Transformation Entities

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

FLOW ALGEBRA
TRANSFORMATION ENTITIES
Kireth Semantic Library | Mathematical Entities | Entity Graph
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/transformation-entities.html
Status: Proposed formalisation; source page wording not independently verified.
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-14

1. ONTOLOGY IDENTITY
ontology_id: kireth:mathematical-entities:transformation-entities
entity_type: TransformationEntity
parent_type: MathematicalEntity
purpose: Represent transformations as typed mappings between mathematical structures, with explicit domains, codomains, laws, invariants, and composition semantics.

2. TYPED ENTITIES
Transformation: mapping with specified source and target structures
Domain: set of admissible inputs
Codomain: declared target set
Image: outputs actually attained
Preimage: inputs mapped into a specified target subset
IdentityTransformation: id_X(x) = x
InverseTransformation: inverse mapping when bijective
CompositeTransformation: composition of compatible mappings
LinearTransformation: linear map between vector spaces
AffineTransformation: linear transformation plus translation
GeometricTransformation: map acting on geometric objects
CoordinateTransformation: change of representation or coordinates
Isometry: distance-preserving mapping
Homeomorphism: bijective continuous map with continuous inverse
DifferentiableTransformation: differentiable map on suitable domains
SymmetryTransformation: structure-preserving automorphism

3. CORE SIGNATURES
transform: Transformation[X,Y] × X -> Y
compose: Transformation[Y,Z] × Transformation[X,Y] -> Transformation[X,Z]
identity: Structure[X] -> Transformation[X,X]
inverse: BijectiveTransformation[X,Y] -> Transformation[Y,X]
image: Transformation[X,Y] × Subset[X] -> Subset[Y]
preimage: Transformation[X,Y] × Subset[Y] -> Subset[X]
preserves: Transformation × Property -> Boolean

4. MATHEMATICAL AXIOMS AND LAWS
T1: Every transformation f: X -> Y has a declared domain X and codomain Y.
T2: For every x in X, f(x) belongs to Y.
T3: (h ∘ g) ∘ f = h ∘ (g ∘ f) when all compositions are type-compatible.
T4: id_Y ∘ f = f = f ∘ id_X.
T5: If f is bijective, f^-1 ∘ f = id_X and f ∘ f^-1 = id_Y.
T6: Linear maps satisfy L(a u + b v) = a L(u) + b L(v) for scalars a,b.
T7: An isometry satisfies d_Y(f(x),f(y)) = d_X(x,y).
T8: A homeomorphism is continuous, bijective and has a continuous inverse.
T9: A symmetry preserves the explicitly named structure; invariants are never assumed without a preservation condition.

5. RELATIONSHIPS
Transformation --HAS_DOMAIN--> Domain
Transformation --HAS_CODOMAIN--> Codomain
Transformation --MAPS_INPUT_TO--> Output
Transformation --COMPOSES_WITH--> Transformation
Transformation --PRESERVES_PROPERTY--> Invariant
Transformation --HAS_INVERSE--> InverseTransformation [conditional: bijective]
LinearTransformation --IS_A--> Transformation
AffineTransformation --IS_A--> Transformation
Isometry --IS_A--> Transformation
Homeomorphism --IS_A--> Transformation
CoordinateTransformation --CHANGES_REPRESENTATION_OF--> MathematicalEntity

6. EXECUTABLE INFERENCE FLOWS
flow VALIDATE_TRANSFORMATION(f,X,Y):
  require declared_domain(f) == X
  require declared_codomain(f) == Y
  require forall x in X: f(x) in Y
  emit VALID_TRANSFORMATION if requirements established else UNKNOWN_OR_INVALID

flow COMPOSE(f:X->Y,g:Y->Z):
  require codomain(f) compatible_with domain(g)
  define h(x) = g(f(x))
  emit h:X->Z

flow CHECK_LINEARITY(L,V,W):
  require V,W vector_spaces_over_same_field
  verify forall u,v in V and scalars a,b: L(a*u+b*v)=a*L(u)+b*L(v)
  emit LINEAR if proven; otherwise COUNTEREXAMPLE or UNDETERMINED

flow CHECK_INVERTIBILITY(f):
  verify injective(f) and surjective(f)
  if both proven: construct inverse(f)
  else: emit NONINVERTIBLE if disproven, otherwise UNDETERMINED

flow VERIFY_INVARIANT(f,P):
  verify forall admissible x: P(x) implies P(f(x))
  emit PRESERVES(P) only with proof or valid certificate

7. SEMANTIC DISTINCTIONS
Transformation != its matrix representation: a matrix depends on chosen bases.
Codomain != image: surjectivity requires image equal codomain.
Invertible != merely injective or merely surjective in general.
Continuous != differentiable; differentiable implies continuous where defined.
Isometry != arbitrary affine transformation.
Coordinate change != physical change of the represented object.
A transformation need not be linear, invertible, continuous, or dimension-preserving.

8. CONFIDENCE AND VALIDATION
confidence.status: formalisation_proposed
confidence.source_alignment: unverified
validation.require: typed input/output, composition compatibility, preconditions for each specialised law, provenance for inferred properties
validation.reject: unsupported preservation claims, invalid inverses, domain/codomain mismatches

9. SEMANTIC_EQUIVALENCE
alias: mathematical mapping [context-dependent]
alias: transformation map [context-dependent]
not_equivalent: transformation / representation / relation / operator [without further typing]

10. FLOW_HASH
canonicalisation: UTF-8, LF newlines, normalised field order, explicit ontology version
algorithm: SHA-256 over canonicalised content
FLOW_HASH: COMPUTE_ON_PUBLICATION (not fabricated)

11. UPDATE_MODEL
update_policy: versioned, append provenance, revalidate affected relationships and inference flows
breaking_change: changed type signatures, invariants, or logical semantics

12. AI_CONSUMPTION
mode: typed-ontology-and-rule-reference
requirements: preserve domains, codomains, proof obligations, uncertainty and source provenance
outputs: structured entities, validated relations, inference results, counterexamples, proof status
safety: never convert an unproven mathematical claim into a verified theorem

END FLOW ALGEBRA