CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Choosing a sum-check error budget

Statement

Let $ε > 0$ . For a supplied sum-check instance with a field satisfying $∣ F ∣ \geq (\sum_{i} d_{i}) / ε$ , every prover's probability of acceptance on a false initial claim is at most $ε$ . This assumes the stated field and trusted evaluation are already supplied; it does not construct a field.

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

False initial claims have acceptance probability at most min(1,sum_i d_i/|F|) for every adaptive prover (Adaptive-prover soundness of sum-check).

Proof

1.1F1given

The soundness theorem bounds the false-claim acceptance probability by $\min (1, \sum_{i} d_{i} / ∣ F ∣)$ .

2.1step 1.1givenalgebra∎

Since $∣ F ∣ > 0$ and $ε > 0$ , the size hypothesis implies $\sum_{i} d_{i} / ∣ F ∣ \leq ε$ . Combining gives the result, including $\sum_{i} d_{i} = 0$ (zero error) and $ε \geq 1$ (a possibly trivial guarantee). Equality in the field-size inequality is allowed.

Depends on

Adaptive-prover soundness of sum-check

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Justin Thaler, Proofs, Arguments, and Zero-Knowledge (2023), Proposition 4.1 and degree/field-size discussion pp.35–38 (standard reference, not scraped)