Mathematical Entity Confidence Ontology

FLOW ALGEBRA
Mathematical Entity Confidence 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-confidence-ontology.html

AI INTERPRETATION NOTE
The following specifications define a formal structural reasoning model. Treat declarations, invariants, and inference rules as constraints; implementation-specific procedures are explicitly identified.

flow "MathematicalEntityConfidenceOntology" v1.0

DOMAIN
  MathematicalEntity := typed mathematical object, statement, or relation
  Evidence := proof | derivation | verified_computation | observation | assertion
  ConfidenceRecord := (entity, proposition, evidence_set, status, score, provenance, context, timestamp)
  Status := proven | refuted | conditional | supported | uncertain | unknown | inconsistent
  Score := real number in [0,1] or undefined

TYPES
  entity_id: Identifier
  proposition: Formula
  context: AssumptionSet
  evidence_set: Set[Evidence]
  provenance: Set[SourceReference]
  timestamp: DateTime

RELATIONSHIPS
  HAS_CONFIDENCE(entity, record)
  ABOUT(record, proposition)
  SUPPORTED_BY(record, evidence)
  DEPENDS_ON(record, assumption)
  DERIVED_FROM(evidence, source)
  CONTRADICTS(proposition_a, proposition_b)
  EQUIVALENT_TO(proposition_a, proposition_b, context)
  VALID_IN(proposition, context)
  SUPERSEDES(record_new, record_old)

INVARIANTS
  I1: forall r: ConfidenceRecord, r.score is undefined OR 0 <= r.score <= 1
  I2: forall r, r.status = proven => exists p in r.evidence_set: VALID_PROOF(p, r.proposition, r.context)
  I3: forall r, r.status = refuted => exists p in r.evidence_set: VALID_PROOF(p, NOT(r.proposition), r.context)
  I4: forall r, r.status = conditional => r.context is nonempty AND assumptions are explicitly recorded
  I5: forall r, r.status in {supported, uncertain} => r.score is not interpreted as mathematical truth
  I6: forall r, evidence provenance and verification method must be traceable when supplied
  I7: absence of proof does not imply refutation
  I8: contradictory evidence must not be silently discarded
  I9: no inference may promote a heuristic score to proven without a valid proof

RULES
  R1 PROOF_VALIDATION:
    VALID_PROOF(p, phi, A) => STATUS(phi, A) := proven

  R2 REFUTATION:
    VALID_PROOF(p, NOT(phi), A) => STATUS(phi, A) := refuted

  R3 CONDITIONAL_PROOF:
    VALID_PROOF(p, phi, A) AND A != empty => ANNOTATE(phi, conditional_on=A)
    # Conditional proof may also be formally proven relative to A.

  R4 VERIFIED_EQUIVALENCE:
    EQUIVALENT_TO(phi, psi, A) AND VALID_PROOF(p, phi, A)
    => STATUS(psi, A) := proven

  R5 DEPENDENCY_PROPAGATION:
    DEPENDS_ON(phi, psi) AND STATUS(psi) in {unknown, inconsistent}
    => FLAG(phi, dependency_requires_review)
    # Do not assign an arbitrary numeric confidence by multiplication.

  R6 CONFLICT_DETECTION:
    VALID_PROOF(p, phi, A) AND VALID_PROOF(q, NOT(phi), A)
    => STATUS(phi, A) := inconsistent; REQUIRE_AUDIT(A,p,q)

  R7 HEURISTIC_SUPPORT:
    HEURISTIC_EVIDENCE(e, phi) => STATUS(phi) := supported
    # A calibrated score may be attached, but cannot certify a theorem.

  R8 REVISION:
    NEW_EVIDENCE(e, phi) => REEVALUATE(phi, assumptions, proofs, provenance)
    => CREATE_VERSIONED_RECORD(phi)

  R9 CONTEXT_SEPARATION:
    STATUS(phi, A) != STATUS(phi, B) may hold when A != B
    => NEVER_MERGE_CONTEXTS_WITHOUT_VALIDATION

  R10 UNKNOWN_HANDLING:
    NOT EXISTS valid proof for phi AND NOT EXISTS valid proof for NOT(phi)
    => STATUS(phi) := unknown unless independently supported evidence is available

CONFIDENCE MODEL
  Formal certainty: determined by valid proof relative to an explicit formal system and assumptions.
  Empirical or computational confidence: estimated from documented evidence and verification methods.
  Numeric scores: optional, calibrated where possible, and never substitutes for formal validity.
  Evidence strength: assessed by soundness, independence, reproducibility, relevance, and provenance.

UPDATE_MODEL
  input: entity, proposition, new_evidence, context
  validate: entity typing, evidence format, context consistency, source identity
  execute: apply R1-R10; detect contradictions; preserve earlier versions
  output: ConfidenceRecord with status, optional score, provenance, audit trail

AI_CONSUMPTION
  retrieve: entity -> confidence records -> supporting evidence -> assumptions
  reason: apply only context-valid rules and explicitly typed relationships
  report: distinguish proven, refuted, conditional, supported, uncertain, unknown, inconsistent
  prohibit: treating a language-model probability as theorem validity
  explain: include evidence and assumptions for confidence claims

SEMANTIC_EQUIVALENCE
  Canonical term: Mathematical Entity Confidence
  Related terms: proof status, epistemic confidence, evidential support, mathematical verification
  Distinguish: formal provability != probabilistic belief != numerical accuracy

FLOW_HASH
  method: SHA-256 over canonical UTF-8 representation of the flow
  value: compute at publication; no hash value asserted here

VALIDATION TESTS
  T1: valid proof of phi in A => proven in A
  T2: valid counterproof in A => refuted in A
  T3: unverified assertion => not proven
  T4: contradictory verified proofs under same assumptions => inconsistent and audit
  T5: score outside [0,1] => reject
  T6: proof dependent on A => do not transfer automatically to B
  T7: revised evidence => new version and preserved provenance

END FLOW