False statementConstruction: Literature-sourcedVerification: 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.

False: a PCP proof is itself a random string

Statement

False claim: in the PCP theorem the proof string must be sampled at random separately on each verifier execution.

Facts & Assumptions

[F1]

For each fixed input and fixed proof, acceptance probability is over the verifier's coins, with the same proof used on every coin string. (PCP verifier resources and deterministic proof strings)

[F2]

A yes input has one fixed proof achieving completeness, while soundness quantifies over every fixed proof for each no input. (PCP classes with completeness and soundness)

Refutation

Given: Let $K = {ε} \subseteq {0, 1}^{*}$ , where $ε$ is the empty input string. Use proof alphabet $Γ = {0, 1}$ and proof length $L (n) = 1$ .

1.1F1F2givenconstruct

Define $V$ to toss one unbiased coin bit, query the sole proof symbol $π_{1}$ , and accept exactly when $x = ε$ and $π_{1} = 1$ ; it ignores the coin. This is a uniform polynomial-time nonadaptive verifier with $r (n) = q (n) = L (n) = 1$ . For the yes input $ε$ , the fixed proof $π = 1$ is accepted on both coin outcomes. For every no input $x \neq ε$ , every fixed proof is rejected on both outcomes. Hence $K \in PCP (1, 1; 1, 0)$ by [F2].

2.1F1step 1.1algebradischarge-construct∎

In this explicit PCP, $π = 1$ is the same deterministic one-bit string for both verifier coin outcomes; only the verifier tosses a random bit, and that bit is unused. Thus the proof need not be resampled on each execution, contradicting the claim.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Irit Dinur, The PCP Theorem by Gap Amplification, §1.1, printed pp. 1–2 (standard reference, not scraped)
Arora and Barak, Computational Complexity: A Modern Approach, §18.1, Definition 18.1, printed pp. 353–354 (standard reference, not scraped)