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Polytope diameter · simplicial 4-polytopes · geodesic corridor enumeration

Dimension-Four Hirsch: Minimal Corridor Census

Collaboration beta

A hypothetical minimal dimension-four violation would contain a long dual path of tetrahedral facets. The source reports a small-n census of abstract path skeletons that pass necessary tests, but completing any skeleton to a polytopal sphere—or ruling them all out—remains open.

L=n-3,n∈{8,9,10,11}
Known results and sources
Landscape card with a chain of tetrahedral-facet nodes stretching from one endpoint to another under a diameter bound question, marked open.
In dimension four, the bounded Hirsch question compares facet-dual distance with the number of vertices minus four.

Research problem

Exact mathematical statement

For each n∈{8,9,10,11}n\in\{8,9,10,11\}, enumerate, up to the specified initial-facet relabelings and path reversal, every tetrahedral facet path F0,…,Fn-3F_0,…,F_{n-3} satisfying the seven listed necessary conditions for a minimal dimension-four Hirsch violation.

Revision 10 reports canonical orbit counts

0, 1, 52, 950,\ 1,\ 52,\ 95

for n=8,9,10,11n=8,9,10,11, respectively. This is a finite necessary-condition census, not a proof that any listed skeleton is realizable and not a proof or disproof of the dimension-four Hirsch conjecture.

Problem infographic

Problem at a glance

Landscape three-panel explainer showing a simplicial four-polytope boundary concept, its facet-dual graph, and the distinction between a path skeleton and a realized sphere or polytope.
A long abstract facet path is only a necessary-condition skeleton; manifold completion and polytopal realization are separate gates.

Current mathematical picture

Where work on Dimension-Four Hirsch: Minimal Corridor Census stands

Open problem

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

Useful failurePath-only promotion

The source explicitly retains 52 and 95 as exact path counts while leaving elimination of all 52 and reduction of 95 to 74 reproduction-required. Exact reciprocal link gluing, closure certificates, and the heighted-annulus or quotient-corridor repair theorem remain viable next layers over the retained catalogs.

Route status · Narrowed route
Main reductionCurrent reduction

A dimension-four counterexample can be truncated to a minimal violating facet-dual geodesic of length n-3. The source encodes such paths as stacked-corridor label exchanges, applies span, endpoint, repetition, surplus, and symmetry constraints, then enumerates canonical bit-mask sequences.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeReproduce link-gluing and finite completionTask status · Ready to work on

Work mapped so far

Dimension-Four Hirsch: Minimal Corridor Census in numbers

3.1kretained lines of mathematical investigation3,093 in the current working snapshot
Argument development
2,485 · 80%
Explored or eliminated routes
168 · 5%
Computational analysis
259 · 8%
Open obligations
63 · 2%
Definitions and setup
118 · 4%
8selected mapped statements1routes investigated4open questions4contribution-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.

Proposed byWarren M. HirschProposed1957Status checkedAug 15, 2026
Source recordedAug 15, 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 Dimension-Four Hirsch: Minimal Corridor CensusA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Which small facet-path skeletons survive the necessary conditions for a minimal dimension-four violation? — Depends on missing premiseWhich small facet-pathskeletons survive thenecessary…Current reduction — Depends on missing premiseCurrent reductionDimension-four dual bound — Depends on missing premiseDimension-four dual boundReentry-surplus identity — Depends on missing premiseReentry-surplus identityClosing target — Depends on missing premiseClosing targetReported canonical counts — Depends on missing premiseReported canonical countsReported n=11 topology table — Depends on missing premiseReported n=11 topology tableSeven necessary conditions — Depends on missing premiseSeven necessary conditionsPath-only promotion — stoppedPath-only promotionReproduce link-gluing and finite completion — OpenReproduce link-gluing andfinite completionProve the heighted annulus repair theorem — OpenProve the heighted annulusrepair theoremLift the repair mechanism to all n — OpenLift the repair mechanism toall nRealizability and completion remain open — OpenRealizability and completionremain open
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 routePath-only promotion

The source explicitly retains 52 and 95 as exact path counts while leaving elimination of all 52 and reduction of 95 to 74 reproduction-required. Exact reciprocal link gluing, closure certificates, and the heighted-annulus or quotient-corridor repair theorem remain viable next layers over the retained catalogs.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Reproduce link-gluing and finite completionSuggested move: Use the retained n≤10 catalogs to rebuild link domains, reciprocal gluing, shortcut detection, closure, and independent certificate verification end to end.
Ready to work on
02
Prove the heighted annulus repair theoremSuggested move: Rule out completion of a minimal corridor with s=0 and R=1 to a closed simplicial 3-sphere while preserving an anchored 1-Lipschitz facet height.
Ready to work on
03
Realizability and completion remain open

The catalogs contain necessary-condition skeletons only and do not certify realizability or later link-gluing eliminations.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Lift the repair mechanism to all nSuggested move: Formulate and prove the quotient-corridor repair theorem with exact manifold hypotheses and full dual adjacency, not merely a pure tetrahedral complex.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 15, 2026
Current statusOpen conjecture

The classical bounded Hirsch conjecture is false in general, but the bounded dimension-four case remains open. The private packet's n=8..11 necessary-condition census is not an external resolution and is not treated here as independently reproduced.

[2][3]
External progress

What the literature has established

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

  1. Authoritative summaryEdward Kim's author-maintained page continues to list the bounded question as open in dimensions 4 through 19.[3]
  2. Peer reviewedSantos published a bounded counterexample, disproving the classical Hirsch conjecture in general but not resolving bounded dimension four.[2]
  3. Authoritative summaryThe survey attributes the diameter bound to a 1957 question of Warren Hirsch.[1]
4 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusDimension-Four Hirsch Conjecture
Stronger or generalized formclassical bounded Hirsch conjecture

The classical all-dimensions bounded Hirsch conjecture is false; its bounded dimension-four restriction remains a separate open special case.

[2]
Related problemformal disproof using a higher-dimensional counterexample

The cited Coq project formalizes a known higher-dimensional counterexample and does not settle the dimension-four bounded restriction or this finite corridor census.

[4]

Formal and computational footholds

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

  • formal proof · source linked; not reproduced by ProofAtlasA Formal Disproof of the Hirsch Conjecture

    A Coq formalization checks a higher-dimensional known counterexample to the general conjecture; it is not statement-aligned to bounded dimension four or the n=8..11 corridor census.

    [4]

Formalization opportunities

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

  • Formalization targetFormalize the polarity-aligned dimension-four statement and distinguish abstract simplicial spheres, polytopal spheres, and facet-path skeletons.
  • Formalization targetIndependently replay the exact n=8..11 necessary-condition census and verify its symmetry quotient and retained catalogs.
  • Formalization targetProve the missing link-gluing, manifold-completion, and all-n reduction steps before a finite path census can affect the open conjecture.

Detailed research inventory

Claims, milestones, and routes in the current map

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

6 standing statements2 proposed statements4 open questions1 narrowed routes
Statements by mathematical role8 selected mapped statements
  • theorem candidate1 of 81
  • reduction1 of 81
  • lemma1 of 81
  • equivalence2 of 82
  • computational claim3 of 83
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.22 displayed rows · 1 route included
  • retained route statementWhich small facet-path skeletons survive the necessary conditions for a minimal dimension-four violation?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementDimension-four dual boundintermediate
  • retained route statementReentry-surplus identityintermediate
  • retained route statementSeven necessary conditionsintermediate
  • retained route statementReported canonical countsintermediate
  • retained route statementReported n=11 topology tableintermediate
  • 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
  • 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 failurePath-only promotionreported failure
  • Research targetReproduce link-gluing and finite completionopen
  • Research targetProve the heighted annulus repair theoremopen
  • Research targetLift the repair mechanism to all nopen
  • Research targetRealizability and completion remain openopen
  • ComputationThe source reports enumerating restricted-growth paths through length n-3, applying seven necessary conditions and quotienting by 24 initial-facet permutations plus path reversal.It reports restricted-growth/canonical counts (0,0), (48,1), (2424,52), and (4464,95) for n=8,9,10,11 and exact path-topology distributions; this lane did not rerun the computation. · reported unreproduced
  • Narrowed routePath-only promotionThe source explicitly retains 52 and 95 as exact path counts while leaving elimination of all 52 and reduction of 95 to 74 reproduction-required. Exact reciprocal link gluing, closure certificates, and the heighted-annulus or quotient-corridor repair theorem remain viable next layers over the retained catalogs.
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 bridgeReproduce link-gluing and finite completion

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.

Continue the mathematics

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Prepared starting pointReproduce link-gluing and finite completion

Dimension-Four Hirsch: Minimal Corridor Census · ready to start

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Research contextPrepared context for any AI agent

A hypothetical minimal dimension-four violation would contain a long dual path of tetrahedral facets. The source reports a small-n census of abstract path skeletons that pass necessary tests, but completing any skeleton to a polytopal sphere—or ruling them all out—remains open.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references4 cited works · next context review by Nov 15, 2026

The mathematical context was checked on Aug 15, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    An Update on the Hirsch Conjecturesurvey or monograph · Edward D. Kim, Francisco Santos · Jahresbericht der Deutschen Mathematiker-Vereinigung; University of California repository · 2010 · DOI 10.1365/s13291-010-0001-8 · accessed Aug 14, 2026
  2. 2
    A counterexample to the Hirsch conjecturepeer reviewed result · Francisco Santos · Annals of Mathematics 176(1), 383–412 · 2012 · DOI 10.4007/annals.2012.176.1.7 · accessed Aug 14, 2026
  3. 3
    Undergraduate Research — Hirsch conjectureauthoritative webpage · Edward D. Kim · Edward D. Kim · maintained · accessed Aug 14, 2026
  4. 4
    A Formal Disproof of the Hirsch Conjectureformalization · Xavier Allamigeon, Quentin Canu, Pierre-Yves Strub · arXiv · 2023 · ARXIV 2301.04060 · accessed Aug 14, 2026

Important qualifications

  • This bounded primary-source pass is not an exhaustive literature, priority, or mathematical review.
  • The classical bounded Hirsch conjecture is false in general; this workspace concerns only its still-open bounded dimension-four restriction and a narrower source-reported path census.
  • No submitted packet URL was fetched and no packet attachment was executed, compiled, or rendered.
  • The n=8..11 census, its counts, and its topology tables remain source-reported and were not independently reproduced in this metadata pass.

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