Tolerance Ontology

FLOW ALGEBRA
Ontology: Tolerance Ontology
Version: 1.0
Date: 2026-10-14
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/measurement-and-dimensional-entities/tolerance-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)

PURPOSE
Represent permissible variation around a specified nominal or target quantity, with explicit units, boundary inclusion, acceptance rules, and traceable provenance. Tolerance describes a specification, not the uncertainty of a measurement.

TYPED ENTITIES
entity Tolerance {id: URI; quantity_kind: QuantityKind; nominal: QuantityValue; lower_deviation: QuantityValue?; upper_deviation: QuantityValue?; lower_limit: QuantityValue; upper_limit: QuantityValue; lower_inclusive: bool = true; upper_inclusive: bool = true; specification: URI?; revision: string?}
entity QuantityValue {magnitude: real; unit: Unit; dimension: Dimension}
entity Measurement {value: QuantityValue; uncertainty: QuantityValue?; method: URI?; traceability: URI?}
entity ConformanceDecision {status: enum(PASS, FAIL, INDETERMINATE); decision_rule: string; evidence: URI?}
entity ToleranceZone {lower: QuantityValue; upper: QuantityValue; inclusivity: tuple(bool,bool)}
entity Specification {id: URI; standard: string?; revision: string?; effective_date: date?}

RELATIONSHIPS
Tolerance --applies_to--> QuantityKind
Tolerance --has_nominal--> QuantityValue
Tolerance --defines_zone--> ToleranceZone
Tolerance --governed_by--> Specification
Measurement --evaluated_against--> Tolerance
ConformanceDecision --uses--> Measurement
ConformanceDecision --uses--> Tolerance
ConformanceDecision --follows--> DecisionRule

FORMAL ALGEBRA
For nominal N, lower deviation d_minus <= 0 and upper deviation d_plus >= 0:
L = N + d_minus; U = N + d_plus; L <= U.
Symmetric tolerance +/-t (t >= 0): L = N - t; U = N + t.
Asymmetric tolerance (+a, -b), a,b >= 0: L = N - b; U = N + a.
One-sided tolerance: only one finite limit is specified; represent the other as unbounded rather than inventing a numerical bound.
Tolerance width W = U - L (for finite bounds); midpoint M = (U + L)/2.
Direct acceptance (closed zone): PASS iff L <= x <= U; FAIL otherwise.
For open endpoints, use strict inequality at that endpoint.

DECISION RULES AND MEASUREMENT UNCERTAINTY
A direct measured-value comparison is valid only when the governing specification selects that decision rule. Measurement uncertainty u is distinct from tolerance width.
Illustrative conservative guard-band rule with expanded uncertainty U_m >= 0:
PASS if [x-U_m, x+U_m] is wholly within [L,U].
FAIL if [x-U_m, x+U_m] is wholly outside [L,U].
INDETERMINATE otherwise.
This illustrative rule is not universally mandated. Document the coverage basis, risk policy, boundary conventions, and applicable standard before using it.

INFERENCE RULES
rule symmetric_bounds: symmetric(N,t) and t>=0 -> lower=N-t, upper=N+t.
rule asymmetric_bounds: asymmetric(N,a,b) and a>=0 and b>=0 -> lower=N-b, upper=N+a.
rule nominal_conforms: decision_rule=direct and lower<=x<=upper -> PASS.
rule dimension_compatibility: comparison(x,L,U) -> dimension(x)=dimension(L)=dimension(U).
rule canonical_conversion: compatible_units -> convert all values to common unit before comparison.
rule specification_precedence: conflicting tolerances -> flag conflict; do not silently select one.

VALIDATION CONSTRAINTS
- Require a declared quantity kind and compatible dimensions for all values.
- For finite bounds require L <= U; for symmetric tolerances require t >= 0.
- Preserve units, sign conventions, endpoint inclusivity, and specification revision.
- Reject NaN, undefined conversion factors, or dimensionally incompatible comparisons.
- Do not equate tolerance with measurement uncertainty, error bound, accuracy, or precision.
- Do not infer acceptance from uncertainty alone without an explicit decision rule.
- Identify whether nominal values are targets or required exact reference values.

WORKED EXAMPLES
1. Symmetric: nominal 20.00 mm +/- 0.10 mm => [19.90, 20.10] mm. Direct comparison: 20.08 mm PASS; 20.12 mm FAIL.
2. Asymmetric: nominal 50.0 mm (+0.2, -0.1) mm => [49.9, 50.2] mm.
3. Unit conversion: [9.9,10.1] cm => [99,101] mm; measured 100 mm PASS under direct comparison.
4. Guard band: limits [9.90,10.10] mm, x=10.08 mm, expanded uncertainty 0.04 mm => uncertainty interval [10.04,10.12] mm, hence INDETERMINATE under the illustrative conservative rule.

SEMANTIC_EQUIVALENCE
permissible variation ~= allowable deviation (context dependent)
tolerance interval ~= specification limits (only when inclusivity and meaning agree)
NOT_EQUIVALENT: tolerance interval != statistical confidence interval
NOT_EQUIVALENT: tolerance != uncertainty
NOT_EQUIVALENT: tolerance != observed error

CONFIDENCE
mathematical_rules: high
example_calculations: high
application_specific_decision_rules: conditional_on_standard_and_specification
source_page_alignment: not_verified

FLOW_HASH
algorithm: SHA-256
canonicalisation: UTF-8, LF line endings, stable entity/relationship ordering
value: COMPUTE_ON_PUBLICATION (do not fabricate a hash)

UPDATE_MODEL
revision_trigger: change in standard, tolerance specification, units, decision rule, or ontology schema
change_control: preserve revision, effective date, provenance, and migration mapping

AI_CONSUMPTION
parse: typed entities and relationships
normalise: dimensions and units
validate: boundary ordering, signs, units, and provenance
infer: allowable range and conformance only under declared decision rule
output: status, evaluated value, applicable limits, rule, uncertainty treatment, and explanation
on_ambiguity: INDETERMINATE with missing requirements

PROVENANCE NOTE
This is a mathematically grounded Flow Algebra draft for the supplied ontology URL; the live page content was not independently verified.