UOR Foundation
type: Ontology
version info: 1.0.0

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 represented as Braille strings. Each address is a sequence of Braille glyphs encoding a unique content-derived identifier.
space: kernel

Address
type: Class
comment: A content-addressable identifier represented as a Braille string. Each Address uniquely identifies a piece of content via its Braille-encoded hash.
subclass of: Thing

Glyph
type: Class
comment: A single Braille cell encoding 6 bits of an address. The bijection between Braille cells and 6-bit values is the foundational encoding of the UOR content-addressing scheme.
subclass of: Thing

glyph
type: DatatypeProperty
type: FunctionalProperty
comment: The Braille string representation of an address. Each character in the string is a Braille glyph (cell).
domain: Address
range: string

codepoint
type: DatatypeProperty
type: FunctionalProperty
comment: The Unicode codepoint of a Braille glyph. Braille glyphs occupy the range U+2800–U+28FF.
domain: Glyph
range: nonNegativeInteger

byteValue
type: DatatypeProperty
type: FunctionalProperty
comment: The 6-bit integer value (0–63) encoded by this Braille glyph. The bijection maps each of the 64 Braille patterns to a unique 6-bit value.
domain: Glyph
range: nonNegativeInteger

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

UOR Schema
type: Ontology
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
imports: UOR Content Addressing

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

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

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

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 quantum 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

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

quantum
type: DatatypeProperty
type: FunctionalProperty
comment: The quantum 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: string

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

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

ringQuantum
type: DatatypeProperty
type: FunctionalProperty
comment: The bit width n of the ring Z/(2^n)Z. Distinct from schema:quantum on Datum — ringQuantum is the container's bit width; datum quantum 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 ringQuantum.
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

π₁
type: NamedIndividual
type: Datum
comment: The unique generator of R_n under successor. Value = 1 at every quantum 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 quantum level. op:add(x, zero) = x for all x in R_n.
value: 0

UOR Operations
type: Ontology
comment: Ring operations, involutions, algebraic identities, and the dihedral symmetry group D_{2^n} generated by neg and bnot.
space: kernel
imports: UOR Schema

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

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

geometricCharacter
type: DatatypeProperty
type: FunctionalProperty
comment: A description of the geometric role of this operation in the UOR ring and hypercube geometry. Examples: 'ring_reflection', 'hypercube_reflection', 'rotation', 'translation', 'scaling'.
domain: Operation
range: string

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 operation of an algebraic identity.
domain: Identity
range: Operation

rhs
type: ObjectProperty
type: FunctionalProperty
comment: The right-hand side operation of an algebraic identity.
domain: Identity
range: Operation

forAll
type: DatatypeProperty
type: FunctionalProperty
comment: The quantifier scope: the variable(s) over which this algebraic identity holds (e.g., 'x ∈ R_n').
domain: Identity
range: string

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). Annotation only — not used for reasoning. Example: ⟨r, s | r^{2^n} = s² = e, srs = r⁻¹⟩
domain: Group
range: string

neg
type: NamedIndividual
type: Involution
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).
arity: 1
geometricCharacter: ring_reflection

bnot
type: NamedIndividual
type: Involution
comment: Hypercube reflection: bnot(x) = (2^n - 1) ⊕ x (bitwise complement). The second generator of D_{2^n}. bnot(bnot(x)) = x.
arity: 1
geometricCharacter: hypercube_reflection

succ
type: NamedIndividual
type: UnaryOp
comment: Successor: succ(x) = neg(bnot(x)) = (x + 1) mod 2^n. The critical identity: succ is the composition neg ∘ bnot.
arity: 1
geometricCharacter: rotation
composedOf: neg
composedOf: bnot
inverse: pred

pred
type: NamedIndividual
type: UnaryOp
comment: Predecessor: pred(x) = bnot(neg(x)) = (x - 1) mod 2^n. The inverse of succ. pred is the composition bnot ∘ neg.
arity: 1
geometricCharacter: rotation_inverse
composedOf: bnot
composedOf: neg
inverse: succ

add
type: NamedIndividual
type: BinaryOp
comment: Ring addition: add(x, y) = (x + y) mod 2^n. Commutative, associative; identity element is 0.
arity: 2
geometricCharacter: translation
commutative: true
associative: true
identity: 0
inverse: sub

sub
type: NamedIndividual
type: BinaryOp
comment: Ring subtraction: sub(x, y) = (x - y) mod 2^n. Not commutative, not associative.
arity: 2
geometricCharacter: translation
commutative: false
associative: false

mul
type: NamedIndividual
type: BinaryOp
comment: Ring multiplication: mul(x, y) = (x × y) mod 2^n. Commutative, associative; identity element is 1.
arity: 2
geometricCharacter: scaling
commutative: true
associative: true
identity: 1

xor
type: NamedIndividual
type: BinaryOp
comment: Bitwise exclusive or: xor(x, y) = x ⊕ y. Commutative, associative; identity element is 0.
arity: 2
geometricCharacter: hypercube_translation
commutative: true
associative: true
identity: 0

and
type: NamedIndividual
type: BinaryOp
comment: Bitwise and: and(x, y) = x ∧ y. Commutative, associative.
arity: 2
geometricCharacter: hypercube_projection
commutative: true
associative: true

or
type: NamedIndividual
type: BinaryOp
comment: Bitwise or: or(x, y) = x ∨ y. Commutative, associative.
arity: 2
geometricCharacter: hypercube_join
commutative: true
associative: true

Critical Identity
type: NamedIndividual
type: Identity
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.
lhs: succ
rhs: neg
rhs: bnot
forAll: x ∈ R_n

D_{2^n}
type: NamedIndividual
type: DihedralGroup
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.
generatedBy: neg
generatedBy: bnot
presentation: ⟨r, s | r^{2^n} = s² = e, srs = r⁻¹⟩

UOR Queries
type: Ontology
comment: Information extraction queries. Users initiate queries; the kernel resolves them against the ring substrate.
space: bridge
imports: UOR Schema
imports: UOR Content Addressing

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

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

coordinate
type: DatatypeProperty
type: FunctionalProperty
comment: Which coordinate the query extracts.
domain: CoordinateQuery
range: string

UOR Resolvers
type: Ontology
comment: Type resolution strategies implementing the partition map Π : T_n → Part(R_n). Resolvers transform type declarations into ring partitions.
space: bridge
imports: UOR Schema
imports: UOR Queries

Resolver
type: Class
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

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: DatatypeProperty
type: FunctionalProperty
comment: A human-readable description of the resolution strategy this resolver implements.
domain: Resolver
range: string

complexity
type: DatatypeProperty
type: FunctionalProperty
comment: The computational complexity of this resolver, expressed as a big-O string (e.g., 'O(n)', 'O(2^n)').
domain: Resolver
range: string

UOR Type System
type: Ontology
comment: Runtime type declarations that parameterize the resolution pipeline. Types are declared by Prism applications and resolved to partitions of the ring.
space: user
imports: UOR Schema
imports: UOR Content Addressing

TypeDefinition
type: Class
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

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

constraint
type: DatatypeProperty
comment: The constraint predicate applied to the base type. Expressed as a string in the Prism constraint language.
domain: ConstrainedType
range: string

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: Address

UOR Partitions
type: Ontology
comment: Irreducibility partitions produced by type resolution. A partition divides the ring into four disjoint components: Irreducible, Reducible, Units, and Exterior.
space: bridge
imports: UOR Schema
imports: UOR Type System

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
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
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
disjoint with: ReducibleSet
disjoint with: UnitSet
disjoint with: ExteriorSet

ReducibleSet
type: Class
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
disjoint with: IrreducibleSet
disjoint with: UnitSet
disjoint with: ExteriorSet

UnitSet
type: Class
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
disjoint with: IrreducibleSet
disjoint with: ReducibleSet
disjoint with: ExteriorSet

ExteriorSet
type: Class
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
disjoint with: IrreducibleSet
disjoint with: ReducibleSet
disjoint with: UnitSet

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: UnitSet

exterior
type: ObjectProperty
type: FunctionalProperty
comment: The exterior component of this partition.
domain: Partition
range: ExteriorSet

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

quantum
type: DatatypeProperty
type: FunctionalProperty
comment: The quantum level n at which this partition was computed. The ring has 2^n elements at this level.
domain: Partition
range: positiveInteger

UOR Observables
type: Ontology
comment: Observable quantities and metrics computed by the UOR kernel. Includes ring-metric, Hamming-metric, curvature, holonomy, and catastrophe-theoretic observables.
space: bridge
imports: UOR Schema
imports: UOR Partitions

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

CascadeObservable
type: Class
comment: An observable measuring cascade 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

CascadeLength
type: Class
comment: The number of operation applications in an operation cascade.
subclass of: CascadeObservable

CascadeCount
type: Class
comment: The number of distinct cascades in a computation.
subclass of: CascadeObservable

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

value
type: DatatypeProperty
type: FunctionalProperty
comment: The numeric value of an observable measurement.
domain: Observable
range: decimal

unit
type: DatatypeProperty
type: FunctionalProperty
comment: The unit of measurement for this observable (e.g., 'bits', 'ring_steps', 'dimensionless').
domain: Observable
range: string

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

UOR Proofs
type: Ontology
comment: Kernel-produced verification proofs attesting to algebraic properties of UOR objects and operations.
space: bridge
imports: UOR Schema
imports: UOR Operations

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

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: Proof

WitnessData
type: Class
comment: Supporting data for a proof: specific examples, counter-examples checked, or intermediate computation results.
subclass of: Thing

quantum
type: DatatypeProperty
type: FunctionalProperty
comment: The quantum level at which this proof was verified.
domain: Proof
range: positiveInteger

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: DatatypeProperty
type: FunctionalProperty
comment: Human-readable statement of the critical identity proven. E.g., 'neg(bnot(x)) = succ(x) for all x in R_n'.
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

UOR Derivations
type: Ontology
comment: Computation witnesses recording term rewriting sequences from original terms to their canonical forms.
space: bridge
imports: UOR Schema
imports: UOR Operations

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

RewriteStep
type: Class
comment: A single rewrite step in a derivation: the application of one rewrite rule to transform a term.
subclass of: Thing

TermMetrics
type: Class
comment: Metrics describing the size and complexity of a term.
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

rule
type: DatatypeProperty
type: FunctionalProperty
comment: The rewrite rule applied in this step (e.g., 'critical_identity', 'involution', 'associativity').
domain: RewriteStep
range: string

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

UOR Computation Traces
type: Ontology
comment: Execution traces recording the sequence of kernel operations, intermediate results, and accumulated metrics for a computation.
space: bridge
imports: UOR Schema
imports: UOR Operations

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

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

UOR Certificates
type: Ontology
comment: Kernel-produced attestation certificates for transforms, isometries, and involutions. Each certificate verifies that a specific structural property holds.
space: bridge
imports: UOR Operations
imports: UOR Proofs

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

transformType
type: DatatypeProperty
type: FunctionalProperty
comment: The type of transform this certificate attests to (e.g., 'isometry', 'embedding', 'action').
domain: TransformCertificate
range: string

method
type: DatatypeProperty
type: FunctionalProperty
comment: The verification method used to produce this certificate (e.g., 'exhaustive_check', 'symbolic_proof', 'sampling').
domain: Certificate
range: string

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

quantum
type: DatatypeProperty
type: FunctionalProperty
comment: The quantum 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

UOR Transforms and Morphisms
type: Ontology
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
imports: UOR Schema
imports: UOR Type System
imports: UOR Operations
imports: UOR Observables
imports: UOR Partitions
imports: UOR Computation Traces

Transform
type: Class
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
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
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

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

UOR State
type: Ontology
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
imports: UOR Content Addressing
imports: UOR Schema
imports: UOR Type System
imports: UOR Computation Traces

Context
type: Class
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
disjoint with: Binding
disjoint with: Frame
disjoint with: Transition

Binding
type: Class
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
disjoint with: Context
disjoint with: Frame
disjoint with: Transition

Frame
type: Class
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
disjoint with: Context
disjoint with: Binding
disjoint with: Transition

Transition
type: Class
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
disjoint with: Context
disjoint with: Binding
disjoint with: Frame

address
type: ObjectProperty
type: FunctionalProperty
comment: The UOR address being bound in this binding.
domain: Binding
range: Address

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

quantum
type: DatatypeProperty
type: FunctionalProperty
comment: The quantum 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: DatatypeProperty
comment: The constraint determining which bindings from the context are visible in this frame.
domain: Frame
range: string

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