Geometric Sequence Ontology

FLOW ALGEBRA
GEOMETRIC SEQUENCE ONTOLOGY
Kireth Semantic Library | Mathematical Entities | Sequence and Series Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/sequence-and-series-entities/geometric-sequence-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-13 | Locale: en-GB
Status: mathematical specification; source page content not independently verified

01. ONTOLOGY IDENTITY
ontology.id: kireth.math.sequence.geometric
ontology.type: MathematicalSequenceOntology
ontology.parent: SequenceAndSeriesEntities
ontology.label: Geometric Sequence
ontology.definition: A sequence with a constant multiplicative ratio between successive terms, expressible as a_n = a_0*r^n for n >= 0.

02. ENTITY SCHEMA
entity GeometricSequence:
  index_set: subset_of(N0)
  first_term: Scalar
  common_ratio: Scalar
  term_function: Index -> Scalar
  coefficient_domain: NumericDomain
  finite_length: NaturalNumber | null
  zero_policy: explicit
  provenance: SourceReference

03. TYPED RELATIONSHIPS
GeometricSequence IS_A Sequence
GeometricSequence HAS_FIRST_TERM Scalar
GeometricSequence HAS_COMMON_RATIO Scalar
GeometricSequence HAS_TERM IndexedValue
GeometricSequence HAS_INDEX_SET IndexSet
GeometricSequence MAY_HAVE PartialSum
GeometricSequence MAY_CONVERGE_TO Limit
GeometricSequence MAY_GENERATE GeometricSeries

04. CORE DEFINITION
parameters: a0 in K, r in K, where K is a commutative field (normally R or C)
index: n in N0
term(n): a0 * r^n
recurrence: a_(n+1) = r*a_n
initial: a_0 = a0
For r = 0, define a_0 separately and a_n = 0 for all n >= 1; do not rely on 0^0.

05. ONE-BASED INDEX CONVENTION
For n >= 1: a_n = a_1*r^(n-1)
Recurrence: a_(n+1) = r*a_n
Index origins MUST be declared before evaluation.

06. COMMON RATIO INFERENCE
if a_n != 0 and a_(n+1) known:
  r = a_(n+1)/a_n
if a_n = 0:
  do not infer r from division
if all terms zero:
  ratio is not uniquely identifiable from observed terms

07. ZERO AND DEGENERATE CASES
if a0 = 0: every term is zero, for every r
if r = 1: sequence constant at a0
if r = 0: terms after first are zero
if r = -1: sequence alternates a0, -a0, ...
if a0 != 0 and r != 0: each term is nonzero

08. REAL-VALUED CLASSIFICATION
r > 1: magnitudes grow if a0 != 0
r = 1: constant
0 < r < 1: magnitudes decay towards zero
r = 0: eventually zero
-1 < r < 0: alternating signs, magnitudes decay
r = -1: alternating constant magnitude
r < -1: alternating signs, magnitudes grow
For a0 = 0 all cases yield the zero sequence.

09. COMPLEX-VALUED CLASSIFICATION
abs(r) < 1: converges to zero
r = 1: converges to a0
abs(r) > 1 and a0 != 0: magnitude diverges to infinity
abs(r) = 1 and r != 1 and a0 != 0: no limit
if a0 = 0: converges to zero for every r

10. FINITE PARTIAL SUM
S_N = sum(n=0..N-1, a0*r^n)
N >= 0
if r != 1: S_N = a0*(1-r^N)/(1-r)
if r = 1: S_N = N*a0
S_0 = 0

11. INFINITE GEOMETRIC SERIES
sum(n=0..infinity, a0*r^n) = a0/(1-r) if abs(r) < 1
if a0 = 0: sum is zero for any finite r
if a0 != 0 and abs(r) >= 1: series diverges
Distinguish convergence of sequence terms from convergence of their series.

12. RECURRENCE AND CLOSED FORM
initial a_0 = a0
step a_(n+1) = r*a_n
inductive invariant: a_n = a0*r^n
proof: base n=0; step multiply by r

13. ORDER AND MONOTONICITY
For real a0 > 0 and r > 1: strictly increasing
For real a0 > 0 and 0 < r < 1: strictly decreasing
For real a0 < 0 and r > 1: strictly decreasing
For real a0 < 0 and 0 < r < 1: strictly increasing
r < 0 with a0 != 0: generally non-monotonic because signs alternate
r in {0,1} or a0 = 0: handle as special cases

14. BOUNDEDNESS
For finite real or complex a0, abs(r) <= 1: bounded
For abs(r) > 1 and a0 != 0: unbounded
For a0 = 0: bounded for all r

15. SUBSEQUENCE RULE
select indices n_k = m + k*d, m >= 0, d >= 1
b_k = a0*r^m*(r^d)^k
new_first_term = a0*r^m
new_ratio = r^d

16. SHIFT AND SCALE
shift by m: a_(n+m) = (a0*r^m)*r^n
scale by c: c*a_n = (c*a0)*r^n
pointwise product of two geometric sequences is geometric with ratio r*s
pointwise sum is not generally geometric

17. GEOMETRIC MEAN PROPERTY
For interior terms of a geometric sequence: a_n^2 = a_(n-1)*a_(n+1)
This condition alone may be insufficient to recover a unique ratio when zero terms occur.
For positive real neighbouring terms, a_n is their positive geometric mean.

18. SEMANTIC EQUIVALENCE
Equivalent when index sets, value domains and all indexed values agree.
Two parameterisations with differing index origins may denote the same indexed values after explicit reindexing.
Do not equate geometric sequence with geometric series.
Do not equate common ratio with additive common difference.

19. KNOWLEDGE GRAPH TRIPLES
(GeometricSequence, subClassOf, Sequence)
(GeometricSequence, characterisedBy, CommonRatio)
(GeometricSequence, definedBy, MultiplicativeRecurrence)
(GeometricSequence, hasRepresentation, ExponentialClosedForm)
(GeometricSequence, generates, GeometricSeries)
(GeometricSequence, relatedTo, ArithmeticSequence)

20. INFERENCE RULES
RULE G01: known a0,r,n => infer term a0*r^n
RULE G02: nonzero consecutive term => infer common ratio
RULE G03: abs(r)<1 => infer term limit zero
RULE G04: r=1 => infer constant sequence
RULE G05: a0=0 => infer zero sequence
RULE G06: abs(r)<1 => infer infinite sum a0/(1-r)
RULE G07: r!=1 => infer finite partial sum formula
RULE G08: equal indexed terms => allow semantic identity only under matched index domains

21. VALIDATION
assert index_origin in {zero_based, one_based, explicitly_custom}
assert n is nonnegative integer for zero-based model
assert arithmetic supports multiplication and exponentiation
assert denominator 1-r != 0 before applying quotient sum formula
assert no division by a_n when a_n = 0
assert complex ratios use modulus for convergence classification
assert partial sums have declared term count

22. EXAMPLES
EXAMPLE A: a0=3, r=2 => 3,6,12,24,...
EXAMPLE B: a0=8, r=1/2 => 8,4,2,1,... ; infinite sum=16
EXAMPLE C: a0=5, r=-1 => 5,-5,5,-5,...
EXAMPLE D: a0=7, r=0 => 7,0,0,0,...
EXAMPLE E: a0=0, r=3 => 0,0,0,...

23. CONFIDENCE
mathematical_definition: 1.0 (standard field-based model)
source_page_alignment: unverified
example_validation: deterministic
inference_confidence: 1.0 when premises and domain constraints hold

24. FLOW HASH
flow_hash.algorithm: SHA-256
flow_hash.canonical_input: UTF-8 normalised ontology payload excluding hash value
flow_hash.value: COMPUTE_ON_PUBLICATION
flow_hash.purpose: detect content revisions, not establish theorem truth

25. UPDATE MODEL
update_model.mode: versioned
update_model.trigger: definition_change | rule_change | example_correction | source_revision
update_model.requirements: preserve provenance, increment version, recompute flow hash, rerun validation

26. AI CONSUMPTION
ai_consumption.mode: typed_symbolic_reasoning
ai_consumption.capabilities: classify, evaluate, derive, validate, compare, map_to_graph
ai_consumption.constraints: preserve index origin, distinguish series, check zero cases, check field and convergence domain
ai_consumption.output: typed assertions with evidence and confidence

27. TEST CASES
TEST T01: term(a0=2,r=3,n=3) == 54
TEST T02: sum(a0=2,r=3,N=4) == 80
TEST T03: infinite_sum(a0=4,r=0.5) == 8
TEST T04: converges(a0=3,r=-0.5) == true
TEST T05: converges(a0=3,r=-1) == false
TEST T06: converges(a0=0,r=10) == true
TEST T07: sum(a0=7,r=1,N=5) == 35
TEST T08: ratio_inference(0,0) == undefined

28. TERMINATION
flow.status: COMPLETE
flow.termination: END_OF_FLOW_ALGEBRA