The precise statements and locators cited in this P versus NP research record.
Author full version dated March 2014; an extended abstract appeared at CCC 2012. The paper explicitly uses a bounded-error randomized formulation of SETH.
The exact passages cited in OpenMath. Each passage identifies how its wording was recorded.
If the stated bounded-error randomized SETH holds, then P ≠ NP.
Passage ID: pnp-entry-seth-implies-separation:v1
Source item: seth-implies-separation
Source record: pnp-seth@arXiv-1112.2275v3
For every fixed α in (0,1), there is a fixed clause width k such that k-CNF-SAT on n variables has no bounded-error randomized O(2^{αn})-time algorithm.
Passage ID: pnp-entry-seth:v1
Source item: seth
Source record: pnp-seth@arXiv-1112.2275v3
Explicit citations in versioned problem publications. A citation records a use; it does not confer proof or independent verification.
Problem publication v14
Source assessment
Proved
Established implication or scoped theorem in cited mathematics
Only a sufficient stronger premise; P ≠ NP gives no such exponent or randomized lower bound.
Observed Oct 10, 2026
Connection
The cited randomized SETH excludes even fixed-base exponential improvements for some clause width, and therefore excludes polynomial deterministic algorithms.
Source assessment
Open
Unresolved assertion in the cited scope
A fine-grained hardness premise stronger than the deterministic version; neither version follows from P ≠ NP.
Observed Oct 10, 2026
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Manuscript / exposition · external
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Resource revision: arXiv-1112.2275v3
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292ff4c93c0834d4eb560ad0f30b6b703a972901682d5af65b65afcd4c4b0ff1Recorded authorship
Marek Cygan · Holger Dell · Daniel Lokshtanov · Dániel Marx · Jesper Nederlof · Yoshio Okamoto · Ramamohan Paturi · Saket Saurabh · Magnus Wahlström
Marek Cygan; Holger Dell; Daniel Lokshtanov; Dániel Marx; Jesper Nederlof; Yoshio Okamoto; Ramamohan Paturi; Saket Saurabh; Magnus Wahlström
AI involvement is not reported by this cited source; this is not a claim of its absence.
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Cygan et al. · On problems as hard as CNF-SAT
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