Constant Entity Graph Ontology

FLOW ALGEBRA
CONSTANT ENTITY GRAPH ONTOLOGY
Version: 1.0
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/ai-semantic-library-mathematical-entities/constant-entity-graph-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Project: Kireth Semantic Library

1. PURPOSE
Represent mathematical constants as typed, context-aware graph entities with precise definitions, identities, relationships, provenance and validation. Distinguish exact mathematical constants from empirical physical constants and context-dependent parameters.

2. ENTITY SCHEMA
entity ConstantEntity {
  id: URI
  canonical_name: string
  symbols: list<Symbol>
  definition: -Expression | Axiom | Reference
  domain: MathematicalDomain
  value_kind: exact | symbolic | measured | conventional
  value: -Expression | Decimal | Interval
  precision: integer?
  uncertainty: Real?
  units: UnitEntity?
  assumptions: set<Constraint>
  aliases: set<string>
  provenance: ProvenanceRecord
  confidence: ConfidenceRecord
  version: string
}

3. GRAPH TYPES
node_types = {Constant, Symbol, Expression, Equation, Number, Domain, Unit, Theorem, Proof, Source, Definition, Context}
edge_types = {DEFINED_BY, DENOTED_BY, EQUALS, APPROXIMATED_BY, OCCURS_IN, DERIVED_FROM, HAS_UNIT, BELONGS_TO, DEPENDS_ON_CONTEXT, PROVED_BY, CITED_FROM, SEMANTIC_EQUIVALENT_TO}

4. TYPING AND IDENTITY
rule R1: Every constant has a stable identifier and at least one authoritative definition.
rule R2: A symbol is not a globally unique identifier; resolve meaning using domain, notation and context.
rule R3: Exact identity requires compatible definitions, domains and assumptions.
rule R4: Numerical proximity does not establish mathematical equivalence.
rule R5: Physical measured constants retain unit, measurement uncertainty and reference epoch or source.

5. EXAMPLE ENTITIES
constant pi {
  definition: circumference / diameter for a Euclidean circle
  domain: EuclideanGeometry
  value_kind: exact
  value: pi
  relations: [pi = 4 * arctan(1)]
}
constant e {
  definition: lim(n -> infinity) (1 + 1/n)^n
  domain: RealAnalysis
  value_kind: exact
  value: e
}
constant golden_ratio {
  definition: positive root of x^2 - x - 1 = 0
  domain: Algebra
  value_kind: exact
  value: (1 + sqrt(5))/2
}

6. RELATIONSHIP CONSTRUCTION
flow ConstantGraphBuild(input: ConstantRecord) -> GraphDelta {
  require validate_schema(input)
  id = canonicalise_identity(input.definition, input.domain, input.assumptions)
  upsert_node(id, type=Constant, properties=input)
  link(id, DEFINED_BY, input.definition)
  for symbol in input.symbols: link(id, DENOTED_BY, symbol)
  for source in input.provenance.sources: link(id, CITED_FROM, source)
  for expression in discover_uses(id): link(id, OCCURS_IN, expression)
  return validated_graph_delta()
}

7. SEMANTIC EQUIVALENCE
flow SemanticEquivalence(a: ConstantEntity, b: ConstantEntity) -> Decision {
  require compatible_domains(a,b)
  require compatible_assumptions(a,b)
  if prove_equal(a.definition,b.definition): return EQUIVALENT
  if disprove_equal(a.definition,b.definition): return DISTINCT
  return UNRESOLVED
}
Do not collapse aliases with unrelated meanings, e.g. c as a speed constant versus c as an arbitrary integration constant.

8. DIMENSIONAL VALIDATION
flow ValidateDimensions(equation: Equation) -> Validation {
  infer dimensions for all operands
  require matching dimensions on both sides of equality
  require dimensionless inputs for exp/log/trigonometric functions unless explicitly generalised
  return PASS or FAIL with explanation
}

9. PROOF AND CONSISTENCY
flow ValidateConstant(c: ConstantEntity) -> ValidationReport {
  check unique id, definition, domain, symbol scope
  check type compatibility and dimensional consistency
  check exact identities using symbolic proofs where available
  check numeric approximations with explicit error bounds
  check graph contradictions and provenance completeness
  return {status: VALID | INVALID | UNRESOLVED, evidence, diagnostics}
}

10. CONFIDENCE
confidence = {definition_quality, proof_status, source_reliability, context_match, uncertainty}
Confidence scores are epistemic annotations, not substitutes for mathematical proof. An unproved identity must never be labelled a theorem based solely on high confidence.

11. PROVENANCE
provenance = {author_or_institution, source_uri, method, recorded_at, version, licence, derivation_chain}
Record source revisions and preserve historical definitions. For measured values, record the measurement standard and applicable uncertainty.

12. FLOW_HASH
FLOW_HASH = SHA256(canonical_serialisation({id, definition, domain, assumptions, value_kind, units, version}))
Canonical serialisation must be deterministic; do not hash display formatting or transient graph traversal order.

13. UPDATE_MODEL
flow UpdateConstant(old: ConstantEntity, proposed: ConstantEntity) -> UpdateResult {
  verify provenance and schema
  calculate semantic_diff(old, proposed)
  if identity_changes: create new version or new entity, preserve links
  else: append auditable revision
  revalidate dependent equations and graph edges
  recompute FLOW_HASH
  return revision_log
}

14. AI_CONSUMPTION
ai_consumption = {
  retrieval: by identifier, symbol plus context, definition, domain
  reasoning: type-checked symbolic substitutions and graph traversal
  outputs: explicit assumptions, exact versus approximate status, citations
  safeguards: avoid symbol collisions, invented proofs and unsupported precision
  uncertainty: expose unresolved identity and source conflicts
}

15. EXECUTION PIPELINE
INPUT -> PARSE -> NORMALISE -> RESOLVE_CONTEXT -> TYPE_CHECK -> BUILD_GRAPH -> VERIFY -> RECORD_PROVENANCE -> COMPUTE_FLOW_HASH -> PUBLISH -> MONITOR_UPDATES

16. INVARIANTS
I1: Stable identifiers remain distinct from surface symbols.
I2: Exact and approximate equality are never conflated.
I3: A measured value never silently becomes an exact mathematical identity.
I4: Unit and domain constraints propagate through graph inferences.
I5: All revisions retain traceable provenance.
I6: Unknown or unproved assertions remain explicitly unresolved.

END FLOW ALGEBRA