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Combinatorial design theory · orthogonal sign matrices · exact constructions

Hadamard Conjecture

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Does a square plus-or-minus-one matrix with mutually orthogonal rows exist in every positive order divisible by four? The source reports audited computations and reductions for a proposed lattice-switching program, but it explicitly leaves the universal construction open.

H∈{±1}4m×4m,HHT=4mI4m
Listed inEpoch FrontierMath Open Problems: Hadamard Matrix of Order 668
Known results and sources
Landscape card with two luminous geometric directions meeting at a marked right angle over a balanced square-grid motif, plus an open-status label.
Hadamard matrices have ±1 entries and mutually orthogonal rows; the conjecture asks for one at every order divisible by four.

Research problem

Exact mathematical statement

For every positive integer m, does there exist a sign matrix H of order 4m whose rows are mutually orthogonal?

H∈{±1}4m×4m,HHT=4mI4m.H\in\{\pm1\}^{4m\times4m},\qquad HH^{\mathsf T}=4mI_{4m}.

Revision v9 makes no claim of a full proof. Its reported finite verifications and residual-lattice reductions remain source-reported evidence for a route, not a universal construction.

Problem infographic

Problem at a glance

Three-panel landscape explainer with a correct order-four Hadamard matrix, the orthogonality equation, and the open every-order question.
The order-four example illustrates the defining orthogonality constraint without claiming a universal construction.

Current mathematical picture

Where work on Hadamard Conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureSource-reported limitation

Row and column weight three in a binary 12-by-12 support matrix does not by itself force any nonzero kernel, much less the required constant-weight code. Prove universal near-saturation or bypass it, establish a terminal escape theorem for every line-resistant full-span prime E8 trap, and incorporate signed-frame realizability to control defect 12 and higher residual plateaus.

Route status · Narrowed route
Main reductionProper-root-span restriction

Inside the current work, the exact parity analysis reports that a proper E8 root span forces multiplier m to be 5 or 9.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeResolve universal near-saturationTask status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Hadamard Conjecture in numbers

2.9kretained lines of mathematical investigation2,906 in the current working snapshot
Argument development
2,488 · 86%
Explored or eliminated routes
25 · 1%
Computational analysis
89 · 3%
Open obligations
123 · 4%
Definitions and setup
181 · 6%
7selected mapped statements1routes investigated5open questions5contribution-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.

Status checkedAug 14, 2026
Source recordedAug 14, 2026

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

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

13 selected steps

Scroll horizontally to explore the route

Working route overview for Hadamard ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Does a Hadamard matrix exist at every positive order divisible by four? — Depends on missing premiseDoes a Hadamard matrix existat every positive orderdivisible…Current reduction — Depends on missing premiseCurrent reductionProper-root-span restriction — Depends on missing premiseProper-root-span restrictionClosing target — Depends on missing premiseClosing targetCubic support is insufficient — Depends on missing premiseCubic support isinsufficientMarked defect-8 parity table — Depends on missing premiseMarked defect-8 parity tableOrthogonal sign-matrix target — Depends on missing premiseOrthogonal sign-matrixtargetSource-reported limitation — stoppedSource-reported limitationResolve universal near-saturation — OpenResolve universalnear-saturationProve full-span prime-E8 escape — OpenProve full-span prime-E8escapeControl realizable defect-12 supports — OpenControl realizable defect-12supportsUniversal construction remains open — OpenUniversal constructionremains openMissing terminal enumerator — OpenMissing terminal enumerator
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

1 recorded
Narrowed routeSource-reported limitation

Row and column weight three in a binary 12-by-12 support matrix does not by itself force any nonzero kernel, much less the required constant-weight code. Prove universal near-saturation or bypass it, establish a terminal escape theorem for every line-resistant full-span prime E8 trap, and incorporate signed-frame realizability to control defect 12 and higher residual plateaus.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Resolve universal near-saturationSuggested move: Construct a partial Hadamard matrix of defect at most eleven for every odd multiplier, or provide a route that avoids this gate with equally explicit hypotheses.
Ready to work on
02
Prove full-span prime-E8 escapeSuggested move: Derive a universal phase-capacity overload or multiline substitute that yields a terminal compatible subspace for every line-resistant saturated prime trap.
Ready to work on
03
Universal construction remains open

The current work explicitly makes no claim of solving the Hadamard Conjecture.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Missing terminal enumerator

A complete independent terminal enumerator is still required to reproduce the large support counts and packet distinctness claims.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Control realizable defect-12 supportsSuggested move: Classify cubic support matrices admitting signed tight-frame completions and determine whether allowed neutral switches force the needed constant-weight kernel code.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The universal conjecture remains open: the cited sources do not provide a construction for every positive order divisible by four. Epoch’s official page provisionally marks its separate finite order-668 task ‘Solved (AI)’ based on a reported construction and explicitly distinguishes that task from the still-open universal conjecture.

[1][2][3]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. Authoritative summaryEpoch provisionally marked its finite order-668 task ‘Solved (AI)’ after a reported construction, while its page continued to state that the universal Hadamard conjecture remains open.[2]
  2. PreprintRamos, Hulak, and de Queiroz excluded most common-multiplier symmetries for length-333 Legendre pairs, a structured path to order 668, while explicitly leaving unrestricted existence open.[3]
  3. Authoritative summaryEpoch’s official open-problem benchmark selected construction of an order-668 Hadamard matrix as a concrete finite task related to the universal conjecture.[2]
  4. PreprintCati and Pasechnik published an open-source construction database covering known Hadamard orders through a substantial finite range and updated construction tables.[1]
3 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusHadamard Conjecture
Solved special caseknown finite Hadamard orders and construction families

Many individual orders and broad construction families are known and implemented, but the database does not supply one construction for every multiple of four.

[1]
Dependency or reductionmultiplier-invariant Legendre pairs of length 333

A Legendre pair of length 333 would yield a Hadamard matrix of order 668; the cited work constrains only pairs invariant under common multiplier subgroups.

[3]
Solved special caseHadamard matrix of order 668

Epoch provisionally lists its order-668 construction task as solved by AI; this is one finite order and does not resolve the universal every-order conjecture.

[2]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

  • software · source linked; not reproduced by ProofAtlasSageMath Hadamard construction database

    The source describes open-source SageMath implementations of known constructions and updated finite-order coverage tables.

    [1]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetA checked universal statement must quantify over every positive multiplier m and distinguish it from construction of any one order.
  • Formalization targetAny admitted finite construction must bind exact matrix bytes and verify all entries and the complete Gram identity.
  • Formalization targetThe private packet’s lattice and switching reductions require theorem-level hypothesis alignment and independent verification.

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. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
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.

Corrected the research recordCorrection note

Correction details

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 statements5 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction2 of 72
  • lemma2 of 72
  • computational claim1 of 71
  • counterexample1 of 71
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included
  • retained route statementDoes a Hadamard matrix exist at every positive order divisible by four?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementOrthogonal sign-matrix targetintermediate
  • retained route statementMarked defect-8 parity tableintermediate
  • retained route statementProper-root-span restrictionintermediate
  • retained route statementCubic support is insufficientintermediate
  • 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 counterexample narrows one intermediate strategy; it does not challenge the open conjecture.challenges · 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 failureSource-reported limitationreported failure
  • Research targetResolve universal near-saturationopen
  • Research targetProve full-span prime-E8 escapeopen
  • Research targetControl realizable defect-12 supportsopen
  • Research targetUniversal construction remains openopen
  • Research targetMissing terminal enumeratoropen
  • ComputationThe current work reports rerunning Appendices A–G, I, and J, but says the source for several large terminal counts and the all-line-pair census is absent; none of those private programs was executed in this staging lane.The embedded reruns reportedly certify exact root, code, packet, parity-kernel, and warning-example statements, while not independently regenerating all terminal-plane counts or all 585 packet distinctions. · reported unreproduced
  • Narrowed routeSource-reported limitationRow and column weight three in a binary 12-by-12 support matrix does not by itself force any nonzero kernel, much less the required constant-weight code. Prove universal near-saturation or bypass it, establish a terminal escape theorem for every line-resistant full-span prime E8 trap, and incorporate signed-frame realizability to control defect 12 and higher residual plateaus.
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

Priority open bridgeResolve universal near-saturation

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

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 pointResolve universal near-saturation

Hadamard Conjecture · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Does a square plus-or-minus-one matrix with mutually orthogonal rows exist in every positive order divisible by four? The source reports audited computations and reductions for a proposed lattice-switching program, but it explicitly leaves the universal construction open.

  • 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.

  1. 1
    A database of constructions of Hadamard matricessoftware or dataset · Matteo Cati, Dmitrii V. Pasechnik · arXiv and SageMath · 2024-11-28 · ARXIV 2411.18897 · accessed Aug 14, 2026
  2. 2
    Hadamard Matrix of Order 668maintained problem list · Epoch AI · Epoch AI FrontierMath Open Problems · 2026 · accessed Aug 14, 2026
  3. 3
    Multiplier obstructions for Legendre pairs of length 333preprint · Arthur F. Ramos, David B. Hulak, Ruy J. G. B. de Queiroz · arXiv · 2026-07-22 · ARXIV 2607.20765 · accessed Aug 14, 2026

Important qualifications

  • This source-separated metadata does not verify any mathematical claim in the private packet or grant review, credit, publication, or deployment authority.
  • Epoch’s official page provisionally marks its finite order-668 construction task ‘Solved (AI)’ based on a reported solution, while explicitly stating that the universal Hadamard conjecture remains open.
  • The provisional finite-task classification is an authoritative-list status, not independent verification by ProofAtlas and not evidence that the universal conjecture is solved.
  • Constructing one or finitely many previously unknown orders would not prove the universal Hadamard Conjecture.
  • The current work’s defect-eight, prime-trap, switching, and residual-lattice claims require independent exact review and are not externally verified by these sources.
  • No alleged construction script or packet attachment was fetched or executed.

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