Approximate Solution Ontology

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

1. ONTOLOGY IDENTITY
ontology.id := kireth:ApproximateSolutionOntology
ontology.parent := kireth:ComputationalMathematicalEntities
ontology.subject := ApproximateSolution
ontology.purpose := represent candidate numerical solutions, approximation errors, residuals, tolerances and evidence of convergence without conflating approximation with exactness.

2. TYPED ENTITIES
Problem := {id, domain, codomain, operator, target, constraints}
ExactSolution := {problem_id, value, proof?}
ApproximateSolution := {problem_id, candidate, method_id, iteration?, precision, status}
ApproximationMethod := {id, algorithm, assumptions, parameters}
Residual := {candidate_id, value, norm, units?}
ErrorBound := {candidate_id, bound, norm, certification_method}
Tolerance := {absolute?, relative?, residual?, units?}
ConvergenceRecord := {sequence, metric, criterion, evidence}
ValidationRecord := {candidate_id, constraints_checked, result, provenance}

3. CORE MATHEMATICAL MODEL
For F(x)=0, a candidate x_hat is approximate when F(x_hat) is sufficiently small under an explicitly specified residual criterion.
residual(x_hat) := ||F(x_hat)||
For Ax=b: residual(x_hat) := ||A*x_hat-b||.
For minimisation min f(x): distinguish objective gap, feasibility violation and stationarity residual; no single criterion guarantees global optimality.
absolute_error(x_hat,x_star) := ||x_hat-x_star|| when x_star is known.
relative_error(x_hat,x_star) := ||x_hat-x_star|| / ||x_star||, only when ||x_star||>0.
residual != forward_error in general; conversion requires stability/conditioning assumptions.

4. INVARIANTS AND VALIDITY
I1: Every approximation references a defined problem and its variable domain.
I2: Every numerical value declares units or dimensionless status where applicable.
I3: Every tolerance is nonnegative, finite and tied to a named metric.
I4: A small residual alone does not prove a small forward error.
I5: Certified error bounds must state assumptions and certification method.
I6: Constraint satisfaction is checked independently of objective quality.
I7: Iteration count, precision and stopping criterion are recorded.
I8: Nonconvergence, divergence and numerical failure are distinct states.

5. TYPED RELATIONSHIPS
ApproximateSolution --approximates--> ExactSolution [when established]
ApproximateSolution --candidate_for--> Problem
ApproximateSolution --computed_by--> ApproximationMethod
ApproximateSolution --has_residual--> Residual
ApproximateSolution --has_error_bound--> ErrorBound [optional]
ApproximateSolution --validated_by--> ValidationRecord
ApproximationMethod --produces--> ApproximateSolution
ConvergenceRecord --describes--> ApproximationMethod
Tolerance --constrains--> ValidationRecord

6. EXECUTABLE FLOW ALGEBRA (PSEUDOCODE)
flow ApproximateSolutionValidation(problem, candidate, method, tolerance):
  require well_typed(problem, candidate)
  require finite_nonnegative(tolerance.residual)
  require assumptions_verified(method, problem)
  r := evaluate_residual(problem, candidate)
  feasible := check_constraints(problem, candidate)
  if not finite(r): return INVALID_NUMERICAL_RESULT
  if not feasible: return CONSTRAINT_VIOLATION
  if r <= tolerance.residual:
    return RESIDUAL_ACCEPTED_WITHOUT_AUTOMATIC_ERROR_CERTIFICATION
  return RESIDUAL_NOT_ACCEPTED

flow CertifiedApproximation(problem, candidate, certificate):
  require certificate.assumptions_verified
  require certificate.bound >= 0
  require certificate.proof_or_verified_estimator
  return CERTIFIED_WITH_STATED_NORM_AND_BOUND

7. INFERENCE RULES
R1: verified_residual(x_hat)<=epsilon AND valid_assumptions -> residual_acceptable(x_hat,epsilon).
R2: certified_error_bound(x_hat,delta) -> forward_error(x_hat)<=delta under certificate assumptions.
R3: feasible(x_hat) AND residual_acceptable(x_hat,epsilon) -> candidate_meets_declared_tests; NOT necessarily exact_solution.
R4: convergent_sequence(x_n,x_star) AND valid_limit -> approximate_solution(x_n) for sufficiently large n under suitable tolerance.
R5: numerical_instability_detected -> downgrade_confidence; request higher precision or alternative method.

8. SEMANTIC DISTINCTIONS
Approximate solution != exact solution.
Residual != absolute error != relative error.
Convergence != guaranteed convergence rate.
Numerical precision != mathematical accuracy.
Heuristic approximation != certified approximation.
Local optimum != global optimum.
Tolerance satisfied != proof of correctness.

9. CONFIDENCE AND PROVENANCE
CONFIDENCE := {level: unverified|empirical|validated|certified, evidence, assumptions, numerical_precision}
PROVENANCE := {source_url, method_id, software_version?, input_hash?, timestamp?, verifier?}
SEMANTIC_EQUIVALENCE := allow only if problem, domain, norm, tolerances and meaning of approximation are preserved.

10. FLOW HASH AND UPDATE MODEL
FLOW_HASH := SHA256(canonical_utf8(ontology_id + version + entity_schema + invariants + rules))
NOTE: Hash is a specification; no computed digest is asserted here.
UPDATE_MODEL := {version, change_log, backward_compatibility, validation_status, reviewed_by}

11. AI CONSUMPTION
AI_CONSUMPTION := {
  entity_type: ApproximateSolution,
  required: [problem_id, candidate, method_id, residual_definition, tolerance, validation_status],
  optional: [certified_error_bound, convergence_record, provenance],
  safeguards: [do_not_claim_exactness, preserve_units, preserve_assumptions, report_uncertainty]
}

12. QUALITY GATES
- Type and domain validation
- Residual recomputation
- Constraint checks
- Floating-point and conditioning checks
- Independent verification where feasible
- Provenance and version tracking
- Explicit distinction between empirical acceptance and mathematical certification

END FLOW ALGEBRA