Coefficient Ontology

FLOW ALGEBRA
COEFFICIENT ONTOLOGY
Kireth Semantic Library | Mathematical Entities > Symbolic Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/symbolic-entities/coefficient-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | 2026-10-13
Status: Mathematics-grounded draft; source page not independently verified.

01. PURPOSE
flow "CoefficientOntology" v1.0
  domain: symbolic_mathematics
  entity_type: Coefficient
  objective: represent, identify, normalise, and reason about coefficients in typed mathematical expressions.

02. CORE DEFINITION
  Coefficient(c, t, E) := c is a scalar or ring element multiplying term basis t in expression E.
  PolynomialCoefficient(a_k, x^k, P) := P(x) = sum(k=0..n, a_k*x^k).
  Context is mandatory: a coefficient is relative to a chosen basis, variable, domain and representation.

03. TYPED ENTITIES
  Coefficient: MathematicalSymbolicEntity
  NumericCoefficient: Coefficient
  SymbolicCoefficient: Coefficient
  PolynomialCoefficient: Coefficient
  LinearCoefficient: Coefficient
  LeadingCoefficient: PolynomialCoefficient
  ConstantCoefficient: PolynomialCoefficient
  MatrixEntryCoefficient: Coefficient
  FourierCoefficient: Coefficient
  RegressionCoefficient: Coefficient
  Term: -ExpressionComponent
  BasisElement: MathematicalEntity
  Expression: MathematicalEntity
  Variable: SymbolicEntity
  Parameter: SymbolicEntity
  ScalarDomain: AlgebraicStructure

04. ENTITY ATTRIBUTES
  id: URI
  value: ExactNumber | SymbolicExpression | ApproximateNumber
  domain: Ring | Field | ModuleScalarRing
  basis_element: -Expression
  variable_order: ordered list of Variable
  exponent_vector: integer vector where polynomial context applies
  position: index | multi_index
  role: leading | constant | general | linear | model_specific
  exactness: exact | approximate | symbolic
  units: optional UnitExpression
  context: -ExpressionContext
  provenance: SourceReference
  confidence: number in [0,1]

05. RELATIONSHIPS
  Coefficient --multiplies--> BasisElement
  Coefficient --belongsTo--> Expression
  Coefficient --hasValue--> NumericOrSymbolicValue
  Coefficient --definedOver--> ScalarDomain
  Coefficient --indexedBy--> Index
  Coefficient --dependsOn--> Parameter [optional]
  Coefficient --hasUnit--> UnitExpression [optional]
  LeadingCoefficient --isA--> PolynomialCoefficient
  ConstantCoefficient --isA--> PolynomialCoefficient
  Coefficient --derivedFrom--> SourceExpression

06. POLYNOMIAL EXTRACTION
  input: P(x) = 4*x^3 - 2*x + 7
  coefficients: a_3=4; a_2=0; a_1=-2; a_0=7
  degree(P): 3
  leading_coefficient(P): 4
  constant_coefficient(P): 7
  rule: absent powers in expanded canonical form have coefficient 0.
  rule: zero polynomial has no uniquely defined degree or leading coefficient under standard conventions.

07. MULTIVARIATE COEFFICIENTS
  P(x,y) = 3*x^2*y + 5*y - 1
  coefficient(P, monomial=x^2*y): 3
  coefficient(P, monomial=y): 5
  coefficient(P, monomial=1): -1
  rule: use exponent vectors and declared variable ordering.

08. CONTEXT SENSITIVITY
  E = (a+1)*x + b
  coefficient(E, x): a+1 when a,b independent of x.
  coefficient(E, a): x when x,b independent of a.
  guard: coefficient extraction requires a specified variable and algebraic independence assumptions.
  guard: basis change can change coefficient values.

09. COEFFICIENT ARITHMETIC
  coefficient(P+Q, x^k) = coefficient(P,x^k)+coefficient(Q,x^k)
  coefficient(c*P, x^k) = c*coefficient(P,x^k)
  coefficient(P*Q, x^k) = sum(i+j=k, coefficient(P,x^i)*coefficient(Q,x^j))
  precondition: P,Q are polynomials over a compatible commutative coefficient ring.

10. LINEAR ALGEBRA
  v = sum(i, c_i*e_i)
  coefficient(v,e_i,B) = c_i for a specified basis B.
  coordinate_vector(v,B) = (c_1,...,c_n).
  guard: coordinate coefficients are basis-dependent.

11. SERIES AND TRANSFORMS
  formal_power_series: F(x)=sum(n>=0,a_n*x^n)
  Fourier_series: f(x) ~ sum(n in Z,c_n*exp(i*n*x)) under applicable conditions.
  coefficient semantics require basis, normalisation convention and domain.

12. STATISTICAL MODELS
  linear_model: y = beta_0 + sum(j=1..p,beta_j*x_j) + epsilon
  beta_j: regression coefficient, conditional on the model specification.
  guard: estimated coefficients are not automatically causal effects.
  guard: coefficient units depend on outcome and predictor units.

13. NORMALISATION
  parse expression into typed abstract syntax tree.
  identify scalar domain and variable/basis context.
  expand or collect terms using algebraically valid transformations.
  combine like terms by exponent vector or basis identifier.
  retain exact symbolic forms when possible.
  preserve assumptions, units and source provenance.

14. SEMANTIC EQUIVALENCE
  2*x + 3*x == 5*x in ordinary commutative arithmetic.
  coefficient(2*x+3*x,x) == coefficient(5*x,x) == 5.
  distinguish structural equality from equality after normalisation.
  avoid treating numeric approximations as exact equivalences.

15. INFERENCE RULES
  IF polynomial(P,x) AND canonical(P) THEN infer coefficient_map(P,x).
  IF coefficient(P,x^k)=c AND c!=0 AND all higher coefficients=0 THEN infer degree(P)=k.
  IF coefficient(P,x^0)=c THEN infer constant_term(P)=c.
  IF P=Q as polynomials over a common ring THEN infer equal corresponding coefficients.
  IF basis changes THEN recompute coordinates; do not reuse old coefficients.

16. VALIDATION
  require valid expression syntax and typed coefficient domain.
  require explicit target variable or basis.
  verify coefficient reconstruction reproduces normalised expression.
  reject conflicting variable orders and incompatible scalar domains.
  check dimensional consistency when quantities have units.
  flag undefined or ambiguous leading coefficient.

17. CONFIDENCE
  confidence := 1.0 for formally verified exact symbolic extraction under stated assumptions.
  confidence := evidence-based score for heuristic parsing or uncertain source interpretation.
  confidence must not substitute for mathematical proof.

18. KNOWLEDGE GRAPH
  node: CoefficientEntity
  edges: multiplies, belongsTo, definedOver, indexedBy, derivedFrom, semanticallyEquivalentTo
  identifiers: stable URI scoped to expression, basis and index.
  avoid global identity for coefficients that happen to share a numeric value.

19. FLOW HASH
  canonical_payload := canonicalise(ontology_version, expression, scalar_domain, basis, variable_order, coefficient_map, assumptions)
  FLOW_HASH := SHA256(UTF8(canonical_payload))
  rule: deterministic serialization and explicit hash algorithm version required.

20. UPDATE MODEL
  trigger: source_change | ontology_revision | basis_change | expression_change | corrected_assumption
  action: reparse -> renormalise -> validate -> recompute graph edges -> regenerate hash.
  preserve prior versions and provenance.

21. AI CONSUMPTION
  output: typed coefficient records + algebraic context + assumptions + evidence.
  support: symbolic manipulation, equation solving, polynomial analysis, modelling and semantic retrieval.
  forbid: treating coefficient as context-free, equating correlation with causation, inventing exactness.

22. EXAMPLE RECORD
  entity_id: coefficient:P:x:2
  expression: P(x)=3*x^2-4*x+9
  basis_element: x^2
  value: 3
  scalar_domain: integers
  role: leading
  exactness: exact
  verification: coefficient reconstruction successful

23. LIMITATIONS
  Noncommutative rings, differential operators, tensors and specialised coefficient conventions require dedicated extension rules.
  This flow is a declarative reasoning specification, not executable source code without an implementation engine.

END FLOW ALGEBRA