The precise statements and locators cited in this P versus NP research record.
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For every fixed ν, all-pairs shortest paths on directed n-vertex graphs with integer edge weights of absolute value at most n^ν and no negative cycles can be computed deterministically in O(n^{2.9995}) time.
Passage ID: pnp-entry-apsp-2026:v1
Source item: apsp-2026
Source record: pnp-claude@arXiv-2610.06783v1
For every fixed ν, 3SUM on n integers with absolute value at most n^ν has a deterministic O(n^{1.9992})-time algorithm in the paper’s integer-operation model.
Passage ID: pnp-entry-three-sum-2026:v1
Source item: three-sum-2026
Source record: pnp-claude@arXiv-2610.06783v1
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
A fine-grained algorithmic improvement within polynomial time; real-weight and randomized variants have separate guarantees.
Observed Oct 10, 2026
Source assessment
Proved
Established implication or scoped theorem in cited mathematics
A fine-grained improvement for a problem already solvable in polynomial time; no SAT speedup or P versus NP resolution is claimed.
Observed Oct 10, 2026
History
AI-assisted algorithms improve 3SUM and APSP
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Manuscript / exposition · external
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Resource revision: arXiv-2610.06783v1
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d91a79c5dbf1fc662f0c5c1bb2b690ac3d83d9ef5bf872bde4cdaf7cc9e76005Formal code
Source-reported evidence
The authors report Lean certification. The linked snapshot is pinned to the latest path commit observed by 10 October 2026. No local compilation, statement-alignment audit or independent reproduction is asserted by OpenMath.
Resource revision: e1a4e6508154ea59f030480661590a9fe3018011
Recorded authorship
Josh Alman · Virginia Vassilevska Williams
Josh Alman; Virginia Vassilevska Williams
The authors credit an internal Anthropic Claude model with algorithm discovery and later Lean formalization. They describe their simplification, strengthening, extensions and mathematical responsibility.
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Alman–Vassilevska Williams · Truly subquadratic 3SUM and subcubic APSP
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