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Internal Anthropic model refuted the 3SUM and APSP hypotheses: first truly subquadratic 3SUM and truly subcubic APSP (Alman & Vassilevska Williams)

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Based on a feed excerpt; full article content has not been acquired. Source evidence retrieved 06/10/2026, 18:47:26 UTC.

Briefings

A preprint by Alman and Vassilevska Williams presents deterministic subquadratic algorithms for 3SUM and APSP, achieving O(n^1.9992) and O(n^2.9995) respectively. The breakthrough relies on a new thin matrix product algorithm formalized in Lean, improving upon textbook n^2 and n^3 bounds. While SETH and Orthogonal Vectors remain unaffected, the paper notes an Anthropic research model discovered the core algorithm independently.

Key points

• The paper presents deterministic O(n^1.9992) 3SUM and O(n^2.9995) APSP algorithms, representing the first polynomial improvements over n^2 and n^3 bounds.
• The core algorithm is implemented in Lean and discovered independently by an Anthropic research model, formalized in the paper's main theorems.

Context

This finding represents a significant theoretical advance in computational complexity, specifically for exact triangle and zero-weight k-clique problems. The results are notable because they improve upon long-standing textbook bounds without contradicting major conjectures like SETH.

Limitations

The evidence is a preprint submitted to r/singularity, not yet peer-reviewed. The paper claims an Anthropic model found the core algorithm, but the specific model and its role remain unverified. No experimental data or computational results are provided, limiting verification of the claimed speedups.

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