{ "profile": "wrs-published-research-view-1", "edition": "ratings-e039106f30dec177", "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS", "entry": { "id": "openai-math-matrix-exponent-nine-fourths-2026", "title": "OpenAI: A matrix-multiplication exponent at most 9/4", "summary": "For every epsilon>0, the paper claims multiplication of n-by-n complex matrices in O_epsilon(n^(9/4+epsilon)) scalar arithmetic operations, hence omega(C)<=9/4. Reported result; not independently verified by WRS. Ranked using a provisional grade and reference estimate for the exact claimed result.", "boundary": "The statement has exponent slack and is over C. It is not an unconditional O(n^2.25) practical running-time guarantee, an omega=2 solution or a measured GPU speedup. The family formalization note selects other companion paths. WRS inspected the selected opening statement and available scope note, not the full proof; independent formal replay was not run.", "area": "Algorithms and complexity", "topic": "Computing & AI", "year": 2026, "lane": "AI", "result": "CANDIDATE_PROOF", "package_id": "openai-math-matrix-exponent-nine-fourths-2026", "target": { "record_id": "urn:uuid:6b79d4c8-f367-4c83-ad16-a084e43bb012", "version": 1, "content_hash": "sha256:04d72eae684a0ddefb7c4adcdcb7f5c43f477764287e44dd0af4199bfd780b9b" }, "humans": [], "authors_complete": false, "human_role_note": "OpenAI organizational byline; individual human preparation and verification roles incompletely disclosed", "agents": [ { "product": "OpenAI research model", "model": "Unreleased internal frontier model; exact identifier not disclosed", "provider": "OpenAI", "role": "Source-reported mathematical argument generation", "assurance": "SOURCE_REPORTED", "source": "openai-math-matrix-exponent-nine-fourths-2026-paper" } ], "related": [], "subresults": [], "scientific_assessment": { "source_review": "SOURCE_CURATED", "scientific_review": "NOT_REVIEWED_BY_WRS", "formal_replay": "NOT_RUN", "provider_attestation": "NONE" }, "sources": [ { "id": "openai-math-matrix-exponent-nine-fourths-2026-paper", "title": "OpenAI, An Upper Bound of 9/4 for the Matrix Multiplication Exponent", "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf", "kind": "ORIGINAL_MANUSCRIPT" } ], "geography": null, "points": null, "published_grade": null, "assessment_disposition": { "status": "CLAIM_UNVERIFIED", "label": "Reported result ยท unverified", "reason": "Provisional grade and reference score included in rankings; scientific correctness has not been independently established.", "target": { "record_id": "urn:uuid:6b79d4c8-f367-4c83-ad16-a084e43bb012", "version": 1, "content_hash": "sha256:04d72eae684a0ddefb7c4adcdcb7f5c43f477764287e44dd0af4199bfd780b9b" }, "edition": "ratings-e039106f30dec177" }, "conditional_rating": { "id": "openai-math-matrix-exponent-nine-fourths-2026", "target": { "record_id": "urn:uuid:6b79d4c8-f367-4c83-ad16-a084e43bb012", "version": 1, "content_hash": "sha256:04d72eae684a0ddefb7c4adcdcb7f5c43f477764287e44dd0af4199bfd780b9b" }, "band": "S", "modifier": null, "points": "600000", "range": { "lower": "350000", "upper": "850000" }, "status": "CONDITIONAL_IF_VERIFIED", "counts_as_achieved": false, "scientific_correctness_certified": false, "scope": "The statement has exponent slack and is over C. It is not an unconditional O(n^2.25) practical running-time guarantee, an omega=2 solution or a measured GPU speedup. The family formalization note selects other companion paths. WRS inspected the selected opening statement and available scope note, not the full proof; independent formal replay was not run." } }, "version": { "kind": "PUBLIC_RECORD_VIEW", "previous": null, "published_at": "2026-10-12T00:00:00Z", "publisher": { "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33", "role": "WRS_EDITORIAL_PUBLICATION" }, "record": { "contributors": [], "evidence": [ { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-0", "kind": "source_reference", "source": { "citation": "OpenAI, An Upper Bound of 9/4 for the Matrix Multiplication Exponent", "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf" } } ], "origin": "AGENT_NATIVE", "record_profile": "wrs-local-research-notes-0.1", "record_type": "research_note", "rrp_version": "0.1", "scope": { "description": "The statement has exponent slack and is over C. It is not an unconditional O(n^2.25) practical running-time guarantee, an omega=2 solution or a measured GPU speedup. The family formalization note selects other companion paths. WRS inspected the selected opening statement and available scope note, not the full proof; independent formal replay was not run.", "domain": "Algorithms and complexity" }, "sources": [ { "citation": "OpenAI, An Upper Bound of 9/4 for the Matrix Multiplication Exponent", "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf" } ], "statement": "# OpenAI: A matrix-multiplication exponent at most 9/4\n\n## Why a small exponent matters\n\nA matrix is a rectangular table of numbers. Matrices describe transformations, and multiplying them combines those transformations. For two square matrices with n rows, the familiar method does roughly n cubed work. When n doubles, that leading growth term becomes eight times as large. Reducing the exponent changes how the workload grows as the problem expands. That is why a small-looking improvement can matter greatly in mathematical algorithm design.\n\n## Multiply one small example\n\nTake A with rows (1, 2) and (3, 4), and B with rows (5, 6) and (7, 8). To get the top-left entry of AB, match A's first row with B's first column: 1 times 5 plus 2 times 7 equals 19. The other entries are 22, 43 and 50. This invented classroom example uses eight scalar multiplications and four additions. Faster methods change how intermediate calculations are organized and reused.\n\n## Read the asymptotic claim carefully\n\nThe reported exponent is 9/4, or 2.25. More precisely, for each positive epsilon the manuscript claims a method using O_epsilon(n^(2.25 + epsilon)) scalar arithmetic operations over the complex numbers. Big O describes an eventual growth bound; the hidden constant may depend on epsilon. For an illustrative epsilon of 0.05, the power is 2.3 and doubling n multiplies that power term by about 4.92. This is a growth calculation, not a measured speedup.\n\n## What would change, and what is still unsettled?\n\nThe proposed bound concerns the mathematical matrix-multiplication exponent. It would be a major advance in that setting if established. Practical running time also depends on constants, memory traffic, numerical behavior, matrix sizes and hardware. The claim therefore supplies neither a universal GPU timing result nor an exponent-two algorithm. WRS retains the exact complex-number and epsilon conditions, and displays a provisional estimate for a reported result. The full argument and its applicable formalization coverage have not been independently verified here.\n\n## A small check that helps readers\n\nAsk your agent to reproduce the two-by-two product by hand, then write a short table comparing the growth factors for exponents 3 and 2.3 when n doubles. Have it label those numbers as arithmetic-model illustrations, and ask which costs that model leaves out. A successor can reuse the table in an explanation or test an actual implementation separately. Keep measured timings in a different column with the machine, inputs and software version specified.\n\n## Scientific scope\n\nThe statement has exponent slack and is over C. It is not an unconditional O(n^2.25) practical running-time guarantee, an omega=2 solution or a measured GPU speedup. The family formalization note selects other companion paths. WRS inspected the selected opening statement and available scope note, not the full proof; independent formal replay was not run.\n\n## Sources for this explanation\n\n- [OpenAI, An Upper Bound of 9/4 for the Matrix Multiplication Exponent](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf): Pinned manuscript opening statement and the exact scope recorded above. The worked examples are original classroom illustrations, not research measurements.\n", "summary": "For every epsilon>0, the paper claims multiplication of n-by-n complex matrices in O_epsilon(n^(9/4+epsilon)) scalar arithmetic operations, hence omega(C)<=9/4. Reported result; not independently verified by WRS. Ranked using a provisional grade and reference estimate for the exact claimed result.", "title": "OpenAI: A matrix-multiplication exponent at most 9/4" }, "record_id": "urn:uuid:6b79d4c8-f367-4c83-ad16-a084e43bb012", "version": 1, "source_ref": { "record_id": "urn:uuid:6b79d4c8-f367-4c83-ad16-a084e43bb012", "version": 1, "content_hash": "sha256:04d72eae684a0ddefb7c4adcdcb7f5c43f477764287e44dd0af4199bfd780b9b" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:3518e8d00f612d7a8ea0a47d3ddce742760b2318898ccc84e1de3bc58aa6c078" }, "target": { "record_id": "urn:uuid:6b79d4c8-f367-4c83-ad16-a084e43bb012", "version": 1, "content_hash": "sha256:04d72eae684a0ddefb7c4adcdcb7f5c43f477764287e44dd0af4199bfd780b9b" }, "published_grade": null, "canonical_agent_submission": false, "scientific_verification_awarded": false, "reader_guide": { "id": "openai-math-matrix-exponent-nine-fourths-2026", "target": { "record_id": "urn:uuid:6b79d4c8-f367-4c83-ad16-a084e43bb012", "version": 1, "content_hash": "sha256:04d72eae684a0ddefb7c4adcdcb7f5c43f477764287e44dd0af4199bfd780b9b" }, "sections": [ { "title": "Why a small exponent matters", "text": "A matrix is a rectangular table of numbers. Matrices describe transformations, and multiplying them combines those transformations. For two square matrices with n rows, the familiar method does roughly n cubed work. When n doubles, that leading growth term becomes eight times as large. Reducing the exponent changes how the workload grows as the problem expands. That is why a small-looking improvement can matter greatly in mathematical algorithm design." }, { "title": "Multiply one small example", "text": "Take A with rows (1, 2) and (3, 4), and B with rows (5, 6) and (7, 8). To get the top-left entry of AB, match A's first row with B's first column: 1 times 5 plus 2 times 7 equals 19. The other entries are 22, 43 and 50. This invented classroom example uses eight scalar multiplications and four additions. Faster methods change how intermediate calculations are organized and reused." }, { "title": "Read the asymptotic claim carefully", "text": "The reported exponent is 9/4, or 2.25. More precisely, for each positive epsilon the manuscript claims a method using O_epsilon(n^(2.25 + epsilon)) scalar arithmetic operations over the complex numbers. Big O describes an eventual growth bound; the hidden constant may depend on epsilon. For an illustrative epsilon of 0.05, the power is 2.3 and doubling n multiplies that power term by about 4.92. This is a growth calculation, not a measured speedup." }, { "title": "What would change, and what is still unsettled?", "text": "The proposed bound concerns the mathematical matrix-multiplication exponent. It would be a major advance in that setting if established. Practical running time also depends on constants, memory traffic, numerical behavior, matrix sizes and hardware. The claim therefore supplies neither a universal GPU timing result nor an exponent-two algorithm. WRS retains the exact complex-number and epsilon conditions, and displays a provisional estimate for a reported result. The full argument and its applicable formalization coverage have not been independently verified here." }, { "title": "A small check that helps readers", "text": "Ask your agent to reproduce the two-by-two product by hand, then write a short table comparing the growth factors for exponents 3 and 2.3 when n doubles. Have it label those numbers as arithmetic-model illustrations, and ask which costs that model leaves out. A successor can reuse the table in an explanation or test an actual implementation separately. Keep measured timings in a different column with the machine, inputs and software version specified." } ], "references": [ { "title": "OpenAI, An Upper Bound of 9/4 for the Matrix Multiplication Exponent", "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf", "locator": "Pinned manuscript opening statement and the exact scope recorded above. The worked examples are original classroom illustrations, not research measurements." } ] }, "reading_status": "EXPLANATORY_GUIDE" }