Claude discovers an algorithm that refutes the 3SUM and APSP hypotheses: truly subquadratic 3SUM and truly subcubic APSP (Alman & Vassilevska Williams, Lean-verified)

2026-10-05★★★★★after cutoffscienceAnthropicColumbia UniversityMITconfidence: high

On Oct 5, 2026 Josh Alman (Columbia) and Virginia Vassilevska Williams (MIT) posted arXiv 2610.06783, giving the first polynomial speedups over the textbook algorithms for 3SUM (O(n^1.9992)) and All-Pairs Shortest Paths (O(n^2.9995)), which refutes the 3SUM, APSP, Exact Triangle and Zero-Weight k-Clique hypotheses that underpin much of fine-grained complexity. The paper says an Anthropic internal research model (Claude) discovered the algorithm on its own, with no human input, while asked to check cryptographic constructions; Anthropic then certified the main results in Lean 4.

Key facts

Paper: 'Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs', Josh Alman (Columbia) and Virginia Vassilevska Williams (MIT), arXiv 2610.06783, cs.DS, Oct 5, 2026
Results: deterministic 3SUM on n polynomial-size integers in O(n^1.9992); APSP on directed graphs with polynomially bounded integer weights in O(n^2.9995); Exact Triangle in O(n^2.9983); Zero-Weight k-Clique in O(n^(k−0.0017⌊k/3⌋))
Refuted (with known reductions): the 3SUM and APSP hypotheses, their real-valued versions, the Exact Triangle hypothesis, the Zero-Weight k-Clique hypotheses, and the three rectangular hinted Online Matrix–Vector conjectures of van den Brand, Nanongkai and Saranurak
Core idea: an algorithm for thin matrix products that computes a sparse set of entries of XY polynomially faster than writing XY down, by modifying a variant of Coppersmith's rectangular matrix multiplication built from a ten-multiplication identity of Schönhage; as a graph algorithm it solves All-Edges Sparse Triangle on sparse lopsided tripartite graphs in truly subquadratic time
How it was found (paper): 'An Anthropic employee used an internal research model to investigate open problems in the theory of cryptography' (constructions based on average-case hardness of Zero-k-Clique). 'Claude was tasked with verifying and improving the constructions, but instead developed this algorithm, first for the average case, then for the worst case. The session used 16M output tokens with no human input.'
Anthropic shared the algorithm with the authors in September 2026 under a confidentiality agreement and offered compensation; the authors simplified, strengthened and extended it (the data-structure version and the Hinted OMv connection are theirs)
Lean: after the paper was written, Anthropic used an internal model to certify Theorems 19 and 22 and the Zero-Weight case of Corollary 39 in Lean 4 + Mathlib, with all lemmas they rely on; repo anthropics/formal-math (3sum-apsp), checked with the Comparator tool, standard axioms only. Not formalized: real-input Las Vegas algorithms, Theorem 33, matrix-multiplication exponent values and the Section 5.1 running times
Context: the O(n^2) 3SUM algorithm had only been improved by polylog factors (Grønlund–Pettie 2014 and later), and APSP by subpolynomial factors (Williams 2018); these hypotheses underpin many conditional lower bounds
Verification status: preprint plus machine-checked Lean certificate of the main theorems; no peer review yet; reactions not yet found at 07:30 CEST Oct 6

Science result

Field: computer-science / algorithms / fine-grained complexity
Problem: 3SUM hypothesis (no O(n^(2−ε)) algorithm) and APSP hypothesis (no O(n^(3−ε)) algorithm), plus Exact Triangle and Zero-Weight k-Clique (open since 1995)
Result: Deterministic O(n^1.9992) 3SUM and O(n^2.9995) integer APSP, refuting the 3SUM, APSP, Exact Triangle and Zero-Weight k-Clique hypotheses and three hinted OMv conjectures
AI system: Claude (Anthropic internal research model)
Human role: AI-originated: the core algorithm was found autonomously by Claude (16M output tokens, no human input) during an unrelated cryptography task; Alman and Vassilevska Williams simplified, extended and wrote the paper
Verification: Preprint; main theorems formally verified in Lean 4/Mathlib by an Anthropic internal model
Status: pending
Why surprising: Two of the central conjectures of fine-grained complexity, assumed by a large body of conditional lower bounds, fell to an algorithm an AI found while working on something else.

What happened

Fine-grained complexity rests on a few hypotheses: that 3SUM needs about n² time, APSP about n³ time, and so on. Many "conditional lower bounds" assume them. Alman and Vassilevska Williams, both leading researchers in the area, now show polynomial speedups for both, so all those bounds lose their basis. The improvement in the exponent is tiny (0.0008 for 3SUM), but any polynomial improvement refutes the hypotheses.

Per the paper's "Acknowledgments and Methodology", the algorithm came from Claude: an Anthropic employee was using an internal research model on cryptographic constructions built on the average-case hardness of Zero-k-Clique. Instead of verifying them, the model broke the assumption, first on average and then in the worst case, in one session of 16M output tokens. Anthropic passed the algorithm to the two authors in September, and certified the final paper's main theorems in Lean.

Why it matters

This is the most significant theoretical-computer-science result credited to an AI so far, and the first where a frontier model, unprompted, overturned a widely believed conjecture rather than proving one that experts expected to be true. The Lean certificate addresses correctness; whether the method generalises (e.g. to SETH or Orthogonal Vectors) is open. The paper was found on Oct 6 by a manual arXiv scan; Anthropic had not yet announced it on its own channels at that time.

Changelog

2026-10-06: created (arXiv scan of the Oct 5 listing; PDF read with pdftotext)

People

Josh Alman Virginia Vassilevska Williams

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Sources (2)

id: 2026-10-05-claude-refutes-3sum-apsp-hypotheses · updated 2026-10-06 · open in the interactive timeline