Divergent Sequence Ontology

FLOW ALGEBRA
TITLE: Divergent Sequence Ontology
SOURCE: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/sequence-and-series-entities/divergent-sequence-ontology.html
VERSION: 4.0
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library

ENTITY: DivergentSequence
TYPE: MathematicalEntity > SequenceEntity > DivergentSequence
DEFINITION: A sequence a:N -> X in a metric space (X,d) is divergent in X if there is no L in X to which it converges.

ENTITIES:
  Sequence: indexed function a:N -> X
  Term: a_n
  Index: n in N
  CandidateLimit: L in X
  Tolerance: epsilon > 0
  Metric: d:X x X -> nonnegative reals
  AmbientSpace: metric space (X,d)
  Subsequence: a_(n_k), where n_k is strictly increasing
  ClusterPoint: limit of a convergent subsequence, if one exists
  UnboundedSequence: real-valued sequence with no finite global bound on absolute values
  OscillatingSequence: sequence exhibiting recurring variation; oscillation alone does not imply divergence

RELATIONSHIPS:
  DivergentSequence IS_A Sequence
  DivergentSequence NOT_A ConvergentSequence [relative to specified ambient space]
  DivergentSequence DEFINED_OVER AmbientSpace
  DivergentSequence MAY_BE UnboundedSequence
  DivergentSequence MAY_BE BoundedSequence
  DivergentSequence MAY_HAVE MultipleSubsequentialLimits
  DivergentSequence MAY_HAVE NoConvergentSubsequence [depending on space]
  UnboundedRealSequence IMPLIES DivergentSequence [in R]
  TwoDistinctSubsequentialLimits IMPLY DivergentSequence [in metric spaces]
  NonCauchySequence IMPLIES DivergentSequence [in metric spaces]

FLOW ALGEBRA:
  INPUT: sequence a:N -> X; metric space (X,d)
  VALIDATE: every a_n belongs to X and d is a metric
  CONVERGENCE_TEST: exists L in X such that forall epsilon > 0, exists N in N, forall n >= N: d(a_n,L) < epsilon
  TRUE -> CONVERGENT(sequence=a, limit=L)
  FALSE -> DIVERGENT(sequence=a, ambient_space=X)
  EQUIVALENT_DIVERGENCE_TEST: forall L in X, exists epsilon > 0 such that forall N in N, exists n >= N with d(a_n,L) >= epsilon
  IMPORTANT: Nonconvergence to one candidate L is not sufficient to prove divergence.

FORMAL RULES:
  R1: DIVERGENT(a,X) <-> NOT EXISTS L IN X: CONVERGES(a,L)
  R2: UNBOUNDED(a) -> DIVERGENT(a,R) [real-valued sequence]
  R3: SUBSEQUENCE(b,a) AND SUBSEQUENCE(c,a) AND CONVERGES(b,L) AND CONVERGES(c,M) AND L != M -> DIVERGENT(a,X)
  R4: NOT_CAUCHY(a) -> DIVERGENT(a,X)
  R5: CAUCHY(a) AND NOT_COMPLETE(X) does not by itself determine whether a converges in X
  R6: BOUNDED(a) does not imply CONVERGENT(a) [without additional conditions]
  R7: DIVERGENT(a,X) does not imply UNBOUNDED(a)
  R8: CONVERGES(a,L) -> every subsequence of a converges to L
  R9: MONOTONE(a) AND UNBOUNDED(a) -> DIVERGENT(a,R) [real-valued]
  R10: CONVERGES_TO_POSITIVE_INFINITY(a) -> DIVERGENT(a,R) [no finite real limit]
  R11: CONVERGES_TO_NEGATIVE_INFINITY(a) -> DIVERGENT(a,R) [no finite real limit]

EXAMPLES:
  E1: a_n=(-1)^n -> divergent in R; subsequences converge to -1 and 1
  E2: a_n=n -> divergent in R; tends to positive infinity
  E3: a_n=(-1)^n*n -> divergent in R; unbounded and alternating
  E4: a_n=sin(n*pi/2) -> divergent in R; recurring subsequential values
  E5: a_n=1/n, n>=1 -> convergent to 0 in R, NOT divergent
  E6: a_n=1/n, n>=1, viewed in X=(0,infinity) -> divergent in X; limit 0 lies outside X
  E7: rational approximations to sqrt(2) may be Cauchy but divergent in Q; they converge in R

SEMANTIC_EQUIVALENCE:
  divergent sequence ~= nonconvergent sequence [relative to ambient metric space]
  divergence in R ~= absence of a finite real limit
  divergence to infinity ~= unbounded growth in a specified direction [not equivalent to all divergence]
  oscillation ~= possible behaviour associated with divergence [not a definition]
  NOTE: Divergence depends on the ambient space and its topology.

CONFIDENCE:
  MathematicalDefinitions: 1.00
  StandardTheorems: 1.00
  SourcePageAlignment: unverified

FLOW_HASH:
  METHOD: SHA-256 of canonical UTF-8 file content, calculated externally when deployed
  STATUS: not computed

UPDATE_MODEL:
  VERSION: 4.0
  REVIEW_TRIGGER: changes to mathematical definitions, ambient-space assumptions, or entity relationships
  PRESERVE: quantifier order, theorem assumptions, provenance, and semantic distinctions

AI_CONSUMPTION:
  MACHINE_READABLE: true
  PURPOSE: semantic retrieval, proof guidance, mathematical entity-graph linkage, classification of sequences
  CAUTION: Do not equate divergence with unboundedness, oscillation, or convergence to infinity.
  CAUTION: A Cauchy sequence may diverge in an incomplete ambient space.
  CAUTION: Verify all convergence and divergence claims relative to the declared ambient space.

END FLOW ALGEBRA