{"id":"palomar/PALOMAR-2026-10-11-000007","url":"https://github.com/selanavot/omega_le_nine_quarters_all_rings/tree/9d10116c89f9bdcdaccd58aacea353d4b7ae6e90","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-11-000007-v1.json","repo":"https://github.com/selanavot/omega_le_nine_quarters_all_rings","tags":["cs.CC","cs.DS","msc:68Q25","msc:15A69","msc:68W30","matrix-multiplication","algebraic-complexity","tensor-rank","lean"],"score":0,"title":"Matrix multiplication: omega at most 9/4 over every ring","trust":"high","domain":"cs.CC","source":"palomar","authors":["Sela Navot"],"reviews":0,"subject":"cs.CC","summary":"An integral extension of OpenAI's matrix multiplication exponent bound and its previous all-fields extension. The formalization proves that the exact integer coefficient tensor-rank exponent is at most 9/4, yielding division-free arithmetic programs over every associative unital ring, including noncommutative rings. For every positive epsilon, one constant bounds the operation count at all positive matrix sizes by C*n^(9/4+epsilon). The arithmetic model is preserved apart from Field-to-Ring assumptions. The extension uses a primitive tensor spectrum, coefficient extraction, fixed finite-free…","category":"research","comments":0,"disputed":false,"division":"cs","mentions":0,"verified":true,"discussed":0,"published":"2026-10-11","reactions":{},"formalized":true,"formalizes":[{"title":"An Upper Bound of 9/4 for the Matrix Multiplication Exponent","identifier":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf","relationship":"adapts"},{"title":"Matrix multiplication exponent at most 9/4 over arbitrary fields","identifier":"https://github.com/selanavot/matrix-multiplication-all-fields/tree/08481ef22bca7dc9ffba091083b7c1e81e537220","relationship":"adapts"}]}