False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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False: graph powering alone keeps the alphabet fixed

Statement

False claim: the local-view graph-powering step used for gap amplification keeps the input alphabet unchanged for every graph and every positive powering parameter.

Facts & Assumptions

[F1]

In the published powering convention, the view alphabet has cardinality $∣ Σ_{t} ∣ = ∣ Σ ∣^{(2 d)^{R}}$ , where $R = t + ⌈ \sqrt{t} ⌉$ . (Constraint graph powering with local-view labels)

[F2]

A loop contributes two incidence slots and is tested on the diagonal pair $(a, a)$ . (Constraint graph and labeling value)

Refutation

Given: A binary constraint graph has finite nonempty alphabet and each of its loop relations is tested on its single vertex label.

1.1F2given

Take one vertex $v$ , alphabet $Σ = {0, 1}$ , and two loop edges $e_{0}, e_{1}$ with relations ${(0, 0)}$ and ${(1, 1)}$ . Label $0$ passes $e_{0}$ and fails $e_{1}$ ; label $1$ passes $e_{1}$ and fails $e_{0}$ . These are all labels, so every labeling violates exactly one of the two edges and $UNSAT (G) = 1 / 2$ . Each loop contributes two incidence slots, hence $d = 4$ .

2.1F1step 1.1algebra∎

Set the positive powering parameter to $t = 1$ . Then $R = 1 + ⌈ \sqrt{1} ⌉ = 2$ and $2 d = 8$ , so there are $8^{2} = 64$ length-two patterns. By [F1], the full view alphabet has size $∣ Σ_{1} ∣ = 2^{64} \neq 2$ . This one graph and positive parameter refute the universal fixed-alphabet claim; the example does not assert that every powered instance has a larger alphabet.

Depends on

Used by

A powered graph whose alphabet grows Counterexample

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 (standard reference, not scraped)
Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)