Integral Ontology

FLOW ALGEBRA
INTEGRAL ONTOLOGY
Version: 1.0
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/calculus-and-analysis-entities/integral-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Date: 2026-10-13

1. ONTOLOGY IDENTITY
ontology_id: kireth:mathematical-entities/calculus-and-analysis/integral
entity_type: MathematicalOperation
preferred_label: Integral
broader_entity: CalculusAndAnalysisEntity
related_entities: Function, Antiderivative, Derivative, Measure, Interval, Partition, Limit, Differential, Domain, Integrand, IntegrationVariable

2. SEMANTIC DEFINITION
An integral is a mathematical construction that assigns an accumulated quantity to a function or other integrable object over a domain, under a specified integration theory. An indefinite integral denotes a family of antiderivatives where they exist; a definite integral denotes a value determined by an integrand, domain, and integration convention.

3. TYPED ENTITIES
Integral: MathematicalOperation
DefiniteIntegral: Integral
IndefiniteIntegral: Integral
RiemannIntegral: DefiniteIntegral
LebesgueIntegral: DefiniteIntegral
ImproperIntegral: DefiniteIntegral
LineIntegral: Integral
SurfaceIntegral: Integral
MultipleIntegral: Integral
Integrand: FunctionOrMeasurableObject
IntegrationDomain: SetOrOrientedGeometricDomain
IntegrationVariable: Variable
IntegrationMeasure: MeasureOrDifferentialElement
Antiderivative: DifferentiableFunction
IntegralValue: ScalarOrVectorOrExtendedReal

4. TYPED RELATIONSHIPS
Integral --HAS_INTEGRAND--> Integrand
Integral --HAS_DOMAIN--> IntegrationDomain [for definite integration]
Integral --USES_VARIABLE--> IntegrationVariable
Integral --USES_MEASURE--> IntegrationMeasure [where applicable]
DefiniteIntegral --PRODUCES--> IntegralValue
IndefiniteIntegral --DENOTES_FAMILY_OF--> Antiderivative
RiemannIntegral --DEFINED_BY--> LimitOfRiemannSums
LebesgueIntegral --DEFINED_USING--> MeasureTheory
Integral --RELATED_TO--> Derivative [via applicable fundamental theorem]
MultipleIntegral --INTEGRATES_OVER--> MultidimensionalDomain
LineIntegral --INTEGRATES_OVER--> Curve
SurfaceIntegral --INTEGRATES_OVER--> Surface

5. FORMAL DEFINITIONS AND CONDITIONS
Riemann: integral_a^b f(x) dx = lim_(mesh(P)->0) sum_i f(x_i*) Delta x_i, provided the limit exists independently of partitions and tags.
Indefinite: integral f(x) dx = F(x) + C on an interval if F'(x)=f(x); C is constant on each connected interval component.
Lebesgue: integral_X f dmu is defined for nonnegative measurable f by supremum of integrals of simple functions below f, and for signed f when at least one of integrals of positive/negative parts is finite; finite integrability requires integral |f| dmu < infinity.
Improper: integral_a^infinity f(x) dx = lim_(b->infinity) integral_a^b f(x) dx, when the limit exists.

6. MATHEMATICAL LAWS (WITH PRECONDITIONS)
LINEARITY: integral (a f + b g) dmu = a integral f dmu + b integral g dmu for integrable f,g and finite scalars a,b.
ADDITIVITY_INTERVAL: integral_a^c f dx = integral_a^b f dx + integral_b^c f dx when integrals exist with consistent orientation.
ORDER: f <= g almost everywhere implies integral f dmu <= integral g dmu for integrable real-valued functions over a positive measure.
FTC_I: If f is continuous on [a,b], F(x)=integral_a^x f(t)dt is differentiable on (a,b) and F'=f.
FTC_II: If f is continuous on [a,b] and F'=f, integral_a^b f(x)dx=F(b)-F(a).
SUBSTITUTION: Under suitable differentiability, integrability, and orientation hypotheses, integral_(g(a))^(g(b)) f(u)du = integral_a^b f(g(t))g'(t)dt.
INTEGRATION_BY_PARTS: integral_a^b u v' dx = [uv]_a^b - integral_a^b u' v dx for sufficiently regular u,v.
FUBINI: Iterated integrals may be interchanged for integrable functions on product measure spaces under Fubini's hypotheses; Tonelli applies to nonnegative measurable functions.

7. INFERENCE RULES
RULE R1: IF differentiable(F,I) AND derivative(F)=f on interval I THEN antiderivative_of(F,f,I).
RULE R2: IF continuous(f,[a,b]) THEN riemann_integrable(f,[a,b]).
RULE R3: IF riemann_integrable(f,[a,b]) AND antiderivative(F,f,[a,b]) THEN integral(f,[a,b])=F(b)-F(a), subject to the relevant FTC hypotheses.
RULE R4: IF measurable(f) AND integral(abs(f),mu)<infinity THEN lebesgue_integrable(f,mu).
RULE R5: IF integrable(f,mu) AND integrable(g,mu) THEN integrable(a*f+b*g,mu) for finite scalars.
RULE R6: IF measurable(f,g) AND nonnegative(f,g) AND f<=g almost_everywhere THEN integral(f,mu)<=integral(g,mu), allowing extended nonnegative values.
RULE R7: IF nonnegative_measurable(f,product_measure) THEN Tonelli_permits_iterated_integration(f).

8. SEMANTIC DISTINCTIONS
Integral != Antiderivative: a definite integral can exist without an elementary antiderivative.
RiemannIntegral != LebesgueIntegral: theories have different definitions and integrability domains.
DefiniteIntegral != IndefiniteIntegral: value versus family of primitives.
ImproperIntegral != LebesgueIntegral: conditional improper convergence need not imply Lebesgue integrability.
IntegrationVariable is bound and may be alpha-renamed without changing a properly formed integral.
Domain orientation matters for signed line, surface, and interval integrals.

9. VALIDATION
CHECK V1: All integral expressions specify integrand and bound variable.
CHECK V2: Definite integrals specify domain/bounds and integration convention.
CHECK V3: Integration theorems include regularity and integrability preconditions.
CHECK V4: Do not infer integrability from measurability alone.
CHECK V5: Do not infer differentiability of an integral at every point without adequate hypotheses.
CHECK V6: Reject interchange of limits, derivatives, and integrals without justification.
CHECK V7: Distinguish scalar, vector, and differential-form outputs.
CHECK V8: Ensure dimensions/units are consistent with integrand and differential element.

10. SEMANTIC EQUIVALENCE
canonical_label: Integral
aliases: Integration; Integral operator [context-dependent]
semantic_equivalence_rule: Two definite-integral expressions are equivalent only when their values agree under established transformations and applicable hypotheses; syntactic resemblance alone is insufficient.

11. CONFIDENCE AND PROVENANCE
CONFIDENCE:
  formal_core: high
  derived_claim: conditional_on_proof_and_assumptions
  application_claim: requires_domain_specific_validation
PROVENANCE:
  source_url: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/calculus-and-analysis-entities/integral-ontology.html
  note: Standalone formalisation of the ontology topic; page-specific wording not asserted.

12. FLOW HASH AND UPDATE MODEL
FLOW_HASH:
  method: SHA-256
  input: canonical UTF-8 serialisation of ontology identity, entity types, relations, rules and version
  generation: compute at ingestion; do not fabricate digest
UPDATE_MODEL:
  version: 1.0
  update_strategy: semantic-versioned, provenance-preserving
  change_detection: compare canonical content hashes
  compatibility: validate dependent ontology references after updates

13. AI CONSUMPTION
AI_CONSUMPTION:
  mode: typed_semantic_reasoning
  parse: entities, domains, operations, hypotheses, laws, inference_rules
  require: theorem_preconditions_before_inference
  output: derivation_with_assumptions_and_provenance
  guardrails: no_unjustified_limit_exchange; no_unjustified_integrability; no_type_conflation
  interoperability: DerivativeOntology, LimitOntology, ContinuityOntology, DifferentialOntology, MeasureOntology, VectorCalculusOntology

END FLOW ALGEBRA