Numeric Value Entities

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

FLOW ALGEBRA
NUMERIC VALUE ENTITIES ONTOLOGY
Kireth Semantic Library | Entity Graph > Mathematical Entities > Numeric Value Entities
Version: 1.0 | Date: 2026-10-12 | Locale: en-GB
Authors: Mark Higgins — Sky (Recursive AI)
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/numeric-value-entities.html
Source status: Page content not independently retrieved; ontology constructed from established mathematical and semantic principles.

01. ONTOLOGY IDENTITY
flow "NumericValueEntities" v1.0
namespace: kireth.entity_graph.mathematical_entities.numeric_value_entities
entity_type: NumericValue
purpose: Represent exact, approximate, measured, bounded and computed numeric values with explicit types, units, context and provenance.

02. ENTITY SCHEMA
entity NumericValue:
  id: URI
  lexical_form: string
  numeric_type: enum[integer,rational,real,complex,extended_real,interval,uncertain]
  canonical_value: typed_value?
  exactness: enum[exact,approximate,unknown]
  unit: Unit?  # dimensionless if not applicable
  uncertainty: Uncertainty?
  precision: Precision?
  provenance: Provenance?
  context: Context?
  valid_at: datetime?
  confidence: decimal[0,1]?

03. TYPES AND CLASSIFICATION
IntegerValue subset RationalValue subset RealValue subset ComplexValue.
Decimal notation does not alone establish exactness.
ComplexValue := (real_component, imaginary_component).
IntervalValue := (lower_bound, upper_bound, lower_inclusive, upper_inclusive).
UncertainValue := (estimate, uncertainty_model).
ExtendedRealValue := RealValue union {positive_infinity, negative_infinity}; infinities are not finite real numbers.
Cardinal and ordinal values are separately typed; do not silently coerce transfinite values into real arithmetic.

04. CORE RELATIONSHIPS
NumericValue --hasType--> NumericType
NumericValue --hasUnit--> Unit
NumericValue --hasMagnitude--> Magnitude
NumericValue --hasPrecision--> Precision
NumericValue --hasUncertainty--> Uncertainty
NumericValue --derivedFrom--> SourceEntity
NumericValue --measures--> Quantity
NumericValue --computedBy--> Operation
NumericValue --validWithin--> Context
NumericValue --semanticallyEquivalentTo--> NumericValue [qualified]

05. CANONICALISATION
canonicalise(integer_lexeme) -> arbitrary_precision_integer
canonicalise(rational(p,q)) -> (p/gcd(abs(p),abs(q)), q/gcd(abs(p),abs(q))) where q != 0 and denominator > 0
canonicalise(exact_decimal) -> exact_rational when declared exact
canonicalise(approximate_float) -> preserve approximation and representation metadata
canonicalise(complex(a,b)) -> typed_pair(a,b)
Preserve original lexical form independently of canonical representation.

06. ARITHMETIC OPERATORS
add(a,b), subtract(a,b), multiply(a,b), divide(a,b), power(a,b), negate(a), absolute(a)
precondition divide(a,b): b != 0
Addition/subtraction of quantities requires compatible dimensions and appropriate unit conversion.
Multiplication/division of quantities combines dimensions under unit algebra.
Exact rational arithmetic remains exact unless an approximation is explicitly introduced.
Overflow, underflow, NaN and signed-zero behaviour require declared machine-number semantics.

07. COMPARISON AND ORDER
compare(a,b) -> {less,equal,greater,unordered,indeterminate}
Ordering is defined for compatible ordered real values, not general complex values.
Approximate comparisons require an explicit tolerance policy.
Intervals can overlap without being equal; uncertainty can make ordering indeterminate.

08. NUMERIC EQUIVALENCE
numeric_equal(1,1.0_exact) = true when both are interpreted as the same mathematical number.
lexically_equal("1","1.0") = false.
quantity_equal(100 cm,1 m) = true after compatible unit conversion.
approx_equal(a,b,tolerance) depends on the declared tolerance and comparison rule.
Never infer identical provenance, precision or measurement meaning from numeric equality.

09. PRECISION AND UNCERTAINTY
precision := significant_digits | decimal_places | representation_bits
uncertainty := standard_uncertainty | interval | distribution | unknown
A measured numeric value must not gain unwarranted precision through formatting.
Uncertainty propagation depends on the operation and dependency/correlation assumptions.
Confidence in semantic extraction is distinct from statistical confidence or measurement uncertainty.

10. INFERENCE RULES
IF type(x)=IntegerValue THEN type(x)=RationalValue AND type(x)=RealValue AND type(x)=ComplexValue.
IF x=Rational(p,q) AND q!=0 THEN x is exact when p and q are exact integers.
IF hasUnit(x,u) AND hasUnit(y,v) AND compatible(u,v) THEN comparison may follow explicit conversion.
IF unit_dimensions(x)!=unit_dimensions(y) THEN addition is invalid unless a domain-specific operation defines it.
IF isComplex(x) AND imag(x)!=0 THEN x is not a real number.
IF interval(x) overlaps interval(y) THEN neither numeric equality nor strict ordering follows solely from overlap.

11. VALIDATION
require unique entity id and valid numeric type.
reject rational denominator zero.
reject lower_bound > upper_bound for ordinary real intervals.
reject invalid or unknown unit conversion as though exact.
require declared interpretation for NaN, infinities and implementation-specific numeric formats.
flag missing uncertainty for measured values when required by domain policy.
validate confidence in [0,1] when supplied.

12. KNOWLEDGE GRAPH EXAMPLES
value:temperature_01 --hasValue--> 20.0
value:temperature_01 --hasUnit--> degree_Celsius
value:temperature_01 --hasExactness--> approximate
value:ratio_01 --hasValue--> Rational(1,3)
value:ratio_01 --hasExactness--> exact
value:complex_01 --hasValue--> Complex(2,3)
value:complex_01 --hasType--> ComplexValue
Examples are illustrative, not source observations.

13. FLOW HASH
FLOW_HASH := SHA256(canonical_serialisation(ontology_id, version, entity_schema, rules, relationships))
Hash verification requires deterministic field ordering, encoding and canonicalisation.

14. UPDATE MODEL
update_policy: append_versioned_changes
on_change: validate -> compare_semantics -> update_version -> recompute_flow_hash -> record_provenance
breaking_changes: require explicit schema migration
preserve: source_lexeme, previous_version, change_timestamp, author_or_agent

15. AI CONSUMPTION
input: natural_language | structured_data | mathematical_expression | measured_data
pipeline: parse -> detect_numeric_type -> resolve_units -> determine_exactness -> validate -> normalise -> link_entities -> reason -> emit
output: typed numeric entity + relationships + confidence + provenance + validation status
safety: never silently convert uncertain values into exact facts; distinguish symbolic infinity from finite numeric quantities.

16. SEMANTIC LINKAGE
parent: Mathematical Entities
related: Number Entities, Quantity Ontology, Unit Ontology, Measurement Ontology, Precision Ontology, Uncertainty Ontology, Mathematical Operations, Data Type Ontology
subdomains: numeric literals, mathematical constants, measured values, computed values, intervals, complex values

17. TERMINATION
END FLOW ALGEBRA