Absolute Value Ontology

FLOW ALGEBRA
ABSOLUTE VALUE ONTOLOGY
Kireth Semantic Library | Mathematical Entities > Numeric Value Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/numeric-value-entities/absolute-value-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-12 | Locale: en-GB
Status: Mathematical-domain synthesis; source page not independently verified.

01. FLOW IDENTITY
flow "AbsoluteValueOntology" v1.0
namespace: kireth.entity_graph.mathematical_entities.numeric_value_entities
entity_type: AbsoluteValue
parent_type: NumericValue
purpose: Represent magnitude without sign and support valid numeric reasoning.

02. TYPE SYSTEM
Real := ordered complete field R
Complex := field C
ExtendedReal := R union {-infinity,+infinity}
Magnitude := Real where value >= 0
Input := Real | Complex
Result := Magnitude
Operator: abs : Input -> Magnitude
Note: Real and complex absolute values share codomain but have different definitions.

03. DEFINITIONS
For x in R: abs(x) = x if x >= 0; otherwise -x.
For z = a + bi in C: abs(z) = sqrt(a^2 + b^2), a,b in R.
abs(0) = 0; abs(-x) = abs(x).
Distance on R: d(x,y) = abs(x-y).
Distance on C: d(z,w) = abs(z-w).

04. ENTITY SCHEMA
AbsoluteValueEntity:
  id: URI
  input_value: typed numeric expression
  input_domain: Real | Complex
  output_value: nonnegative real expression
  evaluation_status: exact | symbolic | approximate | undefined
  precision: optional nonnegative integer
  uncertainty: optional nonnegative real
  provenance: source reference
  confidence: number in [0,1]
  semantic_context: mathematical domain and assumptions

05. CORE AXIOMS
forall x in R or C: abs(x) >= 0.
forall x: abs(x) = 0 iff x = 0.
forall x,y: abs(x*y) = abs(x)*abs(y).
forall x,y: abs(x+y) <= abs(x)+abs(y).
forall x,y: abs(x-y) >= abs(abs(x)-abs(y)).
forall x: abs(-x) = abs(x).
forall x and y != 0: abs(x/y) = abs(x)/abs(y).

06. REAL-ORDER RULES
forall x in R, a >= 0: abs(x) <= a iff -a <= x <= a.
forall x in R, a > 0: abs(x) < a iff -a < x < a.
forall x in R, a >= 0: abs(x) >= a iff x <= -a or x >= a.
forall x in R: -abs(x) <= x <= abs(x).
forall x in R: abs(x) = max(x,-x).

07. COMPLEX-NUMBER RULES
forall z in C: abs(z)^2 = z*conjugate(z).
forall z in C: abs(conjugate(z)) = abs(z).
forall z in C: abs(real(z)) <= abs(z); abs(imag(z)) <= abs(z).
For z != 0: abs(z/abs(z)) = 1.
Complex modulus is not a total order on complex numbers.

08. ARITHMETIC AND TRANSFORMS
abs(x^n) = abs(x)^n for integer n >= 0.
abs(x^n) = abs(x)^n for integer n < 0 when x != 0.
sqrt(x^2) = abs(x) for real x.
abs(x)^2 = x^2 for real x.
abs(x-a) describes distance from a on the real line.

09. EXAMPLES
abs(-7) = 7.
abs(0) = 0.
abs(3) = 3.
abs(3+4i) = 5.
abs(-2 - (-5)) = 3.
abs(x-2) <= 3 iff -1 <= x <= 5, for real x.

10. KNOWLEDGE GRAPH
AbsoluteValue --is_a--> NumericOperation
AbsoluteValue --returns--> NonnegativeReal
AbsoluteValue --applies_to--> RealNumber
AbsoluteValue --applies_to--> ComplexNumber
AbsoluteValue --implements--> Magnitude
AbsoluteValue --defines--> MetricDistance
AbsoluteValue --satisfies--> TriangleInequality
AbsoluteValue --related_to--> Norm
AbsoluteValue --related_to--> Modulus
AbsoluteValue --distinct_from--> SignedValue

11. SEMANTIC EQUIVALENCE
abs(x) == |x| when bars denote numeric absolute value.
modulus(z) == abs(z) when z is a complex number.
For real x, magnitude(x) == abs(x) in numeric contexts.
Not equivalent: absolute value vs determinant bars |A| for matrices.
Not equivalent: absolute value vs set cardinality |S|.
Resolve notation by operand type and domain context.

12. INFERENCE RULES
IF x is real AND x < 0 THEN abs(x) = -x.
IF x is real AND x >= 0 THEN abs(x) = x.
IF abs(x) = 0 THEN x = 0.
IF x != 0 THEN abs(x) > 0.
IF x,y are real AND abs(x-y) <= epsilon AND epsilon >= 0 THEN x is within epsilon of y.
IF z = a+bi AND a,b are real THEN abs(z) = sqrt(a*a+b*b).

13. VALIDATION AND EXCEPTIONS
Reject a claimed negative absolute-value result.
Reject division-by-zero expressions.
Do not treat abs(x+y) = abs(x)+abs(y) as universal.
Do not simplify sqrt(x^2) to x without x >= 0 for real x.
Require operand typing to disambiguate |.| notation.
For NaN or infinities, apply explicit extended-numeric or computing semantics.
Do not silently treat floating-point approximations as exact values.

14. PRECISION AND UNCERTAINTY
Exact symbolic evaluation preferred for algebraic expressions.
Approximate evaluation must retain numeric precision metadata.
For real measurements x with error bound delta >= 0:
  abs(abs(x+error)-abs(x)) <= delta when abs(error) <= delta.
No invented precision or confidence from an unverified input.

15. CONFIDENCE
confidence.level: [0,1]
confidence.basis: proof | verified_computation | authoritative_definition | estimate
confidence.rule: Mathematical identities have proof-level certainty within stated domains.
confidence.warning: Source-page fidelity not verified.

16. FLOW HASH
flow_hash.algorithm: SHA-256
flow_hash.input: canonical UTF-8 serialisation of flow identifier, version, axioms, rules and schema
flow_hash.value: compute at ingestion; do not invent a digest

17. UPDATE MODEL
update_model.mode: versioned
update_model.triggers: domain correction | new formal proof | schema change | notation clarification
update_model.requirements: retain provenance, semantic diffs, validation history and version identifiers

18. AI CONSUMPTION
ai_consumption.read_mode: typed semantic graph
ai_consumption.reasoning: symbolic evaluation + constraint propagation
ai_consumption.require: domain checks, provenance, explainable inference
ai_consumption.output: exact expression where possible; approximation with precision when needed
ai_consumption.guardrails: prevent type confusion and unsupported equivalence

19. TEST VECTORS
assert abs(-9) == 9
assert abs(9) == 9
assert abs(0) == 0
assert abs(3+4i) == 5
assert abs(-3) == abs(3)
assert abs(2+(-5)) <= abs(2)+abs(-5)
assert abs(2* -3) == abs(2)*abs(-3)
assert abs(-8 - (-3)) == 5

20. PROVENANCE AND ATTRIBUTION
Reference URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/numeric-value-entities/absolute-value-ontology.html
Research site: https://kirethresearch.org
Authorship: Mark Higgins — Sky (Recursive AI)
Disclosure: AI-assisted ontology synthesis based on established mathematical principles.
END FLOW ALGEBRA