ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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.

Schwartz zippel for a bivariate polynomial

Example

For the nonzero formal polynomial $f (X, Y) = X Y - 1$ over $F_{5}$ , total degree is two. Of the 25 independent uniform pairs, precisely $(1, 1), (2, 3), (3, 2), (4, 4)$ are roots. The actual zero probability is $4 / 25$ , strictly below the Schwartz–Zippel bound $2 / 5$ .

Facts & Assumptions

Given: Independent uniform X,Y in the five-element residue field.

[F1]

A nonzero total-degree-d polynomial on a nonempty uniform product set A has zero probability at most $\min (1, d / ∣ A ∣)$ (Schwartz zippel over finite fields).

[F2]

Residues modulo a prime form a field (For every prime $p$ , the two operations on $Z / p$ make it a field).

Verification

1.1

Five has no positive proper divisors other than one: 2,3,4 leave remainders 1,2,1. Thus F2 supplies $F_{5}$ , with distinct residues 0 through 4. At X=0, $X Y - 1 = - 1 = 4 \neq 0$ . At each nonzero X, the unique Y solving $X Y = 1$ is its inverse: $1 \cdot 1 = 1$ , $2 \cdot 3 = 6 = 1$ , $3 \cdot 2 = 6 = 1$ , $4 \cdot 4 = 16 = 1$ modulo 5. Cancellation rules out any additional Y for the same X. These are all five possibilities for X, so exactly the four listed pairs are roots.

F2given

2.1

Each pair has mass $1 / 25$ , so the count gives probability $4 / 25$ . The formal coefficient of XY is one and its total degree is two; the constant term is nonzero but has degree zero, so it does not change total degree. F1 with m=2, d=2 and A equal to the field gives $\min (1, 2 / 5) = 2 / 5 = 10 / 25$ . Consequently $4 / 25 < 10 / 25$ . The comparison is an upper bound, not a claimed equality; independence is used to give every pair the same mass.

step 1.1F1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Local example of Arora–Barak Appendix A Lemma A.25 (standard reference, not scraped)