UOR Transforms and Morphisms

IRI: https://uor.foundation/morphism/
Prefix: morphism:
Space: user
Comment: Runtime abstractions for maps between UOR objects: transforms, isometries, embeddings, and group actions. The foundation provides the vocabulary; Prism writes the sentences.

Imports

https://uor.foundation/schema/
https://uor.foundation/type/
https://uor.foundation/op/
https://uor.foundation/observable/
https://uor.foundation/partition/
https://uor.foundation/trace/
https://uor.foundation/homology/
https://uor.foundation/u/
https://uor.foundation/derivation/
https://uor.foundation/cert/

Classes

Name	IRI	Subclass Of	Disjoint With	Comment
Transform	`https://uor.foundation/morphism/Transform`	`http://www.w3.org/2002/07/owl#Thing`	`https://uor.foundation/morphism/CompositionLaw`	A map between UOR objects. The root abstraction: source, target, and optionally what structure (if any) is preserved. This is what cert:TransformCertificate certifies.
Isometry	`https://uor.foundation/morphism/Isometry`	`https://uor.foundation/morphism/Transform`	`https://uor.foundation/morphism/Composition`	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.
Embedding	`https://uor.foundation/morphism/Embedding`	`https://uor.foundation/morphism/Transform`	`https://uor.foundation/morphism/Composition`	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.
Action	`https://uor.foundation/morphism/Action`	`http://www.w3.org/2002/07/owl#Thing`		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.
Composition	`https://uor.foundation/morphism/Composition`	`https://uor.foundation/morphism/Transform`	`https://uor.foundation/morphism/Isometry`, `https://uor.foundation/morphism/Embedding`, `https://uor.foundation/morphism/Identity`	A transform formed by composing two or more transforms sequentially. The categorical composition operation that turns transforms into a category.
Identity	`https://uor.foundation/morphism/Identity`	`https://uor.foundation/morphism/Transform`	`https://uor.foundation/morphism/Composition`	The identity transform on a type: maps every element to itself. The categorical identity morphism.
CompositionLaw	`https://uor.foundation/morphism/CompositionLaw`	`http://www.w3.org/2002/07/owl#Thing`	`https://uor.foundation/morphism/Transform`	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.
GroundingMap	`https://uor.foundation/morphism/GroundingMap`	`https://uor.foundation/morphism/Transform`	`https://uor.foundation/morphism/ProjectionMap`	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.
ProjectionMap	`https://uor.foundation/morphism/ProjectionMap`	`https://uor.foundation/morphism/Transform`	`https://uor.foundation/morphism/GroundingMap`	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.
GroundingCertificate	`https://uor.foundation/morphism/GroundingCertificate`	`https://uor.foundation/cert/Certificate`		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.
TopologicalDelta	`https://uor.foundation/morphism/TopologicalDelta`	`http://www.w3.org/2002/07/owl#Thing`		A topological delta: records changes in topological invariants (Betti numbers, Euler characteristic, nerve structure) before and after a morphism.
ComputationDatum	`https://uor.foundation/morphism/ComputationDatum`	`https://uor.foundation/schema/Datum`		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.
ApplicationMorphism	`https://uor.foundation/morphism/ApplicationMorphism`	`https://uor.foundation/morphism/Transform`		A transform that applies a ComputationDatum to an input datum, producing an output datum. The output inherits the certificate of the ComputationDatum.
PartialApplication	`https://uor.foundation/morphism/PartialApplication`	`https://uor.foundation/morphism/ComputationDatum`		A ComputationDatum formed by fixing some but not all inputs of a multi-argument transform.
TransformComposition	`https://uor.foundation/morphism/TransformComposition`	`https://uor.foundation/morphism/ComputationDatum`		A ComputationDatum representing the composition f ∘ g of two ComputationDatums. Certified iff both components are certified and range(g) = domain(f).
Witness	`https://uor.foundation/morphism/Witness`	`http://www.w3.org/2002/07/owl#Thing`		Abstract supertype for one specific input/output pair witnessing a Transform.
GroundingWitness	`https://uor.foundation/morphism/GroundingWitness`	`https://uor.foundation/morphism/Witness`	`https://uor.foundation/morphism/ProjectionWitness`	One specific surface symbol mapped to one specific grounded address by some GroundingMap.
ProjectionWitness	`https://uor.foundation/morphism/ProjectionWitness`	`https://uor.foundation/morphism/Witness`	`https://uor.foundation/morphism/GroundingWitness`	One specific input partition mapped to one specific surface symbol sequence by some ProjectionMap.
SymbolSequence	`https://uor.foundation/morphism/SymbolSequence`	`http://www.w3.org/2002/07/owl#Thing`		An ordered sequence of surface symbols. Elements are reified as SequenceElement individuals with explicit position indices.
SequenceElement	`https://uor.foundation/morphism/SequenceElement`	`http://www.w3.org/2002/07/owl#Thing`		One position in a SymbolSequence: a reified (value, index) pair.

Properties

Name	Kind	Functional	Domain	Range	Comment
source	Object	true	`https://uor.foundation/morphism/Transform`	`http://www.w3.org/2002/07/owl#Thing`	The domain of the transform.
target	Object	true	`https://uor.foundation/morphism/Transform`	`http://www.w3.org/2002/07/owl#Thing`	The codomain of the transform.
preserves	Object	false	`https://uor.foundation/morphism/Transform`	`http://www.w3.org/2002/07/owl#Thing`	The structure preserved by this transform (if any). E.g., a ring homomorphism preserves addition and multiplication.
preservesMetric	Object	false	`https://uor.foundation/morphism/Isometry`	`https://uor.foundation/observable/MetricObservable`	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.
sourceQuantum	Datatype	true	`https://uor.foundation/morphism/Embedding`	`http://www.w3.org/2001/XMLSchema#positiveInteger`	The quantum level n of the source ring for an embedding.
targetQuantum	Datatype	true	`https://uor.foundation/morphism/Embedding`	`http://www.w3.org/2001/XMLSchema#positiveInteger`	The quantum level n' of the target ring for an embedding. Must satisfy n' > n (embeddings go to larger rings).
group	Object	true	`https://uor.foundation/morphism/Action`	`https://uor.foundation/op/Group`	The group acting in this group action.
actingOn	Object	true	`https://uor.foundation/morphism/Action`	`http://www.w3.org/2002/07/owl#Thing`	The set being acted upon by this group action.
actionIsometry	Datatype	true	`https://uor.foundation/morphism/Action`	`http://www.w3.org/2001/XMLSchema#boolean`	Whether every transform induced by this action is an isometry. True for the dihedral action on 𝒯_n (Frame Theorem).
trace	Object	true	`https://uor.foundation/morphism/Transform`	`https://uor.foundation/trace/ComputationTrace`	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.
composesWith	Object	false	`https://uor.foundation/morphism/Transform`	`https://uor.foundation/morphism/Transform`	A transform that this transform can be composed with. The target of this transform must match the source of the composed transform.
compositionResult	Object	true	`https://uor.foundation/morphism/Composition`	`https://uor.foundation/morphism/Transform`	The transform that results from this composition.
compositionComponents	Object	false	`https://uor.foundation/morphism/Composition`	`https://uor.foundation/morphism/Transform`	A component transform of this composition.
identityOn	Object	true	`https://uor.foundation/morphism/Identity`	`https://uor.foundation/type/TypeDefinition`	The type on which this identity transform acts.
compositionOrder	Datatype	true	`—`	`http://www.w3.org/2001/XMLSchema#nonNegativeInteger`	The number of component transforms in a composition.
isAssociative	Datatype	true	`https://uor.foundation/morphism/CompositionLaw`	`http://www.w3.org/2001/XMLSchema#boolean`	Whether this composition law is associative.
isCommutative	Datatype	true	`https://uor.foundation/morphism/CompositionLaw`	`http://www.w3.org/2001/XMLSchema#boolean`	Whether this composition law is commutative.
lawComponents	Object	false	`https://uor.foundation/morphism/CompositionLaw`	`https://uor.foundation/op/Operation`	An operation that is a component of this composition law.
lawResult	Object	true	`https://uor.foundation/morphism/CompositionLaw`	`https://uor.foundation/op/Operation`	The operation that results from this composition law.
preservedInvariant	Object	true	`https://uor.foundation/morphism/Transform`	`https://uor.foundation/op/Identity`	The identity preserved by this transform (reference to the op:Identity that the transform commutes with).
addressCoherence	Object	true	`https://uor.foundation/morphism/Embedding`	`https://uor.foundation/op/Identity`	Certificate that this embedding's addressing diagram commutes: glyph ∘ ι ∘ addresses is well-defined and injective.
bettisBefore	Object	true	`https://uor.foundation/morphism/TopologicalDelta`	`https://uor.foundation/observable/BettiNumber`	Betti numbers before the morphism.
bettisAfter	Object	true	`https://uor.foundation/morphism/TopologicalDelta`	`https://uor.foundation/observable/BettiNumber`	Betti numbers after the morphism.
eulerBefore	Datatype	true	`https://uor.foundation/morphism/TopologicalDelta`	`http://www.w3.org/2001/XMLSchema#integer`	Euler characteristic before the morphism.
eulerAfter	Datatype	true	`https://uor.foundation/morphism/TopologicalDelta`	`http://www.w3.org/2001/XMLSchema#integer`	Euler characteristic after the morphism.
nerveBefore	Object	true	`https://uor.foundation/morphism/TopologicalDelta`	`https://uor.foundation/homology/SimplicialComplex`	Constraint nerve (simplicial complex) before the morphism.
nerveAfter	Object	true	`https://uor.foundation/morphism/TopologicalDelta`	`https://uor.foundation/homology/SimplicialComplex`	Constraint nerve (simplicial complex) after the morphism.
surfaceSymbol	Object	true	`https://uor.foundation/morphism/GroundingWitness`	`https://uor.foundation/schema/SurfaceSymbol`	The surface symbol that is the source of this grounding witness.
groundedAddress	Object	true	`https://uor.foundation/morphism/GroundingWitness`	`https://uor.foundation/u/Element`	The resolved ring address that is the target of this grounding witness.
groundingDerivation	Object	true	`https://uor.foundation/morphism/GroundingMap`	`https://uor.foundation/derivation/Derivation`	The derivation witnessing the content-addressing computation that produced the grounded address from the surface symbol.
symbolConstraints	Object	false	`https://uor.foundation/morphism/GroundingMap`	`https://uor.foundation/type/Constraint`	A typed attribute preserved by this grounding. Non-functional: one assertion per active constraint axis (vertical, horizontal, diagonal).
projectionFrame	Object	true	`https://uor.foundation/morphism/ProjectionMap`	`https://uor.foundation/state/Frame`	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.
projectionSource	Object	true	`https://uor.foundation/morphism/ProjectionWitness`	`https://uor.foundation/partition/Partition`	The resolved partition (address neighbourhood) that this projection witness projects back to surface symbols.
projectionOrder	Object	true	`https://uor.foundation/morphism/ProjectionMap`	`https://uor.foundation/type/CompositeConstraint`	Ordering constraint determining the output symbol sequence. Domain-specific: syntactic position (NLP), row-major scan (ARC), temporal sequence (music).
roundTripCoherence	Datatype	true	`https://uor.foundation/morphism/ProjectionMap`	`http://www.w3.org/2001/XMLSchema#boolean`	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.
groundingCertMap	Object	true	`https://uor.foundation/morphism/GroundingCertificate`	`https://uor.foundation/morphism/GroundingMap`	The GroundingMap used in this certified round-trip.
groundingCertProjection	Object	true	`https://uor.foundation/morphism/GroundingCertificate`	`https://uor.foundation/morphism/ProjectionMap`	The ProjectionMap used in this certified round-trip.
groundingCertSourceSymbol	Object	true	`https://uor.foundation/morphism/GroundingCertificate`	`https://uor.foundation/schema/Literal`	The surface symbol that entered the grounding boundary.
groundingCertAddress	Object	true	`https://uor.foundation/morphism/GroundingCertificate`	`https://uor.foundation/u/Element`	The ring address the symbol was grounded to.
referencedCertificate	Object	true	`https://uor.foundation/morphism/ComputationDatum`	`https://uor.foundation/cert/TransformCertificate`	The certificate this computation datum encodes.
computationAddress	Object	true	`https://uor.foundation/morphism/ComputationDatum`	`https://uor.foundation/u/Element`	The content address of the referenced certificate.
applicationTarget	Object	true	`https://uor.foundation/morphism/ApplicationMorphism`	`https://uor.foundation/morphism/ComputationDatum`	The computation being applied.
applicationInput	Object	true	`https://uor.foundation/morphism/ApplicationMorphism`	`https://uor.foundation/schema/Datum`	The input datum to the application.
partialBase	Object	true	`https://uor.foundation/morphism/PartialApplication`	`https://uor.foundation/morphism/ComputationDatum`	The base computation being partially applied.
boundArguments	Object	false	`https://uor.foundation/morphism/PartialApplication`	`https://uor.foundation/schema/Datum`	The arguments already bound.
compositionLeft	Object	true	`https://uor.foundation/morphism/TransformComposition`	`https://uor.foundation/morphism/ComputationDatum`	The outer function f in f ∘ g.
compositionRight	Object	true	`https://uor.foundation/morphism/TransformComposition`	`https://uor.foundation/morphism/ComputationDatum`	The inner function g in f ∘ g.
remainingArity	Datatype	true	`https://uor.foundation/morphism/PartialApplication`	`http://www.w3.org/2001/XMLSchema#positiveInteger`	Number of unbound arguments remaining.
inputClass	Object	true	`https://uor.foundation/morphism/Transform`	`http://www.w3.org/2002/07/owl#Class`	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.
outputClass	Object	true	`https://uor.foundation/morphism/Transform`	`http://www.w3.org/2002/07/owl#Class`	The OWL class of outputs this transform produces. Uses OWL2 punning.
outputElementClass	Object	true	`https://uor.foundation/morphism/ProjectionMap`	`http://www.w3.org/2002/07/owl#Class`	When outputClass is a sequence type, the OWL class of individual sequence elements. Uses OWL2 punning.
hasWitness	Object	false	`https://uor.foundation/morphism/Transform`	`https://uor.foundation/morphism/Witness`	Zero or more witness pairs documenting specific input/output bindings of this transform.
hasElement	Object	false	`https://uor.foundation/morphism/SymbolSequence`	`https://uor.foundation/morphism/SequenceElement`	Membership of a SymbolSequence. The sequence is reconstructed by sorting elements by elementIndex.
elementValue	Object	true	`https://uor.foundation/morphism/SequenceElement`	`https://uor.foundation/schema/SurfaceSymbol`	The surface symbol value of this sequence element.
elementIndex	Datatype	true	`https://uor.foundation/morphism/SequenceElement`	`http://www.w3.org/2001/XMLSchema#nonNegativeInteger`	The zero-based position of this element in the sequence.
projectionOutput	Object	true	`https://uor.foundation/morphism/ProjectionWitness`	`https://uor.foundation/morphism/SymbolSequence`	The single SymbolSequence produced by this projection witness.

Named Individuals

Name	Type	Properties	Comment
criticalComposition	`https://uor.foundation/morphism/CompositionLaw`	`lawComponents`: `https://uor.foundation/op/neg` `lawComponents`: `https://uor.foundation/op/bnot` `lawResult`: `https://uor.foundation/op/succ` `isAssociative`: false `isCommutative`: false	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	`https://uor.foundation/morphism/GroundingMap`	`inputClass`: `https://uor.foundation/schema/Literal` `outputClass`: `https://uor.foundation/u/Element`	Grounds integer surface symbols to ring addresses.
Utf8GroundingMap	`https://uor.foundation/morphism/GroundingMap`	`inputClass`: `https://uor.foundation/schema/HostStringLiteral` `outputClass`: `https://uor.foundation/u/Element`	Grounds UTF-8 host strings to ring addresses.
JsonGroundingMap	`https://uor.foundation/morphism/GroundingMap`	`inputClass`: `https://uor.foundation/schema/HostStringLiteral` `outputClass`: `https://uor.foundation/u/Element`	Grounds JSON host strings to ring addresses.
IntegerProjectionMap	`https://uor.foundation/morphism/ProjectionMap`	`inputClass`: `https://uor.foundation/partition/Partition` `outputClass`: `https://uor.foundation/morphism/SymbolSequence` `outputElementClass`: `https://uor.foundation/schema/Literal`	Projects partitions to integer symbol sequences.
Utf8ProjectionMap	`https://uor.foundation/morphism/ProjectionMap`	`inputClass`: `https://uor.foundation/partition/Partition` `outputClass`: `https://uor.foundation/morphism/SymbolSequence` `outputElementClass`: `https://uor.foundation/schema/HostStringLiteral`	Projects partitions to UTF-8 symbol sequences.
JsonProjectionMap	`https://uor.foundation/morphism/ProjectionMap`	`inputClass`: `https://uor.foundation/partition/Partition` `outputClass`: `https://uor.foundation/morphism/SymbolSequence` `outputElementClass`: `https://uor.foundation/schema/HostStringLiteral`	Projects partitions to JSON symbol sequences.