{"what":"Detached certificates for fast matrix-multiplication algorithms. Each entry claims: the rank-r integer factors U (nm x r), V (mp x r), W (np x r) satisfy, over the stated ring and under the stated C-layout, the matmul tensor identity — for every A-index (a,b), B-index (b',c), C-index k: sum_t U[ab][t]*V[b'c][t]*W[k][t] = [b=b']*[k=index(a,c)] (mod 2 when ring is F2). That makes the factors a correct algorithm for ALL matrices, in r < nmp multiplications. Verify with tools/verify_strassen.py — Python stdlib only, no code from this repo.","layoutNote":"layout AC: k = a*p + c; layout CA: k = c*n + a.","generatedBy":"tools/export-strassen-certificate.js @ git 04d4b7c","sourcePins":{"alphatensor_r.npz":"4d59571a2537a9472e8229176d5ebe2925f3e041ad403c095b41853026caec33","alphatensor_f2.npz":"70f09f349d8d2874ef0e0e089459c7320f5aa3eef277df5ffa67f573709db2da","alphaevolve_mathematical_results.ipynb":"2cce2543e48c89aa3e91614272a698a0147dd2548ea11cf92f1292b7435d38ff"},"entries":[{"id":"strassen-1969","source":"Strassen, Numer. Math. 13 (1969) — the textbook transcription; the audit is the check","dims":[2,2,2],"rank":7,"naive":8,"ring":"Q","layout":"AC","statement":"strassen-1969: 2x2 times 2x2 in 7 multiplications VERIFIED over Q — all 64 tensor-identity equations hold exactly (layout AC); rank 7 < 8 naive","U":[[1,0,1,0,1,-1,0],[0,0,0,0,1,0,1],[0,1,0,0,0,1,0],[1,1,0,1,0,0,-1]],"V":[[1,1,0,-1,0,1,0],[0,0,1,0,0,1,0],[0,0,0,1,0,0,1],[1,0,-1,0,1,0,1]],"W":[[1,0,0,1,-1,0,1],[0,0,1,0,1,0,0],[0,1,0,1,0,0,0],[1,-1,1,0,0,1,0]]},{"id":"strassen-squared-4x4x4","source":"generated here: Strassen ⊗ Strassen by exact Kronecker composition, re-decided from scratch","dims":[4,4,4],"rank":49,"naive":64,"ring":"Q","layout":"AC","statement":"strassen-squared-4x4x4: 4x4 times 4x4 in 49 multiplications VERIFIED over Q — all 4096 tensor-identity equations hold exactly (layout AC); rank 49 < 64 naive — generated AND decided 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(DeepMind, Nature 610, 2022), key 2,2,2 of alphatensor_r.npz","dims":[2,2,2],"rank":7,"naive":8,"ring":"Q","layout":"CA","statement":"alphatensor-q-2x2x2: 2x2 times 2x2 in 7 multiplications VERIFIED over Q — all 64 tensor-identity equations hold exactly (layout CA); rank 7 < 8 naive","U":[[0,1,1,0,1,1,0],[0,0,-1,1,0,0,0],[1,1,1,0,1,0,0],[-1,-1,-1,0,0,0,1]],"V":[[0,0,0,0,1,1,0],[1,1,0,0,1,0,1],[0,1,1,1,1,0,0],[0,1,1,0,1,0,1]],"W":[[0,0,0,1,0,1,0],[0,-1,0,0,1,-1,-1],[-1,1,-1,-1,0,0,0],[1,0,0,0,0,0,1]],"sourcePin":{"file":"alphatensor_r.npz","sha256":"4d59571a2537a9472e8229176d5ebe2925f3e041ad403c095b41853026caec33"}},{"id":"alphatensor-q-3x3x3","source":"AlphaTensor (DeepMind, Nature 610, 2022), key 3,3,3 of alphatensor_r.npz","dims":[3,3,3],"rank":23,"naive":27,"ring":"Q","layout":"CA","statement":"alphatensor-q-3x3x3: 3x3 times 3x3 in 23 multiplications VERIFIED over Q — all 729 tensor-identity equations hold exactly (layout CA); rank 23 < 27 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(DeepMind, Nature 610, 2022), key 4,5,5 of alphatensor_r.npz","dims":[4,5,5],"rank":76,"naive":100,"ring":"Q","layout":"CA","statement":"alphatensor-q-4x5x5: 4x5 times 5x5 in 76 multiplications VERIFIED over Q — all 10000 tensor-identity equations hold exactly (layout CA); rank 76 < 100 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over Q — the algorithm genuinely requires characteristic 2","sourcePin":{"file":"alphatensor_f2.npz","sha256":"70f09f349d8d2874ef0e0e089459c7320f5aa3eef277df5ffa67f573709db2da"}},{"id":"alphatensor-f2-5x5x5","source":"AlphaTensor (DeepMind, Nature 610, 2022), key 5,5,5 of alphatensor_f2.npz","dims":[5,5,5],"rank":96,"naive":125,"ring":"F2","layout":"CA","statement":"alphatensor-f2-5x5x5: 5x5 times 5x5 in 96 multiplications VERIFIED over F2 — all 15625 tensor-identity equations hold exactly (layout CA); rank 96 < 125 naive; REFUTED over Q — the algorithm genuinely requires characteristic 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over Q — the algorithm genuinely requires characteristic 2","sourcePin":{"file":"alphatensor_f2.npz","sha256":"70f09f349d8d2874ef0e0e089459c7320f5aa3eef277df5ffa67f573709db2da"}},{"id":"alphaevolve-48-4x4x4","source":"AlphaEvolve (DeepMind, 2025; arXiv:2506.13131), mathematical_results.ipynb of google-deepmind/alphaevolve_results @ commit 4226acb - the first-party byte source","dims":[4,4,4],"rank":48,"naive":64,"ring":"Zi","layout":"CA","scale":8,"statement":"alphaevolve-48-4x4x4: 4x4 times 4x4 in 48 multiplications VERIFIED over Z[i] (doubled half-Gaussian factors; identity = 8*T, denominators cleared) — all 4096 tensor-identity equations hold exactly (layout CA); rank 48 < 64 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