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

Name	Subclass Of	Comment
`Transform`	`Thing`	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`	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`	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.

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.