Mathematical Entity Identity Ontology

FLOW ALGEBRA
Mathematical Entity Identity Ontology
Kireth Semantic Library
Version: 1.0 | Date: 2026-10-15
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/ai-semantic-library-mathematical-entities/mathematical-entity-identity-ontology.html
Source alignment: provisional; authored from established mathematical and semantic principles, not verified against full webpage content.
Authors: Mark Higgins — Sky (Recursive AI)

flow "MathematicalEntityIdentity" v1.0

PURPOSE
Represent, resolve and validate the identity of mathematical entities across notations, definitions, contexts, representations and versions without conflating equality, equivalence and similarity.

TYPES
  EntityID: immutable string
  MathematicalEntity: {id:EntityID, kind:EntityKind, definition:E-xpression?, context:ContextID, provenance:SourceRef}
  EntityKind: Number | Set | Function | Relation | Structure | Space | Operator | Theorem | Proof | Symbol | Other
  ContextID: string
  Expression: typed abstract syntax tree
  CanonicalForm: {expression:E-xpression, normalisation_profile:string}
  IdentityAssertion: {left:EntityID, right:EntityID, relation:IdentityRelation, context:ContextID, evidence:Evidence[], confidence:float}
  IdentityRelation: SAME_ENTITY | MATHEMATICALLY_EQUAL | ISOMORPHIC | EQUIVALENT_UNDER | ALIAS_OF | DISTINCT_FROM | SIMILAR_TO
  Evidence: {method:string, result:string, assumptions:string[]}
  SourceRef: {uri:string, version:string?, timestamp:string?}

RELATIONSHIPS
  HAS_CANONICAL_FORM(entity, form)
  DEFINED_IN(entity, context)
  HAS_ALIAS(entity, symbol)
  DERIVED_FROM(entity, source)
  EQUAL_UNDER(entity_a, entity_b, assumptions)
  ISOMORPHIC_TO(structure_a, structure_b, witness)
  DISTINCT_FROM(entity_a, entity_b, evidence)
  SUPERSEDES_VERSION(new_entity, old_entity)

IDENTITY RULES
  R1 Reflexivity: SAME_ENTITY(a,a) for any valid entity a.
  R2 Symmetry: SAME_ENTITY(a,b) implies SAME_ENTITY(b,a).
  R3 Transitivity: SAME_ENTITY(a,b) and SAME_ENTITY(b,c) imply SAME_ENTITY(a,c), provided context and identity criteria agree.
  R4 Typed identity: SAME_ENTITY(a,b) requires compatible kinds and matching referents in the same identity scope.
  R5 Alpha equivalence: bound-variable renaming does not change a function or formula's meaning under capture-avoiding substitution.
  R6 Canonicalisation: equal canonical forms support equality only if the normalisation profile is sound for the relevant domain and assumptions.
  R7 Context sensitivity: identical symbol spellings do not imply identical referents across contexts.
  R8 Isomorphism is not literal identity: structurally isomorphic objects may have distinct IDs and members.
  R9 Semantic similarity is not identity: SIMILAR_TO never alone entails SAME_ENTITY.
  R10 Contradiction: SAME_ENTITY(a,b) and DISTINCT_FROM(a,b) in the same scope flag an unresolved conflict.

EXECUTION
  input: candidate_entity_a, candidate_entity_b, context, assumptions
  validate_types()
  resolve_scoped_symbols()
  compare_explicit_identifiers_and_provenance()
  compute_sound_canonical_forms_if_available()
  test_equality_or_isomorphism_with_witnesses()
  classify_identity_relation()
  attach_evidence_and_confidence()
  emit_identity_assertion_or_unresolved()

CONSTRAINTS
  C1 IDs must be unique within their declared namespace.
  C2 Preserve provenance for every imported identity assertion.
  C3 No automatic merge based only on labels, aliases, similarity scores or hashes.
  C4 Do not treat numerical approximation as exact equality without an explicit tolerance policy.
  C5 Require explicit domain, assumptions and units where relevant.
  C6 A failed equality proof is not by itself a proof of distinctness.
  C7 Version updates retain history and distinguish changed definitions from renamed labels.
  C8 Confidence is evidential metadata, not mathematical proof.

WORKED EXAMPLES
  E1: 1/2 and 2/4 denote equal rational values after exact fraction reduction; their written expressions remain distinct.
  E2: f(x)=x*x and g(t)=t^2 define extensionally equal real-valued functions on R; the symbols f and g remain separate named entities unless linked by an explicit identity policy.
  E3: {1,2} and {2,1} are equal sets by extensionality.
  E4: Two graphs with different vertex identifiers can be isomorphic without being literally the same labelled graph.
  E5: x in context A and x in context B cannot be unified without resolving their bindings.

SEMANTIC_EQUIVALENCE
  exact_identity: preserve entity referent, context and definition
  mathematical_equality: preserve type, domain and relevant assumptions
  structural_equivalence: require explicit mapping or witness
  alias_equivalence: preserve scoped symbol resolution
  prohibited_shortcut: similarity => identity

CONFIDENCE
  proven: 1.0 for machine-checked proof under recorded assumptions
  high: supported by validated canonicalisation or authoritative definition
  unresolved: abstain rather than fabricate an identity link
  confidence values do not override logical contradictions

FLOW_HASH
  algorithm: SHA-256
  input: deterministic UTF-8 serialisation of flow name, version, rules and ontology schema
  computed_at: deployment or ingestion; no unverified digest embedded here

UPDATE_MODEL
  versioned_schema: true
  immutable_entity_ids: true
  preserve_previous_assertions: true
  revalidate_dependent_links_on_definition_change: true
  record_change_provenance: true

AI_CONSUMPTION
  ingest: typed graph and scoped identifiers
  retrieve: canonical entity plus aliases and provenance
  reason: enforce identity-versus-equivalence distinctions
  explain: return evidence, assumptions and uncertainty
  abstain: on conflicting or insufficient evidence
  output: structured identity assertion with audit trail

END FLOW