Finite Series Ontology

FLOW ALGEBRA
TITLE: Finite Series Ontology
SOURCE: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/sequence-and-series-entities/finite-series-ontology.html
VERSION: 4.0
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library
STATUS: Independently constructed mathematical ontology; source page content not verified.

01. ONTOLOGY IDENTITY
ENTITY: FiniteSeries
TYPE: MathematicalEntity > SequenceAndSeriesEntity > Series > FiniteSeries
DEFINITION: A finite series is the sum of a finite indexed family of terms in a specified additive structure, written sum_{k=m}^{n} a_k for integers m <= n.
SCOPE: Finite indexed sums; not infinite-series convergence.
CANONICAL_NOTATION: S(m,n) = Σ_{k=m}^{n} a_k
INDEX_CONVENTION: Integers m,n with m <= n; empty sums use m > n where explicitly allowed.
AMBIENT_STRUCTURE: Commutative monoid for order-independent finite sums; additive group for subtraction; ring/field for algebraic and closed-form identities as required.

02. CORE ENTITIES
  FiniteSeries: formal sum of finitely many indexed terms
  Term: a_k, an element of ambient additive structure X
  IndexSet: I = {m,m+1,...,n}
  LowerBound: m
  UpperBound: n
  TermCount: N = n-m+1 for m <= n
  SummationOperator: Σ
  SumValue: S(m,n) in X
  EmptySum: 0_X, when the index set is empty and X has an additive identity
  SingletonSum: Σ_{k=m}^{m} a_k = a_m
  PrefixSum: P_j = Σ_{k=m}^{j} a_k
  PartialSum: finite initial segment of a possibly infinite series
  ArithmeticFiniteSeries: finite sum of arithmetic progression
  GeometricFiniteSeries: finite sum of geometric progression
  TelescopingFiniteSeries: finite sum with cancellation of adjacent differences
  WeightedFiniteSeries: Σ_{k=m}^{n} w_k a_k
  AlternatingFiniteSeries: Σ_{k=m}^{n} (-1)^k a_k
  DoubleFiniteSeries: Σ_i Σ_j a_{ij} over finite index sets
  EmptyIndexSet: I = ∅
  ReindexingMap: bijection of finite index sets

03. TYPED RELATIONSHIPS
  FiniteSeries IS_A Series
  FiniteSeries HAS_TERM Term
  FiniteSeries INDEXED_BY IndexSet
  IndexSet HAS_LOWER_BOUND LowerBound
  IndexSet HAS_UPPER_BOUND UpperBound
  IndexSet DETERMINES TermCount
  SummationOperator AGGREGATES Term
  FiniteSeries EVALUATES_TO SumValue
  PrefixSum IS_A FiniteSeries
  PartialSum IS_A FiniteSeries
  FiniteSeries MAY_BE ArithmeticFiniteSeries
  FiniteSeries MAY_BE GeometricFiniteSeries
  FiniteSeries MAY_BE TelescopingFiniteSeries
  FiniteSeries MAY_BE WeightedFiniteSeries
  FiniteSeries MAY_BE AlternatingFiniteSeries
  FiniteSeries RELATED_TO SequenceOntology
  FiniteSeries RELATED_TO SeriesOntology
  FiniteSeries RELATED_TO InfiniteSeriesOntology
  FiniteSeries RELATED_TO SummationOntology

04. INPUT SCHEMA
  terms: finite indexed mapping k -> a_k
  lower_bound: integer m
  upper_bound: integer n
  ambient_space: X
  addition: X × X -> X
  additive_identity: 0_X
  optional_scalar_multiplication: field/ring action
  optional_formula: symbolic a_k
  optional_assumptions: bounds, domains, commutativity, characteristic

05. VALIDATION
  V1: Check bounds and index convention.
  V2: Confirm each a_k belongs to X.
  V3: Confirm addition is closed and associative.
  V4: Require identity 0_X for empty sums.
  V5: Require commutativity for arbitrary permutation invariance.
  V6: Require additive inverses before using subtraction or telescoping differences.
  V7: Require multiplicative/division assumptions for closed-form geometric identities.
  V8: Distinguish exact symbolic sums from numerical approximations.
  V9: Check overflow and precision for machine arithmetic.
  V10: Do not infer convergence of an infinite series from existence of finite sums.

06. EXECUTABLE FLOW ALGEBRA
FLOW FiniteSeriesEvaluation:
  INPUT: (a_k), m, n, X
  ASSERT: finite integer index interval; compatible addition in X
  IF m > n AND empty-sum convention enabled: RETURN 0_X
  accumulator := 0_X
  FOR k FROM m TO n:
    accumulator := accumulator + a_k
  RETURN FiniteSum(value=accumulator, terms=max(0,n-m+1), exactness=declared)

FLOW FormulaSelection:
  INPUT: term formula a_k, index interval [m,n]
  IF a_k = c: APPLY constant-sum rule
  ELSE IF a_k = a + (k-m)d: APPLY arithmetic-sum rule
  ELSE IF a_k = a*r^(k-m): APPLY geometric-sum rule
  ELSE IF a_k = b_k - b_(k+1): APPLY telescoping rule
  ELSE: APPLY direct finite accumulation or symbolic transformations
  VERIFY: result by recurrence or independent expansion when feasible
  OUTPUT: -expression, assumptions, proof trace

FLOW FiniteToInfiniteBoundary:
  INPUT: family S_N = Σ_{k=m}^{N} a_k
  CLASSIFY: each S_N is a finite series
  IF request concerns lim_(N->infinity) S_N:
    DELEGATE: InfiniteSeriesConvergenceOntology
  ELSE:
    RETURN: finite sum only

07. FORMAL AXIOMS AND RULES
  R01 SINGLETON: Σ_{k=m}^{m} a_k = a_m.
  R02 EMPTY: Σ_{k∈∅} a_k = 0_X.
  R03 RECURSION: S(m,n) = S(m,n-1) + a_n, n>m.
  R04 SPLIT: Σ_{k=m}^{n} a_k = Σ_{k=m}^{p} a_k + Σ_{k=p+1}^{n} a_k for m<=p<n.
  R05 ADDITIVITY: Σ(a_k+b_k) = Σa_k + Σb_k in a commutative additive monoid.
  R06 SCALAR: Σ(c a_k) = c Σa_k when scalar action distributes over addition.
  R07 CONSTANT: Σ_{k=m}^{n} c = (n-m+1)c.
  R08 REINDEX: Σ_{k=m}^{n} a_k = Σ_{j=0}^{n-m} a_(m+j).
  R09 PERMUTATION: finite sums are invariant under permutation if addition is commutative.
  R10 TELESCOPING: Σ_{k=m}^{n}(b_k-b_(k+1)) = b_m-b_(n+1) in an additive group.
  R11 GEOMETRIC: Σ_{j=0}^{N-1} ar^j = a(1-r^N)/(1-r), r≠1, in a commutative field.
  R12 GEOMETRIC_ONE: Σ_{j=0}^{N-1} a*1^j = Na.
  R13 ARITHMETIC: Σ_{j=0}^{N-1}(a+jd) = Na + d*N*(N-1)/2 over characteristic-zero scalars.
  R14 INTEGERS: Σ_{k=1}^{N} k = N(N+1)/2.
  R15 SQUARES: Σ_{k=1}^{N} k² = N(N+1)(2N+1)/6.
  R16 CUBES: Σ_{k=1}^{N} k³ = [N(N+1)/2]².
  R17 TRIANGLE: |Σa_k| <= Σ|a_k| for real/complex scalars.
  R18 BOUNDS: if a_k<=b_k for each k, then Σa_k<=Σb_k over ordered additive structures.
  R19 NONNEGATIVE: if all a_k>=0, then Σa_k>=0.
  R20 ZERO_TERMS: adding/removing zero terms does not change sum.
  R21 PREFIX_DIFFERENCE: a_n = P_n-P_(n-1), n>m, in an additive group.
  R22 DOUBLE_SUM: finite nested sums may swap order in a commutative additive monoid.
  R23 DISJOINT_UNION: sum over finite disjoint index sets equals sum of their sums.
  R24 FINITE_EXISTS: a valid finite sum exists in a closed associative additive structure; this alone says nothing about an infinite limit.

08. CONSTRAINTS AND NON-EQUIVALENCES
  C01 FiniteSeries != InfiniteSeries.
  C02 FiniteSum != LimitOfPartialSums.
  C03 A finite series has a value even when the corresponding infinite series diverges.
  C04 Reordering is not generally valid for noncommutative addition.
  C05 A geometric closed form with division by (1-r) is invalid at r=1.
  C06 Arithmetic formulas with division require appropriate algebraic interpretation.
  C07 Approximate floating-point equality != exact symbolic equality.
  C08 Index bounds and inclusivity are part of semantic identity.
  C09 A partial sum S_N is finite for every finite N, regardless of infinite-series convergence.
  C10 A finite sum is not a sequence unless indexed by a varying parameter.

09. SEMANTIC EQUIVALENCE
  EQ01: Σ_{k=1}^{N} a_k ≡ a_1 + ... + a_N.
  EQ02: Σ_{k=m}^{n} a_k ≡ Σ_{j=0}^{n-m} a_(m+j).
  EQ03: finite summation ≡ finite additive aggregation [when the algebraic context matches].
  EQ04: prefix sum ≡ initial finite segment sum [with matching lower bound].
  NOT_EQ01: finite series ≠ convergent infinite series.
  NOT_EQ02: number of terms ≠ upper index unless lower index is 1.
  NOT_EQ03: numerical approximation ≠ exact sum without error specification.

10. EXAMPLES AND EXPECTED RESULTS
  E01: Σ_{k=1}^{4} k = 1+2+3+4 = 10.
  E02: Σ_{k=0}^{3} 2^k = 1+2+4+8 = 15.
  E03: Σ_{k=1}^{5} 3 = 15.
  E04: Σ_{k=1}^{4} (k²-(k-1)²) = 16.
  E05: Σ_{k=1}^{4} (-1)^k = 0.
  E06: Σ_{k=1}^{0} k = 0 under empty-sum convention.
  E07: Σ_{k=1}^{N} 1/k is finite for every positive integer N; the associated infinite harmonic series diverges.
  E08: Σ_{k=0}^{N-1} ar^k is finite for any finite N and defined scalars a,r, including |r|>=1.
  E09: Σ_{k=2}^{4} k = 9; upper bound 4 does not imply four terms.
  E10: Σ_{k=1}^{3} (2k+1) = 3+5+7 = 15.

11. INFERENCE ENGINE
  I01: GIVEN finite indexed family + compatible addition -> CLASSIFY FiniteSeries.
  I02: GIVEN constant terms -> INFER constant-sum formula.
  I03: GIVEN arithmetic progression -> INFER arithmetic finite sum.
  I04: GIVEN geometric progression -> INFER geometric finite sum with r=1 branch.
  I05: GIVEN consecutive differences -> INFER telescoping endpoints.
  I06: GIVEN finite partition -> INFER additive decomposition.
  I07: GIVEN reindexing bijection + commutative addition -> INFER equivalent sum.
  I08: GIVEN finite prefix family indexed by N -> LINK partial-sum sequence.
  I09: GIVEN limit query as N grows -> REQUIRE convergence analysis, not finite-series rules alone.
  I10: GIVEN approximate terms -> PROPAGATE numerical error metadata.

12. KNOWLEDGE GRAPH TRIPLES
  (FiniteSeries, rdf:type, MathematicalEntity)
  (FiniteSeries, rdfs:subClassOf, Series)
  (FiniteSeries, hasOperation, SummationOperator)
  (FiniteSeries, hasIndexSet, FiniteIndexSet)
  (FiniteSeries, hasTerm, Term)
  (FiniteSeries, evaluatesTo, SumValue)
  (FiniteSeries, hasSpecialisation, ArithmeticFiniteSeries)
  (FiniteSeries, hasSpecialisation, GeometricFiniteSeries)
  (FiniteSeries, hasSpecialisation, TelescopingFiniteSeries)
  (PartialSum, rdfs:subClassOf, FiniteSeries)
  (InfiniteSeries, definedThrough, PartialSumSequence)
  (FiniteSeries, distinctFrom, InfiniteSeries)

13. CONFIDENCE
  MathematicalDefinitions: 1.00 [conditional on declared structures and conventions]
  AlgebraicIdentities: 1.00 [when preconditions hold]
  ExampleCalculations: 1.00 [exact arithmetic]
  SourcePageAlignment: UNVERIFIED
  ConfidencePolicy: Never assign empirical confidence to a mathematical theorem; distinguish proof validity from uncertain source alignment.

14. FLOW_HASH
  HASH_INPUT: canonical entity name + version + source URI + ordered rules + assumptions + examples
  HASH_ALGORITHM: SHA-256
  FLOW_HASH: COMPUTE_ON_SERIALISATION
  NOTE: No digest is asserted without hashing final canonical bytes.

15. UPDATE_MODEL
  VERSION: 4.0
  UPDATE_TRIGGERS: corrected mathematical rule; altered scope; improved notation; source-page reconciliation; new cross-ontology links
  CHANGE_CONTROL: preserve previous versions; record date, author, reason and changed rules
  VALIDATION_ON_UPDATE: re-run all examples, precondition checks, and logical consistency checks
  SOURCE_RECONCILIATION: compare with canonical page when retrievable

16. AI_CONSUMPTION
  ENTITY_RESOLUTION: Map "finite sum", "finite series", and "finite summation" to FiniteSeries only with compatible context.
  CONTEXT_REQUIRED: index bounds; term definition; algebraic structure; empty-sum convention.
  REASONING: Apply rules only if preconditions are satisfied; preserve provenance of each transformation.
  ANSWER_FORMAT: value or symbolic form + index interval + assumptions + proof/derivation + exactness.
  ERROR_HANDLING: Report undefined terms, invalid bounds, missing operations, and division-by-zero cases.
  DISAMBIGUATION: Separate finite-sum evaluation from convergence of an infinite series.
  CROSS_ONTOLOGY: SeriesOntology; SequenceOntology; ArithmeticSequenceOntology; GeometricSequenceOntology; ConvergentSequenceOntology; DivergentSequenceOntology.
  OUTPUT_CONTRACT: Return typed FiniteSum with value, term_count, method, assumptions, confidence_class and provenance.

END FLOW ALGEBRA