Trinomial Ontology

FLOW ALGEBRA
TRINOMIAL ONTOLOGY
Kireth Semantic Library | Mathematical Entities > Expression Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/expression-entities/trinomial-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-13
Status: mathematical reference model; source webpage not independently verified

01. PURPOSE
flow "TrinomialOntology" v1.0
  domain: symbolic_algebra
  entity: Trinomial
  superclass: PolynomialExpression
  objective: classify, normalise, transform and reason over three-term polynomials

02. DEFINITIONS
  Polynomial := finite sum of coefficient * monomial over a declared coefficient ring.
  Term := nonzero coefficient times a monomial after canonical combination.
  Trinomial(P) := Polynomial(P) AND count(nonzero_terms(normalise(P))) = 3.
  Zero coefficients do not contribute terms.
  A displayed three-part sum is not necessarily a trinomial after simplification.

03. TYPED ENTITY SCHEMA
  Trinomial:
    id: IRI
    expression: SymbolicExpression
    coefficient_domain: Ring | Field
    variables: ordered list[Symbol]
    terms: list[Monomial]  # exactly three canonical nonzero terms
    degree: NonNegativeInteger
    representation: -ExpressionTree
    normal_form: CanonicalPolynomial
    provenance: SourceReference?
    confidence: [0,1]

04. CORE INVARIANTS
  I1: count(terms) == 3
  I2: forall t in terms: coefficient(t) != 0
  I3: distinct monomial exponent vectors after normalisation
  I4: all exponents are nonnegative integers
  I5: coefficients belong to declared coefficient domain
  I6: degree == max(total_degree(t) for t in terms)
  I7: term ordering is canonical under a declared monomial order

05. CLASSIFICATION
  UnivariateTrinomial := three terms in one indeterminate.
  MultivariateTrinomial := three terms across multiple indeterminates.
  QuadraticTrinomial := degree 2, three canonical terms.
  HomogeneousTrinomial := all three terms have equal total degree.
  SparsePolynomial := polynomial with relatively few nonzero terms.
  Example: x^2 + 3*x + 2 is a quadratic trinomial over Z.
  Example: x^2 + y^2 + z^2 is a homogeneous multivariate trinomial.
  Nonexample: x + x + y simplifies to 2*x + y (binomial).

06. NORMALISATION FLOW
  input: -expression E, coefficient domain D, monomial order O
  parse E -> AST
  validate operators, symbols, exponent domains
  distribute products where polynomial expansion is intended
  collect terms by exponent vector
  add coefficients exactly in D
  eliminate zero coefficients
  sort terms according to O
  output: canonical polynomial P
  classify as Trinomial iff count(terms(P)) == 3

07. OPERATIONS
  addition(P,Q) -> Polynomial
  subtraction(P,Q) -> Polynomial
  multiplication(P,Q) -> Polynomial
  scalar_multiplication(c,P) -> Polynomial
  differentiation(P,x) -> Polynomial
  substitution(P,x,v) -> Expression or Polynomial, subject to types
  evaluation(P,assignment) -> value in compatible domain
  factorisation(P) -> product expression with equivalence proof
  IMPORTANT: trinomial class is not generally closed under these operations.

08. QUADRATIC TRINOMIAL RULES
  For P(x) = a*x^2 + b*x + c, require a,b,c != 0 for three terms.
  Require a != 0 for degree two.
  discriminant(P) := b^2 - 4*a*c, when coefficients support arithmetic.
  If D is an ordered subfield of R:
    discriminant > 0 -> two distinct real roots
    discriminant = 0 -> one repeated real root
    discriminant < 0 -> no real roots
  Quadratic formula in a field extension containing sqrt(discriminant):
    roots := (-b +/- sqrt(discriminant)) / (2*a), char(D) != 2.
  Example: x^2 + 3*x + 2 = (x+1)*(x+2).

09. FACTORISATION
  factor(P,D) must specify coefficient domain D.
  Reducibility depends on D; irreducible over one domain may factor over another.
  Verify expand(factor(P,D)) == normalise(P) exactly.
  A factorised expression need not syntactically contain three additive terms.
  Semantic classification is based on expanded canonical polynomial.

10. SEMANTIC RELATIONSHIPS
  Trinomial --isA--> PolynomialExpression
  Trinomial --hasPart--> Monomial (cardinality 3 in normal form)
  Monomial --hasCoefficient--> Coefficient
  Monomial --hasExponentVector--> ExponentVector
  Trinomial --usesVariable--> Variable
  Trinomial --hasDegree--> PolynomialDegree
  Trinomial --hasNormalForm--> CanonicalPolynomial
  Trinomial --canFactorInto--> PolynomialFactorisation
  Trinomial --isSemanticallyEquivalentTo--> EquivalentExpression
  Trinomial --contrastsWith--> Binomial | Monomial

11. SEMANTIC EQUIVALENCE
  polynomial_equivalent(P,Q,D) := normalise(P,D) == normalise(Q,D)
  alpha_rename permitted only with explicit symbol mapping.
  syntactic equality != semantic equality.
  Example: x^2 + 2*x + 1 == (x+1)^2 as polynomials.
  Expression tree shape may differ without changing polynomial meaning.
  Avoid floating-point comparison for exact symbolic equivalence.

12. KNOWLEDGE GRAPH REPRESENTATION
  node trinomial:example1:
    rdf:type: Trinomial
    expression: "x^2 + 3*x + 2"
    coefficientDomain: IntegerRing
    variables: [x]
    degree: 2
    termCount: 3
    canonicalTerms: [x^2, 3*x, 2]
    factorsAs: "(x+1)*(x+2)"
  graph edges are typed, directed and provenance-aware.

13. INFERENCE RULES
  R1: Polynomial(P) AND nonzeroTermCount(P)=3 -> Trinomial(P).
  R2: Trinomial(P) -> Polynomial(P).
  R3: Trinomial(P) AND degree(P)=2 -> QuadraticTrinomial(P).
  R4: normalise(P)=normalise(Q) -> PolynomialEquivalent(P,Q).
  R5: PolynomialEquivalent(P,Q) AND Trinomial(P) -> Trinomial(Q) semantically,
      provided classification uses canonical polynomial form.
  R6: factorisation claimed -> require exact expansion verification.
  R7: term count changes under normalisation -> reclassify expression.

14. VALIDATION
  V1: parse expression without ambiguity.
  V2: coefficient ring/field specified.
  V3: exponents meet polynomial requirements.
  V4: canonical term count equals three.
  V5: zero coefficients removed.
  V6: no duplicate exponent vectors.
  V7: degree computed consistently.
  V8: claimed identities checked by symbolic expansion.
  V9: real-root claims require ordered real coefficient context.
  V10: transformations preserve semantics within declared domain.

15. EXAMPLES AND EDGE CASES
  PASS: 2*x^3 - x + 7 -> three terms, degree 3.
  PASS: x*y + 2*x + 3 -> three terms, degree 2.
  FAIL: x^2 + x - x -> x^2, one term.
  FAIL: x^2 + 0*x + 1 -> two nonzero terms.
  FAIL: x^(-1) + x + 1 -> Laurent expression, not polynomial over usual ring.
  FAIL: sin(x) + x + 1 -> not a polynomial in x.
  CONTEXT: 1 + x + x^2 over finite fields is still a three-term polynomial,
           unless characteristic or canonical coefficient reduction removes a term.

16. CONFIDENCE MODEL
  mathematical_rule_confidence: 1.0 when assumptions satisfied and verified
  symbolic_parse_confidence: dependent on parser and ambiguity
  source_alignment_confidence: unverified until source content is checked
  proof_status: proved | verified_symbolically | assumed | unresolved
  Never treat confidence as a substitute for formal validation.

17. FLOW HASH
  canonical_payload := serialize(version, domain, AST_normal_form,
                                  coefficient_domain, variables, monomial_order)
  FLOW_HASH := SHA256(UTF8(canonical_payload))
  Hash identifies a versioned canonical representation, not mathematical truth.

18. UPDATE MODEL
  update triggers: new ontology version | revised axioms | corrected examples
                   | changed coefficient domain | parser or normalisation revision
  on_update:
    revalidate invariants
    recompute canonical forms and hashes
    update provenance and timestamps
    preserve version history
    flag dependent graph nodes for review

19. AI CONSUMPTION
  input: natural language | LaTeX | symbolic expression | graph node
  processing: parse -> typecheck -> normalise -> classify -> infer -> validate
  output: typed trinomial node, canonical form, relations, evidence, confidence
  require explicit coefficient domain for domain-dependent conclusions
  prohibit unsupported factorisations or root claims
  return error state for invalid or ambiguous input

20. PROVENANCE AND AUTHORSHIP
  library: Kireth Semantic Library
  source_url: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/expression-entities/trinomial-ontology.html
  authored_by: Mark Higgins — Sky (Recursive AI)
  site: https://kirethresearch.org
  source_page_verified: false
  note: This independently authored formalisation follows established algebra.

END FLOW ALGEBRA