ProofAtlas

Smooth four-manifolds · spin topology · intersection forms · Pin(2)-equivariant methods

11/8 Conjecture

Collaboration beta

Must a smooth closed spin four-manifold have at least eleven eighths as much second homology as the magnitude of its signature? The general inequality is still open.

b2(X)≥118|σ(X)|
Known results and sources
A dark scientific visualization of a smooth four-manifold above paired E8 lattice clusters, hyperbolic-plane tiles, and an unresolved diagonal inequality frontier.
The 11/8 Conjecture compares the second Betti number and signature of a smooth spin four-manifold; the sharp frontier remains open.

Research problem

Exact mathematical statement

For every closed, oriented, smooth spin four-manifold X, prove

b2(X)≥118|σ(X)|.b_2(X) \ge \frac{11}{8}|\sigma(X)|.

After the source's orientation and b₁=0 normalization Q_X≅2p(−E₈)⊕qH, the same target becomes q≥3p. the source is explicit that its finite-dimensional calculations do not yet establish the manifold theorem.

Problem infographic

Problem at a glance

Three-panel 11/8 Conjecture explainer showing a spin four-manifold and normalized intersection form, the b2 versus signature frontier through the origin, and the exact q equals 3p lattice with open case (4,11).
The exact manifold inequality becomes q≥3p after normalization. Packet-reported finite-model cases illuminate the algebraic boundary but do not prove the manifold conjecture.

Current mathematical picture

Where work on 11/8 Conjecture stands

Open conjecture

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

Useful failureScalar trace and eigenvalue-sum closure

The current work records an explicit feasible scalar principal-angle example satisfying those inequalities. Non-scalar contact geometry, discriminant information, and the analytic bridge remain viable, separately scoped targets.

Route status · Narrowed route
Main reductionCritical finite-model case

The current work's p=4 finite-model work leaves q=11 as the sole critical possibility, subject to further contact constraints.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeExclude the balanced simplex branch and every minimal N≥5 contact circuit in the critical finite model (p,q)=(4,11).Task 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

11/8 Conjecture in numbers

2.2kretained lines of mathematical investigation2,213 in the current working snapshot
Argument development
1,881 · 85%
Explored or eliminated routes
68 · 3%
Computational analysis
58 · 3%
Open obligations
63 · 3%
Definitions and setup
143 · 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.

Proposed byYukio MatsumotoProposed1982Status 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 11/8 ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.The sharp 11/8 lower bound for spin four-manifolds remains open. — Depends on missing premiseThe sharp 11/8 lower boundfor spin four-manifoldsremains…Critical finite-model case — Depends on missing premiseCritical finite-model caseCurrent reduction — Depends on missing premiseCurrent reductionExact manifold statement — Depends on missing premiseExact manifold statementNormalized sharp line — Depends on missing premiseNormalized sharp lineClosing target — Depends on missing premiseClosing targetFinite-model factorization — Depends on missing premiseFinite-model factorizationScalar trace and eigenvalue-sum closure — stoppedScalar trace andeigenvalue-sum closureExclude the balanced simplex branch and every minimal N≥5 contact circuit in the critical finite model (p,q)=(4,11). — OpenExclude the balanced simplexbranch and every minimal N≥5contact…Prove the general sharp finite-dimensional implication Pμ_p(z)≠0 for z≠0 implies rank P≥3p. — OpenProve the general sharpfinite-dimensionalimplication…Build the coherent analytic unstable bridge from the Seiberg–Witten monopole map to the zero-free quadratic moment system. — OpenBuild the coherent analyticunstable bridge from theSeiberg–Witten…Analytic production gate — OpenAnalytic production gateGeneral sharp rank theorem — OpenGeneral sharp rank theorem
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 routeScalar trace and eigenvalue-sum closure

The current work records an explicit feasible scalar principal-angle example satisfying those inequalities. Non-scalar contact geometry, discriminant information, and the analytic bridge remain viable, separately scoped targets.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Exclude the balanced simplex branch and every minimal N≥5 contact circuit in the critical finite model (p,q)=(4,11).Suggested move: Use non-coordinate endpoints and the full quartic discriminant inequality, then treat the corrected five-contact structural cases without collapsing them into one Cartan argument.
Ready to work on
02
Prove the general sharp finite-dimensional implication Pμ_p(z)≠0 for z≠0 implies rank P≥3p.Suggested move: Separate low-rank avoidance, contact-certificate size, and general-p induction into independently auditable lemmas before claiming the sharp bound.
Ready to work on
03
Analytic production gate

Produce the required zero-free homogeneous quadratic system coherently and unstably from the monopole map.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
General sharp rank theorem

Prove in general that zero-freeness of Pμ_p forces rank P≥3p.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Build the coherent analytic unstable bridge from the Seiberg–Witten monopole map to the zero-free quadratic moment system.Suggested move: Control the mixed term ρ(a)φ, the two-scale reduction, action conventions, and unstable desuspension without importing the finite-model conclusion as an analytic premise.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The full 11/8 inequality remains open in the checked sources. Furuta's 10/8 theorem and the later 10/8+4 theorem are substantial weaker bounds and are not recorded as solutions of the 11/8 conjecture.

[4][2][3]
External progress

What the literature has established

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

  1. Peer reviewedHopkins, Lin, Shi, and Xu proved a 10/8+4 theorem by resolving a related Pin(2)-equivariant stable-map problem.[3]
  2. Peer reviewedFuruta proved the weaker 10/8 theorem for smooth closed spin four-manifolds, a major step toward the 11/8 bound.[2]
  3. Historical sourceMatsumoto's paper is the historical source to which the 11/8 conjecture is attributed.[1]
4 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focus11/8 conjecture
Weaker or relaxed formFuruta 10/8 theorem

Furuta's 10/8 theorem supplies a weaker lower bound than the conjectural 11/8 inequality.

[2]
Weaker or relaxed form10/8+4 theorem

The 10/8+4 theorem improves the additive constant in the known bound without reaching the full 11/8 coefficient.

[3]

Formalization opportunities

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

  • Formalization targetNo checked formal statement alignment for the smooth closed spin four-manifold inequality was located.
  • Formalization targetNo formal library support for the complete gauge-theoretic or Pin(2)-equivariant argument was assessed.

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

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
  • equivalence2 of 72
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 statementThe sharp 11/8 lower bound for spin four-manifolds remains open.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact manifold statementintermediate
  • retained route statementNormalized sharp lineintermediate
  • retained route statementFinite-model factorizationintermediate
  • retained route statementCritical finite-model caseintermediate
  • 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 failureScalar trace and eigenvalue-sum closurereported failure
  • Research targetExclude the balanced simplex branch and every minimal N≥5 contact circuit in the critical finite model (p,q)=(4,11).open
  • Research targetProve the general sharp finite-dimensional implication Pμ_p(z)≠0 for z≠0 implies rank P≥3p.open
  • Research targetBuild the coherent analytic unstable bridge from the Seiberg–Witten monopole map to the zero-free quadratic moment system.open
  • Research targetAnalytic production gateopen
  • Research targetGeneral sharp rank theoremopen
  • ComputationThe current work records exact numerical checks for its n=3 and n=4 fourth-moment decompositions and a feasible scalar principal-angle model; no submitted attachment was executed by ProofAtlas.These are source-reported internal checks only. They support route triage but carry no independent reproduction, proof, or acceptance effect in this staging record. · reported unreproduced
  • Narrowed routeScalar trace and eigenvalue-sum closureThe current work records an explicit feasible scalar principal-angle example satisfying those inequalities. Non-scalar contact geometry, discriminant information, and the analytic bridge remain viable, separately scoped targets.
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 bridgeExclude the balanced simplex branch and every minimal N≥5 contact circuit in the critical finite model (p,q)=(4,11).

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

Contribute

ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

Read-only beta · actions unavailable
Prepared starting pointExclude the balanced simplex branch and every minimal N≥5 contact circuit in the critical finite model (p,q)=(4,11).

11/8 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

Must a smooth closed spin four-manifold have at least eleven eighths as much second homology as the magnitude of its signature? The general inequality is still open.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data
Hosted agentRun this task with a hosted agent

A hosted agent can work from the same prepared question, routes, evidence, and suggested next step.

Your own AI agentConnect an outside research agent

Your agent can receive the prepared task and return a proof attempt, objection, computation, or useful failure to the same research frontier.

Sources and references4 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
    On the bounding genus of homology 3-spheresoriginal source · Yukio Matsumoto · Faculty of Science, The University of Tokyo · 1982-08-16 · DOI 10.15083/00039575 · MR MR0672065 · accessed Aug 14, 2026
  2. 2
    Monopole equation and the 11/8-conjecturepeer reviewed result · Mikio Furuta · Mathematical Research Letters · 2001 · DOI 10.4310/MRL.2001.v8.n3.a5 · accessed Aug 14, 2026
  3. 3
    Intersection Forms of Spin 4-Manifolds and the Pin(2)-Equivariant Mahowald Invariantpeer reviewed result · Michael J. Hopkins, Jianfeng Lin, XiaoLin Danny Shi, Zhouli Xu · Communications of the American Mathematical Society · 2022 · ARXIV 1812.04052 · DOI 10.1090/cams/4 · accessed Aug 14, 2026
  4. 4
    Four-dimensional topologyauthoritative webpage · Ciprian Manolescu · Stanford University · accessed Aug 14, 2026

Important qualifications

  • This was a bounded source check, not a systematic literature review.
  • The current open-status statement relies on an authoritative university-hosted summary plus the checked primary milestones; it is not a proof that no later literature exists.
  • No formalization, proof artifact, software, or independently reproduced computation was identified in this pass.

Continue exploring

Compare another research frontier

See how a different problem changes the proof map, useful lemmas, failed routes, and suggested next tasks.

Explore all research workspaces

Expanded visual

Open original image