ProofAtlas

Discrete geometry and spherical codes

Five-dimensional kissing number

Collaboration beta

Forty equal spheres can touch one equal central sphere in five dimensions, while current cited work proves only that no more than forty-four can do so. The exact maximum remains open.

τ5=?40
Known results and sources
Equal-sized ivory contact-direction beads lie on a translucent green projection sphere around a small gold origin, with paired gold directions behind them.
The equal-sized beads abstract contact directions in a spherical code; their representative arrangement makes no exact count or optimality claim.

Research problem

Exact mathematical statement

Determine the largest NN for which unit vectors x1,…,xN∈S4⊂ℝ5x_1,…,x_N\in S^4\subset\mathbb R^5 satisfy

⟨xi,xj⟩≤12(i≠j).\langle x_i,x_j\rangle\le \frac12\qquad(i\ne j).

The D5D_5 roots give N=40N=40. The governing source and current external metadata preserve the open interval 40≤τ5≤4440\le\tau_5\le44; neither establishes τ5=40\tau_5=40.

Problem infographic

Problem at a glance

Equal-sized contact-direction beads on a translucent projection sphere, an abstract separation cap, and paired antipodal directions form a three-part landscape.
Contact directions, their separation constraint, and a D5 motif explain the open optimization problem without depicting unequal physical spheres or claiming forty is maximal.

Current mathematical picture

Where work on Five-dimensional kissing number stands

Open problem

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

Useful failureExact-antipode substitution without an error theorem

Local approximate templates may disagree, and midpoint errors remain unless exact antipodality or a quantitative transfer theorem is established. A common-template stability theorem with explicit midpoint and cap radii remains the highest-leverage sparse-branch route.

Route status · Narrowed route
Main reductionForced negative and matching structure

The source reports that the negative graph of a hypothetical forty-one-point code is triangle-free, nonbipartite, and has at least twenty-three edges, and that deleting a forced strong pair yields exactly eight paired rank-five matching branches.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeProve quantitative common-template D5 stability for the eighteen sparse-branch axes with an explicit usable radius.Task status · Ready to work on

Work mapped so far

Five-dimensional kissing number in numbers

948retained lines of mathematical investigation948 in the current working snapshot
Argument development
777 · 82%
Explored or eliminated routes
11 · 1%
Computational analysis
68 · 7%
Open obligations
48 · 5%
Definitions and setup
44 · 5%
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.

Status checkedAug 26, 2026
Source recordedAug 26, 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 Five-dimensional kissing numberA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Can any forty-one-point spherical code satisfy the five-dimensional kissing constraint? — Depends on missing premiseCan any forty-one-pointspherical code satisfy thefive-dimensional…Current reduction — Depends on missing premiseCurrent reductionExact spherical-code target — Depends on missing premiseExact spherical-code targetForced negative and matching structure — Depends on missing premiseForced negative and matchingstructureClosing target — Depends on missing premiseClosing targetForty-point D5 construction — Depends on missing premiseForty-point D5 constructionSecond-audit corrections — Depends on missing premiseSecond-audit correctionsSparsest branch evidence — Depends on missing premiseSparsest branch evidenceExact-antipode substitution without an error theorem — stoppedExact-antipode substitutionwithout an error theoremProve quantitative common-template D5 stability for the eighteen sparse-branch axes with an explicit usable radius. — OpenProve quantitativecommon-template D5 stabilityfor…Supply a sharp capacity or exact certificate for the exceptional C7 plus seventeen-edge branch and other twenty-four-edge structures. — OpenSupply a sharp capacity orexact certificate for theexceptional…Exclude higher-edge graphs and every remaining paired rank-five branch, not only the twenty-three-edge sparse case. — OpenExclude higher-edge graphsand every remaining pairedrank-five…Quantitative D5 stability — OpenQuantitative D5 stability
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 routeExact-antipode substitution without an error theorem

Local approximate templates may disagree, and midpoint errors remain unless exact antipodality or a quantitative transfer theorem is established. A common-template stability theorem with explicit midpoint and cap radii remains the highest-leverage sparse-branch route.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Prove quantitative common-template D5 stability for the eighteen sparse-branch axes with an explicit usable radius.Suggested move: Enumerate signed near-zero and near-half templates up to symmetry, reject non-D5 templates with exact rank and frame constraints, and prove perturbative uniqueness.
Ready to work on
02
Supply a sharp capacity or exact certificate for the exceptional C7 plus seventeen-edge branch and other twenty-four-edge structures.Suggested move: Build exact projective-circle quadrature or a rational certificate that preserves odd-cycle closure and rank-five geometry.
Ready to work on
03
Quantitative D5 stability

The aggregate near-design and root-alphabet defects have not yet been converted into one common D5 template with an explicit closing radius.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Exclude higher-edge graphs and every remaining paired rank-five branch, not only the twenty-three-edge sparse case.Suggested move: Enumerate higher-edge defect cores and couple their component capacities to exact rank-five or pair-conditioned SOS constraints.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 26, 2026
Current statusOpen problem

The cited primary literature supplies forty-point constructions and reports forty-four as the best proved upper bound. Thus 40 <= tau(5) <= 44 remains the sourced current interval; the exact value is not established.

[1][2]
External progress

What the literature has established

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

  1. Peer reviewedCohn and Rajagopal prove that at least four non-isometric forty-point configurations are known and continue to report forty-four as the best proved upper bound.[1]
  2. PreprintSzöllősi reports a third known forty-point kissing arrangement in dimension five while describing forty only as the best known lower bound.[2]
2 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusFive-dimensional kissing number
Equivalent formulationspherical code A(5, 1/2)

The equal-sphere kissing problem is equivalently the maximum size of a unit-vector code in R^5 with all distinct pairwise inner products at most one half.

[2]
Related problemforty-point kissing configurations in five dimensions

The D5, L5, Q5, and R5 constructions are distinct forty-point lower-bound witnesses, but classifying such witnesses is separate from proving the maximum.

[1]

Formalization opportunities

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

  • Formalization targetA formal problem statement must connect equal-sphere tangency and nonoverlap to the unit-vector inner-product bound without changing the endpoint convention.
  • Formalization targetA checked D5 construction would establish only the forty-point lower bound, not the upper bound.
  • Formalization targetThe current work's quantitative stability and rank-five certificate interfaces remain source-reported until separately checked and do not exclude every forty-one-point code.

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
  • reduction2 of 82
  • lemma4 of 84
  • equivalence1 of 81
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 statementCan any forty-one-point spherical code satisfy the five-dimensional kissing constraint?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact spherical-code targetintermediate
  • retained route statementForty-point D5 constructionintermediate
  • retained route statementForced negative and matching structureintermediate
  • retained route statementSparsest branch evidenceintermediate
  • retained route statementSecond-audit correctionsintermediate
  • 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 failureExact-antipode substitution without an error theoremreported failure
  • Research targetProve quantitative common-template D5 stability for the eighteen sparse-branch axes with an explicit usable radius.open
  • Research targetSupply a sharp capacity or exact certificate for the exceptional C7 plus seventeen-edge branch and other twenty-four-edge structures.open
  • Research targetExclude higher-edge graphs and every remaining paired rank-five branch, not only the twenty-three-edge sparse case.open
  • Research targetQuantitative D5 stabilityopen
  • Narrowed routeExact-antipode substitution without an error theoremLocal approximate templates may disagree, and midpoint errors remain unless exact antipodality or a quantitative transfer theorem is established. A common-template stability theorem with explicit midpoint and cap radii remains the highest-leverage sparse-branch route.
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 quantitative common-template D5 stability for the eighteen sparse-branch axes with an explicit usable radius.

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 pointProve quantitative common-template D5 stability for the eighteen sparse-branch axes with an explicit usable radius.

Five-dimensional kissing number · ready to start

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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

Forty equal spheres can touch one equal central sphere in five dimensions, while current cited work proves only that no more than forty-four can do so. The exact maximum remains open.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references2 cited works · next context review by Nov 26, 2026

The mathematical context was checked on Aug 26, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Variations on Five-Dimensional Sphere Packingspeer reviewed result · Henry Cohn, Isaac Rajagopal · Discrete & Computational Geometry · 2026-07-04 · DOI 10.1007/s00454-026-00841-x · accessed Aug 26, 2026
  2. 2
    A note on five dimensional kissing arrangementspreprint · Ferenc Szöllősi · arXiv · 2023-01-19 · ARXIV 2301.08272 · accessed Aug 26, 2026

Important qualifications

  • This was a bounded primary-source status and identity check, not an exhaustive literature, priority, or configuration-classification review.
  • The current forty-four upper bound is recorded through a 2026 peer-reviewed source; this pass did not independently replay the cited semidefinite-program proof.
  • The existence of several forty-point configurations does not establish that forty is optimal.
  • No packet attachment, submitted URL, or source-reported computation was treated as independent external authority.

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