LemmaStatement: AI-adaptedProof: 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.

Constraint expander overlay

Statement

For $G$ with $m > 0$ edges, the full preprocessing graph $G_{2}$ has $2 m$ vertices, degree $387$ , and $387 m$ ordinary edges over the same alphabet. It has loops at every vertex and $α (G_{2}) \leq ρ_{2} : = \frac{259 + 128 ρ_{0}}{387} < 1.$ With $K = 20 / 7$ and $c = 1 / (129 K)$ , $\frac{129 c}{387} UNSAT (G) \leq UNSAT (G_{2}) \leq \frac{UNSAT (G)}{387} .$ For every port labeling $τ$ , ${UNSAT}_{τ} (G_{2}) = (129 / 387) {UNSAT}_{τ} (G_{1})$ and ${UNSAT}_{D τ} (G) \leq 387 K {UNSAT}_{τ} (G_{2})$ . Construction and plurality decoding take polynomial time. The edgeless convention has UNSAT zero.

Facts & Assumptions

Given: the objects and hypotheses in the statement above.

[F1]

Let $G$ have $m = ∣ E ∣ > 0$ ordinary edges, with the fixed nonempty alphabet and paired-loop convention. Its cloud graph $G_{1}$ is degree $129$ , has $2 m$ vertices and $129 m$ ordinary edges, and is constructible in polynomial time without changing the alphabet. Put $K = \max (1, 2 / h_{0}) = 20 / 7$ and $c = 1 / (129 K)$ . Then $c UNSAT (G) \leq UNSAT (G_{1}) \leq UNSAT (G) / 129.$ For every labeling $τ$ of $G_{1}$ , plurality decoding $D τ$ satisfies ${UNSAT}_{D τ} (G) \leq 129 K {UNSAT}_{τ} (G_{1})$ . For an edgeless input use the empty output convention and UNSAT zero. (Regularization preserves value quantitatively).

Proof

1.1

The prescribed overlay adds $128$ slots and $130$ loop slots per vertex to the $129$ -regular cloud graph. Thus its normalized matrix is $(129 M_{1} + 128 M_{H} + 130 I) / 387$ , and ordinary edge count is $387 m$ . For unit mean-zero $f$ , its Rayleigh quotient is at most $(129 + 128 ρ_{0} + 130) / 387$ and at least $(- 129 - 128 ρ_{0} + 130) / 387$ . The absolute value of the lower endpoint is no larger than the positive upper endpoint. The finite-dimensional symmetric spectral decomposition therefore gives $α \leq ρ_{2} < 1$ .

F1algebra

2.1

All added relations are tautological, so for the same labeling the number of bad edges is unchanged while the denominator changes from $129 m$ to $387 m$ . This proves the exact assignment-level factor, hence also its equality after minimizing over the unchanged set of labelings. Combine with the cloud bounds to obtain the displayed two inequalities.

F1step 1.1

3.1

Substitute the exact factor into the cloud decoder inequality to get ${UNSAT}_{D τ} (G) \leq 387 K {UNSAT}_{τ} (G_{2})$ . Counting label frequencies in each finite cloud implements fixed plurality tie breaking in polynomial time; isolated original vertices get the first alphabet symbol. The overlay generator and relation copying are polynomial. For no original edges use the stipulated empty graph instead of a positive-degree assertion.

F1step 2.1∎

Depends on

Regularization preserves value quantitatively

Used by

Constraint cloud rounding and loop counts Example

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

Irit Dinur, The PCP theorem by gap amplification; §4 Lemma4.2 and Corollary4.3, pp14–15. (standard reference, not scraped)