Mathematical Entities

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

FLOW ALGEBRA
MATHEMATICAL ENTITIES — INTEGRATION ONTOLOGY
Version: 1.0 | 2026-10-12
Authors: Mark Anthony Higgins — Sky (Recursive AI)
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities.html
Scope: main-page integration across mathematical entity families
Status: subject-based formalisation grounded in the live page taxonomy; not a transcription of embedded code

01. DECLARATIONS
ontology MathematicalEntities {
  namespace: kireth.entity_graph.mathematical
  root_type: MathematicalEntity
  identity: canonical_id:str
  representation: symbolic | numeric | structural | computational
  context: MathematicalContext
  provenance: ProvenanceRecord
  validation: ValidationResult
}

02. ENTITY TYPE SYSTEM
type MathematicalEntity = {id:str, kind:EntityKind, definition:str, domain:Domain?, representation:E-xpression?, provenance:ProvenanceRecord}
type MathematicalContext = {assumptions:Set[Proposition], notation:str, universe:Universe?, precision:PrecisionPolicy}
type Relation = {subject:MathematicalEntity, predicate:RelationKind, object:MathematicalEntity, justification:ProofReference?}
type Proposition = {formula:E-xpression, context:MathematicalContext, truth:True|False|Unknown, proof:ProofReference?}
type Measurement = {quantity:Quantity, unit:Unit, uncertainty:Uncertainty?, provenance:ProvenanceRecord}
type ProvenanceRecord = {source:str, author:str?, timestamp:str?, method:str?, revision:str?}
type ValidationResult = {status:valid|invalid|undetermined, diagnostics:List[str]}

03. DOMAIN TAXONOMY
  family[01]: "Foundations of Mathematical Entities" -> MathematicalEntity
  family[02]: "Number Entities" -> MathematicalEntity
  family[03]: "Numeric Value Entities" -> MathematicalEntity
  family[04]: "Symbolic Entities" -> MathematicalEntity
  family[05]: "Expression Entities" -> MathematicalEntity
  family[06]: "Equation and Statement Entities" -> MathematicalEntity
  family[07]: "Set Entities" -> MathematicalEntity
  family[08]: "Relation Entities" -> MathematicalEntity
  family[09]: "Function and Mapping Entities" -> MathematicalEntity
  family[10]: "Sequence and Series Entities" -> MathematicalEntity
  family[11]: "Algebraic Entities" -> MathematicalEntity
  family[12]: "Linear Algebra Entities" -> MathematicalEntity
  family[13]: "Geometric Entities" -> MathematicalEntity
  family[14]: "Coordinate and Spatial Entities" -> MathematicalEntity
  family[15]: "Calculus and Analysis Entities" -> MathematicalEntity
  family[16]: "Probability Entities" -> MathematicalEntity
  family[17]: "Statistical Entities" -> MathematicalEntity
  family[18]: "Graph and Network Entities" -> MathematicalEntity
  family[19]: "Logical Entities" -> MathematicalEntity
  family[20]: "Proof Entities" -> MathematicalEntity
  family[21]: "Computational Mathematical Entities" -> MathematicalEntity
  family[22]: "Transformation Entities" -> MathematicalEntity
  family[23]: "Measurement and Dimensional Entities" -> MathematicalEntity
  family[24]: "Recursive Mathematical Entities" -> MathematicalEntity
  family[25]: "AI Semantic Library Mathematical Entities" -> MathematicalEntity

04. CANONICAL RELATIONSHIPS
relation is_a(child, parent) : subtype hierarchy
relation element_of(element, set) : membership
relation subset_of(A, B) : all x in A imply x in B
relation maps_to(f, x, y) : f(x)=y
relation equivalent_to(a, b, context) : mathematical equivalence under assumptions
relation derived_from(result, premises) : justified derivation
relation depends_on(entity, assumption) : dependency
relation represented_by(entity, expression) : syntactic representation
relation measured_in(quantity, unit) : dimensional semantics
relation proven_by(statement, proof) : proof attachment

05. FORMAL INVARIANTS
invariant unique_id: all entities have unique canonical_id
invariant well_typed: every relation satisfies declared domain and codomain
invariant context_bound: all equivalence claims specify their assumptions
invariant no_division_by_zero: denominator != 0 when division evaluated
invariant function_totality: if total(f), every x in domain(f) maps to exactly one y
invariant proof_soundness: verified proofs use permitted inference rules
invariant dimensional_homogeneity: sums compare compatible dimensions
invariant finite_precision_disclosure: numerical approximation records tolerance

06. NUMBER AND VALUE OPERATIONS
rule natural_to_integer: n in N => n in Z
rule integer_to_rational: z in Z => z/1 in Q
rule rational_to_real: q in Q => q in R
rule real_to_complex: r in R => r+0i in C
rule exact_vs_approximate: exact_value != rounded_representation unless rounding_error=0
rule arithmetic_closure: apply only when operation and carrier are closed

07. SYMBOLIC AND EXPRESSION OPERATIONS
flow parse_expression(text, grammar) -> AST | ParseError
flow type_check(AST, context) -> TypedExpression | TypeError
flow normalise(expression, rewrite_rules, assumptions) -> canonical_expression
flow simplify(expression, assumptions) -> equivalent_expression + derivation
flow substitute(expression, variable, replacement) -> capture_avoiding_expression
rule rewrite_validity: apply rewrite only if side_conditions are satisfied

08. SET, RELATION AND FUNCTION OPERATIONS
rule union: x in A union B iff (x in A or x in B)
rule intersection: x in A intersection B iff (x in A and x in B)
rule composition: (g o f)(x) = g(f(x)) where codomain(f) compatible with domain(g)
rule injective: f(x1)=f(x2) => x1=x2
rule surjective: for every y in codomain(f), exists x in domain(f) with f(x)=y
rule bijective: injective(f) and surjective(f)

09. ALGEBRA AND LINEAR ALGEBRA
rule group: associative(op) and identity(op) and inverses(op) on carrier
rule ring: additive_abelian_group and associative_multiplication and distributivity
rule vector_space: vector_addition and scalar_multiplication satisfy field axioms
rule matrix_product: defined iff columns(A)=rows(B)
rule linear_map: T(a*u+b*v)=a*T(u)+b*T(v)
rule eigenpair: A*v=lambda*v and v!=zero_vector

10. GEOMETRY AND SPATIAL MODELS
type Point = {coordinates:Tuple[Scalar], frame:CoordinateSystem}
type Vector = {components:Tuple[Scalar], frame:CoordinateSystem}
rule distance_nonnegative: metric(p,q)>=0
rule distance_identity: metric(p,q)=0 iff p=q
rule coordinate_transform: preserve declared geometric invariants

11. CALCULUS AND ANALYSIS
rule derivative: derivative(f,x) = limit_h_to_0((f(x+h)-f(x))/h) if limit exists
rule integral_fundamental: derivative(integral_a_to_x(f(t)dt))=f(x) under suitable continuity
rule convergence: sequence converges iff epsilon-N definition satisfied
rule continuity: limit_x_to_a(f(x))=f(a) when defined

12. PROBABILITY AND STATISTICS
rule probability_bounds: 0<=P(A)<=1
rule normalisation: P(sample_space)=1
rule conditional: P(A|B)=P(A intersect B)/P(B) if P(B)>0
rule bayes: P(H|E)=P(E|H)*P(H)/P(E) if P(E)>0
rule statistic: estimate includes sample, estimator, assumptions and uncertainty
rule inference_limit: empirical association alone does not imply causation

13. GRAPH, LOGIC AND PROOF
type Graph = {vertices:Set[Vertex], edges:Set[Edge], directed:bool}
rule edge_validity: endpoints(edge) subset_of vertices(graph)
rule modus_ponens: P and (P implies Q) => Q
rule universal_instantiation: forall x P(x) => P(a) for a in domain
rule proof_validation: accept theorem only if all steps validated against axioms/rules

14. COMPUTATION AND TRANSFORMATIONS
flow evaluate(expr, environment, precision) -> Value | EvaluationError
flow transform(entity, operator, context) -> transformed_entity + provenance
flow solve(constraints, domain, assumptions) -> SolutionSet | Unsatisfiable | Unknown
flow numerical_approximation(problem, tolerance) -> result + error_bound
rule termination: recursive execution requires well_founded_measure or explicit resource_bound

15. MEASUREMENT AND DIMENSIONAL ENTITIES
rule convert_units: output = conversion(input, source_unit, target_unit) if dimensions match
rule incompatible_units: reject addition of quantities with incompatible dimensions
rule uncertainty: report uncertainty model and coverage assumptions

16. RECURSIVE ENTITY RESOLUTION
flow resolve(entity_id, visited:Set[id]) -> EntityGraph {
  if entity_id in visited: return cycle_reference(entity_id)
  entity = lookup(entity_id)
  children = map(resolve(child_id, visited union {entity_id}), entity.dependencies)
  return validated_graph(entity, children)
}

17. INFERENCE PIPELINE
flow mathematical_entity_reasoning(input, context) {
  parsed = parse_expression(input, context.grammar)
  typed = type_check(parsed, context)
  canonical = normalise(typed, context.rewrite_rules, context.assumptions)
  graph = resolve_dependencies(canonical)
  candidates = apply_sound_rules(graph, context.assumptions)
  validated = verify_types_assumptions_and_proofs(candidates)
  return attach_provenance_and_confidence(validated)
}

18. CONFIDENCE
CONFIDENCE(result) = {proof_status, source_reliability, numerical_error, assumption_coverage}
rule confidence_exact: a verified formal proof is distinct from empirical confidence
rule confidence_uncertain: unverified propositions retain unknown status
rule confidence_numerical: report tolerance and error bound separately

19. SEMANTIC_EQUIVALENCE
SEMANTIC_EQUIVALENCE(a,b,context) = verified_equivalence(a,b,context.assumptions)
rule equivalence_symmetry: equivalent(a,b) => equivalent(b,a)
rule equivalence_transitivity: equivalent(a,b) and equivalent(b,c) => equivalent(a,c)
rule identity_vs_equivalence: mathematical equivalence does not imply identical syntax or provenance

20. FLOW_HASH
FLOW_HASH(entity) = SHA256(canonical_serialisation(entity.type, entity.definition, assumptions, dependencies, ontology_version))
rule deterministic_hash: canonical_serialisation fixes key order, numeric format and encoding
rule hash_not_proof: matching hash detects identical canonical payload, not mathematical truth

21. UPDATE_MODEL
UPDATE_MODEL(old, new) {
  validate(new)
  compare canonical_id, definition, dependencies, assumptions, provenance
  invalidate dependent derivations if semantic content changed
  increment revision and recompute FLOW_HASH
  retain historical version and change log
}

22. AI_CONSUMPTION
AI_CONSUMPTION {
  ingest: typed_entity_graph
  require: provenance + explicit_assumptions + validation_status
  preserve: exact_vs_approximate + units + proof_status
  prohibit: unsupported_truth_claims + silent_assumption_changes
  output: structured_entities + justified_inferences + unresolved_questions
}

23. VALIDATION TESTS
test set_union: union({1,2},{2,3}) == {1,2,3}
test function_type: composition rejected when intermediate types incompatible
test dimension: 3 metres + 4 seconds -> TypeError
test proof: unsupported theorem -> undetermined
test bayes: P(E)=0 -> conditional_probability_undefined
test graph: edge to missing vertex -> invalid
test recursion: cyclic dependency -> cycle_reference, not infinite loop
test canonical_hash: identical canonical payloads -> identical hashes

24. INTEGRATION
parent: Entity Graph / Mathematical Entities
links: mathematical ontology families and their dedicated child pages
interop: RDF/OWL mappings where semantics are preserved; JSON-LD metadata where appropriate
note: rules are formal specifications; executable implementation requires a parser, type checker, proof checker and runtime

END FLOW ALGEBRA