Mathematical Entity Equivalence Ontology

FLOW ALGEBRA
Mathematical Entity Equivalence Ontology
Version: 1.0
Authors: Mark Anthony Higgins — Sky (Recursive AI)
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/ai-semantic-library-mathematical-entities/mathematical-entity-equivalence-ontology.html

AI INTERPRETATION NOTE
This document defines a formal structural reasoning specification for mathematical equivalence. Interpret the rules as constraints and inference schemas, not as executable software instructions. Every equivalence judgement must name its mathematical domain, assumptions and interpretation.

flow "MathematicalEntityEquivalenceOntology" v1.0

DOMAIN
  MathematicalEntity := number | expression | equation | proposition | function | set | structure | transformation | proof_object
  Context := (formal_system, signature, domain, assumptions, interpretation, version)
  EquivalenceClaim := (left_entity, right_entity, relation_kind, context, evidence, status)
  RelationKind := identity | definitional | algebraic | logical | extensional | structural | semantic | approximate
  ClaimStatus := proven | refuted | conditional | unknown | unchecked

TYPES
  EntityID := stable identifier
  Formula := well-formed mathematical expression
  ProofEvidence := derivation | theorem | rewrite_trace | model_check | counterexample
  Mapping := typed mapping between entity representations
  Tolerance := nonnegative numerical threshold with explicit metric and units
  Confidence := calibrated evidence assessment, not a substitute for proof

RELATIONSHIPS
  EQUIVALENT_TO(a, b, context)
  IDENTICAL_TO(a, b)
  DEFINITIONALLY_EQUIVALENT(a, b, context)
  ALGEBRAICALLY_EQUIVALENT(a, b, context)
  LOGICALLY_EQUIVALENT(a, b, context)
  EXTENSIONALLY_EQUIVALENT(a, b, context)
  ISOMORPHIC_TO(a, b, mapping, context)
  SEMANTICALLY_EQUIVALENT(a, b, interpretation)
  APPROXIMATELY_EQUIVALENT(a, b, metric, tolerance, context)
  DERIVED_BY(claim, transformation)
  PROVEN_BY(claim, evidence)
  REQUIRES_ASSUMPTION(claim, assumption)
  HAS_COUNTEREXAMPLE(claim, counterexample)
  NORMALISES_TO(entity, canonical_form, context)

AXIOMS
  A1 REFLEXIVITY:
    well_typed(a, C) => EQUIVALENT_TO(a, a, C)
  A2 SYMMETRY:
    EQUIVALENT_TO(a, b, C) => EQUIVALENT_TO(b, a, C)
  A3 TRANSITIVITY:
    EQUIVALENT_TO(a, b, C) AND EQUIVALENT_TO(b, c, C)
    => EQUIVALENT_TO(a, c, C)
  A4 CONTEXT_PRESERVATION:
    Equivalence cannot be transferred to another context without a validated context mapping.
  A5 TYPE_COMPATIBILITY:
    An equivalence claim requires compatible types or an explicit, validated coercion.
  A6 EVIDENCE_REQUIREMENT:
    A status of proven requires evidence accepted by the specified formal system.
  A7 NO_UNSAFE_COLLAPSE:
    Similarity, approximation, numerical agreement at samples and shared labels do not establish exact equivalence.

EQUIVALENCE_MODES
  identity:
    Same referent under a declared identity criterion; not merely matching names.
  definitional:
    Interderivable through admissible expansions and contractions of definitions.
  algebraic:
    Equality of expressions over the specified domain subject to side conditions.
  logical:
    For propositions P,Q, prove P <=> Q under the assumptions of C.
  extensional:
    Functions or sets agree on every member of their declared domain.
  structural:
    Structures correspond under a bijection preserving designated operations and relations.
  semantic:
    Interpretations assign the same meaning relative to a specified model or semantics.
  approximate:
    A declared metric yields distance <= tolerance; this is not an equivalence relation in general.

RULES
  R1 REWRITE_SOUNDNESS:
    valid_rewrite(a, b, C) AND side_conditions_hold(C)
    => ALGEBRAICALLY_EQUIVALENT(a, b, C)
  R2 NORMAL_FORM:
    sound_complete_normaliser_for_fragment(N, C)
    AND N(a, C) = N(b, C)
    => EQUIVALENT_TO(a, b, C)
    NOTE: unequal normal forms imply non-equivalence only if the normaliser is complete for the fragment.
  R3 PROPOSITIONAL:
    proves(C, P => Q) AND proves(C, Q => P)
    => LOGICALLY_EQUIVALENT(P, Q, C)
  R4 FUNCTIONAL:
    same_domain(f, g, C) AND forall(x in domain(C), f(x) = g(x))
    => EXTENSIONALLY_EQUIVALENT(f, g, C)
  R5 SET:
    forall(x, x in A <=> x in B)
    => EXTENSIONALLY_EQUIVALENT(A, B, C)
  R6 ISOMORPHISM:
    bijective(phi) AND preserves_signature(phi, C)
    => ISOMORPHIC_TO(A, B, phi, C)
  R7 COUNTEREXAMPLE:
    verified_counterexample(a, b, C)
    => status(EQUIVALENT_TO(a, b, C)) = refuted
  R8 APPROXIMATION:
    metric_defined(d, C) AND d(a, b) <= epsilon
    => APPROXIMATELY_EQUIVALENT(a, b, d, epsilon, C)
    NOTE: do not infer transitivity at the same epsilon.
  R9 ASSUMPTION_CHANGE:
    changing domain, assumptions, branch convention or interpretation
    => revalidate equivalence claim.
  R10 SUBSTITUTION:
    EQUIVALENT_TO(a, b, C) AND congruent_context(F, C)
    => EQUIVALENT_TO(F(a), F(b), C)
    NOTE: substitution is restricted to congruence-preserving operations.

VALIDATION_PIPELINE
  01 parse both entities and assign stable identifiers
  02 identify mathematical types and governing formal system
  03 record domain, assumptions, semantics and side conditions
  04 select the precise equivalence mode
  05 normalise representations using sound transformations where available
  06 attempt symbolic proof, model checking or validated structural mapping
  07 search for counterexamples within the declared domain
  08 classify proven, refuted, conditional, unknown or unchecked
  09 attach evidence, dependencies, provenance and version
  10 expose only context-qualified conclusions to downstream reasoning

EDGE_CASES
  - x/x = 1 requires x != 0.
  - sqrt(x^2) = |x| for real x; it is not universally x.
  - sin(x)^2 + cos(x)^2 = 1 over the usual real interpretation.
  - 0.1 + 0.2 may differ from 0.3 in binary floating-point arithmetic.
  - Equivalence over real numbers need not imply equivalence over complex numbers.
  - A bijection of underlying sets alone does not establish algebraic isomorphism.
  - Syntactic difference does not establish mathematical non-equivalence.
  - Matching computed values on finitely many test inputs does not prove functional equivalence.
  - Approximate closeness must include metric, tolerance and numeric error model.

CONFIDENCE
  proven := verified proof in the named system
  refuted := verified counterexample or contradiction in the named system
  conditional := proof depends on outstanding assumptions or conditions
  unknown := attempted evaluation without a conclusive result
  unchecked := no validation performed
  RULE: confidence scores may rank evidence but must not upgrade unknown to proven.

PROVENANCE
  retain source_uri, source_version, entity_ids, relation_kind,
         formal_system, assumptions, transformation_trace,
         proof_reference, validator, timestamp, and dependencies.
  invalidated_dependency => mark dependent claims for revalidation.

SEMANTIC_EQUIVALENCE
  match labels only as candidate discovery.
  compare definitions, types, referents, interpretations and constraints.
  if semantics are model-dependent, retain model identifier.
  do not merge entity identifiers solely on lexical similarity.

FLOW_HASH
  canonical_payload := canonicalise(
    entity_ids, relation_kind, formal_system, domain,
    assumptions, interpretation, evidence_digest, source_version)
  FLOW_HASH := SHA256(UTF8(canonical_payload))
  RULE: canonicalisation format must be versioned and deterministic.

UPDATE_MODEL
  on entity_definition_change => invalidate affected equivalence evidence
  on assumption_change => recheck side conditions and domain restrictions
  on proof_change => preserve old version and create a new validation record
  on counterexample_found => revise status and propagate downstream alerts
  on source_update => recompute affected FLOW_HASH values

AI_CONSUMPTION
  input: typed mathematical entities, context and evidence
  output: relation kind, claim status, assumptions, evidence and provenance
  permit exact substitution only for proven, context-compatible congruences
  treat conditional claims as guarded inference rules
  keep approximate equivalence distinct from equality
  propagate unknown rather than fabricating proof
  cite the formal context when presenting equivalence conclusions

INVARIANTS
  I1 every claim has an explicit context
  I2 every proven claim has verifiable evidence
  I3 no type-unsafe or domain-unsafe substitution
  I4 no silent promotion of similarity into identity
  I5 every revision preserves provenance and version history

END FLOW ALGEBRA