Equation Ontology

FLOW ALGEBRA
EQUATION ONTOLOGY
Kireth Semantic Library | Mathematical Entities | Equation and Statement Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/equation-and-statement-entities/equation-ontology-2.html
Version: 1.0 | Date: 2026-10-13 | Locale: en-GB
Authors: Mark Higgins — Sky (Recursive AI)
Status: Mathematical reference model; source page not independently verified.

01. PURPOSE
flow "EquationOntology" v1.0
  intent: represent, type, validate, transform and solve mathematical equations
  principle: an equation asserts equality between two expressions under a specified interpretation
  distinction: -expression != equation; equation != identity; solution != proof

02. CORE TYPES
  Equation := Equality(LeftExpression, RightExpression, Context)
  Expression := Constant | Variable | OperatorApplication | FunctionApplication
  Context := {domain, assumptions, bindings, units?, interpretation}
  SolutionSet := {assignment | Equation evaluates true in Context}
  Transformation := {input_equation, operation, side_conditions, output_equation}
  ProofRecord := {premises, inference_rules, conclusion, verification_status}

03. ENTITY SCHEMA
  equation.id: URI
  equation.left: -Expression
  equation.right: -Expression
  equation.variables: Set[Variable]
  equation.domain: Set | ProductDomain
  equation.assumptions: Set[Predicate]
  equation.classification: EquationKind
  equation.solution_set: Set[Assignment] | SymbolicSet | Unknown
  equation.provenance: SourceRecord
  equation.confidence: ConfidenceRecord
  equation.flow_hash: Digest

04. EQUATION KINDS
  algebraic: polynomial, rational, radical
  transcendental: exponential, logarithmic, trigonometric
  differential: ordinary or partial differential equation
  integral: unknown function occurs under an integral operator
  functional: unknown is a function
  linear: unknowns occur to first degree without products of unknowns
  nonlinear: otherwise, relative to specified unknowns
  conditional: true only for some admissible assignments
  identity: true for all admissible assignments in declared context
  inconsistent: no admissible assignment satisfies equality

05. WELL-FORMEDNESS
  validate.syntax: both sides parse as expressions
  validate.types: left and right belong to comparable types
  validate.units: physical dimensions compatible when applicable
  validate.bindings: all free variables are declared or intentionally free
  validate.domain: all partial operations have defined inputs
  validate.assumptions: constraints are explicit and consistent

06. EQUALITY SEMANTICS
  eval(Equality(L,R), assignment, context) :=
    defined(L) AND defined(R) AND equal(eval(L),eval(R))
  solve(E, D) := {a in D | eval(E,a,context) = true}
  identity(E,D) := forall a in D: eval(E,a,context) = true
  unsatisfiable(E,D) := solve(E,D) = empty_set
  NOTE: equality may be exact, symbolic, structural or tolerance-based;
        tolerance comparisons must not silently replace mathematical equality.

07. ALGEBRAIC TRANSFORMATIONS
  add_same: L = R -> L + T = R + T, where terms are defined
  subtract_same: L = R -> L - T = R - T
  multiply_same: L = R -> L*T = R*T (forward implication)
  divide_same: L = R -> L/T = R/T only if T != 0
  cancellation: L*T = R*T -> L = R only if T != 0
  apply_injective: f(L) = f(R) <-> L = R when f is injective on domain
  square_both: L = R -> L^2 = R^2; reverse generally invalid
  root_both: requires sign and branch restrictions
  logarithm: requires positive real arguments and valid base for real logs
  transform.audit: record domain restrictions and whether equivalence is preserved

08. SOLUTION OPERATIONS
  parse -> normalise -> classify -> determine_domain -> transform -> solve -> verify
  verify(candidate): substitute into ORIGINAL equation and check assumptions
  solution.status: exact | implicit | numerical_approximation | conditional | unresolved
  root.multiplicity: defined for polynomial or analytic contexts as appropriate
  extraneous_solution: candidate generated by non-equivalent manipulation

09. SYSTEMS OF EQUATIONS
  System := finite_set[Equation]
  solve(System,D) := intersection of individual solution constraints
  linear_system: A*x = b
  consistency: existence of a common assignment
  uniqueness: solution set has exactly one element
  underdetermined: constraints insufficient for uniqueness; may still be inconsistent
  overdetermined: more equations than unknowns; may be consistent or inconsistent

10. DIFFERENTIAL EQUATION SEMANTICS
  unknown: function with specified domain and differentiability
  equation: relation among unknown function and derivatives
  initial_conditions: constraints at designated input values
  boundary_conditions: constraints on domain boundary
  solution: function satisfying equation and all conditions
  verification: differentiation and substitution under required regularity

11. SYMBOLIC NORMALISATION
  canonicalise numeric constants where exactness is preserved
  normalise commutative operands only where algebraic laws apply
  preserve expression domain and branch semantics
  distinguish syntactic equality from semantic equivalence
  alpha_rename bound variables without changing meaning
  avoid rewriting across singularities or undefined expressions

12. KNOWLEDGE GRAPH
  Equation --hasLeftSide--> Expression
  Equation --hasRightSide--> Expression
  Equation --containsVariable--> Variable
  Equation --hasDomain--> Domain
  Equation --hasAssumption--> Predicate
  Equation --hasSolutionSet--> SolutionSet
  Equation --isPartOf--> EquationSystem
  Equation --derivedBy--> Transformation
  Equation --supportedBy--> ProofRecord
  Equation --relatedTo--> MathematicalStatement
  Equation --usesOperator--> EqualityOperator

13. SEMANTIC_EQUIVALENCE
  E1 equivalent_to E2 iff solution_sets(E1,D) = solution_sets(E2,D)
  compare only under shared domain, variable declarations and assumptions
  identity_equivalence may be stronger than coincident solution sets
  structural_equivalence: canonical AST match under allowed rewrites
  proof_equivalence: certified bidirectional implication
  never infer equivalence solely from matching text or numeric samples

14. INFERENCE RULES
  R01: if a satisfies E and E equivalent_to F, then a satisfies F
  R02: if E implies F, then solutions(E) subset_of solutions(F)
  R03: if E and F are jointly required, solve(E AND F) = solve(E) intersect solve(F)
  R04: if substitution makes both sides equal and defined, candidate is valid
  R05: if E is identity on D, solution_set(E,D) = D
  R06: if E is contradictory on D, solution_set(E,D) = empty_set
  R07: transformations with lost side_conditions cannot be certified equivalent

15. EXAMPLES
  E1: 2*x + 3 = 11; domain real; solution {x=4}
  E2: x^2 = 4; domain real; solutions {x=-2,x=2}
  E3: (x^2-1)/(x-1) = 2; domain real except x=1; solution empty_set
  E4: sin(x) = 0; domain real; solutions {x=k*pi | k integer}
  E5: x + 1 = x + 1; domain real; identity, solution_set = real
  E6: x + 1 = x + 2; domain real; inconsistent, solution_set = empty_set

16. VALIDATION TESTS
  test_01: reject division by zero
  test_02: reject unit-incompatible equality
  test_03: preserve x=1 exclusion in E3 after cancellation
  test_04: verify both signs for x^2=4
  test_05: reject extraneous roots from squaring
  test_06: distinguish exact equality from approximate numerical equality
  test_07: preserve assumptions across symbolic rewrites
  test_08: reject undeclared-variable ambiguity

17. CONFIDENCE
  mathematical_claim: proof_verified | mechanically_checked | literature_supported | unverified
  numerical_result: {method, precision, residual, error_bound?}
  confidence is evidence quality, not a substitute for a mathematical proof
  unknown or unverified results must remain explicitly labelled

18. PROVENANCE
  source_url: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/equation-and-statement-entities/equation-ontology-2.html
  reference_basis: established mathematical principles
  source_page_verified: false
  record: author, date, version, assumptions, transformation_history

19. FLOW_HASH
  canonical_payload := canonicalise(schema, equation, context, assumptions, version)
  FLOW_HASH := SHA256(UTF8(canonical_payload))
  requirement: deterministic key ordering and explicit encoding
  note: hash proves content identity only, not mathematical truth

20. UPDATE_MODEL
  versioning: semantic
  change_triggers: definition_change | rule_correction | schema_extension | proof_update
  migration: preserve prior hashes, provenance and backward references
  revalidate: all dependent inference rules and examples after updates

21. AI_CONSUMPTION
  input: structured equation, variables, domain, assumptions, objective
  output: classification, normalised form, solution, conditions, verification, provenance
  processing: parse -> typecheck -> reason -> solve -> verify -> explain
  safety: do not invent proofs; do not drop excluded values or branch conditions
  interoperability: AST, JSON-LD, RDF triples, symbolic mathematics systems

22. COMPLETION
  ontology_status: reference_flow_ready
  executable_status: declarative specification; implementation required
  citation_policy: retain source URL and explicit verification status
  authorship: Mark Higgins — Sky (Recursive AI)
  site: https://kirethresearch.org
END FLOW ALGEBRA