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.

This is a user-space namespace in the Apply stage of the PRISM pipeline. It provides the application layer — types, morphisms, and state that parameterize the resolution pipeline.

Learn more: Pipeline Overview · Site Bundle Semantics

Class hierarchy

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	Subclass Of	Disjoint With	Comment
`Transform`	`Thing`	`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`	`Transform`	`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`	`Transform`	`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`	`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`	`Transform`	`Isometry`, `Embedding`, `Identity`	A transform formed by composing two or more transforms sequentially. The categorical composition operation that turns transforms into a category.
`Identity`	`Transform`	`Composition`	The identity transform on a type: maps every element to itself. The categorical identity morphism.
`CompositionLaw`	`Thing`	`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`	`Transform`	`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`	`Transform`	`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`	`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`	`Thing`		A topological delta: records changes in topological invariants (Betti numbers, Euler characteristic, nerve structure) before and after a morphism.
`ComputationDatum`	`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`	`Transform`		A transform that applies a ComputationDatum to an input datum, producing an output datum. The output inherits the certificate of the ComputationDatum.
`PartialApplication`	`ComputationDatum`		A ComputationDatum formed by fixing some but not all inputs of a multi-argument transform.
`TransformComposition`	`ComputationDatum`		A ComputationDatum representing the composition f ∘ g of two ComputationDatums. Certified iff both components are certified and range(g) = domain(f).
`Witness`	`Thing`		Abstract supertype for one specific input/output pair witnessing a Transform.
`GroundingWitness`	`Witness`	`ProjectionWitness`	One specific surface symbol mapped to one specific grounded address by some GroundingMap.
`ProjectionWitness`	`Witness`	`GroundingWitness`	One specific input partition mapped to one specific surface symbol sequence by some ProjectionMap.
`SymbolSequence`	`Thing`		An ordered sequence of surface symbols. Elements are reified as SequenceElement individuals with explicit position indices.
`SequenceElement`	`Thing`		One position in a SymbolSequence: a reified (value, index) pair.

Properties

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

Named Individuals

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

UOR Transforms and Morphisms

Imports

Classes

Properties

Named Individuals

Related Concepts