ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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.

Three repetitions of a three-quarters-sound PCP

Example

Let $V$ be a nonadaptive PCP verifier with proof length at most $L (n)$ , randomness bound $r (n)$ , query bound $q (n)$ , and soundness at most $3 / 4$ . Run it three times using independent random tapes, the same fixed proof in all runs, and accept only if all three runs accept. The resulting verifier has soundness at most ${(\frac{3}{4})}^{3} = \frac{27}{64},$ uses at most $3 r (n)$ random bits and $3 q (n)$ symbol queries, and keeps the proof length bound $L (n)$ .

Facts & Assumptions

Given: A fixed nonadaptive verifier $V$ with the stated resource bounds and soundness at most $3 / 4$ .

[F1]

Independent repetition uses the same fixed proof, has acceptance probability $p^{k}$ for each fixed input and proof, multiplies randomness and query bounds by $k$ , and leaves the proof length unchanged. (PCP soundness amplification by independent repetition)

Verification

technique · direct

1.1F1givenalgebra

Fix a no input $x$ and any proof $π$ , and let $p = \Pr [V^{π} (x) accepts]$ . By the soundness premise, $0 \leq p \leq 3 / 4$ . Applying [F1] with $k = 3$ gives repeated acceptance probability $p^{3} \leq (3 / 4)^{3} = 27 / 64$ . Since this holds for every fixed proof, the repeated verifier has the claimed soundness.

2.1F1givenconstruct∎

The three independent runs use at most $3 r (n)$ random bits and concatenate at most three query lists of size $q (n)$ , so the total is at most $3 q (n)$ symbol queries. They all inspect the same proof of length at most $L (n)$ , rather than storing three proofs; the combined query locations are fixed by the input and the full random tape, so the repeated verifier remains nonadaptive.

Depends on

PCP soundness amplification by independent repetition

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Arora and Barak, Computational Complexity: A Modern Approach, §18.1 Note 3 to Theorem 18.2, printed p. 354 (standard reference, not scraped)