LemmaStatement: Literature-sourcedProof: 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.

A violated decoded edge is far from edge-circuit acceptance

Statement

Use the code $C : Σ \to {0, 1}^{ℓ}$ and edge circuit $E_{R_{e}}$ of Shared codeword blocks and edge acceptance circuits, where $Σ$ is finite, ordered, $∣ Σ ∣ = W \geq 2$ , and distinct codewords have relative distance $δ = 1 / 2$ . For any physical block $B_{v} \in {0, 1}^{ℓ}$ at each graph vertex, decode $B_{v}$ to a closest codeword, breaking ties by the fixed alphabet order, and write the decoded label as $a_{v}$ . Regard the $2 ℓ$ formal input bits of $E_{R_{e}}$ as all named inputs, so its accepting set is $SAT (E_{R_{e}}) \subseteq {0, 1}^{2 ℓ}$ . Measure relative Hamming distance on these $2 ℓ$ bits and use distance $1$ when the accepting set is empty, as in Assignment tester and rejection ratio.

If edge $e = (v, w)$ is violated by the decoded labels, then ${dist}_{r e l} ((B_{v}, B_{w}), SAT (E_{R_{e}})) \geq \frac{δ}{4} = \frac{1}{8} .$ For a loop $v = w$ , the displayed input is $(B_{v}, B_{v})$ and the same bound holds.

Facts & Assumptions

Given: A finite ordered alphabet with $W \geq 2$ , its Walsh–Hadamard code blocks, an edge circuit, and arbitrary physical blocks at its vertices.

[F1]

The selected codewords are injective and every two distinct codewords have relative distance $δ = 1 / 2$ . (Shared codeword blocks and edge acceptance circuits)

[F2]

The edge circuit accepts exactly pairs of valid codewords whose decoded labels lie in the ordered edge relation $R_{e}$ . (Shared codeword blocks and edge acceptance circuits)

[F3]

Distance from a named input to a circuit's accepting inputs is relative Hamming distance, with value $1$ when the accepting set is empty. (Assignment tester and rejection ratio)

Proof

1.1F1givenconstructalgebra

For each block $B_{v}$ , the fixed alphabet order makes its nearest valid codeword label $a_{v}$ deterministic; a minimizer exists because $Σ$ is finite and nonempty. For every $a^{'} \neq a_{v}$ , nearestness and the triangle inequality give $δ ℓ \leq d_{H} (C (a_{v}), C (a^{'})) \leq d_{H} (C (a_{v}), B_{v}) + d_{H} (B_{v}, C (a^{'})) \leq 2 d_{H} (B_{v}, C (a^{'}))$ . Hence every changed decoded label has $d_{H} (B_{v}, C (a^{'})) \geq δ ℓ / 2 = ℓ / 4$ .

2.1F2F3step 1.1constructalgebradischarge-construct∎

Suppose $(a_{v}, a_{w}) \notin R_{e}$ . By [F2], every accepting formal input $(X, Y)$ is $(C (a^{'}), C (b^{'}))$ for some $(a^{'}, b^{'}) \in R_{e}$ , so at least one decoded endpoint changes. By step 1.1, the corresponding formal block differs from the actual block by at least $ℓ / 4$ bits; division by the $2 ℓ$ formal input bits gives relative distance at least $δ / 4 = 1 / 8$ . If $v = w$ , the actual formal pair is $(B_{v}, B_{v})$ and the violated loop pair is $(a_{v}, a_{v}) \notin R_{e}$ , so every accepting pair still changes at least one of the two decoded labels and the same bound applies to that formal block. If there are no accepting inputs, [F3] sets the distance to $1 \geq 1 / 8$ .

Depends on

Used by

Composition transfers a constant fraction of unsatisfaction Lemma

Dependency tree · two levels

8 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, §5 proof of Lemma 1.8, printed pp. 18–19 (standard reference, not scraped)