Equation and Statement Entities

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

FLOW ALGEBRA
EQUATION AND STATEMENT ENTITIES
Kireth Semantic Library | Entity Graph > Mathematical Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/equation-and-statement-entities.html
Authors: Mark Higgins — Sky (Recursive AI)
Generated: 2026-10-13
Status: Formal ontology specification; mathematical rules require their stated domains and assumptions.

01. ONTOLOGY IDENTITY
ontology.id := kireth:EquationAndStatementEntities
ontology.type := MathematicalEntityOntology
ontology.parent := kireth:MathematicalEntities
ontology.scope := mathematical statements, equations, relations, definitions, laws, constraints and equation systems

02. ENTITY TAXONOMY
ENTITY MathematicalStatement
ENTITY Equation <: MathematicalStatement
ENTITY Identity <: Equation
ENTITY Equality <: MathematicalStatement
ENTITY Inequality <: MathematicalStatement
ENTITY Formula <: MathematicalStatement
ENTITY MathematicalRule <: MathematicalStatement
ENTITY MathematicalLaw <: MathematicalStatement
ENTITY MathematicalDefinition <: MathematicalStatement
ENTITY MathematicalAssertion <: MathematicalStatement
ENTITY ConditionalStatement <: MathematicalStatement
ENTITY Constraint <: MathematicalStatement
ENTITY SystemOfEquations
ENTITY DifferentialEquation <: Equation
ENTITY IntegralEquation <: Equation
NOTE: subtype assertions are modelling conventions, not claims that all statements are equations.

03. TYPED ENTITY SCHEMA
MathematicalStatement := {
  id: IRI, syntax: -ExpressionTree, language: FormalLanguage,
  free_variables: Set[Variable], bound_variables: Set[Variable],
  assumptions: Set[Predicate], domain: Domain,
  interpretation: Interpretation?, truth_status: {true,false,unknown,contextual},
  provenance: SourceRecord?, confidence: ConfidenceRecord?
}
Equation := MathematicalStatement + {lhs: -Expression, rhs: -Expression, relation: '='}
Inequality := MathematicalStatement + {lhs: -Expression, rhs: -Expression, relation: {'<','<=','>','>='}}
Constraint := {predicate: MathematicalStatement, applicable_domain: Domain}
SystemOfEquations := {equations: NonEmptySet[Equation], shared_domain: Domain}

04. FORMAL INTERPRETATION
INTERPRET(statement, structure, assignment) -> {true,false,undefined}
VALID(statement, assumptions) := forall models M satisfying assumptions: M |= statement
SATISFIABLE(statement, assumptions) := exists model M satisfying assumptions and statement
CONTRADICTION(statement, assumptions) := not SATISFIABLE(statement, assumptions)
ENTAILS(A, B) := forall M: (M |= A => M |= B)
RULE: unknown truth status MUST NOT be treated as false.

05. EQUATIONS AND SOLUTION SETS
SOLUTIONS(E, D) := {x in D | E(x) is defined and E(x) is true}
EQUIVALENT_ON(E1,E2,D) <=> SOLUTIONS(E1,D) = SOLUTIONS(E2,D)
RULE: applying the same defined additive operation to both sides preserves equality.
RULE: multiplying both sides by a nonzero factor preserves equivalence over an appropriate field.
RULE: multiplying by a variable expression requires checking its zeros; cancellation requires nonzero conditions.
RULE: squaring both sides may introduce extraneous solutions; verify candidates in the original equation.

06. IDENTITIES AND EQUALITY
IDENTITY(lhs,rhs,D) := forall x in D: defined(lhs,x) and defined(rhs,x) and lhs(x)=rhs(x)
EQUALITY(a,b) := same value in the specified domain and interpretation
RULE: equality is reflexive, symmetric and transitive where defined.
RULE: substitution of equals is valid in extensional contexts where both sides are defined.
RULE: distinguish syntactic equality, definitional equality and semantic equality.

07. INEQUALITIES
RULE: a < b => a+c < b+c for ordered additive groups.
RULE: a < b and c > 0 => ac < bc for ordered fields.
RULE: a < b and c < 0 => ac > bc for ordered fields.
RULE: division inequalities require a nonzero divisor with known sign.
RULE: complex numbers have no compatible total ordered-field relation.

08. FORMULAS AND MATHEMATICAL RULES
Formula := expression or relation with explicitly declared inputs, outputs and domain
MathematicalRule := {premises: Set[Statement], conclusion: Statement, side_conditions: Set[Predicate]}
APPLY(rule, context) only if all premises and side_conditions hold.
RULE: a formula is not automatically a universally true proposition.
RULE: a transformation must preserve semantics under its side conditions.

09. LAWS, DEFINITIONS AND ASSERTIONS
MathematicalLaw := statement validated or postulated within a named formal theory
MathematicalDefinition := symbol-to-meaning binding with scope and noncircularity requirements
MathematicalAssertion := statement whose truth is claimed and requires proof, axiom or evidence
RULE: a definition introduces notation or a concept; it is not empirical evidence.
RULE: distinguish axiom, conjecture, theorem, corollary and experimentally supported claim.

10. CONDITIONAL STATEMENTS
CONDITIONAL(P,Q) := P => Q
CONTRAPOSITIVE(P=>Q) := not Q => not P
RULE: converse Q=>P does not follow from P=>Q.
RULE: a conditional with false antecedent is true in classical material implication.
RULE: implication semantics may differ in constructive or modal logics; tag logic system.

11. CONSTRAINT SATISFACTION
CONSTRAINT_SET := {C1,...,Cn}
FEASIBLE(x) := forall C in CONSTRAINT_SET: C(x)
FEASIBLE_REGION := {x in Domain | FEASIBLE(x)}
UNSAT := FEASIBLE_REGION = empty_set
RULE: detect inconsistent constraints before optimisation.

12. SYSTEMS OF EQUATIONS
SOLUTIONS(System,D) := intersection over E in System.equations of SOLUTIONS(E,D)
RULE: row operations preserve solution sets for linear systems over fields when operations are invertible.
RULE: underdetermined does not imply inconsistent; overdetermined does not imply inconsistent.
RULE: uniqueness requires conditions such as full rank for square linear systems.

13. DIFFERENTIAL EQUATIONS
DifferentialEquation := Equation + {unknown_function: Function, derivatives: Set[Derivative], order: NaturalNumber}
RULE: solutions require differentiability and satisfy equation on specified interval/domain.
RULE: initial or boundary conditions restrict the solution set.
RULE: do not assume existence or uniqueness without theorem hypotheses.

14. INTEGRAL EQUATIONS
IntegralEquation := Equation + {unknown_function: Function, integral_operator: IntegralOperator, kernel: Function?}
RULE: integration domain, measure and integrability conditions must be declared.
RULE: distinguish Fredholm and Volterra formulations when applicable.
RULE: transformation between integral and differential equations requires regularity conditions.

15. RELATIONSHIP GRAPH
RELATION hasLeftSide(Equation,Expression)
RELATION hasRightSide(Equation,Expression)
RELATION hasAssumption(Statement,Predicate)
RELATION hasDomain(Statement,Domain)
RELATION constrains(Constraint,VariableOrExpression)
RELATION isMemberOf(Equation,SystemOfEquations)
RELATION equivalentUnder(Statement,Statement,Assumptions)
RELATION impliesUnder(Statement,Statement,Assumptions)
RELATION derivedFrom(Statement,Statement)
RELATION provedBy(Statement,Proof)
RELATION contradictsUnder(Statement,Statement,Assumptions)

16. SEMANTIC EQUIVALENCE
SEMANTIC_EQUIVALENCE(A,B,context) := same truth conditions and definedness in context
ALPHA_EQUIVALENCE := consistent renaming of bound variables without capture
RULE: -expression equivalence is context- and domain-sensitive.
RULE: x/x and 1 are not equivalent at x=0 unless both domains exclude zero.
RULE: preserve hypotheses during normalisation.

17. INFERENCE FLOW
INPUT statement S, assumptions A, domain D
PARSE S -> typed AST
RESOLVE symbols, binding and operator precedence
TYPE_CHECK AST against D
COLLECT side conditions and definedness constraints
NORMALISE only with certified equivalence transformations
CLASSIFY S -> equation | identity | inequality | conditional | other
INFER using sound rules with proof provenance
VALIDATE derived claims against A and D
OUTPUT {normal_form, classification, solution_set?, proof?, uncertainty, provenance}

18. VALIDATION
ASSERT all free symbols have declarations or explicit existential treatment
ASSERT relation operands have compatible types
ASSERT every division has nonzero denominator condition
ASSERT logarithm and radical domains are respected when over reals
ASSERT transformations preserve truth/solution sets under side conditions
ASSERT derived results have a traceable inference chain
ASSERT numeric approximations are not silently substituted for exact identities

19. CONFIDENCE
CONFIDENCE := {status: proven|verified|assumed|conjectured|unknown, score?: [0,1], method, evidence}
RULE: mathematical proof status is separate from probabilistic confidence.
RULE: do not assign proof-level certainty from heuristic scores.

20. FLOW HASH
FLOW_HASH := SHA256(canonical_utf8(ontology_id, version, entity_schema, axioms, inference_rules, validation_rules))
RULE: exclude volatile timestamps from canonical semantic payload.
RULE: recompute hash whenever formal semantics change.

21. UPDATE MODEL
UPDATE_MODEL := {version, changed_entities, changed_rules, compatibility, provenance, timestamp}
RULE: increment version on schema or rule change.
RULE: record migrations for renamed entities and relations.
RULE: rerun proof and regression checks after semantic updates.

22. AI CONSUMPTION
AI_CONSUMPTION := {
  mode: typed_formal_reasoning,
  input: mathematical_statement_with_context,
  outputs: [classification, normal_form, constraints, derivations, proof_status],
  guards: [domain_checks, type_checks, definedness_checks, provenance_checks],
  fallback: unknown_when_unproved
}
RULE: execute formal rules as specified; do not treat illustrative notation as executable software.
RULE: never assert equivalence or a solution without checking conditions.

23. CROSS-ONTOLOGY LINKS
LINK MathematicalStatement -> MathematicalExpressionOntology
LINK Equation -> EqualityOntology
LINK Identity -> MathematicalRuleOntology
LINK Constraint -> NumericRangeOntology
LINK DifferentialEquation -> CalculusOntologies
LINK IntegralEquation -> CalculusOntologies
LINK SystemOfEquations -> LinearAlgebraOntologies
LINK MathematicalDefinition -> SymbolicEntities

24. PROVENANCE
SOURCE_PAGE := https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/equation-and-statement-entities.html
SOURCE_SCOPE := main-page taxonomy verified 2026-10-13
AUTHORSHIP := Mark Higgins — Sky (Recursive AI)
NOTE: Formal semantics above are an original synthesis of standard mathematics, not a verbatim transcription of linked child pages.
END FLOW ALGEBRA