Alphabeta Math
LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The affine edge interpolant is the unique degree-at-most-one cardinal polynomial

Statement

Let F be a field, let a,b∈F with a≠b, and let y0,y1∈F. The unique polynomial P∈F[X] of degree at most one with P(a)=y0 and P(b)=y1 is P(X)=y0+(y1−y0)X−ab−a.

Facts & Assumptions

Given: A field F, distinct nodes a,b∈F, and data y0,y1∈F.

[A1]

Addition, subtraction, multiplication, and division by a nonzero field element satisfy the field axioms.

[A2]

a≠b, so b−a is nonzero.

Proof

1.1givenA1A2algebra

The displayed expression has degree at most one. Substitution gives P(a)=y0 and P(b)=y0+(y1−y0)=y1.

2.1givenA1A2step 1.1algebra∎

Write any degree-at-most-one interpolant as Q(X)=α+βX. Its two value equations imply β(b−a)=y1−y0, so β=(y1−y0)/(b−a); then α=y0−βa. These coefficients are uniquely determined and give exactly the polynomial in step 1.1.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources