Convergence Criterion Ontology

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

1. PURPOSE
Represent a convergence criterion as a typed, testable condition for determining whether an iterative numerical process has met a specified stopping tolerance or a mathematically justified convergence guarantee. Distinguish observed termination from proved convergence.

2. ENTITY TYPES
ConvergenceCriterion: {id, metric, comparator, threshold, scale, norm, domain, preconditions, evaluation_window, status}
IterativeProcess: {id, update_operator, state_space, initial_state, iteration_index}
Iterate: {process_id, index, value}
Residual: {iteration_index, value, norm}
ErrorEstimate: {iteration_index, bound, estimator, assumptions}
Tolerance: {absolute, relative, units, precision}
StoppingDecision: {iteration_index, criterion_id, passed, reason}
ConvergenceClaim: {target, mode, proof, assumptions, confidence}

3. TYPED RELATIONSHIPS
HAS_CRITERION(IterativeProcess, ConvergenceCriterion)
PRODUCES(IterativeProcess, Iterate)
MEASURES(ConvergenceCriterion, Residual | ErrorEstimate | Iterate)
USES_TOLERANCE(ConvergenceCriterion, Tolerance)
EVALUATED_AT(StoppingDecision, Iterate)
JUSTIFIED_BY(ConvergenceClaim, Proof)
DEPENDS_ON(ConvergenceClaim, Assumption)

4. FORMAL DEFINITIONS
Let x_(k+1) = T(x_k), with x_k in normed space X.
Sequence convergence: x_k -> x* iff for every epsilon > 0 there exists N such that k >= N implies ||x_k - x*|| < epsilon.
Absolute-step test: ||x_(k+1) - x_k|| <= epsilon_abs.
Relative-step test: ||x_(k+1) - x_k|| / max(||x_(k+1)||, scale_floor) <= epsilon_rel, where scale_floor > 0.
Residual test for F(x)=0: ||F(x_k)|| <= epsilon_res.
Error-bound test: certified_bound(x_k, x*) <= epsilon_err.
Objective-change test: |f(x_(k+1)) - f(x_k)| <= epsilon_obj.
Stopping predicate: STOP(k) = any or all configured tests, subject to specified logical combination and safety guards.

5. MATHEMATICAL GUARANTEES
If T is a contraction on a complete metric space with Lipschitz constant q in [0,1), Banach's fixed-point theorem guarantees a unique fixed point x* and convergence of iterates from any initial point.
Under that contraction hypothesis: ||x_k - x*|| <= q/(1-q) * ||x_k - x_(k-1)|| for k >= 1.
A small step, small objective change or small residual alone does not universally prove closeness to a true solution.
If a residual-to-error bound ||x_k-x*|| <= C||F(x_k)|| is independently established, residual tolerance yields an error guarantee C*epsilon_res.

6. EXECUTABLE FLOW ALGEBRA
flow "ConvergenceCriterionEvaluation" v1.0
  input: process, x_previous, x_current, optional residual, optional certified_error_bound, configuration
  require: finite numeric inputs; nonnegative tolerances; compatible units and norms
  compute: delta = norm(x_current - x_previous)
  compute: relative_delta = delta / max(norm(x_current), configuration.scale_floor)
  evaluate: absolute_step_pass = delta <= configuration.absolute_tolerance
  evaluate: relative_step_pass = relative_delta <= configuration.relative_tolerance
  evaluate: residual_pass = available(residual) AND norm(residual) <= configuration.residual_tolerance
  evaluate: certified_error_pass = available(certified_error_bound) AND certified_error_bound <= configuration.error_tolerance
  combine: pass = configured_boolean_expression(over enabled tests)
  guard: iteration >= configuration.minimum_iterations
  guard: no NaN, infinity, invalid domain state or unresolved numerical failure
  output: StoppingDecision(pass, metrics, guards, iteration, reason)
  assert: pass means configured stopping conditions met, NOT proof of mathematical convergence
end

7. INFERENCE RULES
R1: Certified error bound <= tolerance AND valid assumptions => certified accuracy within tolerance.
R2: Contraction preconditions AND valid iteration => convergence theorem applicable.
R3: Residual <= tolerance WITHOUT residual-to-error guarantee => residual stopping only.
R4: Step size <= tolerance WITHOUT convergence guarantee => possible stagnation; do not infer solution accuracy.
R5: Nonfinite iterate OR invalid evaluation => failure, not convergence.
R6: Maximum iteration count reached AND tests fail => termination by budget, not convergence.

8. SEMANTIC DISTINCTIONS
Convergence criterion != convergence theorem.
Stopping criterion != proof of correctness.
Residual != true error.
Absolute tolerance != relative tolerance.
Convergence rate != convergence status.
Numerical stagnation != mathematical convergence.
Termination due to resource limit != successful convergence.

9. VALIDATION
V1: Thresholds finite and >= 0; scale_floor > 0.
V2: Metric and norm defined on the relevant domain.
V3: Units consistent across measurements and thresholds.
V4: All enabled tests have available valid measurements.
V5: Every convergence guarantee records hypotheses and proof provenance.
V6: Stopping decisions retain iteration number, metric values and reason.
V7: Distinguish unverified, satisfied, failed and indeterminate outcomes.

10. SEMANTIC EQUIVALENCE
"convergence test" ~ "convergence criterion" in iterative numerical stopping context, with scope recorded.
"termination test" overlaps but is not equivalent: termination may be caused by limits or errors.
"error tolerance" is a parameter, not itself a convergence proof.

11. CONFIDENCE AND PROVENANCE
CONFIDENCE: assign evidence-based status {proved, certified, empirically_observed, unverified}; do not fabricate numeric confidence.
PROVENANCE: retain source URI, algorithm, parameter values, norms, arithmetic precision, execution environment and proof references.

12. FLOW HASH AND UPDATE MODEL
FLOW_HASH: compute SHA-256 over canonical UTF-8 serialisation of versioned ontology content and configuration; do not claim a hash until computed.
UPDATE_MODEL: semantic version, change log, validation re-run, compatibility review, preserve historical hashes.

13. AI_CONSUMPTION
Expose entities and relationships as typed records. Preserve assumptions and proof obligations. AI agents must distinguish numerical stopping from mathematical convergence, report tolerance units and failed guards, and avoid upgrading empirical evidence to theorem-level certainty.

END FLOW ALGEBRA