ProofAtlas

Determinants of sign matrices

Hadamard maximal determinant problem, order 23

Collaboration beta

How large can the determinant of a 23 by 23 sign matrix be? The source reports a lower construction and a universal upper endpoint, but still needs exhaustive coverage and exact sign-realization exclusions to establish the maximum.

D23=maxR∈{-1,1}23×23|detR|,662671875≤D23/222≤699323635
Known results and sources
Illustration: A schematic 23-by-23 grid and the permitted entries +1 and −1 frame the question of the largest absolute determinant.
Illustration: Choose a 23-by-23 sign matrix and maximize |det R|; the grid is schematic.

Research problem

Exact mathematical statement

Determine the largest absolute determinant of a 23 by 23 matrix whose entries are all +1 or -1:

D23=maxR∈{-1,1}23×23|detR|.D_{23}=\max_{R\in\{-1,1\}^{23\times23}}|\det R|.

The source reports the interval 662,671,875 <= D23/2^22 <= 699,323,635, with lower value 2,779,447,296,000,000 and upper value 2,933,175,919,575,040 before normalization. Equality is not established. This target concerns the order-23 maximum, not existence of Hadamard matrices in every order divisible by four.

Problem infographic

Problem at a glance

Illustration: A 23-by-23 schematic sign matrix, the maximum-determinant definition, and manuscript-reported normalized bounds show the unresolved optimization gap.
Illustration: The manuscript reports 662671875≤D₂₃/2²²≤699323635; the exact maximum remains unresolved.

Current mathematical picture

Where work on Hadamard maximal determinant problem, order 23 stands

Open problem

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

Useful failureForcing the unchanged coarse 3P3 margin to work at the record

The reported completion has normalized determinant 92,505,999,375,000,000, forcing 5U22 above the record threshold and a margin at most -23,395,982,958,984,375. A genuinely refined class with a complete cover and its own bound, or the margin-free record-level merger route, remains outside this obstruction.

Route status · Narrowed route
Main reductionHD23-T94: conditional small core

Conditional on an independently established cluster-deletion number of at least four, the source reports an induced minimal four-deletion obstruction on at most fifteen vertices. This does not prove the missing global four-deletion hypothesis or eliminate the live three-deletion branch.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeProve complete marked reverse-merger generation with every required restored profile and refinement of other classes.Task status · Ready to work on

Work mapped so far

Hadamard maximal determinant problem, order 23 in numbers

1.2kretained lines of mathematical investigation1,209 in the current working snapshot
Argument development
792 · 66%
Explored or eliminated routes
79 · 7%
Computational analysis
134 · 11%
Open obligations
76 · 6%
Definitions and setup
128 · 11%
9selected 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.

Status checkedSep 7, 2026
Source recordedSep 7, 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

14 selected steps

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

14 selected steps

Scroll horizontally to explore the route

Working route overview for Hadamard maximal determinant problem, order 23A selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Find the exact maximum determinant at order 23. — Depends on missing premiseFind the exact maximumdeterminant at order 23.411 required core classes — Depends on missing premise411 required core classesCurrent reduction — Depends on missing premiseCurrent reductionHD23-T70–T71: merger paths — Depends on missing premiseHD23-T70–T71: merger pathsHD23-T94: conditional small core — Depends on missing premiseHD23-T94: conditional smallcoreClosing target — Depends on missing premiseClosing targetHD23-T92: reported universal endpoint — Depends on missing premiseHD23-T92: reported universalendpointHD23-T93: record-level merger costs — Depends on missing premiseHD23-T93: record-levelmerger costsStrict binary improvement restrictions — Depends on missing premiseStrict binary improvementrestrictionsForcing the unchanged coarse 3P3 margin to work at the record — stoppedForcing the unchanged coarse3P3 margin to work at therecordDiscarding nonsquare intermediates or reversing only one unchanged class partition — stoppedDiscarding nonsquareintermediates or reversingonly…Prove complete marked reverse-merger generation with every required restored profile and refinement of other classes. — OpenProve complete markedreverse-merger generationwith…Turn surviving necessary arithmetic and compatibility conditions into complete simultaneous 23-column sign-realization decisions. — OpenTurn surviving necessaryarithmetic and compatibilityconditions…Resolve a live candidate family without using T80–T83, T86 or T88 as pruning premises before their complete coverage artifacts exist. — OpenResolve a live candidatefamily without usingT80–T83,…
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

2 recorded
Narrowed routeForcing the unchanged coarse 3P3 margin to work at the record

The reported completion has normalized determinant 92,505,999,375,000,000, forcing 5U22 above the record threshold and a margin at most -23,395,982,958,984,375. A genuinely refined class with a complete cover and its own bound, or the margin-free record-level merger route, remains outside this obstruction.

Route status · Narrowed route
Narrowed routeDiscarding nonsquare intermediates or reversing only one unchanged class partition

Only 16 of the 894 above-record clique partitions are squares, but the source requires all 894 roots. Its forward merger can erase distinctions that a complete reverse step must restore. Retain complete relaxed ancestry and restore every permitted profile before applying exact sign-realization tests to terminal candidates.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Prove complete marked reverse-merger generation with every required restored profile and refinement of other classes.Suggested move: Suggested next step: Write the canonical parent and augmentation rules for one exact reverse level, keeping all 894 roots and nonsquare ancestors, then identify the coverage and duplicate-accounting statements that level requires.
Ready to work on
02
Turn surviving necessary arithmetic and compatibility conditions into complete simultaneous 23-column sign-realization decisions.Suggested move: Suggested next step: Specify the full sign-type domain and framed reconstruction equations for one retained candidate, with the record matrix as a positive control and an exact terminal witness required for exclusion.
Ready to work on
03
Resolve a live candidate family without using T80–T83, T86 or T88 as pruning premises before their complete coverage artifacts exist.Suggested move: Suggested next step: Choose one missing family deliberately and enumerate its required input domain, overlap restoration and terminal certificates before deciding whether reconstruction is preferable to the merger route.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026
Current statusOpen problem

The cited maximal-determinant records do not establish the exact optimum for real 23×23 sign matrices. OEIS retains the corresponding order-22 zero-one value 662671875 with a question mark, while the order-21 zero-one value has been confirmed. A construction attaining a lower bound is not a proof of optimality.

[2][1]
External progress

What the literature has established

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

  1. Authoritative summaryBrent's OEIS update confirms the order-21 zero-one maximum, corresponding to sign-matrix order 22. It does not confirm the next order.[2]
  2. PreprintOrrick and coauthors report improved sign-matrix determinant constructions, including order 23, and provide the matrices in their paper.[1]
3 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusHadamard maximal determinant problem, order 23
Equivalent formulationMaximum determinant of 22×22 zero-one matrices

OEIS records g(n)=2^(n−1)a(n−1), so the sign-matrix order-23 optimum equals 2^22 times the zero-one order-22 optimum.

[2]
Related problemHadamard-matrix existence

OEIS distinguishes determinant upper bounds and their equality conditions. The local order-23 maximum is a separate target from existence of Hadamard matrices at orders divisible by four.

[2]

Formal and computational footholds

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

  • dataset · source linked; not reproduced by ProofAtlasPublished sign-matrix determinant constructions

    The 2003 preprint contains explicit matrices, including an order-23 lower-bound construction. Matrix bytes and determinants were not independently checked here.

    [1]

Formalization opportunities

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

  • Formalization targetA proposed formal statement should quantify over all 23×23 real sign matrices and preserve absolute determinant and normalization conventions.
  • Formalization targetAny exclusion proof needs complete candidate coverage and exact determinant or Gram-matrix reasoning; a matrix example alone supplies only a lower bound.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions2 narrowed routes
Statements by mathematical role9 selected mapped statements
  • theorem candidate1 of 91
  • reduction4 of 94
  • lemma3 of 93
  • computational claim1 of 91
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.25 displayed rows · 2 routes included
  • retained route statementFind the exact maximum determinant at order 23.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementHD23-T92: reported universal endpointintermediate
  • retained route statement411 required core classesintermediate
  • retained route statementStrict binary improvement restrictionsintermediate
  • retained route statementHD23-T70–T71: merger pathsintermediate
  • retained route statementHD23-T93: record-level merger costsintermediate
  • retained route statementHD23-T94: conditional small coreintermediate
  • 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
  • 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 failureForcing the unchanged coarse 3P3 margin to work at the recordreported failure
  • Useful failureDiscarding nonsquare intermediates or reversing only one unchanged class partitionreported failure
  • Research targetProve complete marked reverse-merger generation with every required restored profile and refinement of other classes.open
  • Research targetTurn surviving necessary arithmetic and compatibility conditions into complete simultaneous 23-column sign-realization decisions.open
  • Research targetResolve a live candidate family without using T80–T83, T86 or T88 as pruning premises before their complete coverage artifacts exist.open
  • ComputationThe source reports two complete radius-three traversals, each over 641,656,806 descriptions around 350 centers, and exact arithmetic checks on their high outputs.Both traversals reportedly agree on 2,150 above-cutoff descriptions and zero high squares; 293 is the number of distinct high determinant values. The v12 second audit checked input domains, algebra, symmetry and boundary searches, without rerunning both full neighborhoods. · reported unreproduced
  • Narrowed routeForcing the unchanged coarse 3P3 margin to work at the recordThe reported completion has normalized determinant 92,505,999,375,000,000, forcing 5U22 above the record threshold and a margin at most -23,395,982,958,984,375. A genuinely refined class with a complete cover and its own bound, or the margin-free record-level merger route, remains outside this obstruction.
  • Narrowed routeDiscarding nonsquare intermediates or reversing only one unchanged class partitionOnly 16 of the 894 above-record clique partitions are squares, but the source requires all 894 roots. Its forward merger can erase distinctions that a complete reverse step must restore. Retain complete relaxed ancestry and restore every permitted profile before applying exact sign-realization tests to terminal candidates.
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 bridgeProve complete marked reverse-merger generation with every required restored profile and refinement of other classes.

2 approaches have 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 pointProve complete marked reverse-merger generation with every required restored profile and refinement of other classes.

Hadamard maximal determinant problem, order 23 · 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

How large can the determinant of a 23 by 23 sign matrix be? The source reports a lower construction and a universal upper endpoint, but still needs exhaustive coverage and exact sign-realization exclusions to establish the maximum.

  • 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 Dec 7, 2026

The mathematical context was checked on Sep 7, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    New lower bounds for the maximal determinant problempreprint · William P. Orrick, Bruce Solomon, Roland Dowdeswell, Warren D. Smith · arXiv · 2003-04-25 · ARXIV math/0304410 · accessed Sep 7, 2026
  2. 2
    A003432: Hadamard maximal determinant problem for real zero-one matricessoftware or dataset · OEIS Foundation · OEIS A003432 · accessed Sep 7, 2026
  3. 3
    A003433: Hadamard maximal determinant problem for real sign matricessoftware or dataset · OEIS Foundation · OEIS A003433 · accessed Sep 7, 2026

Important qualifications

  • This is a bounded external-context collection, not proof checking, an independent mathematical review, legal clearance, or publication authority.
  • No source programs, datasets, or certificates were executed. A linked resource does not imply reproduction.
  • No exact checked formalization was confirmed in the scoped search; an empty list does not establish nonexistence.
  • Unassigned history and recognition fields mean this collection did not establish them. Proposed formal prerequisites are planning tasks, not an exhaustive library inventory.
  • No original proposer or date is assigned for this particular order. Hadamard's role in the general determinant problem does not establish authorship of the order-23 question.
  • The external snapshot does not independently confirm the current work's numeric upper bound or prove the candidate lower value optimal.
  • The historical Indiana matrix archive was not successfully verified as a live resource. Readiness points to the accessible paper instead; MSC codes come from that paper.

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