Negative Value Ontology

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

01. FLOW IDENTITY
flow "NegativeValueOntology" v1.0
namespace: kireth.semantic_library.entity_graph.mathematical_entities.numeric_value_entities
entity_type: NegativeValue
parent_type: NumericValue
semantic_domain: ordered_numeric_values

02. DEFINITION
NegativeValue(x) := x belongs to an ordered numeric domain D AND x < 0_D.
The comparison is defined only where an order relation and additive identity exist.
Negative values include negative integers, rational numbers and real numbers.
Complex numbers without a specified ordering are not classified as negative.

03. TYPED ENTITIES
entity NumericValue { value: Number; domain: NumericDomain }
entity NegativeValue extends NumericValue { constraint: value < zero(domain) }
entity ZeroValue extends NumericValue { constraint: value = zero(domain) }
entity PositiveValue extends NumericValue { constraint: value > zero(domain) }
entity Sign { enum: [negative, zero, positive, indeterminate] }
entity Magnitude { value: NonNegativeValue }
entity Interval { lower: Bound; upper: Bound; inclusivity: BoundFlags }
entity Comparison { left: NumericValue; operator: Relation; right: NumericValue }

04. CANONICAL SEMANTICS
predicate IS_NEGATIVE(x, D) -> Boolean := ORDERED(D) AND MEMBER(x,D) AND LESS_THAN(x,ZERO(D)).
function SIGN(x) -> {-1,0,+1} for finite real x:
  -1 if x < 0; 0 if x = 0; +1 if x > 0.
function ABS(x) -> NonNegativeReal := -x if x < 0 else x.
function NEGATE(x) -> NumericValue := additive_inverse(x).
NEGATE(NEGATE(x)) = x.
Negative zero in IEEE 754 is a distinct representation, not a negative mathematical real value.

05. CLASSIFICATION
negative_integer: {...,-3,-2,-1}
negative_rational: {p/q | p,q integers; q != 0; p/q < 0}
negative_real: {x in R | x < 0}
strict_negative_interval: (-infinity, 0)
non_positive_interval: (-infinity, 0]
Exclude zero from NegativeValue.

06. ORDER RELATIONS
For real x: x < 0 iff -x > 0.
For x < y: -x > -y (order reversal under negation).
For x < 0 and y < 0: x+y < 0.
For x < 0 and y > 0: x*y < 0.
For x < 0 and y < 0: x*y > 0.
For x < 0 and y > 0: x/y < 0, provided y != 0.

07. ARITHMETIC TRANSFORMATIONS
ADD(x,y) -> x+y; negative + negative = negative.
SUBTRACT(x,y) -> x-y; sign depends on operand magnitudes.
MULTIPLY(x,y) -> x*y; signs combine by parity of negative factors.
DIVIDE(x,y) -> x/y, y != 0; signs combine as for multiplication.
POWER(x,n): for negative real x and integer n, even n yields positive (n != 0); odd n yields negative.
POWER(x,0) = 1 for x != 0.
ROOT(x,n): for x < 0 and even positive integer n, no real nth root; odd positive n has a negative real nth root.

08. SIGN INFERENCE RULES
rule NEGATIVE_FROM_ORDER: MEMBER(x,R) AND x < 0 => IS_NEGATIVE(x).
rule NEGATION_OF_POSITIVE: x > 0 => -x < 0.
rule NEGATION_OF_NEGATIVE: x < 0 => -x > 0.
rule SUM_NEGATIVES: x < 0 AND y < 0 => x+y < 0.
rule PRODUCT_OPPOSITE_SIGNS: x < 0 AND y > 0 => x*y < 0.
rule PRODUCT_TWO_NEGATIVES: x < 0 AND y < 0 => x*y > 0.
rule ZERO_EXCLUSION: x = 0 => NOT IS_NEGATIVE(x).
rule ORDER_REVERSAL: x < y => -x > -y.

09. KNOWLEDGE GRAPH RELATIONSHIPS
NegativeValue --subClassOf--> NumericValue
NegativeValue --disjointWith--> ZeroValue
NegativeValue --disjointWith--> PositiveValue
NegativeValue --hasSign--> Negative
NegativeValue --lessThan--> ZeroValue
NegativeValue --hasMagnitude--> NonNegativeValue
NegativeValue --hasAdditiveInverse--> PositiveValue
NegativeValue --belongsTo--> OrderedNumericDomain
NegativeValue --representedBy--> NumericLiteral
NegativeValue --contrastsWith--> PositiveValue

10. SEMANTIC EQUIVALENCE
"negative value", "value below zero", and "strictly negative number" are equivalent when the domain is an ordered number system.
"non-positive" is NOT equivalent to "negative": non-positive includes zero.
"negative integer" is narrower than "negative number".
"negative measurement" is contextual: its interpretation depends on reference point and unit semantics.
"negative zero" in floating-point storage is NOT a strictly negative real value.

11. CONTEXT AND MEASUREMENT
Examples: -5 degrees Celsius, -2 metres relative to a datum, -10 account balance.
Do not infer that negative indicates harm, debt or physical impossibility.
Preserve measurement unit, reference frame, scale and uncertainty.
A negative absolute temperature in kelvin is invalid for ordinary thermodynamic temperature; negative Celsius temperatures are valid.

12. REPRESENTATION
literal: "-12.5"
datatype: decimal
value: -12.5
sign: negative
magnitude: 12.5
normalisation: preserve exact rational/decimal forms when possible.
Avoid binary floating-point conversion for exact-value equivalence checks.

13. PRECISION AND UNCERTAINTY
A measured estimate x_hat < 0 does not prove the true value is negative when its uncertainty interval crosses zero.
CONFIRMED_NEGATIVE([lower,upper]) := upper < 0.
POSSIBLY_NEGATIVE([lower,upper]) := lower < 0.
NON_NEGATIVE([lower,upper]) := lower >= 0.
For intervals with open endpoints, compare membership at zero precisely.

14. VALIDATION
assert IS_NEGATIVE(-1,Integers) = true.
assert IS_NEGATIVE(-0.25,Rationals) = true.
assert IS_NEGATIVE(0,Reals) = false.
assert IS_NEGATIVE(3,Reals) = false.
assert (-2)*(-3) = 6.
assert (-2)*(3) = -6.
assert -(-7) = 7.
assert NOT IS_NEGATIVE(complex(1,2),Complexes) without an explicit order.
assert SIGN(-0.0) = 0 under mathematical real semantics.

15. CONSTRAINTS AND EXCEPTIONS
Require an explicit ordered numeric domain.
Reject comparisons involving NaN as indeterminate.
Distinguish mathematical infinity from finite numeric values and extended-real -infinity.
Division by zero is undefined.
Do not assert sign of sums with mixed signs without evaluating magnitudes.
Do not conflate negative exponent with negative numeric result.

16. CONFIDENCE
confidence.definition: 1.0
confidence.exact_arithmetic: 1.0 when preconditions are satisfied
confidence.measured_sign: dependent on uncertainty bounds
confidence.contextual_interpretation: dependent on domain metadata
Confidence refers to the assertion and its evidence, not to mathematical truth changing.

17. FLOW HASH
FLOW_HASH.algorithm: SHA-256
FLOW_HASH.input: canonical_utf8(normalised_flow_definition_without_hash_field)
FLOW_HASH.value: computed by implementing system; not fabricated here.
FLOW_HASH.purpose: integrity, version comparison and reproducibility.

18. UPDATE MODEL
UPDATE_MODEL.trigger: mathematical correction, ontology schema revision, source-page revision, or improved semantic alignment.
UPDATE_MODEL.policy: preserve previous versions and provenance; increment semantic version.
UPDATE_MODEL.validation: rerun invariants, examples and relationship constraints.
UPDATE_MODEL.compatibility: flag breaking changes to entity predicates or sign classification.

19. AI CONSUMPTION
AI_CONSUMPTION.read: parse typed entities, relations, rules and preconditions.
AI_CONSUMPTION.reason: evaluate sign only in compatible ordered domains.
AI_CONSUMPTION.retrieve: connect numeric literals, sign predicates and context.
AI_CONSUMPTION.explain: provide sign reasoning and uncertainty caveats.
AI_CONSUMPTION.guardrail: never treat negative as synonymous with undesirable.
AI_CONSUMPTION.output: validated claims with provenance and confidence.

20. PROVENANCE
source_url: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/numeric-value-entities/negative-value-ontology.html
source_verification: not independently retrieved for this version
basis: standard mathematics, ordered fields, numeric representations and semantic modelling
attribution: Mark Higgins — Sky (Recursive AI)
END FLOW ALGEBRA