Solution Ontology

FLOW ALGEBRA
SOLUTION ONTOLOGY
Kireth Semantic Library | Computational Mathematical Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/computational-mathematical-entities/solution-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Status: Proposed mathematical formalisation; source-page wording not independently verified.
Version: 1.0 | Date: 2026-10-14 | Locale: en-GB

1. ONTOLOGY IDENTITY
ontology_id: kireth:computational-mathematical-entities:solution
entity_type: MathematicalSolution
parent_type: ComputationalMathematicalEntity
scope: Solutions to equations, constraints, computational and optimisation problems.

2. TYPED ENTITIES
Problem := {identifier, domain, specification, assumptions, constraints}
CandidateSolution := {problem_id, assignment, representation, provenance}
ExactSolution <: CandidateSolution
ApproximateSolution <: CandidateSolution
FeasibleSolution <: CandidateSolution
OptimalSolution <: FeasibleSolution
SolutionSet := Set[CandidateSolution]
Certificate := {claim, evidence, checker, verification_status}
Residual := {candidate_id, residual_value, norm, units}
Tolerance := {absolute, relative, scale}

3. CORE SEMANTICS
Let P be a problem, D(P) its admissible domain, and C_P(x) its defining condition.
Sol(P) = {x in D(P) | C_P(x) is true}.
Exact(P,x) iff x in D(P) and C_P(x) is satisfied exactly.
Feasible(P,x) iff every required constraint is satisfied.
For an optimisation problem min f(x) over F:
  Optimal(P,x*) iff x* in F and for all y in F, f(x*) <= f(y).
Approximate(P,x,epsilon) requires an explicitly defined error metric and bound;
  small residual alone does not imply small forward error without further assumptions.
Uniqueness(P) iff there exists exactly one x in Sol(P).
Infeasible(P) iff Sol(P) is empty.

4. MATHEMATICAL RULES
R1 [Membership]: x in Sol(P) -> C_P(x).
R2 [Exact verification]: x in D(P) AND verified(C_P(x)) -> Exact(P,x).
R3 [Feasibility]: all constraints verified -> Feasible(P,x).
R4 [Optimality]: Feasible(P,x) AND valid global optimality certificate -> Optimal(P,x).
R5 [Uniqueness]: existence proof AND at-most-one proof -> UniqueSolution(P,x).
R6 [Counterexample]: verified violation of a required constraint -> NOT Feasible(P,x).
R7 [Approximation]: error_metric(x,x*) <= epsilon -> Approximate(P,x,epsilon),
    only when x* is known or a valid error certificate establishes the bound.
R8 [No inference]: convergence of an iteration does not alone prove optimality.
R9 [No inference]: failure to find a solution does not prove infeasibility.

5. TYPED RELATIONSHIPS
SOLVES(CandidateSolution, Problem)
HAS_DOMAIN(Problem, Domain)
SATISFIES(CandidateSolution, Constraint)
APPROXIMATES(ApproximateSolution, ExactSolution)
HAS_RESIDUAL(CandidateSolution, Residual)
HAS_CERTIFICATE(CandidateSolution, Certificate)
GENERATED_BY(CandidateSolution, Algorithm)
VALIDATED_BY(CandidateSolution, VerificationProcedure)
BELONGS_TO(CandidateSolution, SolutionSet)
OPTIMISES(OptimalSolution, ObjectiveFunction)

6. EXECUTABLE FLOW
flow "SolutionValidation" v1.0
input: problem:P, candidate:x, specification:S, optional certificate:K
require: type_check(P,x,S)
step 1: verify domain membership and assumptions
step 2: evaluate constraints using appropriate exact or certified numerical methods
step 3: compute residuals and record numerical precision where relevant
step 4: check supplied proof or error certificate with a trusted checker
step 5: classify as exact, approximate, feasible, optimal, invalid or unverified
step 6: emit evidence, limitations, provenance and validation status
output: {classification, verified_claims, residuals, evidence, status}
rule: never upgrade an unverified numerical result to a proven exact solution

7. SEMANTIC DISTINCTIONS
Solution != algorithm used to obtain a solution.
Candidate != verified solution.
Feasible != optimal.
Local optimum != global optimum.
Numerically small residual != guaranteed small solution error.
Exact symbolic expression != automatically verified exactness.
Solution set != single solution.

8. CONFIDENCE AND VALIDATION
confidence: {level: verified|supported|tentative|unknown, basis: evidence_reference}
validation: {type_correct, domain_checked, constraints_checked,
             certificate_checked, numerical_precision_recorded,
             assumptions_recorded, provenance_recorded}
Do not interpret confidence as mathematical proof.

9. SEMANTIC_EQUIVALENCE
ExactSolution ~= VerifiedSolution only when exact verification is established.
FeasibleSolution != OptimalSolution.
ApproximateSolution != ExactSolution without an exactness certificate.
Synonyms: solution, mathematical solution, admissible answer (context-dependent).

10. FLOW_HASH AND UPDATE_MODEL
FLOW_HASH: SHA-256 of canonical UTF-8 normalised ontology payload, excluding FLOW_HASH value.
UPDATE_MODEL: versioned; preserve source URL, author credits, changes and prior hash.
A real hash must be computed on the final canonical payload; no fabricated digest supplied.

11. AI_CONSUMPTION
consume: typed entities, definitions, premises, rules, certificates and provenance
infer: only claims licensed by stated premises and checked rules
explain: distinguish observed results, numerical estimates and formal proofs
refuse_to_infer: optimality, uniqueness or infeasibility without sufficient evidence
output_format: {entity_id, claim, status, assumptions, evidence, confidence, provenance}

12. EXAMPLE
P: solve x^2 - 4 = 0 over real numbers.
Sol(P) = {-2, 2}.
Verify: (-2)^2 - 4 = 0; 2^2 - 4 = 0.
Factorisation: (x - 2)(x + 2) = 0 proves completeness of this solution set over R.

END FLOW ALGEBRA