equal superposition (|α|² = 0.5). Maximum von Neumann entropy S_vN = ln 2. Maximum Landauer cost per QM_1.
postCollapseLandauerCost: 0.693147
preCollapseEntropy: 0.693147

collapse_biased
type: NamedIndividual
type: MeasurementEvent
comment: Canonical measurement event: collapse of a biased superposition (|α|² = 0.9). Lower entropy than equal superposition. Demonstrates QM_3 bound.
postCollapseLandauerCost: 0.325083
preCollapseEntropy: 0.325083

collapse_classical
type: NamedIndividual
type: MeasurementEvent
comment: Canonical measurement event: collapse of a classical state (|α|² = 1). Zero entropy, zero Landauer cost. Demonstrates QM_4 idempotence.
postCollapseLandauerCost: 0.0
preCollapseEntropy: 0.0

UOR Certificates
type: Ontology
imports: UOR Operations
imports: UOR Proofs
imports: UOR Schema
imports: UOR Type System
comment: Kernel-produced attestation certificates for transforms, isometries, and involutions. Each certificate verifies that a specific structural property holds.
space: bridge

Certificate
type: Class
comment: A kernel-produced attestation. The root class for all certificate types.
subclass of: Thing

TransformCertificate
type: Class
comment: A certificate attesting to the properties of a morphism:Transform. Certifies that the transform maps source to target correctly.
subclass of: Certificate

IsometryCertificate
type: Class
comment: A certificate attesting that a morphism:Isometry preserves metric distances. Certifies the transform is a metric isometry with respect to the specified metric.
subclass of: Certificate

InvolutionCertificate
type: Class
comment: A certificate attesting that an operation is an involution: f(f(x)) = x for all x in R_n.
subclass of: Certificate

CompletenessCertificate
type: Class
comment: A certificate attesting that a type:CompleteType satisfies IT_7d: its constraint nerve has χ = n and all Betti numbers β_k = 0. Issued by the kernel after running the full ψ pipeline on the type's constraint set.
subclass of: Certificate

CompletenessAuditTrail
type: Class
comment: An ordered collection of CompletenessWitness records belonging to a CompletenessCertificate. Provides full provenance of the certification process: every constraint applied, every site closed, in sequence.
subclass of: Thing

GroundingCertificate
type: Class
comment: A certificate attesting that a state:GroundedContext has reached full saturation (σ = 1, freeRank = 0, S = 0, T_ctx = 0) per SC_4. The session-layer dual of CompletenessCertificate.
subclass of: Certificate

GeodesicCertificate
type: Class
comment: A certificate attesting that a trace:GeodesicTrace satisfies both GD_1 conditions: AR_1-ordered and DC_10-selected. Transforms ComputationTrace from descriptive to normative.
subclass of: Certificate

MeasurementCertificate
type: Class
comment: A certificate attesting that a trace:MeasurementEvent respected the von Neumann–Landauer bridge (QM_1): preCollapseEntropy = postCollapseLandauerCost at β* = ln 2.
subclass of: Certificate

GeodesicEvidenceBundle
type: Class
comment: A structured evidence bundle attesting that each sub-predicate of the geodesic condition (GD_6) holds independently: isAR1Ordered and isDC10Selected. Linked from GeodesicCertificate via evidenceBundle.
subclass of: Thing

BornRuleVerification
type: Class
comment: A certificate attesting that a MeasurementEvent outcome probability matches the Born rule: P(outcome k) = |α_k|² (QM_5). Linked from MeasurementCertificate to provide probability distribution verification.
subclass of: Certificate

LiftChainCertificate
type: Class
comment: A kernel-issued certificate attesting that a LiftChain from liftSourceLevel to liftTargetLevel is complete.
subclass of: Certificate

ChainAuditTrail
type: Class
comment: An ordered collection of per-step evidence records for a LiftChainCertificate.
subclass of: Thing

transformType
type: ObjectProperty
type: FunctionalProperty
comment: The type of transform this certificate attests to (e.g., 'isometry', 'embedding', 'action').
domain: TransformCertificate
range: TermExpression

method
type: ObjectProperty
type: FunctionalProperty
comment: The verification method used to produce this certificate (e.g., 'exhaustive_check', 'symbolic_proof', 'sampling').
domain: Certificate
range: ProofStrategy

operation
type: ObjectProperty
type: FunctionalProperty
comment: The operation this certificate applies to.
domain: InvolutionCertificate
range: Operation

verified
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this certificate has been verified by the kernel.
domain: Certificate
range: boolean

wittLength
type: DatatypeProperty
type: FunctionalProperty
comment: The Witt level at which this certificate was produced.
domain: Certificate
range: positiveInteger

timestamp
type: DatatypeProperty
type: FunctionalProperty
comment: The time at which this certificate was issued.
domain: Certificate
range: dateTime

certifies
type: ObjectProperty
type: FunctionalProperty
comment: The resource this certificate attests to. Links a certificate to the observable, transform, or other entity it covers.
domain: Certificate
range: Thing

certifiedType
type: ObjectProperty
type: FunctionalProperty
comment: The TypeDefinition whose completeness this certificate attests. The kernel issues this certificate after running the ψ pipeline on the type's constraint set and confirming IT_7d.
domain: CompletenessCertificate
range: CompleteType

auditTrail
type: ObjectProperty
type: FunctionalProperty
comment: The audit trail attesting the certification provenance. Links a CompletenessCertificate to its ordered sequence of CompletenessWitness records.
domain: CompletenessCertificate
range: CompletenessAuditTrail

witnessCount
type: DatatypeProperty
type: FunctionalProperty
comment: Total number of witness steps in this audit trail.
domain: CompletenessAuditTrail
range: nonNegativeInteger

certifiedGrounding
type: ObjectProperty
type: FunctionalProperty
comment: The GroundedContext whose full saturation this certificate attests. Uses IRI string (cert cannot import state).
domain: GroundingCertificate
range: GroundedContext

groundingWitness
type: ObjectProperty
type: FunctionalProperty
comment: The GroundingWitness providing step-by-step evidence of the saturation process.
domain: GroundingCertificate
range: GroundingWitness

certifiedGeodesic
type: ObjectProperty
type: FunctionalProperty
comment: The GeodesicTrace whose geodesic status this certificate attests. Uses IRI string (cert cannot import trace).
domain: GeodesicCertificate
range: GeodesicTrace

geodesicTrace
type: ObjectProperty
type: FunctionalProperty
comment: The computation trace that this GeodesicCertificate covers. Redundant with certifiedGeodesic but expresses the inverse direction for queryability.
domain: GeodesicCertificate
range: GeodesicTrace

certifiedMeasurement
type: ObjectProperty
type: FunctionalProperty
comment: The MeasurementEvent whose QM_1 compliance this certificate attests. Uses IRI string (cert cannot import trace).
domain: MeasurementCertificate
range: MeasurementEvent

vonNeumannEntropy
type: DatatypeProperty
type: FunctionalProperty
comment: The von Neumann entropy S_vN of the pre-measurement SuperposedSiteState, recorded by this certificate.
domain: MeasurementCertificate
range: decimal

landauerCost
type: DatatypeProperty
type: FunctionalProperty
comment: The Landauer cost incurred by the projective collapse, recorded by this certificate. Equals vonNeumannEntropy at β* = ln 2 per QM_1.
domain: MeasurementCertificate
range: decimal

evidenceBundle
type: ObjectProperty
type: FunctionalProperty
comment: The GeodesicEvidenceBundle attesting to the decomposed sub-predicates (isAR1Ordered, isDC10Selected) of this GeodesicCertificate's geodesic claim.
domain: GeodesicCertificate
range: GeodesicEvidenceBundle

bornRuleVerified
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this BornRuleVerification certificate confirms that all outcome probabilities match the Born rule (QM_5): P(k) = |α_k|² for every site k.
domain: BornRuleVerification
range: boolean

certifiedChain
type: ObjectProperty
type: FunctionalProperty
comment: The LiftChain this certificate attests to.
domain: LiftChainCertificate
range: LiftChain

chainAuditTrail
type: ObjectProperty
type: FunctionalProperty
comment: The ordered per-step evidence for this certificate.
domain: LiftChainCertificate
range: ChainAuditTrail

targetLevel
type: ObjectProperty
type: FunctionalProperty
comment: The quantum level Q_k at which the certificate was issued.
domain: LiftChainCertificate
range: WittLevel

sourceLevel
type: ObjectProperty
type: FunctionalProperty
comment: The quantum level Q_j from which the tower was started.
domain: LiftChainCertificate
range: WittLevel

chainStepCount
type: DatatypeProperty
type: FunctionalProperty
comment: Number of lift steps in this ChainAuditTrail. Must equal chainLength of the certified LiftChain. Distinct from witnessCount (domain-locked to CompletenessAuditTrail).
domain: ChainAuditTrail
range: nonNegativeInteger

isAR1Ordered
type: DatatypeProperty
type: FunctionalProperty
comment: True iff the linked GeodesicTrace is ordered by the AR_1 canonical rewriting rule (smallest lexicographic representative first).
domain: GeodesicEvidenceBundle
range: boolean

isDC10Selected
type: DatatypeProperty
type: FunctionalProperty
comment: True iff each constraint in the trace was selected by the DC_10 Jacobian oracle at the step where it was applied.
domain: GeodesicEvidenceBundle
range: boolean

UOR Transforms and Morphisms
type: Ontology
imports: UOR Schema
imports: UOR Type System
imports: UOR Operations
imports: UOR Observables
imports: UOR Partitions
imports: UOR Computation Traces
imports: UOR Homology
imports: UOR Content Addressing
imports: UOR Derivations
imports: UOR Certificates
comment: Runtime abstractions for maps between UOR objects: transforms, isometries, embeddings, and group actions. The foundation provides the vocabulary; Prism writes the sentences.
space: user

Transform
type: Class
disjoint with: CompositionLaw
comment: A map between UOR objects. The root abstraction: source, target, and optionally what structure (if any) is preserved. This is what cert:TransformCertificate certifies.
subclass of: Thing

Isometry
type: Class
disjoint with: Composition
comment: A transform that preserves metric structure with respect to a specified metric. In UOR, isometry is metric-relative: neg is a ring isometry, bnot is a Hamming isometry. A transform can be an isometry with respect to one metric but not the other. This is what cert:IsometryCertificate certifies.
subclass of: Transform

Embedding
type: Class
disjoint with: Composition
comment: An injective, structure-preserving transform across quantum levels. The canonical instance is the level embedding ι : R_n → R_{n'} (n < n'), which preserves addition, multiplication, and content addressing.
subclass of: Transform

Action
type: Class
comment: The mechanism by which a group applies transforms systematically to a set. Each group element induces a transform of the set. The dihedral action on type space is an action by isometries — every element of D_{2^n} produces an isometric transform of 𝒯_n.
subclass of: Thing

Composition
type: Class
disjoint with: Isometry
disjoint with: Embedding
disjoint with: Identity
comment: A transform formed by composing two or more transforms sequentially. The categorical composition operation that turns transforms into a category.
subclass of: Transform

Identity
type: Class
disjoint with: Composition
comment: The identity transform on a type: maps every element to itself. The categorical identity morphism.
subclass of: Transform

CompositionLaw
type: Class
disjoint with: Transform
comment: A law governing how operations compose. Records whether the composition is associative, commutative, and what the result operation is. The critical composition law (neg ∘ bnot = succ) is the foundational instance.
subclass of: Thing

GroundingMap
type: Class
disjoint with: ProjectionMap
comment: A Transform mapping a surface symbol (any schema:Literal) to its ring datum (a schema:Datum) via the content-addressing bijection. Typed, derivation-witnessed, constraint-preserving map from surface to coordinate. Applies identically across NLP tokens, ARC-AGI grid cells, MIDI notes, pixels, sensor readings, and logical propositions.
subclass of: Transform

ProjectionMap
type: Class
disjoint with: GroundingMap
comment: A Transform mapping a resolved partition:Partition (address neighbourhood) to an ordered sequence of surface symbols. Output ordering determined by the active state:Frame — the same constraint frame that decomposed the input symbol sequence.
subclass of: Transform

GroundingCertificate
type: Class
comment: A certificate attesting that a specific grounding round-trip (P∘Π∘G) satisfied the shared-frame condition and landed in the type-equivalent neighbourhood of the source symbol. Witnesses the Surface Symmetry Theorem (op:surfaceSymmetry) for one specific symbol instance.
subclass of: Certificate

TopologicalDelta
type: Class
comment: A topological delta: records changes in topological invariants (Betti numbers, Euler characteristic, nerve structure) before and after a morphism.
subclass of: Thing

ComputationDatum
type: Class
comment: A datum whose ring value is the content address of a cert:TransformCertificate. Represents a certified computation as a first-class value within the ring.
subclass of: Datum

ApplicationMorphism
type: Class
comment: A transform that applies a ComputationDatum to an input datum, producing an output datum. The output inherits the certificate of the ComputationDatum.
subclass of: Transform

PartialApplication
type: Class
comment: A ComputationDatum formed by fixing some but not all inputs of a multi-argument transform.
subclass of: ComputationDatum

TransformComposition
type: Class
comment: A ComputationDatum representing the composition f ∘ g of two ComputationDatums. Certified iff both components are certified and range(g) = domain(f).
subclass of: ComputationDatum

Witness
type: Class
comment: Abstract supertype for one specific input/output pair witnessing a Transform.
subclass of: Thing

GroundingWitness
type: Class
disjoint with: ProjectionWitness
comment: One specific surface symbol mapped to one specific grounded address by some GroundingMap.
subclass of: Witness

ProjectionWitness
type: Class
disjoint with: GroundingWitness
comment: One specific input partition mapped to one specific surface symbol sequence by some ProjectionMap.
subclass of: Witness

SymbolSequence
type: Class
comment: An ordered sequence of surface symbols. Elements are reified as SequenceElement individuals with explicit position indices.
subclass of: Thing

SequenceElement
type: Class
comment: One position in a SymbolSequence: a reified (value, index) pair.
subclass of: Thing

source
type: ObjectProperty
type: FunctionalProperty
comment: The domain of the transform.
domain: Transform
range: Thing

target
type: ObjectProperty
type: FunctionalProperty
comment: The codomain of the transform.
domain: Transform
range: Thing

preserves
type: ObjectProperty
comment: The structure preserved by this transform (if any). E.g., a ring homomorphism preserves addition and multiplication.
domain: Transform
range: Thing

preservesMetric
type: ObjectProperty
comment: The specific metric this isometry preserves. Points to observable:RingMetric or observable:HammingMetric. A transform that preserves both is an isometry of the full UOR geometry. A transform that preserves one but not the other has nontrivial curvature — observable:CurvatureObservable measures this gap.
domain: Isometry
range: MetricObservable

sourceQuantum
type: DatatypeProperty
type: FunctionalProperty
comment: The quantum level n of the source ring for an embedding.
domain: Embedding
range: positiveInteger

targetQuantum
type: DatatypeProperty
type: FunctionalProperty
comment: The quantum level n' of the target ring for an embedding. Must satisfy n' > n (embeddings go to larger rings).
domain: Embedding
range: positiveInteger

group
type: ObjectProperty
type: FunctionalProperty
comment: The group acting in this group action.
domain: Action
range: Group

actingOn
type: ObjectProperty
type: FunctionalProperty
comment: The set being acted upon by this group action.
domain: Action
range: Thing

actionIsometry
type: DatatypeProperty
type: FunctionalProperty
comment: Whether every transform induced by this action is an isometry. True for the dihedral action on 𝒯_n (Frame Theorem).
domain: Action
range: boolean

trace
type: ObjectProperty
type: FunctionalProperty
comment: The computation trace that realized this transform at runtime. A Transform is an abstraction; a trace is the kernel's record of how it was executed via concrete operations.
domain: Transform
range: ComputationTrace

composesWith
type: ObjectProperty
comment: A transform that this transform can be composed with. The target of this transform must match the source of the composed transform.
domain: Transform
range: Transform

compositionResult
type: ObjectProperty
type: FunctionalProperty
comment: The transform that results from this composition.
domain: Composition
range: Transform

compositionComponents
type: ObjectProperty
comment: A component transform of this composition.
domain: Composition
range: Transform

identityOn
type: ObjectProperty
type: FunctionalProperty
comment: The type on which this identity transform acts.
domain: Identity
range: TypeDefinition

compositionOrder
type: DatatypeProperty
type: FunctionalProperty
comment: The number of component transforms in a composition.
range: nonNegativeInteger

isAssociative
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this composition law is associative.
domain: CompositionLaw
range: boolean

isCommutative
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this composition law is commutative.
domain: CompositionLaw
range: boolean

lawComponents
type: ObjectProperty
comment: An operation that is a component of this composition law.
domain: CompositionLaw
range: Operation

lawResult
type: ObjectProperty
type: FunctionalProperty
comment: The operation that results from this composition law.
domain: CompositionLaw
range: Operation

preservedInvariant
type: ObjectProperty
type: FunctionalProperty
comment: The identity preserved by this transform (reference to the op:Identity that the transform commutes with).
domain: Transform
range: Identity

addressCoherence
type: ObjectProperty
type: FunctionalProperty
comment: Certificate that this embedding's addressing diagram commutes: glyph ∘ ι ∘ addresses is well-defined and injective.
domain: Embedding
range: Identity

bettisBefore
type: ObjectProperty
type: FunctionalProperty
comment: Betti numbers before the morphism.
domain: TopologicalDelta
range: BettiNumber

bettisAfter
type: ObjectProperty
type: FunctionalProperty
comment: Betti numbers after the morphism.
domain: TopologicalDelta
range: BettiNumber

eulerBefore
type: DatatypeProperty
type: FunctionalProperty
comment: Euler characteristic before the morphism.
domain: TopologicalDelta
range: integer

eulerAfter
type: DatatypeProperty
type: FunctionalProperty
comment: Euler characteristic after the morphism.
domain: TopologicalDelta
range: integer

nerveBefore
type: ObjectProperty
type: FunctionalProperty
comment: Constraint nerve (simplicial complex) before the morphism.
domain: TopologicalDelta
range: SimplicialComplex

nerveAfter
type: ObjectProperty
type: FunctionalProperty
comment: Constraint nerve (simplicial complex) after the morphism.
domain: TopologicalDelta
range: SimplicialComplex

surfaceSymbol
type: ObjectProperty
type: FunctionalProperty
comment: The surface symbol that is the source of this grounding witness.
domain: GroundingWitness
range: SurfaceSymbol

groundedAddress
type: ObjectProperty
type: FunctionalProperty
comment: The resolved ring address that is the target of this grounding witness.
domain: GroundingWitness
range: Element

groundingDerivation
type: ObjectProperty
type: FunctionalProperty
comment: The derivation witnessing the content-addressing computation that produced the grounded address from the surface symbol.
domain: GroundingMap
range: Derivation

symbolConstraints
type: ObjectProperty
comment: A typed attribute preserved by this grounding. Non-functional: one assertion per active constraint axis (vertical, horizontal, diagonal).
domain: GroundingMap
range: Constraint

projectionFrame
type: ObjectProperty
type: FunctionalProperty
comment: The active frame — shared with the grounding that produced the query. The shared-frame condition (Surface Symmetry Theorem) requires G and P to reference the same frame.
domain: ProjectionMap
range: Frame

projectionSource
type: ObjectProperty
type: FunctionalProperty
comment: The resolved partition (address neighbourhood) that this projection witness projects back to surface symbols.
domain: ProjectionWitness
range: Partition

projectionOrder
type: ObjectProperty
type: FunctionalProperty
comment: Ordering constraint determining the output symbol sequence. Domain-specific: syntactic position (NLP), row-major scan (ARC), temporal sequence (music).
domain: ProjectionMap
range: CompositeConstraint

roundTripCoherence
type: DatatypeProperty
type: FunctionalProperty
comment: Completeness criterion: does projecting the grounded source address recover a symbol in the same type class as the input? True iff the shared-frame condition holds.
domain: ProjectionMap
range: boolean

groundingCertMap
type: ObjectProperty
type: FunctionalProperty
comment: The GroundingMap used in this certified round-trip.
domain: GroundingCertificate
range: GroundingMap

groundingCertProjection
type: ObjectProperty
type: FunctionalProperty
comment: The ProjectionMap used in this certified round-trip.
domain: GroundingCertificate
range: ProjectionMap

groundingCertSourceSymbol
type: ObjectProperty
type: FunctionalProperty
comment: The surface symbol that entered the grounding boundary.
domain: GroundingCertificate
range: Literal

groundingCertAddress
type: ObjectProperty
type: FunctionalProperty
comment: The ring address the symbol was grounded to.
domain: GroundingCertificate
range: Element

referencedCertificate
type: ObjectProperty
type: FunctionalProperty
comment: The certificate this computation datum encodes.
domain: ComputationDatum
range: TransformCertificate

computationAddress
type: ObjectProperty
type: FunctionalProperty
comment: The content address of the referenced certificate.
domain: ComputationDatum
range: Element

applicationTarget
type: ObjectProperty
type: FunctionalProperty
comment: The computation being applied.
domain: ApplicationMorphism
range: ComputationDatum

applicationInput
type: ObjectProperty
type: FunctionalProperty
comment: The input datum to the application.
domain: ApplicationMorphism
range: Datum

partialBase
type: ObjectProperty
type: FunctionalProperty
comment: The base computation being partially applied.
domain: PartialApplication
range: ComputationDatum

boundArguments
type: ObjectProperty
comment: The arguments already bound.
domain: PartialApplication
range: Datum

compositionLeft
type: ObjectProperty
type: FunctionalProperty
comment: The outer function f in f ∘ g.
domain: TransformComposition
range: ComputationDatum

compositionRight
type: ObjectProperty
type: FunctionalProperty
comment: The inner function g in f ∘ g.
domain: TransformComposition
range: ComputationDatum

remainingArity
type: DatatypeProperty
type: FunctionalProperty
comment: Number of unbound arguments remaining.
domain: PartialApplication
range: positiveInteger

inputClass
type: ObjectProperty
type: FunctionalProperty
comment: The OWL class of inputs this transform accepts. Uses OWL2 punning so the value is a class IRI treated as an individual at this position.
domain: Transform
range: Class

outputClass
type: ObjectProperty
type: FunctionalProperty
comment: The OWL class of outputs this transform produces. Uses OWL2 punning.
domain: Transform
range: Class

outputElementClass
type: ObjectProperty
type: FunctionalProperty
comment: When outputClass is a sequence type, the OWL class of individual sequence elements. Uses OWL2 punning.
domain: ProjectionMap
range: Class

hasWitness
type: ObjectProperty
comment: Zero or more witness pairs documenting specific input/output bindings of this transform.
domain: Transform
range: Witness

hasElement
type: ObjectProperty
comment: Membership of a SymbolSequence. The sequence is reconstructed by sorting elements by elementIndex.
domain: SymbolSequence
range: SequenceElement

elementValue
type: ObjectProperty
type: FunctionalProperty
comment: The surface symbol value of this sequence element.
domain: SequenceElement
range: SurfaceSymbol

elementIndex
type: DatatypeProperty
type: FunctionalProperty
comment: The zero-based position of this element in the sequence.
domain: SequenceElement
range: nonNegativeInteger

projectionOutput
type: ObjectProperty
type: FunctionalProperty
comment: The single SymbolSequence produced by this projection witness.
domain: ProjectionWitness
range: SymbolSequence

criticalComposition
type: NamedIndividual
type: CompositionLaw
isAssociative: false
isCommutative: false
lawComponents: neg
lawComponents: bnot
lawResult: succ
comment: The critical composition law: neg ∘ bnot = succ. This is the operational form of the critical identity theorem. The composition of the two involutions (neg, bnot) yields the successor operation. Non-associative and non-commutative.

IntegerGroundingMap
type: NamedIndividual
type: GroundingMap
inputClass: Literal
outputClass: Element
comment: Grounds integer surface symbols to ring addresses.

Utf8GroundingMap
type: NamedIndividual
type: GroundingMap
inputClass: HostStringLiteral
outputClass: Element
comment: Grounds UTF-8 host strings to ring addresses.

JsonGroundingMap
type: NamedIndividual
type: GroundingMap
inputClass: HostStringLiteral
outputClass: Element
comment: Grounds JSON host strings to ring addresses.

IntegerProjectionMap
type: NamedIndividual
type: ProjectionMap
inputClass: Partition
outputClass: SymbolSequence
outputElementClass: Literal
comment: Projects partitions to integer symbol sequences.

Utf8ProjectionMap
type: NamedIndividual
type: ProjectionMap
inputClass: Partition
outputClass: SymbolSequence
outputElementClass: HostStringLiteral
comment: Projects partitions to UTF-8 symbol sequences.

JsonProjectionMap
type: NamedIndividual
type: ProjectionMap
inputClass: Partition
outputClass: SymbolSequence
outputElementClass: HostStringLiteral
comment: Projects partitions to JSON symbol sequences.

UOR State
type: Ontology
imports: UOR Content Addressing
imports: UOR Schema
imports: UOR Type System
imports: UOR Partitions
imports: UOR Computation Traces
imports: UOR Transforms and Morphisms
imports: UOR Certificates
comment: Parameterized address spaces, context management, binding lifecycle, and state transitions. The user-space overlay onto the kernel's read-only ring substrate.
space: user

Context
type: Class
disjoint with: Binding
disjoint with: Frame
disjoint with: Transition
comment: A bounded set of populated UOR addresses. The parameter space for a resolution cycle. Contexts hold bindings that map addresses to datum values.
subclass of: Thing

Binding
type: Class
disjoint with: Context
disjoint with: Frame
disjoint with: Transition
comment: The association of a datum value with an address in a context. The write primitive: creating a binding populates an address.
subclass of: Thing

Frame
type: Class
disjoint with: Context
disjoint with: Binding
disjoint with: Transition
comment: The visibility boundary determining which bindings are in scope for a given resolution. A frame is a view into a context: it selects which bindings the resolver sees.
subclass of: Thing

Transition
type: Class
disjoint with: Context
disjoint with: Binding
disjoint with: Frame
comment: A state change: the transformation of one context into another through binding or unbinding. The sequence of transitions is the application's computation history.
subclass of: Thing

SessionBoundaryType
type: Class
comment: A typed controlled vocabulary for session boundary reasons. Each individual names a specific reason a context-reset boundary was triggered during a multi-turn session.
subclass of: Thing

Session
type: Class
comment: A bounded sequence of RelationQuery/response pairs sharing a common state:Context. Sessions are the unit of coherent multi-turn reasoning in Prism.
subclass of: Thing

BindingAccumulator
type: Class
comment: The mutable accumulator that appends state:Binding instances to a state:Context as each RelationQuery resolves. Tracks monotonic reduction of aggregate free site space.
subclass of: Thing

SessionBoundary
type: Class
comment: Marks a context-reset event within a session stream. Records why the context was reset and provides a clean state:Context for subsequent queries.
subclass of: Thing

GroundedContext
type: Class
comment: A context that has reached full saturation: σ = 1, freeRank = 0, S = 0, T_ctx = 0 (SC_4). The ground state of the type system. All subsequent queries resolve in O(1) via SC_5.
subclass of: Context

GroundingWitness
type: Class
comment: Step-by-step evidence of the saturation process: records which bindings were applied, in what order, to reach full saturation.
subclass of: Thing

DomainGroundingRecord
type: Class
comment: An informational/monitoring record tracking the saturation progress of a specific domain within a context. Carries no formal authority — purely observational.
subclass of: Thing

GroundingPhase
type: Class
comment: A typed controlled vocabulary for the three phases of context saturation: Open (σ = 0), PartialGrounding (0 < σ < 1), and FullGrounding (σ = 1).
subclass of: Thing

SharedContext
type: Class
comment: A Context visible to more than one Session simultaneously. Holds a set of ContextLease instances that partition its site coordinates among active sessions. Lease disjointness (SR_9) prevents concurrent write conflicts.
subclass of: Context

ContextLease
type: Class
disjoint with: Context
disjoint with: Binding
disjoint with: Frame
disjoint with: Transition
comment: A bounded, exclusive claim on a set of site coordinates within a SharedContext, held by exactly one Session. When the session closes or hits a SessionBoundary, the lease is released and its sites become available for re-leasing.
subclass of: Thing

SessionComposition
type: Class
comment: Records that a Session was formed by merging the binding sets of two or more predecessor sessions. Valid only if all predecessor binding sets pass the cross-session consistency check (SR_8). An invalid composition attempt produces a ContradictionBoundary on the target session.
subclass of: Thing

address
type: ObjectProperty
type: FunctionalProperty
comment: The UOR address being bound in this binding.
domain: Binding
range: Element

content
type: ObjectProperty
type: FunctionalProperty
comment: The datum value bound to the address in this binding.
domain: Binding
range: Datum

boundType
type: ObjectProperty
comment: The type under which this binding's datum is resolved.
domain: Binding
range: TypeDefinition

timestamp
type: DatatypeProperty
type: FunctionalProperty
comment: The time at which this binding was created.
domain: Binding
range: dateTime

binding
type: ObjectProperty
comment: A binding held in this context.
domain: Context
range: Binding

capacity
type: DatatypeProperty
type: FunctionalProperty
comment: The maximum number of bindings this context can hold.
domain: Context
range: positiveInteger

contentAddress
type: DatatypeProperty
type: FunctionalProperty
comment: The content-derived address of this context, uniquely identifying its current state in the UOR address space.
domain: Context
range: string

wittLength
type: DatatypeProperty
type: FunctionalProperty
comment: The Witt level of this context's address space.
domain: Context
range: positiveInteger

activeBindings
type: ObjectProperty
comment: The bindings currently in scope for this frame.
domain: Frame
range: Binding

context
type: ObjectProperty
type: FunctionalProperty
comment: The context this frame is a view of.
domain: Frame
range: Context

constraint
type: ObjectProperty
type: FunctionalProperty
comment: The type:Constraint determining which bindings from the context are visible in this frame. The resolver applies this constraint to filter the context's binding set, producing the frame's active bindings. An absent constraint means all bindings are visible.
domain: Frame
range: Constraint

from
type: ObjectProperty
type: FunctionalProperty
comment: The context before this transition.
domain: Transition
range: Context

to
type: ObjectProperty
type: FunctionalProperty
comment: The context after this transition.
domain: Transition
range: Context

addedBindings
type: ObjectProperty
comment: Bindings added to the context in this transition.
domain: Transition
range: Binding

removedBindings
type: ObjectProperty
comment: Bindings removed from the context in this transition.
domain: Transition
range: Binding

trace
type: ObjectProperty
type: FunctionalProperty
comment: The computation trace recording the kernel operations that effected this state transition.
domain: Transition
range: ComputationTrace

topologicalSnapshot
type: ObjectProperty
type: FunctionalProperty
comment: A snapshot of topological invariants at this transition point.
domain: Transition
range: TopologicalDelta

sessionBindings
type: ObjectProperty
type: FunctionalProperty
comment: The shared context holding all bindings accumulated across the queries in this session.
domain: Session
range: Context

sessionQueries
type: DatatypeProperty
type: FunctionalProperty
comment: The number of RelationQuery evaluations completed in this session.
domain: Session
range: nonNegativeInteger

aggregateSiteDeficit
type: ObjectProperty
type: FunctionalProperty
comment: The aggregate FreeRank deficit across all accumulated bindings: the total remaining free sites that have not yet been closed by resolution. Decreases monotonically as the session progresses.
domain: BindingAccumulator
range: FreeRank

accumulatedBindings
type: ObjectProperty
comment: A binding accumulated by this accumulator from a resolved RelationQuery.
domain: BindingAccumulator
range: Binding

boundaryReason
type: DatatypeProperty
type: FunctionalProperty
comment: A human-readable description of why this session boundary was triggered.
domain: SessionBoundary
range: string

boundaryType
type: ObjectProperty
type: FunctionalProperty
comment: The typed reason category for this session boundary.
domain: SessionBoundary
range: SessionBoundaryType

priorContext
type: ObjectProperty
type: FunctionalProperty
comment: The state:Context that was active before this boundary reset.
domain: SessionBoundary
range: Context

freshContext
type: ObjectProperty
type: FunctionalProperty
comment: The clean state:Context produced after this boundary reset, ready for subsequent queries.
domain: SessionBoundary
range: Context

groundingDegree
type: DatatypeProperty
type: FunctionalProperty
comment: The saturation degree σ ∈ \[0, 1\] of this context. Defined by SC_2: σ = (n − freeRank) / n.
domain: Context
range: decimal

contextTemperature
type: DatatypeProperty
type: FunctionalProperty
comment: The context temperature T_ctx ∈ \[0, ln 2\]. Defined by SC_1: T_ctx = freeRank × ln 2 / n. At σ = 1, T_ctx = 0.
domain: Context
range: decimal

isGrounded
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this context has reached full saturation (σ = 1). Equivalent to freeRank = 0, S = 0, T_ctx = 0 per SC_4.
domain: Context
range: boolean

groundingPhase
type: ObjectProperty
type: FunctionalProperty
comment: The current saturation phase of this context: Open, PartialGrounding, or FullGrounding.
domain: Context
range: GroundingPhase

groundingCertificate
type: ObjectProperty
type: FunctionalProperty
comment: The GroundingCertificate attesting that this context has reached full saturation.
domain: GroundedContext
range: GroundingCertificate

witnessBinding
type: ObjectProperty
comment: A binding that contributed to the saturation process, recorded in this GroundingWitness.
domain: GroundingWitness
range: Binding

witnessStep
type: DatatypeProperty
type: FunctionalProperty
comment: The step index at which a particular binding was applied during the saturation process.
domain: GroundingWitness
range: nonNegativeInteger

residualFreeCount
type: DatatypeProperty
type: FunctionalProperty
comment: The number of free (unbound) sites remaining in this context. At saturation, residualFreeCount = 0.
domain: Context
range: nonNegativeInteger

groundedContext
type: ObjectProperty
type: FunctionalProperty
comment: The GroundedContext that this DomainGroundingRecord monitors.
domain: DomainGroundingRecord
range: GroundedContext

groundedDomain
type: ObjectProperty
type: FunctionalProperty
comment: The domain within the context being tracked by this DomainGroundingRecord.
domain: DomainGroundingRecord
range: TypeDefinition

domainFreeCount
type: DatatypeProperty
type: FunctionalProperty
comment: The number of free sites remaining in the specific domain tracked by this DomainGroundingRecord.
domain: DomainGroundingRecord
range: nonNegativeInteger

leasedSites
type: ObjectProperty
type: FunctionalProperty
comment: The subset of sites claimed by this lease. Must be disjoint from all other active leases on the same SharedContext (SR_9).
domain: ContextLease
range: FreeRank

leaseHolder
type: ObjectProperty
type: FunctionalProperty
comment: The Session that holds this lease.
domain: ContextLease
range: Session

leaseSet
type: ObjectProperty
comment: A currently active ContextLease on this SharedContext.
domain: SharedContext
range: ContextLease

composedFrom
type: ObjectProperty
comment: A predecessor session contributing bindings to this composition. Non-functional: one composition may merge two or more sessions.
domain: SessionComposition
range: Session

compositionCompatible
type: DatatypeProperty
type: FunctionalProperty
comment: Whether all predecessor binding sets passed the SR_8 consistency check. If false, the composition is invalid and must not be used as a session context.
domain: SessionComposition
range: boolean

compositionResult
type: ObjectProperty
type: FunctionalProperty
comment: The merged Context produced by a valid composition. Only present when compositionCompatible = true.
domain: SessionComposition
range: Context

towerConsistencyVerified
type: DatatypeProperty
type: FunctionalProperty
comment: Whether the LiftChain tower consistency check (SR_8 parametric extension) was performed across all Q_0 through Q_k levels. Required for compositions involving sessions at Q_1 or higher.
domain: SessionComposition
range: boolean

ExplicitReset
type: NamedIndividual
type: SessionBoundaryType
comment: The caller explicitly requested a context reset. All accumulated bindings are discarded.

ConvergenceBoundary
type: NamedIndividual
type: SessionBoundaryType
comment: The session resolver determined that no further queries can reduce the aggregate site deficit.

ContradictionBoundary
type: NamedIndividual
type: SessionBoundaryType
comment: A new query produced a type contradiction with an accumulated binding. Context must reset before resolution can continue.

Open
type: NamedIndividual
type: GroundingPhase
comment: The context has σ = 0: no bindings accumulated, all sites are free. The initial phase of every session.

PartialGrounding
type: NamedIndividual
type: GroundingPhase
comment: The context has 0 < σ < 1: some sites are pinned by accumulated bindings, but free sites remain. The accumulation phase.

FullGrounding
type: NamedIndividual
type: GroundingPhase
comment: The context has σ = 1: all sites are pinned, freeRank = 0. The ground state. All subsequent queries resolve in O(1) via SC_5.

ground_state
type: NamedIndividual
type: GroundedContext
comment: The canonical ground-state witness: a GroundedContext at σ = 1, freeRank = 0, T_ctx = 0, S = 0 (SC_4). Demonstrates that full saturation is achievable and O(1) resolution (SC_5) is realized.
contextTemperature: 0.0
groundingDegree: 1.0
groundingPhase: FullGrounding
isGrounded: true
residualFreeCount: 0

UOR Euler Reduction
type: Ontology
imports: UOR Operations
imports: UOR State
imports: UOR Partitions
imports: UOR Resolvers
imports: UOR Transforms and Morphisms
imports: UOR Observables
imports: UOR Predicates and Dispatch
imports: UOR Effect Algebra
imports: UOR Schema
imports: UOR Content Addressing
comment: Sequential composition of ψ-maps into a parameterized reduction ψ = ψ_9 ∘ … ∘ ψ_1. Defines stages, phase gates, rollback, and epochs.
space: kernel

EulerReduction
type: Class
comment: The composite endofunctor ψ = ψ_9 ∘ … ∘ ψ_1, parameterized by Ω = e^{iπ/6}.
subclass of: Thing

PhaseRotationScheduler
type: Class
comment: Schedule Ω⁰, Ω¹, …, Ω⁵ assigning a phase angle to each stage of the reduction.
subclass of: Thing

TargetConvergenceAngle
type: Class
comment: The angle at which the reduction terminates (default: π).
subclass of: Thing

PhaseGateAttestation
type: Class
comment: Validation at each stage boundary checking that the accumulated phase angle matches the expected Ω^k.
subclass of: Thing

ComplexConjugateRollback
type: Class
comment: Recovery operation when a phase gate fails: z → z̄. Involutory: applying twice yields the original value.
subclass of: Thing

ReductionStep
type: Class
comment: A named stage of the reduction. The standard reduction has six stages (Initialization through Convergence).
subclass of: Thing

ReductionState
type: Class
comment: State of reduction execution at a specific point, including the current stage, phase angle, and pinned mask.
subclass of: Thing

ReductionRule
type: Class
comment: Guard-effect pair governing stage transitions in the reduction. The guard must be satisfied before the effect is applied.
subclass of: Thing

Epoch
type: Class
comment: Temporal segment of reduction execution. Each epoch represents one complete pass through the reduction stages.
subclass of: Thing

EpochBoundary
type: Class
comment: Transition between epochs. Carries metadata about the epoch boundary crossing.
subclass of: Thing

PredicateExpression
type: Class
comment: A Boolean expression over the reduction state. The atomic guard unit.
subclass of: Thing

GuardExpression
type: Class
comment: A conjunction of PredicateExpressions that must hold for a transition to fire.
subclass of: Thing

TransitionEffect
type: Class
comment: State changes applied when a transition fires. Contains PropertyBind steps.
subclass of: Thing

PropertyBind
type: Class
comment: A single site pinning: target site + value.
subclass of: Thing

ReductionAdvance
type: Class
comment: Advancement from one ReductionStep to the next.
subclass of: Thing

ServiceWindow
type: Class
comment: A sliding window over recent epochs providing BaseContext.
subclass of: Thing

ReductionTransaction
type: Class
comment: An atomic group of state changes within the reduction.
subclass of: Thing

PipelineSuccess
type: Class
comment: Successful termination (FullGrounding).
subclass of: Thing

PipelineFailureReason
type: Class
comment: Typed failure: DispatchMiss, GroundingFailure, ConvergenceStall, etc.
subclass of: Thing

PreflightCheck
type: Class
comment: A pre-execution validation: feasibility, dispatch coverage, coherence.
subclass of: Thing

FeasibilityResult
type: Class
comment: Result of a preflight check: feasibility witness or infeasibility witness.
subclass of: Thing

LeaseState
type: Class
comment: Lifecycle of a partitioned context lease: Pending → Active → Released/Expired/Suspended.
subclass of: Thing

ManagedLease
type: Class
comment: A context lease with lifecycle tracking.
subclass of: Thing

LeaseCheckpoint
type: Class
comment: Snapshot of lease state at a point in time.
subclass of: Thing

BackPressureSignal
type: Class
comment: Flow control when reduction produces faster than downstream can consume.
subclass of: Thing

DeferredQuerySet
type: Class
comment: Queries postponed to a future epoch.
subclass of: Thing

SubleaseTransfer
type: Class
comment: Transfer of a lease from one computation to another.
subclass of: Thing

ComparisonPredicate
type: Class
comment: Predicate comparing a state field against a value.
subclass of: PredicateExpression

ConjunctionPredicate
type: Class
comment: Conjunction (AND) of multiple predicates.
subclass of: PredicateExpression

DisjunctionPredicate
type: Class
comment: Disjunction (OR) of multiple predicates.
subclass of: PredicateExpression

NegationPredicate
type: Class
comment: Negation (NOT) of a single predicate.
subclass of: PredicateExpression

MembershipPredicate
type: Class
comment: Predicate testing membership of an element in a set.
subclass of: PredicateExpression

GroundingPredicate
type: Class
comment: Predicate testing whether grounding exceeds a threshold.
subclass of: PredicateExpression

SiteCoveragePredicate
type: Class
comment: Predicate testing whether a site coverage target is met.
subclass of: PredicateExpression

EqualsPredicate
type: Class
comment: Predicate testing equality of two expressions.
subclass of: PredicateExpression

NonNullPredicate
type: Class
comment: Predicate testing that a field is non-null.
subclass of: PredicateExpression

QuerySubtypePredicate
type: Class
comment: Predicate testing whether a query is a subtype of a given type.
subclass of: PredicateExpression

CompileUnit
type: Class
comment: The typed input graph submitted to the reduction pipeline. Packages a root Term, target quantum level, verification domains, and thermodynamic budget. Stage 0 accepts exactly one CompileUnit and initializes the reduction state vector from it.
subclass of: Thing

phaseParameter
type: DatatypeProperty
type: FunctionalProperty
comment: The base phase parameter Ω for this reduction (e.g., e^{iπ/6}).
domain: EulerReduction
range: decimal

stageCount
type: DatatypeProperty
type: FunctionalProperty
comment: The number of stages in this reduction.
domain: EulerReduction
range: nonNegativeInteger

convergenceAngle
type: DatatypeProperty
type: FunctionalProperty
comment: The cumulative phase angle at which the reduction converges.
domain: EulerReduction
range: decimal

composedOfMaps
type: ObjectProperty
comment: The ordered list of ψ-maps that compose this reduction.
domain: EulerReduction
range: TermExpression

rotationSchedule
type: DatatypeProperty
type: FunctionalProperty
comment: String representation of the rotation schedule Ω⁰, Ω¹, …, Ω⁵.
domain: PhaseRotationScheduler
range: decimal

baseAngle
type: DatatypeProperty
type: FunctionalProperty
comment: The base angle π/6 from which the schedule is derived.
domain: PhaseRotationScheduler
range: decimal

targetAngle
type: DatatypeProperty
type: FunctionalProperty
comment: The target convergence angle (default: π).
domain: TargetConvergenceAngle
range: decimal

gateStage
type: ObjectProperty
type: FunctionalProperty
comment: The reduction stage at which this gate is applied.
domain: PhaseGateAttestation
range: ReductionStep

gateExpectedPhase
type: DatatypeProperty
type: FunctionalProperty
comment: The expected phase angle Ω^k at this gate.
domain: PhaseGateAttestation
range: decimal

gateResult
type: DatatypeProperty
type: FunctionalProperty
comment: Whether the phase gate check passed or failed.
domain: PhaseGateAttestation
range: boolean

rollbackTarget
type: ObjectProperty
type: FunctionalProperty
comment: The reduction stage to which execution rolls back on gate failure.
domain: ComplexConjugateRollback
range: ReductionStep

stageIndex
type: DatatypeProperty
type: FunctionalProperty
comment: Zero-based index of this stage in the reduction.
domain: ReductionStep
range: nonNegativeInteger

stageName
type: DatatypeProperty
type: FunctionalProperty
comment: Human-readable name of this reduction stage.
domain: ReductionStep
range: string

expectedPhase
type: DatatypeProperty
type: FunctionalProperty
comment: The expected phase angle Ω^k at this stage.
domain: ReductionStep
range: decimal

entryGuard
type: ObjectProperty
type: FunctionalProperty
comment: A typed predicate evaluated on the current ReductionState. Must be satisfied to enter this stage.
domain: ReductionStep
range: StatePredicate

exitGuard
type: ObjectProperty
type: FunctionalProperty
comment: A typed predicate that must be satisfied before the reduction advances past this stage.
domain: ReductionStep
range: StatePredicate

stageEffect
type: ObjectProperty
type: FunctionalProperty
comment: The effect applied by this stage upon successful exit.
domain: ReductionStep
range: Effect

currentStage
type: ObjectProperty
type: FunctionalProperty
comment: The reduction stage at which execution is currently positioned.
domain: ReductionState
range: ReductionStep

phaseAngle
type: DatatypeProperty
type: FunctionalProperty
comment: The accumulated phase angle at the current point.
domain: ReductionState
range: decimal

pinnedMask
type: DatatypeProperty
type: FunctionalProperty
comment: Bit mask of sites that are pinned (resolved) at this point.
domain: ReductionState
range: nonNegativeInteger

freeRank
type: DatatypeProperty
type: FunctionalProperty
comment: The number of free (unresolved) sites at this point.
domain: ReductionState
range: nonNegativeInteger

transitionGuard
type: ObjectProperty
type: FunctionalProperty
comment: A typed GuardedTransition from predicate/ governing the stage transition.
domain: ReductionRule
range: GuardedTransition

transitionEffect
type: ObjectProperty
type: FunctionalProperty
comment: The effect applied when this transition fires.
domain: ReductionRule
range: TermExpression

transitionAdvance
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this transition advances to the next stage.
domain: ReductionRule
range: boolean

epochIndex
type: DatatypeProperty
type: FunctionalProperty
comment: Zero-based index of this epoch in the reduction execution.
domain: Epoch
range: nonNegativeInteger

epochDatum
type: ObjectProperty
type: FunctionalProperty
comment: Metadata or summary datum for this epoch.
domain: Epoch
range: TermExpression

epochBoundaryType
type: ObjectProperty
type: FunctionalProperty
comment: The type of epoch boundary crossing (e.g., normal, forced, timeout).
domain: EpochBoundary
range: TermExpression

predicateField
type: DatatypeProperty
type: FunctionalProperty
comment: The state field this predicate tests.
domain: PredicateExpression
range: string

predicateOperator
type: DatatypeProperty
type: FunctionalProperty
comment: The comparison operator (e.g., '=', '<', '>=').
domain: PredicateExpression
range: string

predicateValue
type: DatatypeProperty
type: FunctionalProperty
comment: The value against which the field is compared.
domain: PredicateExpression
range: string

guardPredicates
type: ObjectProperty
comment: The predicate expressions that compose this guard.
domain: GuardExpression
range: PredicateExpression

effectBindings
type: ObjectProperty
comment: The property bind steps applied by this effect.
domain: TransitionEffect
range: PropertyBind

bindTarget
type: DatatypeProperty
type: FunctionalProperty
comment: The target site identifier for this binding.
domain: PropertyBind
range: string

bindValue
type: DatatypeProperty
type: FunctionalProperty
comment: The value to pin the target site to.
domain: PropertyBind
range: string

advanceFrom
type: ObjectProperty
type: FunctionalProperty
comment: The source stage of the advancement.
domain: ReductionAdvance
range: ReductionStep

advanceTo
type: ObjectProperty
type: FunctionalProperty
comment: The target stage of the advancement.
domain: ReductionAdvance
range: ReductionStep

windowSize
type: DatatypeProperty
type: FunctionalProperty
comment: The number of recent epochs in this service window.
domain: ServiceWindow
range: nonNegativeInteger

windowOffset
type: DatatypeProperty
type: FunctionalProperty
comment: The starting epoch offset of this service window.
domain: ServiceWindow
range: nonNegativeInteger

transactionPolicy
type: DatatypeProperty
type: FunctionalProperty
comment: The execution policy for this transaction (e.g., AllOrNothing, BestEffort).
domain: ReductionTransaction
range: string

transactionOutcome
type: ObjectProperty
type: FunctionalProperty
comment: The outcome of this transaction (e.g., committed, rolled back).
domain: ReductionTransaction
range: TermExpression

failureKind
type: DatatypeProperty
type: FunctionalProperty
comment: The kind of pipeline failure (e.g., DispatchMiss, ConvergenceStall).
domain: PipelineFailureReason
range: string

preflightKind
type: DatatypeProperty
type: FunctionalProperty
comment: The kind of preflight check (e.g., feasibility, dispatch coverage).
domain: PreflightCheck
range: string

preflightResult
type: ObjectProperty
type: FunctionalProperty
comment: The result of the preflight check (e.g., pass, fail).
domain: PreflightCheck
range: TermExpression

groundingReached
type: DatatypeProperty
type: FunctionalProperty
comment: Whether full grounding was achieved.
domain: PipelineSuccess
range: boolean

feasibilityKind
type: DatatypeProperty
type: FunctionalProperty
comment: The kind of feasibility result (e.g., Feasible, Infeasible).
domain: FeasibilityResult
range: string

feasibilityWitness
type: DatatypeProperty
type: FunctionalProperty
comment: The witness justifying the feasibility or infeasibility result.
domain: FeasibilityResult
range: string

leasePhase
type: DatatypeProperty
type: FunctionalProperty
comment: The lifecycle phase of a lease (e.g., Pending, Active, Released).
domain: LeaseState
range: string

managedLeaseId
type: ObjectProperty
type: FunctionalProperty
comment: Unique identifier for this managed lease.
domain: ManagedLease
range: TermExpression

leaseLifecycle
type: ObjectProperty
type: FunctionalProperty
comment: The current lifecycle state of this managed lease.
domain: ManagedLease
range: LeaseState

checkpointEpoch
type: DatatypeProperty
type: FunctionalProperty
comment: The epoch index at which this checkpoint was taken.
domain: LeaseCheckpoint
range: nonNegativeInteger

checkpointState
type: ObjectProperty
type: FunctionalProperty
comment: The reduction state captured at this checkpoint.
domain: LeaseCheckpoint
range: ReductionState

pressureLevel
type: DatatypeProperty
type: FunctionalProperty
comment: The current back-pressure level (e.g., Low, Medium, High).
domain: BackPressureSignal
range: string

pressureThreshold
type: DatatypeProperty
type: FunctionalProperty
comment: The threshold at which back-pressure activates.
domain: BackPressureSignal
range: decimal

deferredCount
type: DatatypeProperty
type: FunctionalProperty
comment: The number of queries in this deferred set.
domain: DeferredQuerySet
range: nonNegativeInteger

deferralEpoch
type: DatatypeProperty
type: FunctionalProperty
comment: The epoch in which these queries were deferred.
domain: DeferredQuerySet
range: nonNegativeInteger

sourceLeaseRef
type: ObjectProperty
type: FunctionalProperty
comment: The lease being transferred from.
domain: SubleaseTransfer
range: TermExpression

targetLeaseRef
type: ObjectProperty
type: FunctionalProperty
comment: The lease being transferred to.
domain: SubleaseTransfer
range: TermExpression

transferredBudget
type: DatatypeProperty
type: FunctionalProperty
comment: The site budget transferred between leases.
domain: SubleaseTransfer
range: nonNegativeInteger

transferCompleted
type: DatatypeProperty
type: FunctionalProperty
comment: Whether the sublease transfer has been completed.
domain: SubleaseTransfer
range: boolean

comparisonField
type: ObjectProperty
type: FunctionalProperty
comment: The state field tested by this comparison predicate.
domain: ComparisonPredicate
range: TermExpression

comparisonOperator
type: ObjectProperty
type: FunctionalProperty
comment: The comparison operator (e.g., '=', '<', '>=').
domain: ComparisonPredicate
range: TermExpression

comparisonValue
type: ObjectProperty
type: FunctionalProperty
comment: The value against which the comparison is made.
domain: ComparisonPredicate
range: TermExpression

conjuncts
type: ObjectProperty
comment: A conjunct predicate in a conjunction.
domain: ConjunctionPredicate
range: TermExpression

disjuncts
type: ObjectProperty
comment: A disjunct predicate in a disjunction.
domain: DisjunctionPredicate
range: TermExpression

negatedPredicate
type: ObjectProperty
type: FunctionalProperty
comment: The predicate being negated.
domain: NegationPredicate
range: TermExpression

membershipSet
type: ObjectProperty
type: FunctionalProperty
comment: The set against which membership is tested.
domain: MembershipPredicate
range: TermExpression

membershipElement
type: ObjectProperty
type: FunctionalProperty
comment: The element being tested for set membership.
domain: MembershipPredicate
range: TermExpression

groundingThreshold
type: DatatypeProperty
type: FunctionalProperty
comment: The grounding threshold above which the predicate holds.
domain: GroundingPredicate
range: nonNegativeInteger

coverageTarget
type: DatatypeProperty
type: FunctionalProperty
comment: The site coverage target expression.
domain: SiteCoveragePredicate
range: nonNegativeInteger

equalityLeft
type: ObjectProperty
type: FunctionalProperty
comment: The left-hand side of an equality test.
domain: EqualsPredicate
range: TermExpression

equalityRight
type: ObjectProperty
type: FunctionalProperty
comment: The right-hand side of an equality test.
domain: EqualsPredicate
range: TermExpression

nonNullField
type: DatatypeProperty
type: FunctionalProperty
comment: The field that must be non-null.
domain: NonNullPredicate
range: nonNegativeInteger

queryTypeRef
type: ObjectProperty
type: FunctionalProperty
comment: The query type reference for subtype testing.
domain: QuerySubtypePredicate
range: TermExpression

siteState
type: ObjectProperty
type: FunctionalProperty
comment: The site state descriptor within a reduction state.
domain: ReductionState
range: TermExpression

transactionScope
type: ObjectProperty
type: FunctionalProperty
comment: The scope of sites affected by this transaction.
domain: ReductionTransaction
range: TermExpression

transactionStatus
type: ObjectProperty
type: FunctionalProperty
comment: Current status of this transaction (e.g., pending, committed).
domain: ReductionTransaction
range: TermExpression

baseContextRef
type: ObjectProperty
type: FunctionalProperty
comment: Reference to the base context provided by this service window.
domain: ServiceWindow
range: TermExpression

leaseRemainingBudget
type: DatatypeProperty
type: FunctionalProperty
comment: The remaining site budget at this checkpoint.
domain: LeaseCheckpoint
range: nonNegativeInteger

expiryEpoch
type: DatatypeProperty
type: FunctionalProperty
comment: The epoch at which this managed lease expires.
domain: ManagedLease
range: nonNegativeInteger

leaseBudget
type: DatatypeProperty
type: FunctionalProperty
comment: The total site budget allocated to this managed lease.
domain: ManagedLease
range: nonNegativeInteger

sourceStage
type: ObjectProperty
type: FunctionalProperty
comment: The source stage emitting back-pressure.
domain: BackPressureSignal
range: TermExpression

targetStage
type: ObjectProperty
type: FunctionalProperty
comment: The target stage receiving back-pressure.
domain: BackPressureSignal
range: TermExpression

deferralReason
type: ObjectProperty
type: FunctionalProperty
comment: The reason for deferring these queries.
domain: DeferredQuerySet
range: TermExpression

infeasibilityKind
type: ObjectProperty
type: FunctionalProperty
comment: The kind of infeasibility detected.
domain: FeasibilityResult
range: TermExpression

failureStage
type: ObjectProperty
type: FunctionalProperty
comment: The reduction stage at which the pipeline failure occurred.
domain: PipelineFailureReason
range: TermExpression

finalGrounding
type: DatatypeProperty
type: FunctionalProperty
comment: The final grounding level achieved on pipeline success.
domain: PipelineSuccess
range: decimal

preservedGrounding
type: DatatypeProperty
type: FunctionalProperty
comment: Whether grounding was preserved across the epoch boundary.
domain: EpochBoundary
range: boolean

rootTerm
type: ObjectProperty
type: FunctionalProperty
comment: The root term expression of a CompileUnit — the top-level syntactic node from which all sub-expressions descend.
domain: CompileUnit
range: TermExpression

unitWittLevel
type: ObjectProperty
type: FunctionalProperty
comment: The quantum level at which this CompileUnit operates.
domain: CompileUnit
range: WittLevel

thermodynamicBudget
type: DatatypeProperty
type: FunctionalProperty
comment: The Landauer-bounded energy budget for this CompileUnit's resolution, measured in k_B T ln 2 units.
domain: CompileUnit
range: decimal

targetDomains
type: ObjectProperty
comment: The verification domain(s) targeted by this CompileUnit.
domain: CompileUnit
range: VerificationDomain

unitAddress
type: ObjectProperty
type: FunctionalProperty
comment: Content-addressable identifier computed as the u:Element of the root term’s transitive closure. Computed by stage_initialization, not declared by the submitter. Excludes budget, domains, and quantum level to enable memoization.
domain: CompileUnit
range: Element

preflightOrder
type: DatatypeProperty
type: FunctionalProperty
comment: Zero-based execution order for preflight checks. Lower indices execute first. BudgetSolvencyCheck (order 0) must precede all others.
domain: PreflightCheck
range: nonNegativeInteger

Initialization
type: NamedIndividual
type: ReductionStep
expectedPhase: Ω⁰
stageIndex: 0
stageName: Initialization
comment: Stage 0: initialize state vector to identity.

Declare
type: NamedIndividual
type: ReductionStep
expectedPhase: Ω¹
stageIndex: 1
stageName: Declare
comment: Stage 1: dispatch resolver (δ selects).

Factorize
type: NamedIndividual
type: ReductionStep
expectedPhase: Ω²
stageIndex: 2
stageName: Factorize
comment: Stage 2: produce valid ring address (G grounds).

Resolve
type: NamedIndividual
type: ReductionStep
expectedPhase: Ω³
stageIndex: 3
stageName: Resolve
comment: Stage 3: resolve constraints (Π terminates).

Attest
type: NamedIndividual
type: ReductionStep
expectedPhase: Ω⁴
stageIndex: 4
stageName: Attest
comment: Stage 4: accumulate without contradiction (α consistent).

Extract
type: NamedIndividual
type: ReductionStep
expectedPhase: Ω⁵
stageIndex: 5
stageName: Extract
comment: Stage 5: extract coherent output (P projects).

Convergence
type: NamedIndividual
type: ReductionStep
expectedPhase: π
stageIndex: 6
stageName: Convergence
comment: Terminal stage: reduction has reached the convergence angle π.

gate_initialization
type: NamedIndividual
type: PhaseGateAttestation
gateExpectedPhase: Ω⁰
gateResult: true
gateStage: Initialization
comment: Phase gate at stage 0 boundary: checks Ω⁰ = 1.

gate_declare
type: NamedIndividual
type: PhaseGateAttestation
gateExpectedPhase: Ω¹
gateResult: true
gateStage: Declare
comment: Phase gate at stage 1 boundary: checks Ω¹.

gate_factorize
type: NamedIndividual
type: PhaseGateAttestation
gateExpectedPhase: Ω²
gateResult: true
gateStage: Factorize
comment: Phase gate at stage 2 boundary: checks Ω².

gate_resolve
type: NamedIndividual
type: PhaseGateAttestation
gateExpectedPhase: Ω³
gateResult: true
gateStage: Resolve
comment: Phase gate at stage 3 boundary: checks Ω³.

gate_attest
type: NamedIndividual
type: PhaseGateAttestation
gateExpectedPhase: Ω⁴
gateResult: true
gateStage: Attest
comment: Phase gate at stage 4 boundary: checks Ω⁴.

gate_extract
type: NamedIndividual
type: PhaseGateAttestation
gateExpectedPhase: Ω⁵
gateResult: true
gateStage: Extract
comment: Phase gate at stage 5 boundary: checks Ω⁵.

euler_reduction_instance
type: NamedIndividual
type: EulerReduction
convergenceAngle: π
phaseParameter: e^{iπ/6}
stageCount: 6
comment: The canonical Euler reduction instance with Ω = e^{iπ/6} and 6 stages.

phase_schedule
type: NamedIndividual
type: PhaseRotationScheduler
baseAngle: π/6
rotationSchedule: Ω⁰, Ω¹, Ω², Ω³, Ω⁴, Ω⁵
comment: The canonical phase rotation schedule for the 6-stage reduction.

convergence_target
type: NamedIndividual
type: TargetConvergenceAngle
targetAngle: π
comment: The default convergence target angle π.

conjugate_rollback
type: NamedIndividual
type: ComplexConjugateRollback
rollbackTarget: Initialization
comment: The canonical complex conjugate rollback operation: z → z̄.

DispatchMiss
type: NamedIndividual
type: PipelineFailureReason
failureKind: DispatchMiss
comment: Failure: no resolver matched the dispatch query.

GroundingFailure
type: NamedIndividual
type: PipelineFailureReason
failureKind: GroundingFailure
comment: Failure: grounding to a valid ring address failed.

ConvergenceStall
type: NamedIndividual
type: PipelineFailureReason
failureKind: ConvergenceStall
comment: Failure: reduction stalled before reaching convergence angle.

ContradictionDetected
type: NamedIndividual
type: PipelineFailureReason
failureKind: ContradictionDetected
comment: Failure: accumulation detected a logical contradiction.

CoherenceViolation
type: NamedIndividual
type: PipelineFailureReason
failureKind: CoherenceViolation
comment: Failure: coherence constraint violated during extraction.

FullGroundingSuccess
type: NamedIndividual
type: PipelineSuccess
groundingReached: true
comment: Successful termination: all sites saturated.

FeasibilityCheck
type: NamedIndividual
type: PreflightCheck
preflightKind: Feasibility
preflightOrder: 1
comment: Preflight: checks that the reduction can reach convergence.

DispatchCoverageCheck
type: NamedIndividual
type: PreflightCheck
preflightKind: DispatchCoverage
preflightOrder: 2
comment: Preflight: checks that every dispatch query has a resolver.

PackageCoherenceCheck
type: NamedIndividual
type: PreflightCheck
preflightKind: PackageCoherence
preflightOrder: 3
comment: Preflight: checks package-level coherence constraints.

default_service_window
type: NamedIndividual
type: ServiceWindow
windowOffset: 0
windowSize: 3
comment: The default service window: 3 epochs, zero offset.

advance_init_to_declare
type: NamedIndividual
type: ReductionAdvance
advanceFrom: Initialization
advanceTo: Declare
comment: Advancement from Initialization to Declare.

atomic_transaction
type: NamedIndividual
type: ReductionTransaction
transactionPolicy: AllOrNothing
comment: An all-or-nothing atomic reduction transaction.

empty_guard
type: NamedIndividual
type: GuardExpression
comment: A trivially satisfied guard with no predicates.

identity_effect
type: NamedIndividual
type: TransitionEffect
comment: The identity effect: no state changes.

true_predicate
type: NamedIndividual
type: PredicateExpression
predicateField: *
predicateOperator: true
predicateValue: *
comment: A predicate that always evaluates to true.

noop_bind
type: NamedIndividual
type: PropertyBind
bindTarget: none
bindValue: unchanged
comment: A no-op property binding that changes nothing.

Pending
type: NamedIndividual
type: LeaseState
leasePhase: Pending
comment: Lease is pending activation.

Active
type: NamedIndividual
type: LeaseState
leasePhase: Active
comment: Lease is active and resources are allocated.

Released
type: NamedIndividual
type: LeaseState
leasePhase: Released
comment: Lease has been explicitly released.

Expired
type: NamedIndividual
type: LeaseState
leasePhase: Expired
comment: Lease has expired due to timeout.

Suspended
type: NamedIndividual
type: LeaseState
leasePhase: Suspended
comment: Lease is temporarily suspended.

FeasibilityWitness
type: NamedIndividual
type: FeasibilityResult
feasibilityKind: Feasible
comment: Preflight result: reduction is feasible.

InfeasibilityWitness
type: NamedIndividual
type: FeasibilityResult
feasibilityKind: Infeasible
feasibilityWitness: obstruction detected
comment: Preflight result: reduction is infeasible.

PreflightTiming
type: NamedIndividual
type: PreflightCheck
preflightKind: Timing
preflightOrder: 4
comment: Preflight: timing feasibility check.

RuntimeTiming
type: NamedIndividual
type: PreflightCheck
preflightKind: RuntimeTiming
preflightOrder: 5
comment: Preflight: runtime timing bounds check.

BudgetSolvencyCheck
type: NamedIndividual
type: PreflightCheck
preflightKind: BudgetSolvency
preflightOrder: 0
comment: Preflight: verifies thermodynamicBudget ≥ bitsWidth(unitWittLevel) × ln 2. Rejects the CompileUnit if the budget is absent or insufficient. Must execute before all other preflights (preflightOrder 0). Cost is O(1) per CS_4.

BackPressureDefault
type: NamedIndividual
type: BackPressureSignal
pressureLevel: Medium
pressureThreshold: 0.75
comment: Default back-pressure signal with medium threshold.

UOR Convergence Tower
type: Ontology
imports: UOR Operations
imports: UOR Euler Reduction
imports: UOR Observables
imports: UOR Homology
comment: Hopf convergence tower: four levels R, C, H, O corresponding to the four normed division algebras of dimensions 1, 2, 4, 8. Each level carries a Hopf fibration fiber and Betti signature.
space: kernel

ConvergenceLevel
type: Class
comment: A level in the convergence tower. Four instances: R (dim 1), C (dim 2), H (dim 4), O (dim 8).
subclass of: Thing

HopfFiber
type: Class
comment: The fiber of the Hopf fibration at a convergence level. Four instances: S⁰, S¹, S³, S⁷.
subclass of: Thing

ConvergenceResidual
type: Class
comment: The unresolved structure at a convergence level. The β_{2^k−1} = 1 Betti number that persists.
subclass of: Thing

CommutativeSubspace
type: Class
comment: The subspace U(1) ⊂ SU(2) selected when pairwise interaction converges.
subclass of: Thing

AssociativeSubalgebra
type: Class
comment: The subspace H ⊂ O selected when triple interaction converges.
subclass of: Thing

algebraDimension
type: DatatypeProperty
type: FunctionalProperty
comment: The dimension of the division algebra at this level (1, 2, 4, or 8).
domain: ConvergenceLevel
range: nonNegativeInteger

bettiSignature
type: DatatypeProperty
type: FunctionalProperty
comment: The Betti number signature at this convergence level.
domain: ConvergenceLevel
range: string

fiberType
type: ObjectProperty
type: FunctionalProperty
comment: The Hopf fiber associated with this convergence level.
domain: ConvergenceLevel
range: HopfFiber

characteristicIdentity
type: DatatypeProperty
type: FunctionalProperty
comment: The characteristic identity at this convergence level (existence, feedback, choice, self-reference).
domain: ConvergenceLevel
range: string

levelName
type: DatatypeProperty
type: FunctionalProperty
comment: Human-readable name of this convergence level.
domain: ConvergenceLevel
range: string

fiberDimension
type: DatatypeProperty
type: FunctionalProperty
comment: The dimension of the Hopf fiber sphere.
domain: HopfFiber
range: nonNegativeInteger

totalSpace
type: DatatypeProperty
type: FunctionalProperty
comment: The total space of the Hopf fibration.
domain: HopfFiber
range: string

baseSpace
type: DatatypeProperty
type: FunctionalProperty
comment: The base space of the Hopf fibration.
domain: HopfFiber
range: string

fiberSphere
type: DatatypeProperty
type: FunctionalProperty
comment: The fiber sphere designation (e.g. S⁰, S¹).
domain: HopfFiber
range: string

residualBetti
type: DatatypeProperty
type: FunctionalProperty
comment: The persistent Betti number at this residual.
domain: ConvergenceResidual
range: nonNegativeInteger

residualDimension
type: DatatypeProperty
type: FunctionalProperty
comment: The dimension at which the residual persists.
domain: ConvergenceResidual
range: nonNegativeInteger

subspaceRef
type: ObjectProperty
type: FunctionalProperty
comment: The commutative subspace selected by pairwise convergence.
domain: CommutativeSubspace
range: CommutativeSubspace

subalgebraRef
type: ObjectProperty
type: FunctionalProperty
comment: The associative subalgebra selected by triple convergence.
domain: AssociativeSubalgebra
range: AssociativeSubalgebra

commutatorRef
type: ObjectProperty
type: FunctionalProperty
comment: Reference to the commutator pair for this convergence.
domain: CommutativeSubspace
range: Commutator

associatorRef
type: ObjectProperty
type: FunctionalProperty
comment: Reference to the associator triple for this convergence.
domain: AssociativeSubalgebra
range: AssociatorTriple

L0_State
type: NamedIndividual
type: ConvergenceLevel
algebraDimension: 1
bettiSignature: [1]
characteristicIdentity: existence
fiberType: hopf_S0
levelName: R
comment: Level 0: R (reals), dimension 1, existence.

L1_Memory
type: NamedIndividual
type: ConvergenceLevel
algebraDimension: 2
bettiSignature: [1,1]
characteristicIdentity: feedback
fiberType: hopf_S1
levelName: C
comment: Level 1: C (complex), dimension 2, feedback.

L2_Agency
type: NamedIndividual
type: ConvergenceLevel
algebraDimension: 4
bettiSignature: [1,0,0,1]
characteristicIdentity: choice
fiberType: hopf_S3
levelName: H
comment: Level 2: H (quaternions), dimension 4, choice.

L3_Self
type: NamedIndividual
type: ConvergenceLevel
algebraDimension: 8
bettiSignature: [1,0,0,0,0,0,0,1]
characteristicIdentity: self-reference
fiberType: hopf_S7
levelName: O
comment: Level 3: O (octonions), dimension 8, self-reference.

hopf_S0
type: NamedIndividual
type: HopfFiber
baseSpace: pt
fiberDimension: 0
fiberSphere: S⁰
totalSpace: S¹
comment: Hopf fiber S⁰: dimension 0, total space S¹, base pt.

hopf_S1
type: NamedIndividual
type: HopfFiber
baseSpace: S²
fiberDimension: 1
fiberSphere: S¹
totalSpace: S³
comment: Hopf fiber S¹: dimension 1, total space S³, base S².

hopf_S3
type: NamedIndividual
type: HopfFiber
baseSpace: S⁴
fiberDimension: 3
fiberSphere: S³
totalSpace: S⁷
comment: Hopf fiber S³: dimension 3, total space S⁷, base S⁴.

hopf_S7
type: NamedIndividual
type: HopfFiber
baseSpace: S⁸
fiberDimension: 7
fiberSphere: S⁷
totalSpace: S¹⁵
comment: Hopf fiber S⁷: dimension 7, total space S¹⁵, base S⁸.

UOR Division Algebras
type: Ontology
imports: UOR Operations
imports: UOR Convergence Tower
comment: The four normed division algebras R, C, H, O and the Cayley-Dickson construction.
space: kernel

NormedDivisionAlgebra
type: Class
comment: An algebra over R that is a division ring with multiplicative norm. Exactly four exist (Hurwitz theorem): R, C, H, O.
subclass of: Thing

CayleyDicksonConstruction
type: Class
comment: The doubling construction that builds each division algebra from the previous: R → C → H → O.
subclass of: Thing

MultiplicationTable
type: Class
comment: The explicit product rules for a division algebra’s basis elements.
subclass of: Thing

AlgebraCommutator
type: Class
comment: The commutator [a,b] = ab − ba. Zero for R and C; non-zero for H and O.
subclass of: Thing

AlgebraAssociator
type: Class
comment: The associator [a,b,c] = (ab)c − a(bc). Zero for R, C, H; non-zero for O.
subclass of: Thing

algebraDimension
type: DatatypeProperty
type: FunctionalProperty
comment: The dimension of this division algebra (1, 2, 4, or 8).
domain: NormedDivisionAlgebra
range: nonNegativeInteger

isCommutative
type: DatatypeProperty
type: FunctionalProperty
comment: Whether multiplication in this algebra is commutative.
domain: NormedDivisionAlgebra
range: boolean

isAssociative
type: DatatypeProperty
type: FunctionalProperty
comment: Whether multiplication in this algebra is associative.
domain: NormedDivisionAlgebra
range: boolean

basisElements
type: DatatypeProperty
type: FunctionalProperty
comment: The basis elements of this division algebra.
domain: NormedDivisionAlgebra
range: string

algebraMultiplicationTable
type: ObjectProperty
type: FunctionalProperty
comment: The multiplication table for this algebra.
domain: NormedDivisionAlgebra
range: MultiplicationTable

cayleyDicksonSource
type: ObjectProperty
type: FunctionalProperty
comment: The source algebra of the Cayley-Dickson doubling.
domain: CayleyDicksonConstruction
range: NormedDivisionAlgebra

cayleyDicksonTarget
type: ObjectProperty
type: FunctionalProperty
comment: The target algebra of the Cayley-Dickson doubling.
domain: CayleyDicksonConstruction
range: NormedDivisionAlgebra

adjoinedElement
type: DatatypeProperty
type: FunctionalProperty
comment: The new basis element adjoined by this doubling step.
domain: CayleyDicksonConstruction
range: string

conjugationRule
type: DatatypeProperty
type: FunctionalProperty
comment: The conjugation and multiplication rule for the adjoined element.
domain: CayleyDicksonConstruction
range: string

RealAlgebra
type: NamedIndividual
type: NormedDivisionAlgebra
algebraDimension: 1
basisElements: {1}
isAssociative: true
isCommutative: true
comment: The real numbers R: dimension 1, commutative, associative.

ComplexAlgebra
type: NamedIndividual
type: NormedDivisionAlgebra
algebraDimension: 2
basisElements: {1, i}
isAssociative: true
isCommutative: true
comment: The complex numbers C: dimension 2, commutative, associative.

QuaternionAlgebra
type: NamedIndividual
type: NormedDivisionAlgebra
algebraDimension: 4
basisElements: {1, i, j, k}
isAssociative: true
isCommutative: false
comment: The quaternions H: dimension 4, non-commutative, associative.

OctonionAlgebra
type: NamedIndividual
type: NormedDivisionAlgebra
algebraDimension: 8
basisElements: {1, i, j, k, l, il, jl, kl}
isAssociative: false
isCommutative: false
comment: The octonions O: dimension 8, non-commutative, non-associative.

cayleyDickson_R_to_C
type: NamedIndividual
type: CayleyDicksonConstruction
adjoinedElement: i
cayleyDicksonSource: RealAlgebra
cayleyDicksonTarget: ComplexAlgebra
conjugationRule: i² = −1
comment: Cayley-Dickson doubling R → C: adjoin i with i² = −1.

cayleyDickson_C_to_H
type: NamedIndividual
type: CayleyDicksonConstruction
adjoinedElement: j
cayleyDicksonSource: ComplexAlgebra
cayleyDicksonTarget: QuaternionAlgebra
conjugationRule: ij = k, ji = −k
comment: Cayley-Dickson doubling C → H: adjoin j with j² = −1, ij = k, ji = −k.

cayleyDickson_H_to_O
type: NamedIndividual
type: CayleyDicksonConstruction
adjoinedElement: l
cayleyDicksonSource: QuaternionAlgebra
cayleyDicksonTarget: OctonionAlgebra
conjugationRule: non-associative Fano plane products
comment: Cayley-Dickson doubling H → O: adjoin l, non-associative Fano plane products.

UOR Interaction Algebra
type: Ontology
imports: UOR Operations
imports: UOR Convergence Tower
imports: UOR State
imports: UOR Observables
imports: UOR Schema
comment: Multi-entity interaction: commutator states, associator triples, negotiation convergence.
space: bridge

InteractionContext
type: Class
comment: Two entities sharing sites through composed operations. Properties: entityA, entityB, sharedSiteMask, commutatorNorm.
subclass of: Thing

CommutatorState
type: Class
comment: The norm ‖[δ_A, ι_B]‖ on shared sites. Zero iff operators commute on the shared domain.
subclass of: Thing

AssociatorState
type: Class
comment: The norm of the three-way associator on shared sites.
subclass of: Thing

AssociatorTriple
type: Class
comment: Three entities whose interaction exhibits non-associativity due to read-write interleaving.
subclass of: Thing

ThreeWaySite
type: Class
comment: A site shared by all three entities in an AssociatorTriple.
subclass of: Thing

NegotiationTrace
type: Class
comment: Sequence of CommutatorStates across interaction steps.
subclass of: Thing

MutualModelTrace
type: Class
comment: Sequence of AssociatorStates across interaction steps.
subclass of: Thing

InteractionNerve
type: Class
comment: Simplicial complex of N-entity coupling.
subclass of: Thing

InteractionComposition
type: Class
comment: IC(A,B) = κ(session(A,B)). Combined interaction-composition operator.
subclass of: Thing

entityA
type: ObjectProperty
type: FunctionalProperty
comment: First entity in the interaction context.
domain: InteractionContext
range: TermExpression

entityB
type: ObjectProperty
type: FunctionalProperty
comment: Second entity in the interaction context.
domain: InteractionContext
range: TermExpression

sharedSiteMask
type: DatatypeProperty
type: FunctionalProperty
comment: Bitmask identifying which sites are shared between the two entities.
domain: InteractionContext
range: nonNegativeInteger

commutatorNorm
type: DatatypeProperty
type: FunctionalProperty
comment: The norm of the commutator on shared sites. Zero iff the operators commute.
domain: InteractionContext
range: decimal

commutatorValue
type: DatatypeProperty
type: FunctionalProperty
comment: The computed commutator norm value.
domain: CommutatorState
range: decimal

associatorNorm
type: DatatypeProperty
type: FunctionalProperty
comment: The norm of the three-way associator on shared sites.
domain: AssociatorState
range: decimal

tripleComponentA
type: ObjectProperty
type: FunctionalProperty
comment: First component datum in the associator triple.
domain: AssociatorTriple
range: Datum

tripleComponentB
type: ObjectProperty
type: FunctionalProperty
comment: Second component datum in the associator triple.
domain: AssociatorTriple
range: Datum

tripleComponentC
type: ObjectProperty
type: FunctionalProperty
comment: Third component datum in the associator triple.
domain: AssociatorTriple
range: Datum

sitePosition
type: DatatypeProperty
type: FunctionalProperty
comment: The position index of the shared site.
domain: ThreeWaySite
range: nonNegativeInteger

leftGroupingValue
type: DatatypeProperty
type: FunctionalProperty
comment: Value under left-associative grouping (AB)C.
domain: ThreeWaySite
range: decimal

rightGroupingValue
type: DatatypeProperty
type: FunctionalProperty
comment: Value under right-associative grouping A(BC).
domain: ThreeWaySite
range: decimal

isPinned
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this site is pinned by a lease constraint.
domain: ThreeWaySite
range: boolean

pinningPair
type: ObjectProperty
type: FunctionalProperty
comment: Identifier of the entity pair that pins this site.
domain: ThreeWaySite
range: TermExpression

negotiationSteps
type: DatatypeProperty
type: FunctionalProperty
comment: Number of steps in the negotiation trace.
domain: NegotiationTrace
range: nonNegativeInteger

isConvergent
type: DatatypeProperty
type: FunctionalProperty
comment: Whether the trace converges to zero commutator or zero associator.
domain: NegotiationTrace
range: boolean

modelConvergent
type: DatatypeProperty
type: FunctionalProperty
comment: Whether the mutual model trace converges.
domain: MutualModelTrace
range: boolean

nerveDimension
type: DatatypeProperty
type: FunctionalProperty
comment: Maximum dimension of the interaction nerve simplicial complex.
domain: InteractionNerve
range: nonNegativeInteger

nerveBettiNumbers
type: DatatypeProperty
type: FunctionalProperty
comment: Betti number sequence of the interaction nerve.
domain: InteractionNerve
range: nonNegativeInteger

reificationDepth
type: DatatypeProperty
type: FunctionalProperty
comment: Depth of the interaction composition reification.
domain: InteractionComposition
range: nonNegativeInteger

traceLength
type: DatatypeProperty
type: FunctionalProperty
comment: The number of steps in a negotiation trace.
domain: NegotiationTrace
range: nonNegativeInteger

convergenceRate
type: DatatypeProperty
type: FunctionalProperty
comment: The rate at which the negotiation trace converges.
domain: NegotiationTrace
range: decimal

terminalValue
type: ObjectProperty
type: FunctionalProperty
comment: The terminal value of the negotiation trace.
domain: NegotiationTrace
range: TermExpression

associatorProfileRef
type: ObjectProperty
type: FunctionalProperty
comment: Reference to the observable describing the associativity profile.
domain: AssociatorTriple
range: Observable

associatorNormValue
type: DatatypeProperty
type: FunctionalProperty
comment: The maximum norm of the associator for this triple.
domain: AssociatorTriple
range: decimal

UOR Monoidal Composition
type: Ontology
imports: UOR Operations
imports: UOR Euler Reduction
imports: UOR Certificates
imports: UOR Transforms and Morphisms
imports: UOR Schema
comment: Sequential composition of computations via monoidal product A ⊗ B.
space: kernel

MonoidalProduct
type: Class
comment: A ⊗ B: the sequential composition of two computations. Output of A feeds input of B.
subclass of: Thing

MonoidalUnit
type: Class
comment: The identity computation I: passes input through unchanged. I ⊗ A ≅ A ≅ A ⊗ I.
subclass of: Thing

MonoidalAssociator
type: Class
comment: The witness that (A⊗B)⊗C ≅ A⊗(B⊗C). The associativity isomorphism.
subclass of: Thing

leftOperand
type: ObjectProperty
type: FunctionalProperty
comment: The left operand in the monoidal product A ⊗ B.
domain: MonoidalProduct
range: ComputationDatum

rightOperand
type: ObjectProperty
type: FunctionalProperty
comment: The right operand in the monoidal product A ⊗ B.
domain: MonoidalProduct
range: ComputationDatum

composedResult
type: ObjectProperty
type: FunctionalProperty
comment: The result datum of the composed computation A ⊗ B.
domain: MonoidalProduct
range: Datum

saturationValue
type: DatatypeProperty
type: FunctionalProperty
comment: σ(A⊗B) relationship: saturation of the sequential composition.
domain: MonoidalProduct
range: decimal

unitWitnessRef
type: ObjectProperty
type: FunctionalProperty
comment: Certificate witnessing I ⊗ A ≅ A ≅ A ⊗ I.
domain: MonoidalUnit
range: Certificate

associatorLeft
type: ObjectProperty
type: FunctionalProperty
comment: The left-grouped product (A⊗B)⊗C.
domain: MonoidalAssociator
range: MonoidalProduct

associatorRight
type: ObjectProperty
type: FunctionalProperty
comment: The right-grouped product A⊗(B⊗C).
domain: MonoidalAssociator
range: MonoidalProduct

associatorWitnessRef
type: ObjectProperty
type: FunctionalProperty
comment: Certificate witnessing the associativity isomorphism (A⊗B)⊗C ≅ A⊗(B⊗C).
domain: MonoidalAssociator
range: Certificate

UOR Operad Composition
type: Ontology
imports: UOR Type System
imports: UOR Carry Algebra
imports: UOR Transforms and Morphisms
comment: Structural type nesting via operad composition. Governs how types compose: Table(Tuple(Sequence(Symbol(...)))).
space: kernel

StructuralOperad
type: Class
comment: The composition structure on the eight structural types. Governs how types nest.
subclass of: Thing

OperadComposition
type: Class
comment: A specific nesting: outer type F applied to inner type G.
subclass of: Thing

operadStructure
type: ObjectProperty
type: FunctionalProperty
comment: The structural operad defining this composition.
domain: StructuralOperad
range: StructuralOperad

outerType
type: ObjectProperty
type: FunctionalProperty
comment: The outer type F in the nesting F(G).
domain: OperadComposition
range: TypeDefinition

innerType
type: ObjectProperty
type: FunctionalProperty
comment: The inner type G in the nesting F(G).
domain: OperadComposition
range: TypeDefinition

composedType
type: ObjectProperty
type: FunctionalProperty
comment: The resulting composed type F(G).
domain: OperadComposition
range: TypeDefinition

composedSiteCount
type: DatatypeProperty
type: FunctionalProperty
comment: Site count of the composed type F(G).
domain: OperadComposition
range: nonNegativeInteger

composedGrounding
type: ObjectProperty
type: FunctionalProperty
comment: Grounding of the composed type F(G).
domain: OperadComposition
range: GroundingMap

UOR Effect Algebra
type: Ontology
imports: UOR Operations
imports: UOR State
imports: UOR Partitions
imports: UOR Type System
imports: UOR Certificates
comment: Typed endomorphisms on state:Context classified by site target. Formalizes what reduction guard-effect pairs do to the site budget.
space: kernel

Effect
type: Class
comment: A typed endomorphism on state:Context. Maps one site-budget configuration to another. The atomic unit of state mutation in the kernel.
subclass of: Thing

ReversibleEffect
type: Class
comment: An effect that has a well-defined inverse. PinningEffect and PhaseEffect are reversible; ExternalEffect and CompositeEffect are not in general.
subclass of: Effect

PinningEffect
type: Class
comment: Pins a single free site to a definite value. Decrements freeRank by exactly 1. The effect produced by constraint resolution.
subclass of: ReversibleEffect

UnbindingEffect
type: Class
comment: Releases a pinned site back to free state. Increments freeRank by exactly 1. The effect produced by session boundary reset.
subclass of: Effect

PhaseEffect
type: Class
comment: Rotates the reduction phase angle by Ω^k. Does not alter the site budget. The effect produced by reduction step transitions.
subclass of: ReversibleEffect

CompositeEffect
type: Class
comment: An ordered sequence of effects applied atomically. The composition E₁ ; E₂ applies E₁ then E₂.
subclass of: Effect

ExternalEffect
type: Class
comment: An opaque effect declared by a Prism implementation. The kernel treats it as a site-budget transformation satisfying the declared commutation contract. Must carry an effect:externalEffectShape linking to a conformance:EffectShape.
subclass of: Effect

EffectTarget
type: Class
comment: The set of site coordinates that an effect reads or writes. Determines commutation.
subclass of: Thing

DisjointnessWitness
type: Class
comment: A certificate that two EffectTargets have empty intersection, enabling commutative reordering.
subclass of: Thing

effectTarget
type: ObjectProperty
type: FunctionalProperty
comment: The site coordinates this effect touches.
domain: Effect
range: EffectTarget

targetSites
type: ObjectProperty
comment: The individual site coordinates in this target set.
domain: EffectTarget
range: SiteIndex

compositeHead
type: ObjectProperty
type: FunctionalProperty
comment: The first effect in the composite sequence.
domain: CompositeEffect
range: Effect

compositeTail
type: ObjectProperty
comment: Subsequent effects in the composite sequence (ordered by effect:compositeIndex).
domain: CompositeEffect
range: Effect

disjointnessLeft
type: ObjectProperty
type: FunctionalProperty
comment: The left target in the disjointness claim.
domain: DisjointnessWitness
range: EffectTarget

disjointnessRight
type: ObjectProperty
type: FunctionalProperty
comment: The right target in the disjointness claim.
domain: DisjointnessWitness
range: EffectTarget

preContext
type: ObjectProperty
type: FunctionalProperty
comment: The context before effect application.
domain: Effect
range: Context

postContext
type: ObjectProperty
type: FunctionalProperty
comment: The context after effect application.
domain: Effect
range: Context

externalEffectShape
type: ObjectProperty
type: FunctionalProperty
comment: The conformance shape that this external effect satisfies.
domain: ExternalEffect
range: EffectShape

freeRankDelta
type: DatatypeProperty
type: FunctionalProperty
comment: Change in freeRank: −1 for PinningEffect, +1 for UnbindingEffect, 0 for PhaseEffect.
domain: Effect
range: integer

phaseAngleDelta
type: DatatypeProperty
type: FunctionalProperty
comment: The phase rotation applied, expressed as Ω^k.
domain: PhaseEffect
range: decimal

compositeIndex
type: DatatypeProperty
type: FunctionalProperty
comment: Position within a CompositeEffect sequence.
domain: Effect
range: nonNegativeInteger

isCommutativeWith
type: DatatypeProperty
type: FunctionalProperty
comment: Computed: true iff effectTargets are disjoint with the compared effect. Derived from DisjointnessWitness existence.
domain: Effect
range: boolean

targetCardinality
type: DatatypeProperty
type: FunctionalProperty
comment: Number of site coordinates in this target set.
domain: EffectTarget
range: nonNegativeInteger

UOR Predicates and Dispatch
type: Ontology
imports: UOR Operations
imports: UOR Schema
imports: UOR Type System
imports: UOR State
imports: UOR Effect Algebra
imports: UOR Partitions
comment: Boolean-valued functions on kernel objects. Formalizes resolver dispatch, reduction guard evaluation, and conditional resolution paths.
space: kernel

Predicate
type: Class
comment: A total, pure, boolean-valued function on a kernel object. Evaluation terminates for all inputs and produces no side effects.
subclass of: Thing

TypePredicate
type: Class
comment: A predicate over type:TypeDefinition. Used for resolver dispatch.
subclass of: Predicate

StatePredicate
type: Class
comment: A predicate over state:Context or reduction:ReductionState. Used for reduction step guards.
subclass of: Predicate

SitePredicate
type: Class
comment: A predicate over partition:SiteIndex. Used for site-level selection in geodesic resolution.
subclass of: Predicate

DispatchRule
type: Class
comment: A pair (Predicate, Target) where Target is a resolver:Resolver class. The kernel evaluates the predicate; if true, the target resolver is selected.
subclass of: Thing

DispatchTable
type: Class
comment: An ordered set of DispatchRules for a single dispatch point. Must satisfy exhaustiveness and mutual exclusion.
subclass of: Thing

GuardedTransition
type: Class
comment: A triple (StatePredicate, effect:Effect, reduction:ReductionStep). The guard is a StatePredicate; if true, the effect is applied and the reduction advances to the target step.
subclass of: Thing

MatchArm
type: Class
comment: A single case in a pattern match: a Predicate and a result Term. The match evaluates predicates in order and returns the result of the first matching arm.
subclass of: Thing

MatchExpression
type: Class
comment: A term formed by evaluating a sequence of MatchArms. Extends the term language with deterministic conditional evaluation.
subclass of: Term

evaluatesOver
type: ObjectProperty
type: FunctionalProperty
comment: The OWL class of objects this predicate accepts as input.
domain: Predicate
range: Class

dispatchPredicate
type: ObjectProperty
type: FunctionalProperty
comment: The predicate that triggers this dispatch rule.
domain: DispatchRule
range: Predicate

dispatchTarget
type: ObjectProperty
type: FunctionalProperty
comment: The resolver selected when the predicate is satisfied.
domain: DispatchRule
range: Resolver

dispatchRules
type: ObjectProperty
comment: The ordered set of rules in this table.
domain: DispatchTable
range: DispatchRule

guardPredicate
type: ObjectProperty
type: FunctionalProperty
comment: The guard predicate for this transition.
domain: GuardedTransition
range: StatePredicate

guardEffect
type: ObjectProperty
type: FunctionalProperty
comment: The effect applied when the guard is satisfied.
domain: GuardedTransition
range: Effect

guardTarget
type: ObjectProperty
type: FunctionalProperty
comment: The reduction step to advance to.
domain: GuardedTransition
range: ReductionStep

matchArms
type: ObjectProperty
comment: The ordered arms of this match expression.
domain: MatchExpression
range: MatchArm

armPredicate
type: ObjectProperty
type: FunctionalProperty
comment: The predicate guarding this arm.
domain: MatchArm
range: Predicate

armResult
type: ObjectProperty
type: FunctionalProperty
comment: The result term if this arm matches.
domain: MatchArm
range: Term

boundedEvaluator
type: ObjectProperty
comment: A termination witness for user-declared predicates. Kernel predicates are total by construction; user-declared predicates must carry a descent measure certifying termination.
domain: Predicate
range: DescentMeasure

dispatchIndex
type: DatatypeProperty
type: FunctionalProperty
comment: Position in the dispatch table (evaluation order).
domain: DispatchRule
range: nonNegativeInteger

isExhaustive
type: DatatypeProperty
type: FunctionalProperty
comment: True iff the disjunction of all dispatch predicates is a tautology over the input class.
domain: DispatchTable
range: boolean

isMutuallyExclusive
type: DatatypeProperty
type: FunctionalProperty
comment: True iff no two dispatch predicates can be simultaneously true for any input.
domain: DispatchTable
range: boolean

armIndex
type: DatatypeProperty
type: FunctionalProperty
comment: Position in the match expression (evaluation order).
domain: MatchArm
range: nonNegativeInteger

always
type: NamedIndividual
type: Predicate
evaluatesOver: Datum
comment: True on every Datum. Match-arm default catch-all.

never
type: NamedIndividual
type: Predicate
evaluatesOver: Datum
comment: False on every Datum. Disabled-arm marker.

isZero
type: NamedIndividual
type: TypePredicate
evaluatesOver: Datum
comment: True iff the Datum is the additive identity of its ring.

isUnit
type: NamedIndividual
type: TypePredicate
evaluatesOver: Datum
comment: True iff the Datum is the multiplicative identity.

isOdd
type: NamedIndividual
type: TypePredicate
evaluatesOver: Datum
comment: True iff the Datum parity bit is 1.

isEven
type: NamedIndividual
type: TypePredicate
evaluatesOver: Datum
comment: True iff the Datum parity bit is 0.

isInvolution
type: NamedIndividual
type: TypePredicate
evaluatesOver: Datum
comment: True iff op(op(x)) = x for the bound op.

sitePinned
type: NamedIndividual
type: SitePredicate
evaluatesOver: SiteIndex
comment: True iff the named site coordinate is currently pinned.

siteFree
type: NamedIndividual
type: SitePredicate
evaluatesOver: SiteIndex
comment: True iff the named site coordinate is currently free.

contradictionReached
type: NamedIndividual
type: StatePredicate
evaluatesOver: ContradictionBoundary
comment: True iff the resolver has entered a ContradictionBoundary.

budgetExhausted
type: NamedIndividual
type: StatePredicate
evaluatesOver: FreeRank
comment: True iff the FreeRank deficit is zero.

reductionConverged
type: NamedIndividual
type: StatePredicate
evaluatesOver: ReductionState
comment: True iff the reduction fixpoint has been reached.

UOR Parallel Composition
type: Ontology
imports: UOR Operations
imports: UOR Effect Algebra
imports: UOR Monoidal Composition
imports: UOR Partitions
imports: UOR Certificates
imports: UOR Computation Traces
comment: Independent computations over provably disjoint site budgets. Formalizes when execution order is irrelevant.
space: kernel

ParallelProduct
type: Class
comment: A ∥ B: two computations with provably disjoint site targets. Execution order does not affect the result.
subclass of: Thing

DisjointnessCertificate
type: Class
comment: A kernel-produced certificate attesting that the site targets of two computations are disjoint.
subclass of: Certificate

SynchronizationPoint
type: Class
comment: A point where two parallel computations must agree on shared state before proceeding. Only applicable when parallelism is partial.
subclass of: Thing

ParallelTrace
type: Class
comment: A computation trace recording interleaved steps from two parallel computations. Valid iff every interleaving produces the same final context.
subclass of: ComputationTrace

SitePartitioning
type: Class
comment: A partition of the total site budget into disjoint subsets, one per parallel component. The sum of subset cardinalities equals the total site budget.
subclass of: Thing

leftComputation
type: ObjectProperty
type: FunctionalProperty
comment: The left parallel component (itself a sequential computation).
domain: ParallelProduct
range: MonoidalProduct

rightComputation
type: ObjectProperty
type: FunctionalProperty
comment: The right parallel component.
domain: ParallelProduct
range: MonoidalProduct

disjointnessCert
type: ObjectProperty
type: FunctionalProperty
comment: The certificate proving site disjointness.
domain: ParallelProduct
range: DisjointnessCertificate

certLeftTarget
type: ObjectProperty
type: FunctionalProperty
comment: The site target of the left computation.
domain: DisjointnessCertificate
range: EffectTarget

certRightTarget
type: ObjectProperty
type: FunctionalProperty
comment: The site target of the right computation.
domain: DisjointnessCertificate
range: EffectTarget

syncSites
type: ObjectProperty
comment: The shared sites requiring synchronization.
domain: SynchronizationPoint
range: SiteIndex

partitionComponents
type: ObjectProperty
comment: The disjoint subsets composing the full site budget.
domain: SitePartitioning
range: EffectTarget

componentCount
type: DatatypeProperty
type: FunctionalProperty
comment: Number of parallel components.
domain: SitePartitioning
range: positiveInteger

isFullyDisjoint
type: DatatypeProperty
type: FunctionalProperty
comment: True iff site targets have zero overlap (no SynchronizationPoints required).
domain: ParallelProduct
range: boolean

disjointnessCommutation
type: DatatypeProperty
type: FunctionalProperty
comment: Declares whether this parallel product commutes with disjoint effects per FX_4.
domain: ParallelProduct
range: boolean

UOR Productive Streams
type: Ontology
imports: UOR Operations
imports: UOR Euler Reduction
imports: UOR State
imports: UOR Monoidal Composition
imports: UOR Certificates
imports: UOR Transforms and Morphisms
imports: UOR Schema
comment: Coinductive sequences of reduction epochs. Each epoch terminates independently; the stream is the unbounded composition of terminating epochs.
space: kernel

ProductiveStream
type: Class
comment: An unbounded sequence of reduction epochs where each epoch terminates and produces a well-typed output. The coinductive dual of a finite computation.
subclass of: Thing

Epoch
type: Class
comment: A single bounded iteration within a productive stream. Each epoch is a complete reduction execution from Initialization through Convergence.
subclass of: Thing

EpochBoundary
type: Class
comment: The transition point between consecutive epochs. Carries the continuation context and the epoch output.
subclass of: Thing

StreamPrefix
type: Class
comment: A finite prefix of a productive stream: the first k epochs and their outputs. Every finite prefix is computable in finite time.
subclass of: Thing

StreamMorphism
type: Class
comment: A transform between productive streams: maps each epoch of the source to an epoch of the target while preserving the productive property.
subclass of: Transform

Unfold
type: Class
comment: The coinductive constructor: given an initial context and a step function (a morphism:ComputationDatum), produces a ProductiveStream. The step function must be certified to always reach reduction convergence.
subclass of: Thing

epochReduction
type: ObjectProperty
type: FunctionalProperty
comment: The reduction execution for this epoch.
domain: Epoch
range: EulerReduction

epochOutput
type: ObjectProperty
type: FunctionalProperty
comment: The output datum produced by this epoch.
domain: Epoch
range: Datum

epochContext
type: ObjectProperty
type: FunctionalProperty
comment: The context at the start of this epoch.
domain: Epoch
range: Context

boundaryFrom
type: ObjectProperty
type: FunctionalProperty
comment: The epoch that just completed.
domain: EpochBoundary
range: Epoch

boundaryTo
type: ObjectProperty
type: FunctionalProperty
comment: The epoch about to begin.
domain: EpochBoundary
range: Epoch

continuationContext
type: ObjectProperty
type: FunctionalProperty
comment: The context carried across the boundary.
domain: EpochBoundary
range: Context

prefixEpochs
type: ObjectProperty
comment: The epochs in this prefix (ordered by stream:epochIndex).
domain: StreamPrefix
range: Epoch

unfoldSeed
type: ObjectProperty
type: FunctionalProperty
comment: The initial context for the stream.
domain: Unfold
range: Context

unfoldStep
type: ObjectProperty
type: FunctionalProperty
comment: The certified step function producing each epoch.
domain: Unfold
range: ComputationDatum

unfoldResult
type: ObjectProperty
type: FunctionalProperty
comment: The stream produced by this unfold.
domain: Unfold
range: ProductiveStream

epochIndex
type: DatatypeProperty
type: FunctionalProperty
comment: Zero-based index of this epoch in the stream.
domain: Epoch
range: nonNegativeInteger

prefixLength
type: DatatypeProperty
type: FunctionalProperty
comment: Number of epochs in this prefix.
domain: StreamPrefix
range: positiveInteger

isProductive
type: DatatypeProperty
type: FunctionalProperty
comment: Always true by construction: every epoch terminates. Invariant, not computed.
domain: ProductiveStream
range: boolean

UOR Failure Algebra
type: Ontology
imports: UOR Operations
imports: UOR Schema
imports: UOR Effect Algebra
imports: UOR State
imports: UOR Euler Reduction
imports: UOR Certificates
imports: UOR Computation Traces
imports: UOR Proofs
comment: Partial computations, typed failure propagation, and recovery. Formalizes how computations that cannot reach convergence are represented, composed, and recovered from within the ring substrate.
space: kernel

ComputationResult
type: Class
comment: The outcome of a computation that may fail: either a Success carrying a datum, or a Failure carrying a typed reason. The coproduct of the success and failure cases.
subclass of: Thing

Success
type: Class
comment: A computation that reached convergence and produced a valid datum. Carries the output datum and the computation certificate.
subclass of: ComputationResult

Failure
type: Class
comment: A computation that could not reach convergence. Carries a typed FailureReason and the reduction state at the point of failure.
subclass of: ComputationResult

FailureReason
type: Class
comment: A typed classification of why a computation failed.
subclass of: Thing

GuardFailure
type: Class
comment: A reduction step guard evaluated to false and no alternative transition exists.
subclass of: FailureReason

ConstraintContradiction
type: Class
comment: Two constraints in the type’s constraint set are jointly unsatisfiable.
subclass of: FailureReason

SiteExhaustion
type: Class
comment: The linear budget was exhausted before resolution completed.
subclass of: FailureReason

LiftObstructionFailure
type: Class
comment: A WittLift encountered a non-trivial obstruction that could not be resolved.
subclass of: FailureReason

PartialComputation
type: Class
comment: A computation that produces a ComputationResult rather than a guaranteed datum. The general case of all computations.
subclass of: Thing

TotalComputation
type: Class
comment: A computation where the FailureReason type is empty — failure is structurally impossible.
subclass of: PartialComputation

Recovery
type: Class
comment: A strategy for converting a Failure into a Success by modifying the computation path.
subclass of: Thing

FailurePropagation
type: Class
comment: The rule for how failures compose under monoidal and parallel products.
subclass of: Thing

resultDatum
type: ObjectProperty
type: FunctionalProperty
comment: The output datum of a successful computation.
domain: Success
range: Datum

resultCertificate
type: ObjectProperty
type: FunctionalProperty
comment: The certificate attesting the computation’s correctness.
domain: Success
range: ComputationCertificate

failureReason
type: ObjectProperty
type: FunctionalProperty
comment: The typed reason for failure.
domain: Failure
range: FailureReason

failureState
type: ObjectProperty
type: FunctionalProperty
comment: The reduction state at the point of failure.
domain: Failure
range: ReductionState

failureStage
type: ObjectProperty
type: FunctionalProperty
comment: The reduction step where failure occurred.
domain: Failure
range: ReductionStep

recoveryStrategy
type: ObjectProperty
comment: Available recovery strategies for this failure.
domain: Failure
range: Recovery

recoveryEffect
type: ObjectProperty
type: FunctionalProperty
comment: The effect applied to recover from failure.
domain: Recovery
range: Effect

recoveryTarget
type: ObjectProperty
type: FunctionalProperty
comment: The reduction step to retry after recovery.
domain: Recovery
range: ReductionStep

propagationRule
type: ObjectProperty
comment: Which failure reasons propagate through this composition.
domain: FailurePropagation
range: FailureReason

isTotal
type: DatatypeProperty
type: FunctionalProperty
comment: True iff the computation is a TotalComputation (no possible failure path).
domain: PartialComputation
range: boolean

failureDepth
type: DatatypeProperty
type: FunctionalProperty
comment: The reduction step index at which failure occurred.
domain: Failure
range: nonNegativeInteger

UOR Linear Resources
type: Ontology
imports: UOR Operations
imports: UOR Effect Algebra
imports: UOR Partitions
imports: UOR Type System
imports: UOR State
imports: UOR Computation Traces
comment: Linear discipline on site consumption. Formalizes that each site in a complete resolution path is targeted by exactly one effect.
space: kernel

LinearSite
type: Class
comment: A site index annotated with a linearity constraint: must be pinned exactly once in any complete resolution path.
subclass of: SiteIndex

LinearEffect
type: Class
comment: A PinningEffect that consumes its target LinearSite. After application, the site is no longer available for pinning by any subsequent effect.
subclass of: PinningEffect

LinearTrace
type: Class
comment: A computation trace where every site in the budget is targeted by exactly one LinearEffect.
subclass of: ComputationTrace

LinearBudget
type: Class
comment: The multiset of LinearSites available at a given point in resolution. Starts as the full site budget; each LinearEffect removes exactly one element.
subclass of: Thing

LeaseAllocation
type: Class
comment: A binding between a state:ContextLease and a subset of LinearSites. Formalizes what resources a lease claims.
subclass of: Thing

AffineSite
type: Class
comment: A site that may be pinned at most once (but need not be pinned). Relaxation of LinearSite for incomplete resolution paths.
subclass of: SiteIndex

linearTarget
type: ObjectProperty
type: FunctionalProperty
comment: The single site consumed by this effect.
domain: LinearEffect
range: LinearSite

budgetSites
type: ObjectProperty
comment: The sites remaining in the budget.
domain: LinearBudget
range: LinearSite

budgetContext
type: ObjectProperty
type: FunctionalProperty
comment: The context associated with this budget state.
domain: LinearBudget
range: Context

leaseTarget
type: ObjectProperty
comment: The sites claimed by this lease.
domain: LeaseAllocation
range: LinearSite

leaseSource
type: ObjectProperty
type: FunctionalProperty
comment: The ContextLease individual that owns this allocation.
domain: LeaseAllocation
range: ContextLease

leaseAllocation
type: ObjectProperty
type: FunctionalProperty
comment: Links a state:ContextLease to its LinearSite allocation.
domain: ContextLease
range: LeaseAllocation

remainingCount
type: DatatypeProperty
type: FunctionalProperty
comment: Number of unconsumed sites. Equals freeRank on the associated context.
domain: LinearBudget
range: nonNegativeInteger

leaseCardinality
type: DatatypeProperty
type: FunctionalProperty
comment: Number of sites claimed by this lease.
domain: LeaseAllocation
range: positiveInteger

UOR Bounded Recursion
type: Ontology
imports: UOR Operations
imports: UOR Schema
imports: UOR Predicates and Dispatch
imports: UOR Transforms and Morphisms
imports: UOR Effect Algebra
imports: UOR Failure Algebra
imports: UOR Computation Traces
comment: Self-referential computations with well-founded descent measures guaranteeing termination.
space: kernel

BoundedRecursion
type: Class
comment: A self-referential computation parameterized by a descent measure. Each recursive step strictly decreases the measure; the computation terminates when the base case predicate is satisfied.
subclass of: Thing

DescentMeasure
type: Class
comment: A well-founded function from computation state to xsd:nonNegativeInteger. Strictly decreases on every recursive step.
subclass of: Thing

BaseCase
type: Class
comment: The match arm selected when the descent measure reaches zero or the base predicate is satisfied. Produces a direct result without further recursion.
subclass of: MatchArm

RecursiveCase
type: Class
comment: The match arm selected when the descent measure is positive. Applies a decomposition step, decreases the measure, and invokes the recursion.
subclass of: MatchArm

RecursiveStep
type: Class
comment: A single step in a bounded recursion: the decomposition applied before the recursive invocation. Must strictly decrease the DescentMeasure.
subclass of: Thing

RecursionTrace
type: Class
comment: A computation trace recording the sequence of recursive steps, measure values, and the base case result. The trace length is bounded by the initial measure value.
subclass of: ComputationTrace

StructuralRecursion
type: Class
comment: A bounded recursion where the descent measure is the structural size of the input type (site count, constraint count, or operad nesting depth).
subclass of: BoundedRecursion

measure
type: ObjectProperty
type: FunctionalProperty
comment: The descent measure guaranteeing termination.
domain: BoundedRecursion
range: DescentMeasure

baseCase
type: ObjectProperty
type: FunctionalProperty
comment: The base case match arm.
domain: BoundedRecursion
range: BaseCase

recursiveCase
type: ObjectProperty
type: FunctionalProperty
comment: The recursive case match arm.
domain: BoundedRecursion
range: RecursiveCase

stepDecomposition
type: ObjectProperty
type: FunctionalProperty
comment: The transform applied to decompose the input before recursion.
domain: RecursiveStep
range: Transform

stepMeasurePre
type: ObjectProperty
type: FunctionalProperty
comment: The measure value before this step.
domain: RecursiveStep
range: DescentMeasure

stepMeasurePost
type: ObjectProperty
type: FunctionalProperty
comment: The measure value after this step.
domain: RecursiveStep
range: DescentMeasure

basePredicate
type: ObjectProperty
type: FunctionalProperty
comment: The predicate that identifies the base case.
domain: BoundedRecursion
range: Predicate

recursionBody
type: ObjectProperty
type: FunctionalProperty
comment: The computation applied at each recursive step (may reference itself via its content address).
domain: BoundedRecursion
range: ComputationDatum

initialMeasure
type: DatatypeProperty
type: FunctionalProperty
comment: The measure value at the start of recursion. Bounds the maximum recursion depth.
domain: BoundedRecursion
range: nonNegativeInteger

measureValue
type: DatatypeProperty
type: FunctionalProperty
comment: The current value of the descent measure.
domain: DescentMeasure
range: nonNegativeInteger

UOR Address Regions
type: Ontology
imports: UOR Operations
imports: UOR Content Addressing
imports: UOR Schema
imports: UOR Type System
imports: UOR Partitions
imports: UOR Observables
comment: Spatial locality of content-addressed ring elements. Defines computable working sets for resolver stages, independent of physical memory layout.
space: kernel

AddressRegion
type: Class
comment: A contiguous range of u:Element values accessible during a single reduction step. Defines the resolver’s working set.
subclass of: Thing

RegionBound
type: Class
comment: The boundary of an AddressRegion: a pair (lowerAddress, upperAddress) in the content-address ordering.
subclass of: Thing

LocalityMetric
type: Class
comment: A metric on u:Element values determining which addresses are near each other for resolution purposes.
subclass of: MetricObservable

WorkingSet
type: Class
comment: The set of AddressRegions needed by a resolver at a specific reduction step for a specific type. Computable from the type’s constraint nerve.
subclass of: Thing

RegionAllocation
type: Class
comment: An assignment of AddressRegions to reduction steps for a given computation. Enables Prism to pre-compute memory layout.
subclass of: Thing

regionLower
type: ObjectProperty
type: FunctionalProperty
comment: The lower bound of the address range.
domain: RegionBound
range: Element

regionUpper
type: ObjectProperty
type: FunctionalProperty
comment: The upper bound of the address range.
domain: RegionBound
range: Element

regionBound
type: ObjectProperty
type: FunctionalProperty
comment: The boundary of this region.
domain: AddressRegion
range: RegionBound

localityMetric
type: ObjectProperty
type: FunctionalProperty
comment: The metric defining contiguity for this region.
domain: AddressRegion
range: LocalityMetric

workingSetRegions
type: ObjectProperty
comment: The address regions composing this working set.
domain: WorkingSet
range: AddressRegion

workingSetStage
type: ObjectProperty
type: FunctionalProperty
comment: The reduction step this working set applies to.
domain: WorkingSet
range: ReductionStep

workingSetType
type: ObjectProperty
type: FunctionalProperty
comment: The type being resolved.
domain: WorkingSet
range: TypeDefinition

allocationStage
type: ObjectProperty
comment: Reduction steps in this allocation.
domain: RegionAllocation
range: ReductionStep

allocationWorkingSet
type: ObjectProperty
comment: The working sets assigned to each stage.
domain: RegionAllocation
range: WorkingSet

regionCardinality
type: DatatypeProperty
type: FunctionalProperty
comment: Number of addresses in this region.
domain: AddressRegion
range: nonNegativeInteger

workingSetSize
type: DatatypeProperty
type: FunctionalProperty
comment: Total addresses across all regions in the working set.
domain: WorkingSet
range: nonNegativeInteger

UOR IO Boundary
type: Ontology
imports: UOR Operations
imports: UOR Schema
imports: UOR Effect Algebra
imports: UOR Transforms and Morphisms
imports: UOR Type System
imports: UOR State
imports: UOR Certificates
comment: Typed interface between kernel computation and the external world. Formalizes how data enters and exits the ring substrate.
space: bridge

IOBoundary
type: Class
comment: A typed interface point between the kernel and the external world. Every data flow into or out of the ring crosses exactly one IOBoundary.
subclass of: Thing

Source
type: Class
comment: A typed source of external data entering the ring. Carries an expected TypeDefinition describing the shape of incoming data.
subclass of: IOBoundary

Sink
type: Class
comment: A typed destination for data leaving the ring. Carries an expected TypeDefinition describing the shape of outgoing data.
subclass of: IOBoundary

BoundaryEffect
type: Class
comment: An effect that crosses the kernel/external boundary. Specializes effect:ExternalEffect with explicit source or sink binding.
subclass of: ExternalEffect

IngestEffect
type: Class
comment: A BoundaryEffect that reads from a Source and produces a datum in the ring.
subclass of: BoundaryEffect

EmitEffect
type: Class
comment: A BoundaryEffect that writes a ring datum to a Sink.
subclass of: BoundaryEffect

BoundaryProtocol
type: Class
comment: A specification of the data shape, ordering, and framing constraints for data crossing a boundary.
subclass of: Thing

BoundarySession
type: Class
comment: A Session that includes BoundaryEffects. Extends the session model to track which boundaries were crossed.
subclass of: Session

sourceType
type: ObjectProperty
type: FunctionalProperty
comment: The expected type of data arriving from this source.
domain: Source
range: TypeDefinition

sinkType
type: ObjectProperty
type: FunctionalProperty
comment: The expected type of data departing through this sink.
domain: Sink
range: TypeDefinition

sourceGrounding
type: ObjectProperty
type: FunctionalProperty
comment: The grounding map that transforms incoming surface data to ring datums.
domain: Source
range: GroundingMap

sinkProjection
type: ObjectProperty
type: FunctionalProperty
comment: The projection map that transforms ring datums to outgoing surface data.
domain: Sink
range: ProjectionMap

effectBoundary
type: ObjectProperty
type: FunctionalProperty
comment: The boundary this effect crosses.
domain: BoundaryEffect
range: IOBoundary

ingestSource
type: ObjectProperty
type: FunctionalProperty
comment: The source being read.
domain: IngestEffect
range: Source

emitSink
type: ObjectProperty
type: FunctionalProperty
comment: The sink being written to.
domain: EmitEffect
range: Sink

protocolType
type: ObjectProperty
type: FunctionalProperty
comment: The type specification for boundary data.
domain: BoundaryProtocol
range: TypeDefinition

protocolOrdering
type: ObjectProperty
type: FunctionalProperty
comment: Sequencing constraints on boundary data.
domain: BoundaryProtocol
range: CompositeConstraint

sessionBoundaries
type: ObjectProperty
comment: The boundaries crossed during this session.
domain: BoundarySession
range: IOBoundary

isIdempotent
type: DatatypeProperty
type: FunctionalProperty
comment: True iff applying the boundary effect twice produces the same result as applying it once.
domain: BoundaryEffect
range: boolean

crossingCount
type: DatatypeProperty
type: FunctionalProperty
comment: Total number of boundary crossings in this session.
domain: BoundarySession
range: nonNegativeInteger

UOR Conformance Shapes
type: Ontology
imports: UOR Schema
imports: UOR Type System
imports: UOR Operations
imports: UOR Effect Algebra
imports: UOR Predicates and Dispatch
imports: UOR Parallel Composition
imports: UOR Productive Streams
imports: UOR Linear Resources
imports: UOR Address Regions
imports: UOR Failure Algebra
imports: UOR Bounded Recursion
imports: UOR IO Boundary
imports: UOR Euler Reduction
imports: UOR Certificates
imports: UOR Computation Traces
imports: UOR State
imports: UOR Transforms and Morphisms
comment: SHACL-equivalent constraint shapes defining what a Prism implementation must provide at each extension point. Machine-verifiable contracts.
space: bridge

Shape
type: Class
comment: A constraint shape that a Prism-declared extension must satisfy. Analogous to sh:NodeShape in SHACL.
subclass of: Thing

PropertyConstraint
type: Class
comment: A single required property within a shape: the property URI, its expected range, minimum and maximum cardinality.
subclass of: Thing

WittLevelShape
type: Class
comment: Shape for declaring a new WittLevel beyond Q3.
subclass of: Shape

EffectShape
type: Class
comment: Shape for declaring an ExternalEffect.
subclass of: Shape

ParallelShape
type: Class
comment: Shape for declaring a ParallelProduct.
subclass of: Shape

StreamShape
type: Class
comment: Shape for declaring a ProductiveStream (targets stream:Unfold, the coinductive constructor).
subclass of: Shape

DispatchShape
type: Class
comment: Shape for declaring a new DispatchRule in a DispatchTable.
subclass of: Shape

LeaseShape
type: Class
comment: Shape for declaring a Lease with LinearSite allocation.
subclass of: Shape

GroundingShape
type: Class
comment: Shape for declaring a GroundingMap from surface data to the ring.
subclass of: Shape

ValidationResult
type: Class
comment: The result of validating an extension against a shape: conforms (boolean), and violation details if non-conformant.
subclass of: Thing

PredicateShape
type: Class
comment: Shape for user-declared predicates. Requires a bounded evaluator (termination witness) and input type declaration.
subclass of: Shape

WitnessDatum
type: Class
comment: Opaque ring element witness. Cannot be constructed outside the foundation crate — only produced by reduction evaluation or the two-phase minting boundary.
subclass of: Thing

GroundedCoordinate
type: Class
comment: Boundary crossing intermediate for a single grounded coordinate value. Not a WitnessDatum — must be validated and minted by the foundation.
subclass of: Thing

GroundedTuple
type: Class
comment: Boundary crossing intermediate for a fixed-size array of GroundedCoordinate values. Stack-resident, no heap allocation.
subclass of: Thing

GroundedValueMarker
type: Class
comment: Sealed marker trait class. Implemented only for GroundedCoordinate and GroundedTuple. Prevents downstream crates from substituting arbitrary types.
subclass of: Thing

ValidatedWrapper
type: Class
comment: Generic validation-proof wrapper. Proves that the inner value was produced by the conformance checker, not fabricated by Prism code.
subclass of: Thing

WitnessDerivation
type: Class
comment: Opaque derivation trace that can only be extended by the rewrite engine. Records rewrite step count and root term content address.
subclass of: Thing

WitnessSiteBudget
type: Class
comment: Opaque site budget that can only be decremented by PinningEffect and incremented by UnbindingEffect — never by direct mutation.
subclass of: Thing

ShapeViolationReport
type: Class
comment: Structured violation diagnostic carrying the shape IRI, constraint IRI, property IRI, expected range, cardinality bounds, and violation kind.
subclass of: Thing

ViolationKind
type: Class
comment: The kind of shape violation: Missing, TypeMismatch, CardinalityViolation, ValueCheck, or LevelMismatch.
subclass of: Thing

CompileUnitBuilder
type: Class
comment: Builder for CompileUnit admission. Collects rootTerm, quantumLevelCeiling, thermodynamicBudget, and targetDomains. Validates against CompileUnitShape.
subclass of: Thing

EffectDeclaration
type: Class
comment: Builder for EffectShape. Collects effect name, target sites, budget delta, and commutation flag.
subclass of: Thing

GroundingDeclaration
type: Class
comment: Builder for GroundingShape. Collects source type, ring mapping, and invertibility contract.
subclass of: Thing

DispatchDeclaration
type: Class
comment: Builder for DispatchShape. Collects predicate, target resolver, and dispatch priority.
subclass of: Thing

LeaseDeclaration
type: Class
comment: Builder for LeaseShape. Collects linear site and lease scope.
subclass of: Thing

StreamDeclaration
type: Class
comment: Builder for StreamShape. Collects unfold seed, step term, and productivity witness.
subclass of: Thing

PredicateDeclaration
type: Class
comment: Builder for PredicateShape. Collects input type, evaluator term, and termination witness.
subclass of: Thing

ParallelDeclaration
type: Class
comment: Builder for ParallelShape. Collects site partition and disjointness witness.
subclass of: Thing

WittLevelDeclaration
type: Class
comment: Builder for WittLevelShape. Collects declared bit width, cycle size, and predecessor level.
subclass of: Thing

MintingSession
type: Class
comment: Boundary session state tracker. Records crossing count and idempotency flag for the two-phase minting boundary.
subclass of: Thing

targetClass
type: ObjectProperty
type: FunctionalProperty
comment: The OWL class that instances of this shape must belong to.
domain: Shape
range: Class

requiredProperty
type: ObjectProperty
comment: A required property in this shape.
domain: Shape
range: PropertyConstraint

constraintProperty
type: ObjectProperty
type: FunctionalProperty
comment: The property URI that must be present.
domain: PropertyConstraint
range: Thing

constraintRange
type: ObjectProperty
type: FunctionalProperty
comment: The expected range of the required property.
domain: PropertyConstraint
range: Class

validationShape
type: ObjectProperty
type: FunctionalProperty
comment: The shape that was validated against.
domain: ValidationResult
range: Shape

validationTarget
type: ObjectProperty
type: FunctionalProperty
comment: The instance that was validated.
domain: ValidationResult
range: Thing

minCount
type: DatatypeProperty
type: FunctionalProperty
comment: Minimum cardinality of the required property.
domain: PropertyConstraint
range: nonNegativeInteger

maxCount
type: DatatypeProperty
type: FunctionalProperty
comment: Maximum cardinality (0 = unbounded).
domain: PropertyConstraint
range: nonNegativeInteger

conforms
type: DatatypeProperty
type: FunctionalProperty
comment: True iff the target satisfies all constraints of the shape.
domain: ValidationResult
range: boolean

witnessLevel
type: DatatypeProperty
type: FunctionalProperty
comment: The quantum level at which this witness datum was minted.
domain: WitnessDatum
range: nonNegativeInteger

witnessBytes
type: DatatypeProperty
type: FunctionalProperty
comment: The raw byte representation of this witness datum.
domain: WitnessDatum
range: hexBinary

coordinateLevel
type: ObjectProperty
type: FunctionalProperty
comment: The quantum level tag of this grounded coordinate.
domain: GroundedCoordinate
range: WittLevel

validatedInner
type: ObjectProperty
type: FunctionalProperty
comment: The validated inner value wrapped by this proof.
domain: ValidatedWrapper
range: Thing

shapeIri
type: DatatypeProperty
type: FunctionalProperty
comment: IRI of the conformance:Shape that was validated against.
domain: ShapeViolationReport
range: string

constraintIri
type: DatatypeProperty
type: FunctionalProperty
comment: IRI of the specific PropertyConstraint that failed.
domain: ShapeViolationReport
range: string

propertyIri
type: DatatypeProperty
type: FunctionalProperty
comment: IRI of the property that was missing or invalid.
domain: ShapeViolationReport
range: string

expectedRange
type: DatatypeProperty
type: FunctionalProperty
comment: The expected range class IRI for the violated property.
domain: ShapeViolationReport
range: string

violationMinCount
type: DatatypeProperty
type: FunctionalProperty
comment: The minimum cardinality from the violated constraint.
domain: ShapeViolationReport
range: nonNegativeInteger

violationMaxCount
type: DatatypeProperty
type: FunctionalProperty
comment: The maximum cardinality from the violated constraint (0 = unbounded).
domain: ShapeViolationReport
range: nonNegativeInteger

violationKind
type: ObjectProperty
type: FunctionalProperty
comment: The kind of violation that occurred.
domain: ShapeViolationReport
range: ViolationKind

builderRootTerm
type: ObjectProperty
type: FunctionalProperty
comment: The root term expression for the CompileUnit.
domain: CompileUnitBuilder
range: Term

builderWittLevelCeiling
type: ObjectProperty
type: FunctionalProperty
comment: The widest quantum level the computation may reference.
domain: CompileUnitBuilder
range: WittLevel

builderThermodynamicBudget
type: DatatypeProperty
type: FunctionalProperty
comment: Landauer-bounded energy budget in kBT ln 2 units.
domain: CompileUnitBuilder
range: decimal

builderTargetDomains
type: ObjectProperty
comment: Verification domains targeted by the CompileUnit.
domain: CompileUnitBuilder
range: VerificationDomain

effectName
type: DatatypeProperty
type: FunctionalProperty
comment: The name of the declared effect.
domain: EffectDeclaration
range: string

targetSites
type: DatatypeProperty
comment: Site coordinates this effect reads or writes.
domain: EffectDeclaration
range: nonNegativeInteger

budgetDelta
type: DatatypeProperty
type: FunctionalProperty
comment: The site budget delta (positive = increment, negative = decrement).
domain: EffectDeclaration
range: integer

commutationFlag
type: DatatypeProperty
type: FunctionalProperty
comment: Whether this effect commutes with effects on disjoint sites.
domain: EffectDeclaration
range: boolean

groundingSourceType
type: ObjectProperty
type: FunctionalProperty
comment: The source type of incoming external data.
domain: GroundingDeclaration
range: TypeDefinition

ringMapping
type: DatatypeProperty
type: FunctionalProperty
comment: Description of the mapping from surface data to ring.
domain: GroundingDeclaration
range: string

invertibilityContract
type: DatatypeProperty
type: FunctionalProperty
comment: Whether the grounding map is invertible.
domain: GroundingDeclaration
range: boolean

dispatchPredicate
type: ObjectProperty
type: FunctionalProperty
comment: The predicate expression guarding this dispatch rule.
domain: DispatchDeclaration
range: PredicateExpression

targetResolver
type: ObjectProperty
type: FunctionalProperty
comment: The resolver to dispatch to when the predicate holds.
domain: DispatchDeclaration
range: Resolver

dispatchPriority
type: DatatypeProperty
type: FunctionalProperty
comment: Priority ordering for this dispatch rule (lower = first).
domain: DispatchDeclaration
range: nonNegativeInteger

linearSite
type: DatatypeProperty
type: FunctionalProperty
comment: The site coordinate allocated linearly by this lease.
domain: LeaseDeclaration
range: nonNegativeInteger

leaseScope
type: DatatypeProperty
type: FunctionalProperty
comment: The scope within which this lease is valid.
domain: LeaseDeclaration
range: string

unfoldSeed
type: ObjectProperty
type: FunctionalProperty
comment: The seed term for the stream unfold constructor.
domain: StreamDeclaration
range: Term

stepTerm
type: ObjectProperty
type: FunctionalProperty
comment: The step function term for the stream unfold.
domain: StreamDeclaration
range: Term

productivityWitness
type: DatatypeProperty
type: FunctionalProperty
comment: Evidence that the stream is productive (always produces a next element).
domain: StreamDeclaration
range: string

predicateInputType
type: ObjectProperty
type: FunctionalProperty
comment: The input type for the declared predicate.
domain: PredicateDeclaration
range: TypeDefinition

evaluatorTerm
type: ObjectProperty
type: FunctionalProperty
comment: The evaluator term for the declared predicate.
domain: PredicateDeclaration
range: Term

terminationWitness
type: DatatypeProperty
type: FunctionalProperty
comment: Evidence that the predicate evaluator terminates on all inputs.
domain: PredicateDeclaration
range: string

sitePartition
type: ObjectProperty
type: FunctionalProperty
comment: The site partition for the parallel composition.
domain: ParallelDeclaration
range: Partition

disjointnessWitness
type: DatatypeProperty
type: FunctionalProperty
comment: Evidence that the site partition components are pairwise disjoint.
domain: ParallelDeclaration
range: string

declaredBitWidth
type: DatatypeProperty
type: FunctionalProperty
comment: The declared bit width for this quantum level.
domain: WittLevelDeclaration
range: positiveInteger

declaredCycleSize
type: DatatypeProperty
type: FunctionalProperty
comment: The declared number of ring states at this level.
domain: WittLevelDeclaration
range: nonNegativeInteger

predecessorLevel
type: ObjectProperty
type: FunctionalProperty
comment: The predecessor quantum level in the chain.
domain: WittLevelDeclaration
range: WittLevel

sessionCrossingCount
type: DatatypeProperty
type: FunctionalProperty
comment: Total boundary crossings in this minting session.
domain: MintingSession
range: nonNegativeInteger

sessionIsIdempotent
type: DatatypeProperty
type: FunctionalProperty
comment: Whether applying this session's boundary effect twice equals applying it once.
domain: MintingSession
range: boolean

CompileUnitShape
type: NamedIndividual
type: Shape
requiredProperty: compileUnit_rootTerm_constraint
requiredProperty: compileUnit_unitWittLevel_constraint
requiredProperty: compileUnit_thermodynamicBudget_constraint
requiredProperty: compileUnit_targetDomains_constraint
targetClass: CompileUnit
comment: Shape validating that a CompileUnit carries all required properties before reduction admission. The unitAddress property is NOT required — it is computed by stage_initialization after shape validation passes.

compileUnit_rootTerm_constraint
type: NamedIndividual
type: PropertyConstraint
constraintProperty: rootTerm
constraintRange: Term
maxCount: 1
minCount: 1
comment: Exactly one root term is required. Range is schema:Term.

compileUnit_unitWittLevel_constraint
type: NamedIndividual
type: PropertyConstraint
constraintProperty: unitWittLevel
constraintRange: WittLevel
maxCount: 1
minCount: 1
comment: Exactly one quantum level is required. Range is schema:WittLevel.

compileUnit_thermodynamicBudget_constraint
type: NamedIndividual
type: PropertyConstraint
constraintProperty: thermodynamicBudget
constraintRange: decimal
maxCount: 1
minCount: 1
comment: Exactly one thermodynamic budget is required. Shape validates presence and type; the BudgetSolvencyCheck preflight validates the value against the Landauer bound.

compileUnit_targetDomains_constraint
type: NamedIndividual
type: PropertyConstraint
constraintProperty: targetDomains
constraintRange: VerificationDomain
maxCount: 0
minCount: 1
comment: At least one target verification domain is required. maxCount 0 means unbounded.

Missing
type: NamedIndividual
type: ViolationKind
comment: Required property was not set on the builder.

TypeMismatch
type: NamedIndividual
type: ViolationKind
comment: Property was set but its value is not an instance of the constraintRange.

CardinalityViolation
type: NamedIndividual
type: ViolationKind
comment: Cardinality violated: too few or too many values provided.

ValueCheck
type: NamedIndividual
type: ViolationKind
comment: Value-dependent check failed (Tier 2). For example, thermodynamic budget insufficient for Landauer bound.

LevelMismatch
type: NamedIndividual
type: ViolationKind
comment: A term's quantum level annotation exceeds the CompileUnit ceiling, or binary operation operands are at different levels without an intervening lift or project.