CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Nonconstructive expanders suffice for uniform reductions

Statement refuted

There is a family of degree- $256$ expanders on every positive size with a uniform absolute gap but without any polynomial-time uniform generator, refuting Nonconstructive expanders suffice for uniform reductions.

Facts & Assumptions

Given: the objects and hypotheses in the statement above.

[F2]

For every integer $N \geq 1$ there is a polynomial-time constructible reverse-paired $128$ -regular multigraph $H_{N}$ on exactly $N$ vertices with $α (H_{N}) \leq ρ_{0} : = 1 - \frac{49}{1638400} < 1.$ For $N \geq 2$ every $S$ satisfies $cut (S) \geq (7 / 10) \min (∣ S ∣, N - ∣ S ∣)$ . Every vertex has loops. (Expander size adjustment and laziness).

Counterexample

1.1

From $H_{N}$ form $C_{N}^{0} = A (H_{N}) + 128 I$ and $C_{N}^{1} = A (H_{N}) + 128 P_{N}$ , where $P_{N}$ swaps vertices one and two for $N \geq 2$ . Their entry $(1, 2)$ differs by $128$ , and both normalized mean-zero norms are at most $(1 + ρ_{0}) / 2 < 1$ . A centered indicator then bounds every normalized cut ratio below by $(1 - ρ_{0}) / 4$ on sets of size at most half. At size one take only loops.

F2algebra

2.1

Assign the $i$ -th polynomially clocked candidate generator size $i + 2$ . Choose the second matrix if its parsed output equals the first, and choose the first otherwise. The resulting family meets the same gap bound at all sizes but differs from every candidate on its assigned input. This is a counterexample to automatic constructibility of an arbitrary chosen family; the explicitly constructible family $H_{N}$ continues to exist.

step 1.1algebra∎

Depends on

Used by

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Dependency tree · two levels

6 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

Hoory–Linial–Wigderson, Expander Graphs and Their Applications, May 2006 draft; Definition2.3, p19; diagonal counterexample to constructibility inference, with spectral verification. (standard reference, not scraped)