The APSP hypothesis: are all-pairs shortest paths computable in truly subcubic time?

Yes: deterministic $O(n^{2.9995})$ algorithms for APSP and for the (min,+)-product with polynomially bounded integer weights on the word RAM (the Lean development states $n^{2.99942}$ ), and Exact Triangle in $O(n^{2.9983})$ , refuting the APSP and Exact Triangle hypotheses; a Las Vegas $O(n^{2.998})$ version works for real weights. The core is a new thin-matrix-product algorithm solving Lopsided All-Edges Sparse Triangle, with known reductions giving the rest. The same paper refutes the 3SUM hypothesis, catalogued separately. The constants are enormous; the impact is on the theory of subcubic equivalences, not on practice. SETH and Orthogonal Vectors are unaffected.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Oct 5, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: preprint. 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: The APSP hypothesis: are all-pairs shortest paths computable in truly subcubic time? · APSP hypothesis

Confidence: Not scored

Registry verification: lean checked · preprint · candidate

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Attribution

VibeMathed
registry · event recorded by

Claude (internal Anthropic research model, unnamed)
model · ai model contributor · Anthropic

Josh Alman
human · human collaborator

Virginia Vassilevska Williams
human · human collaborator

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SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

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VibeMathed record: The APSP hypothesis: are all-pairs shortest paths computable in truly subcubic time?

specialist registry · passed

No independent mathematician has checked this yet. Read here on 6 October 2026, not rebuilt. The paper's Theorem 2 was compared with the standard hypotheses as its own introduction states them (word RAM with O(log n)-bit words, integers of absolute value n^O(1)) and matches. The Lean development (anthropics/formal-math, folder 3sum-apsp, commit e1a4e65; Lean 4.33.1 with mathlib, about 102,000 lines) proves correctness and running time of explicit programs on a word-RAM machine defined in Lean, a weaker machine than the standard word RAM, which only strengthens upper bounds. Its five headline claims are stated in a 139-line file with no imports. No sorry outside the deliberate Challenge folder, no axiom declarations, native_decide, implemented_by or unsafe. The statement file was written by the same party as the proof and has not been audited independently, and no Comparator run is recorded. No uninvolved expert has commented yet; the authors are the leading researchers of the area and are not independent reviewers.