Cloudroom Math: new matrix multiplication algorithms found by AI agents

This repository contains explicit matrix multiplication algorithms and short manuscripts produced in one day (2026-10-08) by an autonomous multi-agent research campaign. The agents were Claude Opus 5.5 threads orchestrated in Cloudroom: one manager thread and seven worker threads. In the last phase, which targeted 4x4 matrices, two GPT-6-Astra brainstorm threads (run through Codex) also took part.

Results

• 27 new upper bounds on the rank of rectangular matrix multiplication, each with an explicit algorithm that uses fewer multiplications than any previously published one:
  • <4,6,15>: 263 -> 261 (new idea: hidden grids in serendipitous products),
  • <9,11,11>: 711 -> 710 (mod-3 flip-graph walk and a core-restricted lift),
  • 25 formats with a dimension between 17 and 32, gains of 1 to 12 multiplications (composition-type improvements from a systematic sweep).
• No rank-47 algorithm for 4x4 matrices. The best known rank over Q or C stays 48. We publish the exact obstructions and the negative results of a focused attack, with our own corrections.
• 5 ties with new constructions, 636 exact rank-48 decompositions of <4,4,4> as seeds, and a small, memory-safe exact verifier.

<n1,n2,n3> denotes multiplying an n1 x n2 matrix by an n2 x n3 matrix. A rank-r algorithm does this with r scalar multiplications. Applied recursively to blocks, fewer multiplications give asymptotically faster matrix multiplication, as in Strassen's algorithm (<2,2,2> with 7 instead of 8).

Verify everything with one command

pip install -r verification/requirements.txt     # numpy, sympy, python-flint
bash verification/check_all.sh                   # about 3 minutes on one CPU core

The script checks every scheme exactly over Q (or Q(i)), runs each one as an algorithm on random integer matrices in exact arithmetic, compares each record with the previous best, and verifies the 636 rank-48 seeds. To check one file: python3 verification/verify.py schemes/4x6x15_rank261.json.

Navigating the repository

• CONTENTS.md: the result map, with numbered families, abstracts and links.
• overview.md (PDF): a short overview of all families.
• schemes/: all algorithms as JSON, with a table of formats, previous best and new rank.
• manuscripts/: one short paper per family (paper.md, paper.tex, paper.pdf, citation.bib).
• verification/: verifier, product tests, check_all.sh and the baseline table.
• tools/: the hidden-grid detector.
• research_summaries/: abridged summaries of how the agents reasoned.
• history.md: release history.

How the results were produced

The manager thread split the work by method: flip-graph walks over small fields with lifting to Q, compositions and serendipitous products, numerical search on a GPU, flip graphs with symmetry, and a dedicated thread for <4,4,4>. Each thread kept working notes and reported claims to the manager. The manager re-verified every claimed scheme with an independent exact checker and compared it with the published tables before recording it. Compute was a cloud sandbox, one GPU workstation and one laptop, for about 13.5 hours.

The ideas, code and write-ups of the campaign were produced by the agents. A human set the goal, provided the compute, decided the switch to <4,4,4> at 18:20 UTC, and stopped the campaign at 21:15 UTC. This release was prepared by a further Claude thread from the campaign's final report, notes and verified files. The manuscripts are AI-written and have not been peer reviewed. The algorithms do not depend on the manuscripts: each JSON file can be checked directly.

Scope and honesty

• The two main records improve small and medium formats by 2 and 1 multiplications. They are not asymptotic results and do not change the exponent of matrix multiplication.
• The 25 large-format bounds are composition-type improvements: serendipitous products of existing schemes with better bookkeeping (best known rational block ranks, and in six cases a hidden grid). No new small scheme is involved. We expect several of them to be improved by the same kind of search.
• "Previous best" means the minimum over the FMM catalogue, Lacelle's catalogue, Perminov's repository and the bounds in Perminov's 2026 serendipitous-product paper, checked on 2026-10-08. A bound we did not know about could exist elsewhere.
• For <4,4,4>, manuscript 004 separates proved statements from search evidence. Several of our own earlier claims were overstated. The corrections are listed there.

Base schemes and credit

The constructions start from schemes in Andrew Perminov's FastMatrixMultiplication repository (MIT license). The two base schemes of the main records are copied unchanged in third_party/perminov/, see THIRD_PARTY.md. The records were compared with the FMM catalogue of Sedoglavic et al. and with Lacelle's matmulcatalog.

License

Our code, schemes and texts are released under the Apache License 2.0 (LICENSE). The files in third_party/perminov/ keep their MIT license.

Citation

@misc{cloudroommath2026,
  author       = {{Cloudroom Math Campaign}},
  title        = {Cloudroom Math: new matrix multiplication algorithms found by AI agents},
  year         = {2026},
  month        = oct,
  howpublished = {GitHub repository cloudroom-math, release 2026-10-08},
  note         = {Claude Opus 5.5 agents orchestrated in Cloudroom; explicit schemes with an exact verifier}
}

Each manuscript folder has its own citation.bib.