Verification Metrics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Verification Metrics Ontology within the Kireth Research Semantic Library. Verification metrics are quantitative measures used to evaluate computational correctness, numerical accuracy, residual behaviour, convergence, stability, agreement with trusted references and compliance with declared acceptance criteria. This ontology supports absolute and relative error, residual norms, convergence rates, observed order, tolerance checks, norm-based metrics, conservation and invariant metrics, reproducibility comparisons, performance-linked verification measures, uncertainty-aware thresholds, aggregation rules and evidence-based verification decisions.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Verification Metrics Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "Verification & Validation / Verification Metrics"
 CANONICAL_PAGE = "https://kirethframeworkold.co.uk/books/semantic-library/ai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/verification-validation/verification-metrics-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = VerificationMetrics
 ENTITY_CLASS = ComputationalVerificationMeasurementFramework
 PRIMARY_CONTEXT = VerificationValidation
 DEFINITION = "Verification metrics are quantitative measures used to assess computational correctness, numerical error, equation satisfaction, convergence, stability and compliance with verification criteria."
 SYMBOLIC_FORMS = [
  "e_abs = |x_hat-x_ref|",
  "e_rel = |x_hat-x_ref|/|x_ref|",
  "r = ||F(x_hat)||",
  "pass if metric <= tolerance"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for error metrics, residual metrics, convergence metrics, invariant checks, thresholds, aggregation and verification decision rules."
}

SCOPE {
 INCLUDE = [
  VerificationMetrics,
  VerificationMetric,
  NumericalVerificationMetric,
  ComputationalVerificationMetric,
  ErrorMetric,
  ResidualMetric,
  ConvergenceMetric,
  StabilityMetric,
  AccuracyMetric,
  CorrectnessMetric,
  AgreementMetric,
  ConsistencyMetric,
  InvariantMetric,
  ConservationMetric,
  ReproducibilityMetric,
  RegressionMetricRelationship,
  BenchmarkMetricRelationship,
  AbsoluteError,
  RelativeError,
  PercentageErrorRelationship,
  SignedError,
  ForwardError,
  BackwardError,
  ComponentwiseError,
  NormwiseError,
  PointwiseError,
  GlobalError,
  LocalError,
  MeanAbsoluteError,
  MAE,
  MeanSquaredError,
  MSE,
  RootMeanSquaredError,
  RMSE,
  MaximumAbsoluteError,
  MaximumRelativeError,
  MedianAbsoluteErrorRelationship,
  QuantileErrorRelationship,
  ErrorNorm,
  L1Norm,
  L2Norm,
  InfinityNorm,
  FrobeniusNorm,
  EnergyNormRelationship,
  WeightedNormRelationship,
  RelativeL2NormRelationship,
  Residual,
  ResidualVector,
  ResidualNorm,
  RelativeResidual,
  NormalisedResidual,
  BackwardResidualRelationship,
  ConstraintResidual,
  KKTResidualRelationship,
  EquationResidual,
  BoundaryResidual,
  InitialConditionResidual,
  ConservationResidual,
  MassResidual,
  EnergyResidual,
  MomentumResidual,
  ProbabilityNormalisationResidualRelationship,
  GradientNorm,
  ProjectedGradientNormRelationship,
  StationarityMeasure,
  OptimalityGap,
  PrimalResidual,
  DualResidual,
  ComplementarityResidualRelationship,
  FeasibilityViolation,
  ConstraintViolation,
  Convergence,
  ConvergenceRate,
  LinearConvergence,
  SublinearConvergence,
  SuperlinearConvergenceRelationship,
  QuadraticConvergence,
  OrderOfAccuracy,
  FormalOrderOfAccuracy,
  ObservedOrderOfAccuracy,
  ExperimentalOrderOfConvergenceRelationship,
  RefinementRatio,
  ErrorReductionFactor,
  RichardsonExtrapolation,
  GridConvergenceIndex,
  GCI,
  IterationCount,
  FunctionEvaluationCount,
  GradientEvaluationCount,
  SolverIterationMetricRelationship,
  Tolerance,
  AbsoluteTolerance,
  RelativeTolerance,
  ResidualTolerance,
  ConstraintTolerance,
  ConvergenceTolerance,
  StabilityToleranceRelationship,
  AcceptanceCriterion,
  PassCriterion,
  FailCriterion,
  WarningCriterion,
  Threshold,
  LowerThreshold,
  UpperThreshold,
  Margin,
  ConfidenceAdjustedThresholdRelationship,
  MetricScale,
  Normalisation,
  MetricNormalisation,
  DimensionlessMetric,
  UnitAwareMetricRelationship,
  WeightedMetric,
  AggregateMetric,
  CompositeMetric,
  WeightedScoreRelationship,
  MaxMetricAggregationRelationship,
  MeanMetricAggregationRelationship,
  RMSMetricAggregationRelationship,
  ParetoMetricRelationship,
  MultiObjectiveVerificationRelationship,
  InvariantCheck,
  ConservationCheck,
  SymmetryCheck,
  PositivityCheck,
  MonotonicityCheck,
  OrthogonalityCheck,
  NormalisationCheck,
  BoundaryConditionCheck,
  InitialConditionCheck,
  ConstraintCheck,
  ShapeCheck,
  TypeCheck,
  DimensionCheck,
  IndexCheckRelationship,
  DeterminismMetricRelationship,
  BitwiseEquality,
  NumericalEquality,
  ApproximateEquality,
  ToleranceAwareEquality,
  HashEqualityRelationship,
  ChecksumEqualityRelationship,
  DistributionalAgreementRelationship,
  StatisticalDistanceRelationship,
  MeanDifference,
  VarianceDifference,
  KolmogorovSmirnovRelationship,
  WassersteinDistanceRelationship,
  KullbackLeiblerDivergenceRelationship,
  ConfidenceIntervalOverlapRelationship,
  RepeatedRunVariability,
  StandardDeviation,
  Variance,
  CoefficientOfVariation,
  ConfidenceInterval,
  ErrorBar,
  RobustnessMetric,
  PerturbationResponseMetricRelationship,
  StabilityMarginRelationship,
  ConditionNumber,
  SensitivityMetric,
  LipschitzEstimateRelationship,
  SpectralRadiusRelationship,
  EnergyGrowthMetricRelationship,
  RuntimeRelationship,
  MemoryUsageRelationship,
  PerformanceRegressionMetricRelationship,
  AccuracyPerformanceTradeoffRelationship,
  ReferenceSolution,
  AnalyticalSolution,
  ExactSolution,
  ManufacturedSolution,
  KnownAnswer,
  GoldenReference,
  BaselineResult,
  BenchmarkResult,
  ReferenceImplementation,
  ComparisonPair,
  MetricInput,
  MetricOutput,
  ScalarMetric,
  VectorMetric,
  MatrixMetricRelationship,
  TimeSeriesMetricRelationship,
  SpatialFieldMetricRelationship,
  DistributionMetricRelationship,
  VerificationDecision,
  Accept,
  Reject,
  ReviewRequiredRelationship,
  EvidenceStrength,
  Confidence,
  Uncertainty,
  NumericalUncertainty,
  MeasurementUncertaintyRelationship,
  UncertaintyAdjustedMetricRelationship,
  ErrorBound,
  UpperErrorBound,
  LowerErrorBound,
  APrioriErrorEstimate,
  APosterioriErrorEstimate,
  MetricUncertaintyRelationship,
  Verification,
  NumericalVerification,
  ComputationalValidation,
  ErrorAnalysis,
  Benchmark,
  RegressionTesting,
  Reproducibility,
  VerificationValidation,
  NumericalStability,
  FloatingPointArithmetic,
  NumericalLinearAlgebra,
  DifferentialEquationSolver,
  ComputationalOptimisation,
  ComputationalStatistics,
  ScientificSimulation,
  SymbolicComputation,
  AIComputationalMathematics,
  Provenance,
  MetricDefinition,
  MetricVersion,
  ReferenceVersion,
  DatasetVersion,
  SolverVersion,
  LibraryVersion,
  ImplementationVersion
 ]
 EXCLUDE_AS_PRIMARY = [
  BusinessKPIsWithoutComputationalCorrectnessContext,
  PerformanceMetricsWithoutVerificationSemantics,
  ValidationMetricsWhenModelAdequacyRatherThanComputationalCorrectnessIsPrimary,
  StatisticalScoresWithoutReferenceToVerificationDecision
 ]
 DISAMBIGUATION_RULE = "When a quantitative measure is used to judge computational correctness, numerical accuracy, convergence, equation satisfaction or verification acceptance, interpret the context as a verification metric."
}

CORE_CONCEPTS {
 VerificationMetrics = "Quantitative measures for computational verification."
 ErrorMetric = "Measure of difference between computed and reference quantities."
 ResidualMetric = "Measure of equation, constraint or balance violation."
 ConvergenceMetric = "Measure of approach toward a limiting or stationary solution."
 InvariantMetric = "Measure checking preservation of a known mathematical property."
 AcceptanceCriterion = "Rule mapping metric values to verification decisions."
 AggregateMetric = "Combination of multiple verification measures into a summary score."
 UncertaintyAdjustedMetric = "Metric interpreted together with uncertainty or confidence bounds."
}

RELATIONS {
 VerificationMetrics --is_child_of--> VerificationValidation
 ErrorMetric --compares--> ComputedAndReferenceValues
 ResidualMetric --measures--> EquationSatisfaction
 ConvergenceMetric --measures--> SolutionProgress
 ObservedOrderOfAccuracy --compares_to--> FormalOrderOfAccuracy
 InvariantMetric --checks--> PreservedProperty
 Tolerance --defines--> VerificationAcceptance
 AggregateMetric --combines--> VerificationMetrics
 Uncertainty --qualifies--> MetricInterpretation
 VerificationDecision --depends_on--> MetricAndAcceptanceCriterion
 Provenance --supports--> MetricTraceability
}

OPERATORS {
 ABS_ERROR(x,x_ref) := abs(x-x_ref)
 REL_ERROR(x,x_ref) := abs(x-x_ref)/max(abs(x_ref),epsilon)
 L2_ERROR(x,x_ref) := norm(x-x_ref,2)
 RESIDUAL_NORM(F,x) := norm(F(x))
 OBSERVED_ORDER(e1,e2,r) := log(e1/e2)/log(r)
 NORMALISE_METRIC(m,scale) := m/scale
 ACCEPT(metric,tolerance) := metric <= tolerance
 AGGREGATE(metrics,weights) := declared_metric_combination(metrics,weights)
}

AXIOMS {
 A1 = "Every verification metric requires a clearly defined mathematical quantity."
 A2 = "Metric interpretation depends on scale, units, norm and reference choice."
 A3 = "Residual and solution error are distinct verification quantities."
 A4 = "A single metric rarely captures every relevant failure mode."
 A5 = "Acceptance thresholds must be declared and justified."
 A6 = "Aggregated metrics can hide local or componentwise failures."
 A7 = "Uncertainty can affect whether a metric difference is practically significant."
 A8 = "Verification-metric conclusions require explicit definitions, references, thresholds, aggregation and provenance semantics."
}

REASONING {
 R1_CONTEXT = identify(computational_quantity,reference,verification_goal,units,scale,norm,uncertainty_and_risk)
 R2_METRIC_CLASS = classify(error,residual,convergence,invariant,stability,reproducibility_or_composite_metric)
 R3_REFERENCE = verify_reference_definition_quality_and_provenance)
 R4_SCALE = choose_absolute_relative_normalised_dimensionless_or_unit_aware_form)
 R5_COMPUTE = calculate_metric_using_declared_formula)
 R6_UNCERTAINTY = propagate_or_attach_metric_uncertainty_where_relevant)
 R7_THRESHOLD = compare_metric_with_acceptance_warning_or_failure_criteria)
 R8_AGGREGATE = if_multiple_metrics combine_only_with_explicit_weighting_or_decision_logic)
 R9_DIAGNOSE = inspect_componentwise_local_or_regime_specific_failures)
 R10_REPRODUCE = document_metric_definition_versions_reference_thresholds_and_provenance)
}

CONSTRAINTS {
 C1 = "Relative error is unstable when the reference value is zero or near zero."
 C2 = "Low residual does not guarantee low forward error for ill-conditioned problems."
 C3 = "Aggregate norms can hide localised large discrepancies."
 C4 = "Changing norm or scaling can materially change metric interpretation."
 C5 = "Arbitrary weighting of composite metrics can bias verification conclusions."
 C6 = "Observed convergence rates require compatible refinement levels."
 C7 = "Statistical variability must be separated from deterministic numerical discrepancy."
 C8 = "Tolerance thresholds that are too loose can conceal real failures."
 C9 = "Thresholds that are too tight can create false failures."
 C10 = "Metric values without reference provenance or context are not sufficient verification evidence."
}

SEMANTIC_EQUIVALENCE {
 "RootMeanSquaredError" <=> "RMSE"
 "MeanSquaredError" <=> "MSE"
 "MeanAbsoluteError" <=> "MAE"
 "GridConvergenceIndex" <=> "GCI"
}

VALIDATION {
 V1 = "Metric formula is documented."
 V2 = "Reference quantity and provenance are documented."
 V3 = "Norm, units and normalisation are documented."
 V4 = "Tolerance or acceptance threshold is explicit."
 V5 = "Residual and error metrics are not conflated."
 V6 = "Composite metrics document weights and aggregation rules."
 V7 = "Local or componentwise diagnostics are retained where aggregate scores can hide failures."
 V8 = "Uncertainty is incorporated where material."
 V9 = "Metric definitions are versioned when they change."
 V10 = "Results are reproducible under documented metric, reference, threshold and implementation versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 VERIFICATION_METRICS_DEFINITION = 0.99
 ERROR_METRIC_REASONING = 0.99
 RESIDUAL_METRIC_REASONING = 0.99
 CONVERGENCE_METRIC_REASONING = 0.99
 AGGREGATE_METRIC_REASONING = 0.98
 UNCERTAINTY_METRIC_REASONING = 0.98
 POLICY = "Reduce confidence when formula, norm, reference, scaling, threshold, uncertainty or aggregation rules are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,VerificationMetricsReasoning,
  ErrorMetricReasoning,ResidualMetricReasoning,
  ConvergenceMetricReasoning,InvariantMetricReasoning,
  StabilityMetricReasoning,NormalisationReasoning,
  ThresholdReasoning,AcceptanceCriterionReasoning,
  AggregateMetricReasoning,UncertaintyAdjustedMetricReasoning,
  DiagnosticMetricReasoning,ReproducibilityReasoning,
  ProvenanceReasoning,KnowledgeGraphConstruction,
  RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "verification metrics" -> VerificationMetrics
  "verification metric" -> VerificationMetric
  "RMSE" -> RootMeanSquaredError
  "MSE" -> MeanSquaredError
  "MAE" -> MeanAbsoluteError
  "GCI" -> GridConvergenceIndex
 }
}

EXECUTION_RULES {
 E1 = "If a quantitative measure is used to assess computational correctness, activate VerificationMetricsOntology."
 E2 = "Define reference, scale, norm and units before computing error metrics."
 E3 = "Use residual metrics for equation satisfaction and error metrics for reference discrepancy."
 E4 = "Use convergence metrics only across compatible refinement or iteration levels."
 E5 = "Use invariant and conservation metrics when known structure should be preserved."
 E6 = "Use multiple metrics when one aggregate measure could hide important failures."
 E7 = "Attach uncertainty and statistical variability when they affect interpretation."
 E8 = "Compare metrics with explicit pass, warning and fail criteria."
 E9 = "Document metric definitions, thresholds, references and versions for reproducibility."
}

RECURSION {
 ENTRY = VerificationMetrics
 EXPAND(VerificationMetrics) -> [
  VerificationMetric,NumericalVerificationMetric,ErrorMetric,ResidualMetric,
  ConvergenceMetric,StabilityMetric,AccuracyMetric,CorrectnessMetric,
  AbsoluteError,RelativeError,ForwardError,BackwardError,
  ComponentwiseError,NormwiseError,MAE,MSE,RMSE,
  ErrorNorm,L1Norm,L2Norm,InfinityNorm,FrobeniusNorm,
  Residual,ResidualNorm,RelativeResidual,ConstraintResidual,
  GradientNorm,OptimalityGap,PrimalResidual,DualResidual,
  ConvergenceRate,OrderOfAccuracy,ObservedOrderOfAccuracy,
  GridConvergenceIndex,GCI,IterationCount,Tolerance,
  AbsoluteTolerance,RelativeTolerance,AcceptanceCriterion,
  PassCriterion,FailCriterion,WarningCriterion,
  Normalisation,DimensionlessMetric,WeightedMetric,
  AggregateMetric,CompositeMetric,InvariantCheck,
  ConservationCheck,SymmetryCheck,PositivityCheck,
  NumericalEquality,ToleranceAwareEquality,
  RepeatedRunVariability,ConfidenceInterval,
  RobustnessMetric,ConditionNumber,SensitivityMetric,
  ReferenceSolution,KnownAnswer,GoldenReference,
  VerificationDecision,Uncertainty,ErrorBound,
  NumericalVerification,ComputationalValidation,
  ErrorAnalysis,Benchmark,RegressionTesting,
  Reproducibility,Provenance
 ]
 EXPAND(VerificationMetricFamilies) -> [
  AbsoluteRelativeAndNormBasedMetrics,
  ResidualConstraintAndBalanceMetrics,
  ConvergenceAndOrderMetrics,
  InvariantConservationAndStructureMetrics,
  ReproducibilityAndRegressionComparisonMetrics,
  StabilitySensitivityAndRobustnessMetrics,
  CompositeAggregateAndDecisionMetrics,
  UncertaintyThresholdAndProvenanceSemantics
 ]
 STOP_CONDITION = "Stop when metric definition, reference, scale, threshold, uncertainty, aggregation and verification decision are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Verification & Validation"
 RELATED_ONTOLOGIES = [
  "Verification & Validation",
  "Numerical Verification Ontology",
  "Computational Validation Ontology",
  "Error Analysis Ontology",
  "Benchmark Ontology",
  "Regression Testing Ontology",
  "Reproducibility Ontology",
  "Numerical Stability Ontology 2",
  "Floating Point Arithmetic Ontology",
  "Numerical Linear Algebra Ontology 2",
  "Differential Equation Solver Ontology",
  "Computational Optimisation Ontology",
  "Computational Statistics Ontology",
  "Scientific Simulation Ontology",
  "AI Computational Mathematics Ontology"
 ]
 ROUTE_RULES = [
  "Verification Metrics -> Verification & Validation for parent decision semantics",
  "Verification Metrics -> Numerical Verification for error, residual and convergence measures",
  "Verification Metrics -> Computational Validation for model-agreement metric relationships",
  "Verification Metrics -> Error Analysis for forward, backward, absolute and relative error semantics",
  "Verification Metrics -> Benchmark for comparative scoring and aggregation",
  "Verification Metrics -> Regression Testing for baseline difference and tolerance metrics",
  "Verification Metrics -> Reproducibility for cross-run comparison semantics",
  "Verification Metrics -> Numerical Stability and Floating Point Arithmetic for sensitivity to precision and rounding",
  "Verification Metrics -> Numerical Linear Algebra, Differential Equation Solvers and Optimisation for domain-specific residual and convergence measures",
  "Verification Metrics -> Computational Statistics for statistical uncertainty and confidence metrics",
  "Verification Metrics -> Scientific Simulation and AI Computational Mathematics for model-specific verification measures"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible verification metrics, aggregations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve error-versus-residual distinction.",
  "Preserve absolute-versus-relative metric distinction.",
  "Preserve local-versus-global metric distinction.",
  "Preserve scalar-versus-aggregate metric distinction.",
  "Preserve metric-value-versus-verification-decision distinction.",
  "Preserve deterministic-versus-statistical metric distinction.",
  "Preserve raw-metric-versus-normalised-metric distinction.",
  "Preserve machine-readable verification-metrics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Verification metrics are quantitative measures used to assess computational correctness, numerical accuracy, equation satisfaction, convergence and compliance with acceptance criteria."
 ERROR_SUMMARY = "Absolute, relative, forward, backward and norm-based errors quantify disagreement with trusted references."
 RESIDUAL_SUMMARY = "Residual metrics quantify equation, constraint and conservation-law satisfaction."
 CONVERGENCE_SUMMARY = "Convergence rates, observed order and refinement metrics quantify numerical approach toward a limiting solution."
 DECISION_SUMMARY = "Verification decisions require explicit pass, warning and fail thresholds together with uncertainty and context."
 VALIDATION_SUMMARY = "Verification metrics should be versioned, reproducible, traceable to trusted references and interpreted with enough local diagnostics to avoid misleading aggregate conclusions."
}

FLOW {
 INPUT
 -> detect_verification_metrics_context
 -> identify_computational_quantity_reference_verification_goal_units_scale_norm_uncertainty_and_risk
 -> classify_error_residual_convergence_invariant_stability_reproducibility_or_composite_metric
 -> verify_reference_definition_quality_and_provenance
 -> choose_absolute_relative_normalised_dimensionless_or_unit_aware_form
 -> compute_metric_using_declared_formula
 -> attach_uncertainty_or_variability_when_relevant
 -> compare_metric_with_pass_warning_or_fail_thresholds
 -> if_multiple_metrics_apply_explicit_aggregation_or_decision_logic
 -> inspect_local_componentwise_or_regime_specific_failures
 -> document_metric_definition_reference_threshold_versions_and_provenance
 -> route_to_related_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-VERIFICATION-VALIDATION-VERIFICATION-METRICS-ONTOLOGY-V1"
}

}

END_FLOW