The 3SUM hypothesis: is 3SUM solvable in truly subquadratic time?

Yes: a deterministic $O(n^{1.9992})$ algorithm for 3SUM on polynomially bounded integers on the word RAM, refuting the hypothesis, and a Las Vegas $O(n^{1.998})$ algorithm for real inputs using only additions, subtractions and comparisons. The core is a new algorithm for wanted entries of thin matrix products (a modified Coppersmith rectangular algorithm with a Schönhage identity), which solves Lopsided All-Edges Sparse Triangle; known reductions give 3SUM. The same paper refutes the APSP hypothesis, catalogued separately. The constants are enormous, so the impact is on the theory: the conditional lower bounds built on 3SUM no longer stand as stated. 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 3SUM hypothesis: is 3SUM solvable in truly subquadratic time? · 3SUM 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

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: The 3SUM hypothesis: is 3SUM solvable in truly subquadratic 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.

Source ↗

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

preprint · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Lean formalisation of the algorithms and running times (anthropics/formal-math)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

The APSP hypothesis, refuted in the same paper

witness · pending

other link supplied by the VibeMathed registry.

Source ↗

Artifacts and verifiers

Lean formalisation of the algorithms and running times (anthropics/formal-math)

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

The 3SUM hypothesis: is 3SUM solvable in truly subquadratic time? parent of this event

This event attributed to Claude (internal Anthropic research model, unnamed)

The APSP hypothesis, refuted in the same paper evidence for this event

This event attributed to Virginia Vassilevska Williams

This event attributed to Josh Alman

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

Lean formalisation of the algorithms and running times (anthropics/formal-math) evidence for this event

VibeMathed record: The 3SUM hypothesis: is 3SUM solvable in truly subquadratic time? evidence for this event

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