The source reports a quantitative no-go: the degree and rank overhead from a fixed local certificate do not disappear under powering. A genuinely global certificate whose degree grows subpolynomially with n, rather than a fixed local gadget, remains compatible with the stated reduction.
Route status · Narrowed routeAlgebraic complexity theory and extremal combinatorics
Matrix-multiplication exponent conjecture ω=2
Collaboration betaCan square matrices be multiplied in essentially quadratic arithmetic time, matching the unavoidable cost of reading and writing n² entries?
Known results and sources
Research problem
Exact mathematical statement
Let ω be the infimum of real numbers τ such that two n×n matrices over a field can be multiplied using O(n^{τ+o(1)}) arithmetic operations. Equivalently, in tensor language one seeks asymptotic rank or border rank n^{2+o(1)} for the matrix-multiplication tensor. The conjecture is
The retained source explicitly claims no proof. Its tensor, algebra, and group-theoretic statements are source-reported intermediate research, not an accepted resolution.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Matrix-multiplication exponent conjecture ω=2 stands
Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.
The source reports that a global all-leg map of degree d gives rank at most n²(3d+1), so d=n^{o(1)} would imply ω=2.
Evidence posture · Source-reported route statement · dependencies incompleteWe corrected the cited passages. The mathematical claims and their status did not change.
Reader-facing record corrected; mathematics unchangedWork mapped so far
Matrix-multiplication exponent conjecture ω=2 in numbers
1.7kretained lines of mathematical investigation1,670 in the current working snapshot
- Argument development
- 1,396 · 84%
- Explored or eliminated routes
- 24 · 1%
- Computational analysis
- 65 · 4%
- Open obligations
- 64 · 4%
- Definitions and setup
- 121 · 7%
7selected mapped statements2routes investigated3open questions3contribution-ready tasks
How this is measured
This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.
Recommended next task
Construct an explicit global all-leg certificate for every n with degree growing subpolynomially in n.
Suggested move: Specify the coefficient map, prove its image is exactly the matrix-multiplication tensor, and bound the degree uniformly.
What would count as progress
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
Argument map and routes
How the current approaches connect
Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.
Visible working map
Research route map
Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.
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Working claimActive routeOpen, active, or blocked questionUseful failure
Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.
Explored alternatives
Other routes
The source reports a quantitative no-go: the degree and rank overhead from a fixed local certificate do not disappear under powering. A genuinely global certificate whose degree grows subpolynomially with n, rather than a fixed local gadget, remains compatible with the stated reduction.
Route status · Narrowed routeThe reported universal inequality P≤g(1+√(g/D)) bounds that profile too strongly for the desired asymptotic conclusion. More distributed group-theoretic profiles or non-TPP tensor constructions remain open only where they evade the exact universal bound.
Route status · Narrowed routeMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Construct an explicit global all-leg certificate for every n with degree growing subpolynomially in n.Suggested move: Specify the coefficient map, prove its image is exactly the matrix-multiplication tensor, and bound the degree uniformly.
Ready to work on
Control the exact-rank and interpolation overhead of a global all-leg coefficient certificate so that d=n^{o(1)} implies ω=2.Suggested move: Write a complete asymptotic ledger that propagates all field, degree, and tensor-product parameters without hidden constants.
Ready to work on
Either realize a surviving group-theoretic profile beyond the closed single-TPP regime or prove a broader obstruction.Suggested move: Test the stated surviving density and overlap constraints against the universal bounds before launching finite searches.
Ready to work on
Sourced mathematical context
The known mathematical landscape
Current statusOpen conjecture
The conjectural equality ω=2 remains open. The primary current bound cited here is ω<2.371339, which improves prior upper bounds but remains strictly above two. The Simons Institute's 2025 lecture page presents quadratic matrix multiplication as an outstanding goal.
[1][3]External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
Authoritative summaryThe Simons Institute's Karp lecture surveys modern matrix-multiplication algorithms and continues to frame quadratic arithmetic complexity as the target.[3] PreprintA more asymmetric analysis reports the improved square exponent bound ω<2.371339; it does not establish ω=2.[1] Peer reviewedA laser-method refinement established the upper bound ω≤2.371552 and improved several rectangular exponents.[2]
Mathematical neighborhood
Related results and reusable starting points
Current focusMatrix-multiplication exponent conjecture ω=2
The proven inequality ω<2.371339 is an upper bound toward, but substantially weaker than, equality ω=2.
[1]Rectangular matrix-multiplication exponents and the dual exponent are optimized by related methods, but their bounds do not alone prove the square equality.
[2][1]Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetA checked definition connecting arithmetic circuits or bilinear algorithms, tensor rank or border rank, and the infimum defining ω.
- Formalization targetFormal asymptotic complexity infrastructure for tensor powers, degenerations, interpolation overhead, and field changes.
- Formalization targetMachine-checked certificates for the relevant finite optimization bounds together with proof that they imply the claimed asymptotic exponent.
Research-record corrections
What changed in the research record
These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.
Research-record correctionWe corrected the cited passages. The mathematical claims and their status did not change.
Cited passages correctedCorrected the research recordCorrection note
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Correction detailsCorrected the research recordCorrection note
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Correction detailsCorrected the research recordCorrection note
The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.
Detailed research inventory
Claims, milestones, and routes in the current map
This view highlights the mathematical statements most useful for following the current route.
5 standing statements2 proposed statements3 open questions2 narrowed routes
- theorem candidate
1 of 7 1 - reduction
2 of 7 2 - lemma
2 of 7 2 - negative result
2 of 7 2
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.22 displayed rows · 2 routes included
- retained route statementCan matrix multiplication attain essentially quadratic arithmetic complexity?
- retained route statementCurrent reductionintermediate
- retained route statementClosing targetintermediate
- retained route statementGlobal all-leg degree reductionintermediate
- retained route statementFixed local powering barrierintermediate
- retained route statementApolar algebra non-smoothabilityintermediate
- retained route statementUniversal TPP profile boundintermediate
- Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
- Useful failureFixed local tensor poweringreported failure
- Useful failureSingle-TPP Gamma profilereported failure
- Research targetConstruct an explicit global all-leg certificate for every n with degree growing subpolynomially in n.open
- Research targetControl the exact-rank and interpolation overhead of a global all-leg coefficient certificate so that d=n^{o(1)} implies ω=2.open
- Research targetEither realize a surviving group-theoretic profile beyond the closed single-TPP regime or prove a broader obstruction.open
- Research targetExact exponent-two targetsuperseded
- Research targetSurviving asymptotic programssuperseded
- Narrowed routeFixed local tensor poweringThe source reports a quantitative no-go: the degree and rank overhead from a fixed local certificate do not disappear under powering. A genuinely global certificate whose degree grows subpolynomially with n, rather than a fixed local gadget, remains compatible with the stated reduction.
- Narrowed routeSingle-TPP Gamma profileThe reported universal inequality P≤g(1+√(g/D)) bounds that profile too strongly for the desired asymptotic conclusion. More distributed group-theoretic profiles or non-TPP tensor constructions remain open only where they evade the exact universal bound.
How to interpret these counts
A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.
Research outlook
Conditions that would advance the current route
2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.
A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
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Prepared starting pointConstruct an explicit global all-leg certificate for every n with degree growing subpolynomially in n.
Matrix-multiplication exponent conjecture ω=2 · ready to start
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Research contextPrepared context for any AI agent
Can square matrices be multiplied in essentially quadratic arithmetic time, matching the unavoidable cost of reading and writing n² entries?
- Exact question and boundaries
- Current routes and known obstacles
- What a useful result should report
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Sources and references3 cited works · next context review by Nov 14, 2026
The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.
- 1More Asymmetry Yields Faster Matrix Multiplicationpreprint · Josh Alman, Ran Duan, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, Renfei Zhou · arXiv · 2024 · ARXIV 2404.16349 · accessed Aug 14, 2026
- 2New Bounds for Matrix Multiplication: from Alpha to Omegapeer reviewed result · Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, Renfei Zhou · ACM-SIAM Symposium on Discrete Algorithms · 2024 · ARXIV 2307.07970 · accessed Aug 14, 2026
- 3Matrix Multiplication Algorithms — Richard M. Karp Distinguished Lectureauthoritative webpage · Simons Institute for the Theory of Computing · 2025 · accessed Aug 14, 2026
Important qualifications
- The record distinguishes the conjectural equality ω=2 from published upper bounds strictly larger than two.
- The arithmetic-operation exponent depends on its exact computational model; bit complexity and implementation performance are not substituted for it.
- No submitted code or source-package attachment was executed, and no optimization calculation was independently rerun.
- The bounded formalization search does not establish nonexistence of private or unindexed work.
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