Transcendental Number Ontology

FLOW ALGEBRA
TRANSCENDENTAL NUMBER ONTOLOGY
Kireth Semantic Library | Entity Graph > Mathematical Entities > Number Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/number-entities/transcendental-number-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-12 | Locale: en-GB

01. ONTOLOGY IDENTITY
flow "TranscendentalNumberOntology" v1.0
ontology.id: kireth:number:transcendental
ontology.type: MathematicalNumberOntology
ontology.parent: NumberEntities
ontology.domain: NumberTheory
ontology.status: formal_semantic_model

02. CORE DEFINITION
sets:
  Z: integers
  Q: rationals
  R: reals
  C: complex_numbers
  A: {z in C | exists nonzero p in Z[x] such that p(z)=0}
  T: C \ A
rules:
  transcendental(z) <=> z in C and not algebraic(z)
  algebraic(z) <=> exists nonzero polynomial p with integer coefficients and p(z)=0
  transcendental_real(z) <=> transcendental(z) and z in R
  transcendental_complex(z) <=> transcendental(z) and z in C
  transcendental(z) => irrational(z) only if z in R
  transcendental(z) => not rational(z)

03. ENTITY TYPES
entity TranscendentalNumber extends ComplexNumber:
  properties: [symbol, representation, real_part, imaginary_part, proof_status, provenance]
entity TranscendentalReal extends TranscendentalNumber, RealNumber
entity NonRealTranscendental extends TranscendentalNumber
entity AlgebraicNumber extends ComplexNumber
entity IntegerPolynomial:
  properties: [degree, coefficients, nonzero]
entity TranscendenceProof:
  properties: [method, assumptions, theorem, conclusion, citation]
entity TranscendenceStatus:
  values: [proven_transcendental, proven_algebraic, unknown]

04. EXAMPLES AND CLASSIFICATION
examples:
  pi: proven_transcendental
  e: proven_transcendental
  Liouville_constant: proven_transcendental
  sqrt(2): proven_algebraic
  i: proven_algebraic
  e^pi: proven_transcendental
  pi+e: unknown
  pi*e: unknown
notes:
  - An irrational number need not be transcendental.
  - A complex transcendental number need not be real or classified as irrational.
  - Decimal approximations alone cannot establish transcendence.

05. SET RELATIONSHIPS
axioms:
  Q subset A
  Z subset Q
  A subset C
  T intersection A = empty_set
  T union A = C
  T intersection R subset (R \ Q)
  cardinality(A) = countably_infinite
  cardinality(T) = uncountable
  cardinality(T intersection R) = uncountable
  almost_every_real_number_is_transcendental: true  # Lebesgue measure

06. ARITHMETIC AND CLOSURE
rules:
  for t in T, a in A, a != 0:
    t + a in T
    t - a in T
    a - t in T
    a * t in T
    t / a in T
    a / t in T
  for t in T:
    -t in T
    1/t in T
  for t1,t2 in T:
    transcendental(t1+t2): not_guaranteed
    transcendental(t1*t2): not_guaranteed
  counterexamples:
    pi + (1-pi) = 1
    pi * (1/pi) = 1

07. POLYNOMIAL RELATIONS
rules:
  nonconstant_integer_polynomial_root(z) => algebraic(z)
  transcendental(z) => forall nonzero p in Z[x], p(z) != 0
  polynomial_with_algebraic_coefficients_root(z) => algebraic(z)
  nonconstant_polynomial_over_algebraic_numbers_of_transcendental(z) => transcendental(result)
constraints:
  polynomial_nonzero: required
  coefficient_domain: integers_or_algebraic_numbers
  complex_domain: required

08. TRANSCENDENCE THEOREMS
knowledge:
  Hermite_1873: e is transcendental
  Lindemann_1882: pi is transcendental
  Lindemann_Weierstrass:
    statement: if alpha is nonzero algebraic then exp(alpha) is transcendental
  Gelfond_Schneider:
    statement: if a is algebraic, a not in {0,1}, and b is algebraic irrational, then every value of a^b is transcendental
  Liouville_theorem:
    statement: algebraic irrational numbers admit a degree-dependent lower bound on rational approximation
    implication: Liouville numbers are transcendental

09. SEMANTIC RELATIONSHIPS
relationships:
  TranscendentalNumber --is_a--> ComplexNumber
  TranscendentalNumber --disjoint_with--> AlgebraicNumber
  TranscendentalReal --is_a--> IrrationalNumber
  AlgebraicNumber --root_of--> IntegerPolynomial
  TranscendentalNumber --not_root_of--> NonzeroIntegerPolynomial
  TranscendenceProof --establishes--> TranscendenceStatus
  TranscendentalNumber --studied_in--> NumberTheory
  TranscendentalNumber --studied_in--> TranscendenceTheory

10. KNOWLEDGE GRAPH TRIPLES
triples:
  (pi, rdf:type, TranscendentalReal)
  (e, rdf:type, TranscendentalReal)
  (sqrt(2), rdf:type, AlgebraicNumber)
  (TranscendentalNumber, rdfs:subClassOf, ComplexNumber)
  (TranscendentalNumber, owl:disjointWith, AlgebraicNumber)
  (pi, transcendenceStatus, proven_transcendental)
  (pi+e, transcendenceStatus, unknown)

11. INFERENCE ENGINE
infer:
  IF z in C AND verified_nonzero_integer_polynomial_root(z)
    THEN status(z) = proven_algebraic
  IF verified_transcendence_proof(z)
    THEN status(z) = proven_transcendental
  IF transcendental(z) AND algebraic(a) AND a != 0
    THEN transcendental(z+a), transcendental(z*a)
  IF transcendental(z) AND real(z)
    THEN irrational(z)
  IF no_known_proof(z)
    THEN status(z) = unknown  # never infer algebraicity or transcendence

12. CONFIDENCE AND EVIDENCE
confidence:
  formal_proof_verified: 1.0
  theorem_with_verified_hypotheses: 1.0
  numerical_heuristic: not_sufficient
  unverified_assertion: not_sufficient
requirements:
  theorem_name: optional
  proof_reference: required_for_proven_claim
  assumptions_checked: true
  unknown_status_preserved: true

13. SEMANTIC EQUIVALENCE
semantic_equivalence:
  "transcendental complex number": "complex number not algebraic over Q"
  "non-algebraic number": "transcendental number"  # domain C
  "transcendental real number": "real number not root of any nonzero integer polynomial"
semantic_difference:
  irrational_vs_transcendental: irrational is real and not rational; transcendental is complex and not algebraic
  algebraic_irrational_vs_transcendental: sqrt(2) is irrational but algebraic

14. VALIDATION
validate:
  - ensure numeric domain is C
  - distinguish exact expression from approximation
  - reject zero polynomial as transcendence test
  - verify polynomial coefficients and proof conditions
  - prevent irrational_implies_transcendental inference
  - prevent unknown_implies_transcendental inference
  - preserve distinctions between real and nonreal complex numbers
  - verify theorem assumptions before rule execution

15. FLOW EXECUTION
pipeline:
  input: number_expression
  parse: exact_symbolic_representation
  normalise: semantic_equivalent_expression
  classify_domain: [integer, rational, real, complex]
  search_algebraic_certificate: nonzero_integer_polynomial
  search_transcendence_certificate: verified_theorem_or_proof
  resolve_status: [proven_algebraic, proven_transcendental, unknown]
  emit: typed_entity_and_evidence_graph
  preserve: provenance_and_uncertainty

16. FLOW HASH AND UPDATE MODEL
flow_hash:
  method: SHA-256
  input: canonical_serialisation_of_ontology_version
  purpose: change_detection_and_integrity
update_model:
  versioned: true
  immutable_prior_versions: true
  update_triggers: [new_proof, theorem_correction, ontology_revision]
  require_provenance: true
  revalidate_dependent_inferences: true

17. AI CONSUMPTION
ai_consumption:
  machine_readable: true
  typed_entities: true
  explicit_relationships: true
  inference_safe: true
  uncertainty_aware: true
  supports: [semantic_search, knowledge_graph, theorem_assisted_reasoning, classification]
  prohibitions:
    - do_not_treat_numerical_approximation_as_proof
    - do_not_claim_unknown_constants_are_transcendental
    - do_not_conflate_irrationality_with_transcendence

18. OUTPUT CONTRACT
output:
  entity_type: TranscendentalNumber | AlgebraicNumber | UnresolvedNumber
  transcendence_status: proven_transcendental | proven_algebraic | unknown
  proof_reference: string_or_null
  assumptions: list
  confidence: evidence_based
  provenance: source_records
  flow_version: 1.0

END FLOW ALGEBRA