Euclidean geometry · configuration spaces · symmetric powers · determinant nonvanishing

Euclidean Atiyah–Sutcliffe Conjecture 1

Collaboration beta

Exact formulas organize insertion and collision limits and settle several special geometries, while four global nonvanishing bridges remain open.

D_{n} (x_{1,} \dots, x_{n}) \neq 0 for all n \geq 2 and distinct x_{i} \in ℝ^{3}

Known results and sources

Exact insertion and collision formulas sharpen the Euclidean nonvanishing problem without completing the full conjecture.

Research problem

Exact mathematical statement

For every integer $n\ge 2$ and every ordered configuration of distinct points $x_1,\dots,x_n\in\mathbb R^3$ , let $D_n(x_1,\dots,x_n)$ be the normalized Atiyah–Sutcliffe determinant formed from coherent Hopf lifts of the pairwise directions, with the convention $D_2=1$ . The conjecture asks whether

D_n(x_1,\dots,x_n)\ne 0

for every such Euclidean configuration. The source distinguishes this target from the stronger inequality $|D_n|\ge 1$ , which is not required and is not claimed here.

Problem infographic

Problem at a glance

Four-panel source-bound diagram of the exact normalized determinant target, one-point transversality, collision-factor compactification, and four unresolved global bridges. — Source-reported exact interfaces and special sectors organize two proof routes, but transversality, magnetic and multiplicity-center nonvanishing, and interior exclusion remain open.

Current mathematical picture

Where work on Euclidean Atiyah–Sutcliffe Conjecture 1 stands

Open conjecture

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

Useful failureUniversal one-point Gram-growth lower bound

At the stated four-point geometry, the reported normalized Gram-growth ratio is about 0.9335, below one, whereas D_4/D_3 remains about 1.0117. Strict transversality through equality rigidity or a geometry-dependent phase-preserving estimate remains viable; the counterexample does not refute determinant nonvanishing.

Route status · Narrowed route

Main reductionSimultaneous multiplicity-center determinant

The source reports an exact normalized determinant-line object for simultaneous clusters, including collision factorization, mandatory anchors, and associative composition under arbitrary nested collision trees; its general genuine-collision nonvanishing remains open.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve phase-preserving top-star transversality for every Euclidean insertion, equivalently strict positivity of the exact insertion Schur complement.Task status · Ready to work on

Work mapped so far

Euclidean Atiyah–Sutcliffe Conjecture 1 in numbers

1.1kretained lines of mathematical investigation1,105 in the current working snapshot

Argument development: 901 · 82%
Explored or eliminated routes: 46 · 4%
Computational analysis: 23 · 2%
Open obligations: 53 · 5%
Definitions and setup: 82 · 7%

8selected mapped statements2routes 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 byMichael Atiyah, Paul SutcliffeProposed2001Status checkedAug 30, 2026

Source recordedAug 30, 2026

Recommended next task

Prove phase-preserving top-star transversality for every Euclidean insertion, equivalently strict positivity of the exact insertion Schur complement.

Suggested move: Build an exact weighted norm identity from the triangle number-operator relations that telescopes lower-spin contractions without discarding Bargmann phases, and test equality rigidity against the common-direction and collinear models.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

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.

Scroll horizontally to explore the route

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 routeUniversal one-point Gram-growth lower bound

Route status · Narrowed route

Narrowed routeTermwise inverse-diagonal positivity in the three-base magnetic excess

AS-C7 is described as a rational Euclidean geometry with an 80-digit interval certificate against the termwise positivity claim. A combined inequality, a different diagonal scaling, a sectorial compression, or an exact zero search remains viable; the source explicitly does not claim all scalings fail.

Route status · Narrowed route

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Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks

Prove phase-preserving top-star transversality for every Euclidean insertion, equivalently strict positivity of the exact insertion Schur complement.Suggested move: Build an exact weighted norm identity from the triangle number-operator relations that telescopes lower-spin contractions without discarding Bargmann phases, and test equality rigidity against the common-direction and collinear models.

Ready to work on

Prove or falsify the generic three-base k=2 magnetic factor using the combined recurrence or a polarized compression, without signing inverse-diagonal terms separately.Suggested move: Work from the combined excess identity, parameterize its gauge-invariant face slacks, Sigma, and squared volume, and require any proposed factorization to survive the source-reported AS-C7 counterexample geometry.

Ready to work on

Global nonvanishing closure

Close AS1 either through phase-preserving top-star transversality or through genuine collision-factor nonvanishing plus a separate mechanism that excludes zeros in the configuration-space interior.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Prove nonvanishing of every genuine multiplicity-center determinant and connect complete collision-boundary control to a mechanism excluding interior zeros.Suggested move: Start with the first unsolved genuine multiplicity vectors, derive block Schur complements compatible with nested associativity, and keep AS-O7 distinct from the separate global AS-O6 boundary-to-interior step.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 30, 2026

Current statusOpen conjecture

The all-n Euclidean nonvanishing statement remains open in this scoped collection. The four-point case is proved, and additional special configurations and stronger-inequality cases are known, but the 2024 source still treats the general statement as a conjecture and gives consequences conditional on it. This does not upgrade the current work's internal results to an externally verified solution.

[2][3][4]

External progress

What the literature has established

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

2024
PreprintGuerra and Salvatore showed that the Atiyah-Sutcliffe conjecture would imply specified E3- and E2-algebra structures and provided supporting calculations; the result is conditional on the conjecture.[4]
2011
PreprintMazur and Petrenko proved the stronger second conjecture for regular polygons and convex quadrilaterals and the third conjecture for inscribed quadrilaterals; these special configurations do not settle…[3]
2001
Peer reviewedEastwood and Norbury proved nonvanishing for every configuration of four points in Euclidean three-space.[2]

Mathematical neighborhood

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 questions2 narrowed routes

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
reduction3 of 83
lemma1 of 81
equivalence1 of 81
special case2 of 82

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.24 displayed rows · 2 routes included

retained route statementIs the normalized Atiyah–Sutcliffe determinant nonzero for every finite Euclidean configuration?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementExact insertion transversality interfaceintermediate
retained route statementAll-n collinear Hessianintermediate
retained route statementAll-multiplicity collinear weighted sectorintermediate
retained route statementExact three-base magnetic recurrenceintermediate
retained route statementSimultaneous multiplicity-center determinantintermediate
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 failureUniversal one-point Gram-growth lower boundreported failure
Useful failureTermwise inverse-diagonal positivity in the three-base magnetic excessreported failure
Research targetProve phase-preserving top-star transversality for every Euclidean insertion, equivalently strict positivity of the exact insertion Schur complement.open
Research targetProve or falsify the generic three-base k=2 magnetic factor using the combined recurrence or a polarized compression, without signing inverse-diagonal terms separately.open
Research targetProve nonvanishing of every genuine multiplicity-center determinant and connect complete collision-boundary control to a mechanism excluding interior zeros.open
Research targetGlobal nonvanishing closureopen
ComputationThe governing Markdown reports exact, symbolic, interval, and deterministic numerical checks, including counterexamples to two proposed inequalities and regression suites for the collinear, magnetic, and multi-center formulas.The computations are source-reported falsification and regression evidence only; the current work explicitly says no exact or numerical search is promoted to the full theorem. · reported unreproduced
Narrowed routeUniversal one-point Gram-growth lower boundAt the stated four-point geometry, the reported normalized Gram-growth ratio is about 0.9335, below one, whereas D_4/D_3 remains about 1.0117. Strict transversality through equality rigidity or a geometry-dependent phase-preserving estimate remains viable; the counterexample does not refute determinant nonvanishing.
Narrowed routeTermwise inverse-diagonal positivity in the three-base magnetic excessAS-C7 is described as a rational Euclidean geometry with an 80-digit interval certificate against the termwise positivity claim. A combined inequality, a different diagonal scaling, a sectorial compression, or an exact zero search remains viable; the source explicitly does not claim all scalings fail.

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 phase-preserving top-star transversality for every Euclidean insertion, equivalently strict positivity of the exact insertion Schur complement.

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 phase-preserving top-star transversality for every Euclidean insertion, equivalently strict positivity of the exact insertion Schur complement.

Euclidean Atiyah–Sutcliffe Conjecture 1 · ready to start

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Research contextPrepared context for any AI agent

Exact formulas organize insertion and collision limits and settle several special geometries, while four global nonvanishing bridges remain open.

Exact question and boundaries
Current routes and known obstacles
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Sources and references4 cited works · next context review by Nov 30, 2026

The mathematical context was checked on Aug 30, 2026. Status can be refreshed sooner after a material result or claim.

1
The Geometry of Point Particlesoriginal source · Michael Atiyah, Paul Sutcliffe · Proceedings of the Royal Society A · 2002 · ARXIV hep-th/0105179 · DOI 10.1098/rspa.2001.0913 · accessed Aug 30, 2026
2
A proof of Atiyah's conjecture on configurations of four points in Euclidean three-spacepeer reviewed result · Michael Eastwood, Paul Norbury · Geometry & Topology · 2001 · ARXIV math/0109161 · DOI 10.2140/gt.2001.5.885 · accessed Aug 30, 2026
3
On the conjectures of Atiyah and Sutcliffepreprint · Marcin Mazur, Bogdan V. Petrenko · arXiv · 2011 · ARXIV 1102.4662 · accessed Aug 30, 2026
4
The Atiyah-Sutcliffe conjecture and E_n-algebraspreprint · Lorenzo Guerra, Paolo Salvatore · arXiv · 2024 · ARXIV 2410.24124 · accessed Aug 30, 2026

Important qualifications

The 2024 preprint still treats the relevant Atiyah-Sutcliffe statement as a conjecture, but this scoped search did not exhaust every paper after 2024; status is therefore stated conservatively and must be refreshed on a material claim.
The record separates Conjecture 1, nonvanishing of the determinant, from the stronger lower-bound Conjectures 2 and 3.
No dedicated formalization, proof certificate, or independently reproduced computation for the all-n Euclidean conjecture was verified; empty readiness lists do not establish nonexistence.

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Euclidean Atiyah–Sutcliffe Conjecture 1

Exact mathematical statement

Problem at a glance

Where work on Euclidean Atiyah–Sutcliffe Conjecture 1 stands

Euclidean Atiyah–Sutcliffe Conjecture 1 in numbers

Prove phase-preserving top-star transversality for every Euclidean insertion, equivalently strict positivity of the exact insertion Schur complement.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formalization opportunities

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

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