[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"kXv3QhJTxH":3},"# Matrix multiplication: ω ≤ 9/4 over every ring\n\n[![DOI](https://zenodo.org/badge/DOI/10.5281/zenodo.23268534.svg)](https://doi.org/10.5281/zenodo.23268534)\n\nOpenAI proved ω ≤ 9/4 over the complex numbers in its\n[matrix multiplication preprint](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf).\nOur [earlier all-fields extension](https://github.com/selanavot/matrix-multiplication-all-fields)\nestablishes the same bound over every field.\nThis repository extends the result to arbitrary associative unital\nrings, including noncommutative rings.\nThe full theorem, including the trivial ring, passed compilation, axiom audit,\nfresh-kernel replay, and frozen-specification Comparator with all six bundled\ncheckers on Lean 4.35.0-rc3. Both deliberate negative controls also passed.\nThe rc3 pin avoids a Challenge-rendering failure in the rc4 renderer; all 160\nLean files under `lean/` are unchanged from the previously verified rc4 version. See\n[verification details](verification/comparator/README.md) and [current status](docs/STATUS.md).\n\n**Both the extension's Lean proof development and the [paper](paper/paper.pdf)\nare AI-generated with Codex under human direction.** The numerical exponent\nand main proof architecture are OpenAI's. This is an extension of scope,\nwith no claim of a better numerical bound or independent human peer review.\n\n## The theorem\n\n[AllRings.lean](lean/OAI/LinearAlgebra/MatrixMultiplication/AllRings.lean):\n\n```lean\ntheorem omega_le_nine_quarters_all_rings\n    (R : Type u) [Ring R] :\n    Arithmetic.omega R ≤ (9 : ℝ) / 4\n```\n\nThe substantive statement change from the prior theorem is small:\n\n```diff\n- (F : Type u) [Field F] : Arithmetic.omega F ≤ (9 : ℝ) / 4\n+ (R : Type u) [Ring R] : Arithmetic.omega R ≤ (9 : ℝ) / 4\n```\n\nThe arithmetic model has exactly seven `Field` → `Ring` substitutions.\nIts gate costs, evaluation, correctness quantifiers, positive exponent slack,\nand infimum definition are unchanged. Nontriviality is retained for the lower\nbound 2 and boundedness of the admissible-exponent set.\n\nFor the trivial ring, one free constant-zero register supplies every output.\nEvery real exponent is therefore admissible. The unchanged real-valued\ndefinition gives `omega R = 0` by Mathlib's convention `sInf Set.univ = 0`;\nthe admissible set in this case has no genuine real infimum. The explicit\nzero-cost program and the direct operation-count theorem avoid reliance on\nthat convention for the algorithmic claim.\n\nThe stronger intermediate theorem is `exactRankExponent_int_le_nine_quarters`:\nthe exponent of exact **integer coefficient tensor rank** is at most 9/4.\nThose identities give programs over every ring, since integer constants are\ncentral and each left-input factor stays before its right-input factor.\n\nThe direct operation-count theorem says that for every ε > 0 there is a\nconstant C > 0 and, for every matrix size n ≥ 1, a correct program costing\nat most C n^(9/4 + ε) additions, subtractions and multiplications. This also\nincludes the zero ring. It is an existence theorem for arithmetic programs;\nit does not establish an exact O(n^(9/4)) endpoint, an efficient uniform\ncircuit generator, or a bit-complexity bound.\n\n## Where the bound applies\n\n![One connected algebraic hierarchy: green ring classes satisfy the nine-fourths bound; orange semiring classes contain the listed cubic examples.](docs/assets/algebraic-structures.svg)\n\nGreen means the `9/4` upper bound holds throughout the class. Orange means the\nclass contains cubic examples, not that every member is cubic; rings are also\nsemirings. The comparison uses addition, multiplication and constants within\nthe chosen structure, with no division gates. Over rings, negative constants\nsimulate subtraction with constant-factor overhead.\n\nFor the nonnegative reals, matrix multiplication requires `n³` multiplication\ngates in this model: Jerrum and Snir (1982),\n[§4.1, p. 886](https://snir.cs.illinois.edu/listed/J7.pdf#page=13),\n[DOI: 10.1145/322326.322341](https://doi.org/10.1145/322326.322341).\nThat counterexample rules out a universal semiring extension. This classical\nlower bound is cited background, not formalized in this repository. See\n[definitions, model details and full citation](docs/ALGEBRAIC-STRUCTURES.md)\nor the [PNG diagram](docs/assets/algebraic-structures.png).\n\n## What the extension adds\n\n- A suitable integral tensor spectrum, avoiding the false assumption that\n  every nonzero integer tensor restricts to the unit tensor.\n- Division-free coefficient extraction and descent through one fixed finite\n  free integer algebra after taking powers.\n- Consecutive unnormalized Fourier periods, with their scalar multipliers\n  cancelled by Bezout before character evaluation.\n- Integral convolution schemes obtained by finite-field specialization,\n  Vandermonde adjugates, integer norms, and a finite coprime patch.\n- The inherited determinant, sector and profile argument over these inputs,\n  followed by the bridge to arbitrary ring arithmetic.\n\nSee the [paper PDF](paper/paper.pdf), [LaTeX source](paper/paper.tex), and\n[provenance](UPSTREAM.md). The paper has no author byline or date and is\nassociated with its repository commit.\n\n## Citation and archival release\n\nVersion **0.1.0** is published on [Zenodo](https://zenodo.org/records/23268534):\n**[DOI 10.5281/zenodo.23268534](https://doi.org/10.5281/zenodo.23268534)**.\nThe archive preserves source snapshot\n[`2d2cc89859d17d3143cd40c4a4b3df49801aa533`](https://github.com/selanavot/omega_le_nine_quarters_all_rings/tree/2d2cc89859d17d3143cd40c4a4b3df49801aa533),\nincluding the Lean development, paper, verification records, Git history bundle\nand checksums. Subsequent documentation and the rc3 compatibility changes are\noutside that frozen release.\nUse the version DOI to cite this exact archive; the\n[concept DOI](https://doi.org/10.5281/zenodo.23268533) identifies the evolving project.\n\n[CITATION.cff](CITATION.cff) supplies the software citation, and\n[.zenodo.json](.zenodo.json) retains the archive metadata and attribution.\nThe creator credit describes human direction of the project; the proof\ndevelopment and manuscript remain explicitly disclosed as AI-generated.\n\nSee the [Zenodo release record](docs/ZENODO.md) and\n[Palomar submission status](docs/PALOMAR.md). Palomar runs its own\nverification against a fixed public commit; the local verification receipt\ndoes not substitute for that registry workflow.\nThe [full Palomar mechanical preflight and registry verification passed](verification/palomar/README.md)\nfor the submitted rc4 snapshot, but its registry rendering stage failed.\nThe unchanged Challenge has since rendered locally with the unmodified rc3\nrenderer; see the [rendering check](verification/palomar/RC3-RENDER.md).\nThe [official hosted rc3 mechanical preflight also passed](https://github.com/selanavot/omega_le_nine_quarters_all_rings/actions/runs/38024128253)\nfor candidate commit `9d10116c89f9bdcdaccd58aacea353d4b7ae6e90`, with empty report\nerror and warning lists. [PR #6](https://github.com/selanavot/omega_le_nine_quarters_all_rings/pull/6)\ncontains the compatibility change and its receipts; subsequent receipt and\ndocumentation commits are outside the exact snapshot checked by that workflow.\nThe original submission remains tied to its rc4 commit. No new submission or\nregistry acceptance is claimed.\n\n## Build and verify\n\nLean **4.35.0-rc3** and Mathlib\n`c55e6e786f49471c72fbddbec5415808896aec1e` are pinned on the compatibility branch.\nThe arithmetic model and all six frozen Challenge statements are unchanged.\n\n```sh\nlake exe cache get\nLEAN_NUM_THREADS=1 lake build\nLEAN_NUM_THREADS=1 lake build OAI.LinearAlgebra.MatrixMultiplication.FinalAudit\npython3 scripts/check-specification.py\nbash scripts/check-kernel.sh\npython3 scripts/check-comparator.py --trusted-local --negative-controls\n```\n\nThe specification checker freezes the original model plus only the seven\napproved typeclass substitutions. Comparator checks six statements, including\nBddBelow, the lower bound, direct costs, and exact integer coefficients.\nThe [rc3 verification receipt](verification/comparator/RESULTS-RC3.md) records\nthe successful full build, fresh-kernel replay and six-checker Comparator run.\nThe two negative controls rejected changed multiplication cost and `sorry`\nagain under rc3. The [archived rc4 receipt](verification/comparator/RESULTS.md)\nand [earlier nontrivial-ring receipt](verification/comparator/RESULTS-NONTRIVIAL.md)\nretain their original versions and scope.\nThe local Comparator command disables its build sandbox; it is not a claim\nof sandboxed source-provenance verification. See [audit details](verification/comparator/README.md).\n\nDevelopment uses a serialized build queue on this machine:\n`python3 scripts/lean-queue.py submit --owner NAME Module.Name`.\nOnly the coordinator starts `worker`; `stop` drains the current request.\nStop the worker before kernel/Comparator checks; the scripts share its lock.\nThe [working agreements](AGENTS.md) document ownership and resource limits.\n",1792350356025]