Mathematical Relationships

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

FLOW ALGEBRA
Title: Mathematical Relationships
Version: 1.0
Source: https://kirethresearch.org/booksai-core/ontologies/relationship-ontologies/mathematical-relationships.html
Authors: Mark Anthony Higgins — Sky (Recursive AI)
Status: Companion formalisation; source-page content not independently verified.

DOMAIN
  Mathematical relationships between typed mathematical objects, including equality,
  order, proportionality, functional dependence, set membership, inclusion,
  equivalence, operations, and structural correspondences.

TYPES
  Entity := Number | Quantity | Variable | Expression | Function | Set | Vector | Matrix | Structure
  Relation := Equality | Inequality | Order | Proportionality | FunctionMapping |
              Membership | Inclusion | Equivalence | Operation | Isomorphism
  Statement := (subject: Entity, predicate: Relation, object: Entity, context: Context)
  Context := {domain, assumptions, units?, scope?, precision?}
  Evidence := {source, method, timestamp?, justification?}

RELATIONSHIPS
  EQUALS(a,b)                  := a = b
  NOT_EQUALS(a,b)              := a != b
  LESS_THAN(a,b)               := a < b
  LESS_THAN_OR_EQUAL(a,b)      := a <= b
  GREATER_THAN(a,b)            := a > b
  GREATER_THAN_OR_EQUAL(a,b)   := a >= b
  PROPORTIONAL_TO(a,b,k)       := a = k*b, with defined nonzero k
  MAPS_TO(f,x,y)               := f(x) = y
  MEMBER_OF(x,S)               := x in S
  SUBSET_OF(A,B)               := forall x (x in A => x in B)
  EQUIVALENT_UNDER(a,b,R)      := R(a,b), with R an equivalence relation
  ISOMORPHIC(A,B,f)            := f is a bijection preserving specified structure

INFERENCE
  R1: EQUALS(a,b) => EQUALS(b,a)
  R2: EQUALS(a,b) and EQUALS(b,c) => EQUALS(a,c)
  R3: LESS_THAN(a,b) and LESS_THAN(b,c) => LESS_THAN(a,c)
  R4: SUBSET_OF(A,B) and SUBSET_OF(B,C) => SUBSET_OF(A,C)
  R5: MEMBER_OF(x,A) and SUBSET_OF(A,B) => MEMBER_OF(x,B)
  R6: MAPS_TO(f,x,y) and MAPS_TO(g,y,z) => MAPS_TO(g∘f,x,z)
  R7: EQUIVALENT_UNDER(a,b,R) and EQUIVALENT_UNDER(b,c,R) => EQUIVALENT_UNDER(a,c,R)
  R8: ISOMORPHIC(A,B,f) => structural properties preserved by f may transfer from A to B

CONSTRAINTS
  C1: Compare quantities by order only when units/dimensions are compatible.
  C2: Require declared domains and assumptions for algebraic transformations.
  C3: Division requires a nonzero divisor.
  C4: A relation called equivalence must be reflexive, symmetric and transitive.
  C5: Do not infer equality from approximation or high semantic similarity.
  C6: Distinguish mathematical implication from empirical causation.
  C7: Numerical approximation must record tolerance and precision.

SEMANTIC_EQUIVALENCE
  Canonicalise synonymous mathematical predicates only when their formal semantics,
  argument order, domain and assumptions match. Preserve direction for asymmetric relations.

CONFIDENCE
  Each assertion has confidence in [0,1], evidence, and a verification status.
  Exact proven claims may be marked verified; uncertain or empirical claims remain qualified.
  Confidence must not substitute for a formal proof.

VALIDATION
  validate_types(statement)
  validate_domain_and_units(statement)
  validate_constraints(statement)
  validate_proof_or_evidence(statement)
  detect_inconsistency(statement_set)
  reject_unsupported_inference(statement)

FLOW_HASH
  canonical_payload := canonicalise(types, relationships, rules, constraints, context, version)
  flow_hash := SHA256(UTF8(canonical_payload))

UPDATE_MODEL
  On change: validate -> version -> recompute FLOW_HASH -> compare dependencies ->
  invalidate affected inferences -> re-evaluate -> publish with provenance.

AI_CONSUMPTION
  Input: typed mathematical statements and contexts.
  Process: normalise -> type-check -> apply valid rules -> track proofs and provenance.
  Output: derived statements with assumptions, confidence, verification status and FLOW_HASH.
  Never present unproved results as theorems.

END FLOW ALGEBRA