Significant Figure Ontology

FLOW ALGEBRA
Significant Figure Ontology
Kireth Semantic Library | Mathematical Entities | Measurement and Dimensional Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/measurement-and-dimensional-entities/significant-figure-ontology.html
Version: 1.0 | Date: 2026-10-14 | Locale: en-GB
Authors: Mark Higgins — Sky (Recursive AI)

1. PURPOSE
Model significant figures as a property of a numerical representation and its declared measurement/reporting precision. Distinguish significant-figure count from decimal places, numerical accuracy, uncertainty, and the intrinsic mathematical value.

2. TYPED ENTITIES
entity SignificantFigure { digit: 0..9; position: integer; significant: boolean }
entity NumericRepresentation { lexeme: string; radix: integer = 10; notation: enum(decimal, scientific); sign: enum(positive,negative); exponent: integer? }
entity SignificantFigureCount { count: integer >= 1; status: enum(explicit, inferred, ambiguous) }
entity DecimalPlaceCount { count: integer >= 0 }
entity MeasuredQuantity { magnitude: real; unit: Unit?; representation: NumericRepresentation; uncertainty: Uncertainty? }
entity ExactQuantity { value: rational|real; exact: true }
entity RoundedResult { representation: NumericRepresentation; rounding_mode: RoundingMode; source: NumericRepresentation }
entity RoundingMode { name: enum(half_even, half_up, toward_zero, floor, ceiling) }
entity PrecisionSpecification { significant_digits: integer?; decimal_places: integer?; absolute_increment: positive_real? }
entity TrailingZeroAmbiguity { representation: NumericRepresentation; candidate_counts: set<integer> }
entity Provenance { source_uri: string; method: string?; timestamp: datetime? }

3. RELATIONSHIPS
HAS_REPRESENTATION(MeasuredQuantity, NumericRepresentation)
HAS_SIGNIFICANT_COUNT(NumericRepresentation, SignificantFigureCount)
HAS_DECIMAL_PLACES(NumericRepresentation, DecimalPlaceCount)
ROUNDED_FROM(RoundedResult, NumericRepresentation)
USES_ROUNDING_MODE(RoundedResult, RoundingMode)
REPORTS_PRECISION(MeasuredQuantity, PrecisionSpecification)
HAS_UNCERTAINTY(MeasuredQuantity, Uncertainty)
HAS_PROVENANCE(MeasuredQuantity, Provenance)
SEMANTIC_EQUIVALENCE(lexical_forms, mathematical_value) only when context permits; equivalent numeric values need not have equivalent precision.

4. CORE RULES
R1: Non-zero decimal digits are significant.
R2: Zeros between significant non-zero digits are significant.
R3: Leading zeros before the first non-zero digit are not significant.
R4: Trailing zeros to the right of a displayed decimal point after a non-zero digit are significant when deliberately reported.
R5: Trailing zeros in an integer without a decimal point may be ambiguous unless notation or metadata declares precision.
R6: In a non-zero scientific-notation coefficient, the written coefficient digits determine significant-figure count; the power-of-ten exponent does not add significant figures.
R7: Exact counts and defined conversion factors have no finite significant-figure limitation; distinguish them from measured inputs.
R8: The number 0 without further precision metadata is ambiguous for significant-figure purposes; use explicit decimal or scientific notation to indicate reporting precision.
R9: Preserve the input lexeme. Parsing to a floating-point value alone can discard significant trailing zeros.
R10: A reported number of significant figures describes notation/precision, not proof of accuracy or a probability distribution.

5. MATHEMATICAL OPERATIONS
For multiplication/division of independently measured inputs, a conventional classroom reporting rule is to round to the smallest input significant-figure count; perform intermediate calculations at higher precision.
For addition/subtraction, the conventional classroom rule rounds to the least precise decimal place (or, more generally, least precise absolute place value) among measured inputs.
These are simplified reporting conventions, not substitutes for uncertainty propagation or metrological reporting requirements.
For mixed operations, retain guard digits and round once at the final reporting stage.
For exact factors, do not reduce significant-figure count.

6. ROUNDING
round_sig(x,n,mode) requires finite x and integer n >= 1.
For x != 0, let p = floor(log10(abs(x))); increment = 10^(p-n+1); round x to a multiple of increment under the declared rounding mode.
For x = 0, an explicit precision increment or representation is required to communicate significant figures.
Binary floating-point approximations and halfway ties require a documented numeric implementation and rounding mode.
Rounding does not automatically produce a valid uncertainty statement.

7. VALIDATION CONSTRAINTS
V1: significant_digits >= 1 when declared.
V2: radix=10 for the rules in this ontology; other radices require separate rules.
V3: Scientific notation coefficient and exponent must parse without losing lexical zeros.
V4: Precision inferred from ambiguous integer trailing zeros must be marked ambiguous, not asserted as certain.
V5: Do not treat significant-figure count as a unit or physical dimension.
V6: Do not infer uncertainty, calibration, or accuracy solely from significant figures.
V7: If declared precision conflicts with the representation, flag for review.
V8: Preserve original representation, rounding mode and provenance when transforming or serialising data.

8. WORKED EXAMPLES
E1: 0.00450 has 3 significant figures (4, 5, final 0).
E2: 1002 has 4 significant figures (interior zeros count).
E3: 1.200 × 10^3 has 4 significant figures.
E4: 1200 without an explicit decimal or precision metadata has an ambiguous significant-figure count.
E5: 2.5 × 3.42 = 8.55; conventional multiplication reporting to 2 significant figures gives 8.6 (half-up rounding).
E6: 12.11 + 0.3 = 12.41; conventional addition reporting to tenths gives 12.4.
E7: 3.00 and 3 are mathematically equal but differ in declared decimal precision (3 and 1 significant figures respectively).
E8: 2.500 rounded to 3 significant figures under half-even gives 2.50; retain the trailing zero in the output lexeme.

9. INFERENCE RULES
IF decimal_lexeme has leading zeros THEN exclude them from significant-figure count.
IF zero lies between non-zero digits THEN classify as significant.
IF decimal_lexeme has trailing fractional zeros THEN include them as significant.
IF integer_lexeme has trailing zeros AND no explicit precision THEN classify count as ambiguous.
IF quantity.exact=true THEN exclude it from measured-input precision limitation.
IF reporting_rule=addition THEN select least precise place value.
IF reporting_rule=multiplication THEN select minimum measured-input significant count.
IF uncertainty metadata exists THEN prefer uncertainty-aware calculation/reporting over heuristic digit rules.

10. CONFIDENCE
CONFIDENCE:
  explicit_scientific_notation: high
  explicit_decimal_trailing_zeros: high when reporting convention applies
  unqualified_integer_trailing_zeros: ambiguous
  precision_to_accuracy_inference: invalid
Confidence labels express inference support, not measurement coverage probability.

11. UPDATE_MODEL
On revision: preserve original source text; version changed parsing/rounding rules; revalidate dependent entities; regenerate derived precision annotations; log changes and provenance.

12. FLOW_HASH
FLOW_HASH: SHA-256(canonical UTF-8 normalisation of versioned ontology payload, excluding the hash field itself); compute at publication, do not fabricate a digest.

13. AI_CONSUMPTION
Input: numerical lexeme, quantity kind, exactness, optional uncertainty, reporting context, rounding mode.
Output: parsed value, significant count or ambiguity, decimal-place count, applicable reporting rule, rounded lexical output, validation flags, provenance.
AI systems MUST preserve lexical precision, flag ambiguity, distinguish uncertainty from significant figures, and avoid asserting unverified accuracy.

14. RELATED ONTOLOGIES
Mathematical Quantity; Quantity Value; Unit; Dimension; Measurement Value; Measurement Interval; Uncertainty; Error Bound; Tolerance; Precision; Accuracy.

END FLOW ALGEBRA