NP-hardness of approximating Uniform Sparsest Cut within every constant factor

Theorem 1.1: for every fixed $C>1$ there is a polynomial-time reduction from 3SAT producing uniform-demand instances with a multiplicative gap $C$ , so approximating Uniform Sparsest Cut within any constant factor is NP-hard. The constants and exponents of the reduction depend on $C$ . It does not give a superconstant (for instance $\sqrt{\log N}$ -type) hardness factor, does not address Balanced Separator directly, and says nothing about the Small-Set Expansion hypothesis itself.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: NP-hardness of approximating Uniform Sparsest Cut within every constant factor · Uniform Sparsest Cut constant-factor hardness

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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Unreleased internal OpenAI model
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VibeMathed record: NP-hardness of approximating Uniform Sparsest Cut within every constant factor

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the question. For every fixed $C>1$ it gives a polynomial-time reduction from 3CNF satisfiability to graphs with nonnegative rational capacities and unit demand on every pair, with $\Phi(G)\le a$ for satisfiable and $\Phi(G)>Ca$ for unsatisfiable formulas, using no unproved complexity hypothesis. That answers the question as posed (every constant factor, exact uniform demands, deterministic Karp reduction). The PCP-style proof (Garcia-Stichtenoth tower tests, rare-event extraction, cut-variance analysis) was not refereed and is not formalised; the Lean formalization in this family covers only the integrality-gap companion.

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Companion: Constant-factor hardness of uniform sparsest cut

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The Goemans-Linial integrality gap for uniform sparsest cut is nearly square-root logarithmic

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The Goemans-Linial integrality gap for uniform sparsest cut is nearly square-root logarithmic evidence for this event

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Companion: Constant-factor hardness of uniform sparsest cut evidence for this event

VibeMathed record: NP-hardness of approximating Uniform Sparsest Cut within every constant factor evidence for this event

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