TheoremStatement: 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.

Adaptive-prover soundness of sum-check

Statement

Fix a sum-check instance over a finite field $F$ , with $n \geq 1$ , trusted individual bounds $d_{1}, \dots, d_{n}$ , trusted evaluation, and false initial claim $H \neq \sum_{b} g (b)$ . For every adaptive prover strategy the acceptance probability is at most $\min (1, \frac{\sum_{i = 1}^{n} d_{i}}{∣ F ∣}) .$ In particular it is at most $n d / ∣ F ∣$ if all $d_{i} \leq d$ . A randomized prover is allowed; its coins are independent of future verifier challenges.

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

At each reached false-claim prefix, invalid messages reject and the probability that the next scalar becomes true is at most min(1,d_i/|F|), also for randomized messages (A false sum-check claim rarely becomes true).

Proof

1.1

Let $B_{i}$ be the event that the protocol reaches round $i$ with all earlier scalar claims false, passes that round’s message checks, and its updated scalar claim becomes true. For each positive-probability prefix with those earlier claims false, the one-round lemma bounds the conditional probability of this transition by $d_{i} / ∣ F ∣$ . Averaging over prefixes (and messages) gives $\Pr (B_{i}) \leq d_{i} / ∣ F ∣$ .

F1given

2.1

Acceptance forces the final scalar claim to equal the trusted value $g (r)$ , hence to be true. The initial claim is false. There is therefore a first transition from false to true, so acceptance is contained in $⋃_{i = 1}^{n} B_{i}$ . Pointwise, the indicator of this union is at most the sum of its event indicators; averaging gives acceptance probability at most $\sum_{i} d_{i} / ∣ F ∣$ . This requires no independence between the $B_{i}$ .

step 1.1givenalgebra

3.1

Probabilities are also at most one, giving the stated cap, and $\sum_{i} d_{i} \leq n d$ gives the uniform-degree bound. The reasoning works for $n = 1$ and for zero degree bounds (all-zero bounds force zero acceptance of a false claim). Randomized adaptive strategies were already covered by conditional averaging.

step 1.1step 2.1algebra∎

Depends on

A false sum-check claim rarely becomes true

Used by

Choosing a sum-check error budget Corollary
One point evaluation for an exponentially large cube Counterexample
A complete sum-check transcript over F₁01 Example
Verifying formula counts over a supplied field Proposition

Dependency tree · two levels

3 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 non-inductive proof, pp.36–37 (standard reference, not scraped)