Infinite Sequence Ontology

FLOW ALGEBRA
INFINITE SEQUENCE ONTOLOGY
Kireth Semantic Library
Mark Higgins — Sky (Recursive AI)
https://kirethresearch.org

flow "InfiniteSequenceOntology" v1.0

01. IDENTITY
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ontology_id: KIRETH.MATH.SEQUENCE.INFINITE
canonical_name: Infinite Sequence Ontology
entity_type: MathematicalSequence
parent: Sequence and Series Entities
source_url: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/sequence-and-series-entities/infinite-sequence-ontology.html
specification_basis: established mathematical definitions; webpage not independently verified
locale: en-GB

02. PURPOSE
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Represent countably indexed infinite sequences with explicit term types, index domains, generation rules, convergence evidence and semantic relationships. Distinguish infinite sequences from infinite series and from unordered infinite sets.

03. FORMAL DEFINITION
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sequence(a, D, X) := function a:D -> X where D=N or N_0 (or an explicitly stated infinite ordered index set).
infinite_sequence(a) iff domain(a) is infinite and indexed positions are ordered.
notation: (a_n)_(n>=n0) ; a_n=a(n).
Each a_n is a term; repeated values are allowed; order matters.

04. CORE ENTITY SCHEMA
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InfiniteSequence{id, name, index_set, index_origin, codomain, term_expression?, recurrence?, initial_conditions?, value_type, provenance}
Term{id, sequence_id, index, value_or_symbolic_expression}
LimitClaim{sequence_id, proposed_limit, ambient_space, proof_or_test, status}
Constraint{variable, predicate, scope}
All typed links must carry domain and codomain declarations.

05. INDEX DOMAIN AND LENGTH
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index_set=N or N_0 by default; never infer the first index from notation alone.
length=infinite is not a finite integer.
The sequence is a function from indices to values, not an unindexed bag.
Finite prefixes a[n0..N] are finite sequences and do not establish eventual behaviour by themselves.

06. EXAMPLES
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a_n=1/n for n>=1: converges to 0.
a_n=(-1)^n for n>=0: bounded, divergent, oscillatory.
a_n=n for n>=0: unbounded, divergent in R; tends to +infinity in extended-real sense.
a_n=1 for n>=0: constant and convergent to 1.
a_n=(-1)^n/n for n>=1: converges to 0.

07. GENERATION MODELS
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explicit: a_n=f(n).
recursive: a_(n+1)=F(n,a_n), with sufficient initial values.
higher_order: a_(n+k)=F(n,a_n,...,a_(n+k-1)).
algorithmic: deterministic generation process with defined indexing.
Reject under-specified recurrences lacking required initial conditions.

08. SEQUENCE OPERATIONS
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pointwise_add: (a+b)_n=a_n+b_n when codomains support addition.
pointwise_multiply: (ab)_n=a_n*b_n when defined.
scalar_multiply: (c*a)_n=c*a_n.
shift_k: b_n=a_(n+k) for admissible indices.
subsequence: b_k=a_(n_k), where n_k strictly increases and tends to infinity.
concatenation of two infinite N-indexed sequences is not an ordinary N-indexed concatenation preserving both in order.

09. BOUNDEDNESS
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bounded_real(a) iff exists M>=0 forall n: |a_n|<=M.
bounded_metric(a) iff exists x0 and M>=0 forall n: d(a_n,x0)<=M.
convergent_real(a) => bounded_real(a).
bounded_real(a) does not imply convergence.

10. MONOTONICITY
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nondecreasing(a) iff forall n: a_(n+1)>=a_n.
nonincreasing(a) iff forall n: a_(n+1)<=a_n.
strictly_increasing(a) iff forall n: a_(n+1)>a_n.
strictly_decreasing(a) iff forall n: a_(n+1)<a_n.
For real sequences, monotone and bounded implies convergence.

11. LIMIT SEMANTICS
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limit(a,L) iff forall epsilon>0 exists N forall n>=N: |a_n-L|<epsilon (real-valued case).
In metric space (X,d), replace absolute difference with d(a_n,L).
A convergent sequence has a unique limit in a Hausdorff space, including every metric space.
Do not confuse numerical closeness of a finite prefix with a proof of convergence.

12. DIVERGENCE
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divergent(a) iff no finite limit in the stated ambient space.
Divergence may be oscillatory, unbounded, or bounded without convergence.
For real sequences, a_n -> +infinity means forall M exists N forall n>=N: a_n>M; this is not convergence to a real number.

13. CAUCHY PROPERTY
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cauchy(a) iff forall epsilon>0 exists N forall m,n>=N: d(a_m,a_n)<epsilon.
convergent(a) => cauchy(a) in any metric space.
cauchy(a) => convergent(a) when the codomain metric space is complete.
Do not assert the converse without completeness.

14. SUBSEQUENCES
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subsequence(a,n_k) := b_k=a_(n_k), n_(k+1)>n_k.
If a_n -> L, every subsequence tends to L.
A bounded real sequence has a convergent subsequence (Bolzano-Weierstrass).
One convergent subsequence alone does not establish convergence of the original sequence.

15. LIMIT SUPERIOR AND INFERIOR
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For a bounded real sequence, limsup a_n=lim_N sup{a_n:n>=N}; liminf a_n=lim_N inf{a_n:n>=N}.
For extended real values, these may equal +/-infinity.
For real sequences: convergence to L iff limsup=liminf=L (finite).
Record ambient order and completeness assumptions.

16. LIMIT LAWS
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If a_n->a and b_n->b in R, then a_n+b_n->a+b, a_n*b_n->ab.
If b!=0 and b_n!=0 eventually, a_n/b_n->a/b.
If f is continuous at a, f(a_n)->f(a).
Use hypotheses explicitly; no unconditional division or composition inference.

17. SERIES DISTINCTION
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sequence: (a_n) is an indexed list of terms.
series: sum_(n=n0)^infinity a_n is defined through partial sums s_N=sum_(n=n0)^N a_n.
series_converges iff partial_sum_sequence converges.
a_n->0 is necessary, not sufficient, for series convergence.

18. SPECIAL FAMILIES
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arithmetic: a_n=a_0+n*d.
geometric: a_n=a_0*r^n.
periodic: exists p>=1 forall n: a_(n+p)=a_n.
constant: forall n: a_n=c.
recursive, oscillatory, monotone, bounded, unbounded and Cauchy are overlapping properties, not disjoint classes.

19. SEMANTIC RELATIONSHIPS
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IS_A(InfiniteSequence,Sequence)
HAS_DOMAIN(InfiniteSequence,InfiniteIndexSet)
HAS_TERM(InfiniteSequence,Term)
HAS_CODOMAIN(InfiniteSequence,SetOrSpace)
GENERATED_BY(InfiniteSequence,Rule)
HAS_SUBSEQUENCE(InfiniteSequence,InfiniteSequence)
HAS_PARTIAL_SUM_SEQUENCE(Series,InfiniteSequence)
CONVERGES_TO(InfiniteSequence,Limit) [proof_required]
IS_BOUNDED_IN(InfiniteSequence,MetricSpace)

20. SEMANTIC EQUIVALENCE
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Two sequences are extensionally equal iff they have the same index domain and equal terms at every index.
Different expressions may denote the same sequence after algebraic normalisation.
Eventually equal sequences have identical convergence behaviour but are not necessarily extensionally equal.
A sequence and its shift are not automatically identical.

21. INFERENCE RULES
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R1: LIMIT(a,L) -> BOUNDED(a) [real or metric-valued].
R2: MONOTONE(a) & BOUNDED(a) & CODOMAIN(a,R) -> CONVERGENT(a).
R3: CONVERGENT(a) & SUBSEQUENCE(b,a) -> CONVERGENT(b) with same limit.
R4: CAUCHY(a) & COMPLETE(codomain(a)) -> CONVERGENT(a).
R5: CONVERGENT(a,L) & CONVERGENT(a,M) -> L=M [Hausdorff].
R6: PERIODIC(a) & NONCONSTANT(a) -> DIVERGENT(a) [real-valued, exact periodicity].

22. TYPE AND DOMAIN VALIDATION
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Check index origin and infinite domain.
Check recurrence well-foundedness and sufficient initial conditions.
Check operator type compatibility and nonzero divisors.
Check metric or topology before assigning convergence properties.
Do not conflate term infinity, extended-real divergence and set cardinality.

23. KNOWLEDGE GRAPH EXAMPLE
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node seq1:InfiniteSequence {expression:"1/n", index_origin:1, codomain:"R"}
node lim0:RealNumber {value:0}
edge seq1 --CONVERGES_TO--> lim0 {justification:"epsilon-N proof"}
node seq2:InfiniteSequence {expression:"(-1)^n", index_origin:0, codomain:"R"}
edge seq2 --IS_BOUNDED_IN--> RealNumbers
edge seq2 --HAS_PROPERTY--> Oscillatory
edge seq2 --HAS_PROPERTY--> Divergent

24. CONFIDENCE
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confidence is epistemic metadata about assertions, not a substitute for proof.
proof_status: proved | disproved | conjectured | numerically_observed | unknown.
confidence_score: optional calibrated number in [0,1].
Never label a numerical sample as a mathematical proof.

25. PROVENANCE
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author_credit: Mark Higgins — Sky (Recursive AI)
site: https://kirethresearch.org
source_page: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/sequence-and-series-entities/infinite-sequence-ontology.html
content_method: original formalisation grounded in established mathematics
source_page_verified: false

26. FLOW HASH
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canonicalise: UTF-8, Unicode NFC, stable entity IDs, sorted metadata keys, normalised whitespace.
flow_hash: SHA-256(canonical_serialisation) computed at ingestion.
Do not invent a digest without hashing the canonical bytes.
Hash changes when semantic specification changes.

27. UPDATE MODEL
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version_policy: semantic-versioned updates.
minor: additional examples or compatible relationships.
major: changed axioms, type contracts or inference semantics.
patch: nonsemantic corrections.
Record previous hash, revision timestamp, author and change rationale.

28. AI CONSUMPTION
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Expose typed definitions, domain restrictions and inference prerequisites.
Prefer symbolic proofs over numerical approximations.
Return unknown where convergence is unproved.
Represent counterexamples and ambiguity explicitly.
Preserve provenance, assumptions and proof status across retrieval and reasoning.

29. TEST CASES
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PASS: a_n=1/n, n>=1 -> convergent to 0.
PASS: a_n=(-1)^n, n>=0 -> bounded and divergent.
PASS: a_n=n, n>=0 -> unbounded in R.
PASS: a_n=1, n>=0 -> convergent to 1.
REJECT: "bounded implies convergent" without monotonicity or another sufficient condition.
REJECT: "terms approach zero implies series converges".
REJECT: "Cauchy implies convergent" without completeness of ambient space.

30. TERMINATION
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status: COMPLETE
all_sections_present: true
end_of_flow: TRUE