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Simplicial manifolds · normalization · polytope diameter

Dimension-Four Hirsch: Normalization-Fiber Audit

Collaboration beta

Normalization can split singular local sheets and turn a labeled pseudomanifold into a genuine sphere with more vertices. The source reports reversing two such fibers to diagnose the original singular objects, but neither object is a dimension-four Hirsch counterexample.

S193,S213⟵K13sing
Known results and sources
Landscape card showing a singular pinched tetrahedral complex splitting into a smooth triangulated sphere with more vertices, marked as a diagnostic rather than a counterexample.
Normalization splits singular local sheets; reversing the retained fibers diagnoses why the smaller labeled source is not a manifold.

Research problem

Exact mathematical statement

Reverse the retained consecutive normalization fibers of the two source-reported diameter-11 simplicial 3-spheres on 19 and 21 vertices, reconstruct the corresponding 13-label source complexes, and audit their exact face vectors, dual diameters, edge links, vertex links, and normalization collision behavior.

Revision 10 reports source complexes with

(f0,f1,f2,f3)=(13,66,106,53)and(13,68,110,55),(f_0,f_1,f_2,f_3)=(13,66,106,53) \quad\text{and}\quad (13,68,110,55),

each of dual diameter 11, but with disconnected edge links and nonspherical vertex links. These reconstructed degree-two pseudomanifolds are not combinatorial 3-manifolds, not polytopal counterexamples, and not a proof or disproof of the dimension-four conjecture.

Problem infographic

Problem at a glance

Landscape three-panel explainer distinguishing a degree-two pseudomanifold, normalization into a genuine sphere, and the separate sphere and polytopality gates for a Hirsch counterexample.
High dual diameter survives normalization in the reported examples, but the original 13-label sources fail manifold-link tests and cannot be polytopal counterexamples.

Current mathematical picture

Where work on Dimension-Four Hirsch: Normalization-Fiber Audit stands

Open problem

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

Useful failureFacet-injective quotienting and local Pachner repair

The edge-normal 13-vertex quotient retains six pinched vertex links, and the current work states the rigidity theorem prevents removing all six without collapse or duplication. Normalization-cost optimization, deeper radius-four or contract-then-recover searches on the 16-vertex seed, and different cavity or completion routes remain open.

Route status · Narrowed route
Main reductionCurrent reduction

The route treats normalization fibers as a reversible combinatorial map: split local sheets to genuine larger spheres, retain the consecutive fibers, then reverse them to recover and diagnose the smaller labeled singular sources.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeIndependently replay the reconstruction auditTask status · Ready to work on
Research-record correctionResearch-record correction

We 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.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Dimension-Four Hirsch: Normalization-Fiber Audit in numbers

613retained lines of mathematical investigation613 in the current working snapshot
Argument development
525 · 86%
Explored or eliminated routes
18 · 3%
Computational analysis
52 · 8%
Open obligations
12 · 2%
Definitions and setup
6 · 1%
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.

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: Normalization-Fiber AuditA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.What singular local sheets were split when the two diameter-11 spheres were normalized? — Depends on missing premiseWhat singular local sheetswere split when the twodiameter-11…Current reduction — Depends on missing premiseCurrent reductionClosing target — Depends on missing premiseClosing targetEdge-normal quotient remains nonmanifold — Depends on missing premiseEdge-normal quotient remainsnonmanifoldFirst singular source — Depends on missing premiseFirst singular sourceReversed normalization fibers — Depends on missing premiseReversed normalizationfibersSecond singular source — Depends on missing premiseSecond singular sourceFacet-injective quotienting and local Pachner repair — stoppedFacet-injective quotientingand local Pachner repairIndependently replay the reconstruction audit — OpenIndependently replay thereconstruction auditMinimize normalization cost at construction time — OpenMinimize normalization costat construction timeReach and realize a valid 13-vertex candidate — OpenReach and realize a valid13-vertex candidateNo dimension-four resolution — OpenNo dimension-four resolutionFirst unexplored local layer — OpenFirst unexplored local layer
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 routeFacet-injective quotienting and local Pachner repair

The edge-normal 13-vertex quotient retains six pinched vertex links, and the current work states the rigidity theorem prevents removing all six without collapse or duplication. Normalization-cost optimization, deeper radius-four or contract-then-recover searches on the 16-vertex seed, and different cavity or completion routes remain open.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Independently replay the reconstruction auditSuggested move: Reverse each retained normalization fiber in an independent implementation and verify exact tetrahedra, face vectors, dual diameter, edge links, vertex links, and collision behavior.
Ready to work on
02
Minimize normalization cost at construction timeSuggested move: Search singular diameter-10 or diameter-11 completions while directly penalizing the number of local sheet components and disconnected edge links.
Ready to work on
03
No dimension-four resolution

The source reports neither a 13-vertex polytopal counterexample nor a proof of the full dimension-four conjecture.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
First unexplored local layer

For the current 16-vertex seed, the source reports exhaustive radius-three failure and identifies radius four as the first unexplored direct layer.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Reach and realize a valid 13-vertex candidateSuggested move: Find a 13-vertex diameter-at-least-10 combinatorial 3-sphere and then separately provide an exact rank-5 oriented-matroid or rational-coordinate polytopality certificate.
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 current work's reconstructed singular sources and normalized spheres are not external counterexamples, are not independently reproduced here, and do not change open status.

[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 bounded dimension four or validate a normalization-fiber reconstruction.

[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 this normalization-fiber audit.

    [4]

Formalization opportunities

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

  • Formalization targetFormalize the dimension-four boundary-sphere statement and separate pseudomanifold, combinatorial-sphere, and polytopal-realization gates.
  • Formalization targetIndependently replay each retained normalization fiber and verify the reconstructed singular complexes, link defects, and sphere certificates.
  • Formalization targetAny counterexample claim needs a 13-vertex diameter-at-least-10 combinatorial 3-sphere plus a separate exact polytopality certificate.

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 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
  • reduction1 of 71
  • lemma1 of 71
  • computational claim3 of 73
  • negative result1 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 statementWhat singular local sheets were split when the two diameter-11 spheres were normalized?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementReversed normalization fibersintermediate
  • retained route statementFirst singular sourceintermediate
  • retained route statementSecond singular sourceintermediate
  • retained route statementEdge-normal quotient remains nonmanifoldintermediate
  • 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 failureFacet-injective quotienting and local Pachner repairreported failure
  • Research targetIndependently replay the reconstruction auditopen
  • Research targetMinimize normalization cost at construction timeopen
  • Research targetReach and realize a valid 13-vertex candidateopen
  • Research targetNo dimension-four resolutionopen
  • Research targetFirst unexplored local layeropen
  • ComputationThe source reports reconstructing the two 13-label source complexes and independently checking their closed degree-two structure, face vectors, dual diameters, local links, and collision-free normalization.The recovered objects are reported as singular pseudomanifolds, not manifolds: one has six disconnected two-cycle edge links and the other eight, with six and five nonspherical vertex links respectively. · reported unreproduced
  • Narrowed routeFacet-injective quotienting and local Pachner repairThe edge-normal 13-vertex quotient retains six pinched vertex links, and the current work states the rigidity theorem prevents removing all six without collapse or duplication. Normalization-cost optimization, deeper radius-four or contract-then-recover searches on the 16-vertex seed, and different cavity or completion routes remain open.
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 bridgeIndependently replay the reconstruction audit

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 pointIndependently replay the reconstruction audit

Dimension-Four Hirsch: Normalization-Fiber Audit · ready to start

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Research contextPrepared context for any AI agent

Normalization can split singular local sheets and turn a labeled pseudomanifold into a genuine sphere with more vertices. The source reports reversing two such fibers to diagnose the original singular objects, but neither object is a dimension-four Hirsch counterexample.

  • 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 reconstruction audit.
  • No submitted packet URL was fetched and no packet attachment was executed, compiled, or rendered.
  • The reconstructed singular sources, normalization fibers, sphere certificates, and local searches remain source-reported and were not independently reproduced in this metadata pass.

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