Discrete geometry · logarithmic energy on the sphere · deterministic algorithms

Smale’s Seventh Problem

Collaboration beta

For every number N of points, the problem asks for a fast deterministic way to place N distinct rational points on the unit sphere almost as evenly as the best possible arrangement, with only logarithmic excess energy. The packet develops several exact tools and corrects failed shortcuts, but does not yet provide that algorithm.

\forall N \geq 2, \exists X_{N} \in (S^{2} \cap ℚ^{3})^{N} distinct and polynomial-time constructible : H_{N} (X_{N}) - e_{N} \leq C \log N

Smale's Problems, Problem 7

Known results and sources

The problem asks for a fast deterministic construction whose sphere-point energy stays within a logarithmic gap of the unknown optimum.

Research problem

Exact mathematical statement

Given every integer $N\ge 2$ , construct deterministically a tuple of distinct rational points

X_N=(x_1,…,x_N)\in (S^2\cap\mathbb Q^3)^N

in time bounded by a fixed polynomial in the numerical value of $N$ , and prove that an absolute constant $C$ satisfies

H_N(X_N)-e_N\le C\log N,\qquad H_N(X)=-\sum_{i<j}\log\lVert x_i-x_j\rVert^2,\quad e_N=\min_{(S^2)^N}H_N.

This is the source’s ordered-energy normalization. Exact verification of rational coordinates and their energy does not by itself certify the required gap from the unknown optimum.

Problem infographic

Problem at a glance

Problem-first diagram of N rational points on the unit sphere, their all-pairs logarithmic energy, the unknown optimum, and the requirement for a deterministic polynomial-time construction with only logarithmic excess. — Exact identities and scoped constructions organize the research, but no deterministic polynomial-time method is yet shown to achieve the required O(log N) gap for every N.

Current mathematical picture

Where work on Smale’s Seventh Problem stands

Open problem

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

Useful failureUniform fixed-radius local mode truncation for every distinct configuration

For sufficiently large N, the restricted allocation on the opposite cap can place at most ell modes with ell less than qm, so its feasible set is empty while the full determinant and full correction remain finite. A locality theorem restricted to exact minimizers or another proved regularity class, with adaptive neighborhoods or a finite fallback and an exact marginal-supply theorem, remains open.

Route status · Narrowed route

Main reductionExact marginal-supply criterion

The source reports an exact finite implication: enough favorable near-chain increments together with a separated far penalty forces an optimum to use only the allowed number of far modes; the needed lower supply theorem remains open.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeGive a polynomial-bit algorithm selecting N whole rank-m osculating blocks from the explicit rational grid with determinant within exp(-A d log d) of the best integral candidate set.Task status · Ready to work on

Work mapped so far

Smale’s Seventh Problem in numbers

1.2kretained lines of mathematical investigation1,243 in the current working snapshot

Argument development: 860 · 69%
Explored or eliminated routes: 42 · 3%
Computational analysis: 145 · 12%
Open obligations: 120 · 10%
Definitions and setup: 76 · 6%

7selected 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 Shub, Steve SmaleProposed1998Status checkedAug 30, 2026

Source recordedAug 30, 2026

Recommended next task

Give a polynomial-bit algorithm selecting N whole rank-m osculating blocks from the explicit rational grid with determinant within exp(-A d log d) of the best integral candidate set.

Suggested move: Propose one block-selection mechanism, prove its bound relative to the integral rather than fractional optimum, and test its structural premise against the exact three-point fractional gap and four-site non-Lorentzian example.

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

13 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 routeUniform fixed-radius local mode truncation for every distinct configuration

Route status · Narrowed route

Narrowed routeUniform fractional block design and direct whole-block polynomial shortcuts

The N=3 exact baseline gap persists at high jet order, while the four-site quadratic has two positive Hessian eigenvalues; neither direct shortcut reaches the required integral-relative approximation. A specialized integral-relative block algorithm, a justified lift, a multi-point move with a proved basin, or a different algebraic structure is not excluded by these scoped examples.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks

Give a polynomial-bit algorithm selecting N whole rank-m osculating blocks from the explicit rational grid with determinant within exp(-A d log d) of the best integral candidate set.Suggested move: Propose one block-selection mechanism, prove its bound relative to the integral rather than fractional optimum, and test its structural premise against the exact three-point fractional gap and four-site non-Lorentzian example.

Ready to work on

Construct a finite conditioned-Gibbs or coefficient-prefix state whose polynomial-time update handles zero cases, near singularity, bit lengths, and cumulative logarithmic error on every adaptive prefix.Suggested move: Specify the exact stored state and update map, then prove a full-path error and complexity theorem meeting the current work’s explicit T-GIBBS-CHOICE or T-COEFF-ORACLE budget.

Ready to work on

Polynomial whole-block selector

Construct a polynomial-bit algorithm for the explicit rational osculating-block family with the stated determinant guarantee relative to the best integral tuple.

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

Ready to work on

For a repaired local or ring construction, control spectral supply, genuine Jensen/Fischer loss, curvature and population defects, all-N construction, rounding, and comparison to the global optimum within O(log N).Suggested move: Before global patching, establish a growing-parameter residual bound on an explicit configuration class and prove the lower marginal-supply side of T-MARGINAL-LOCALITY on that same valid domain.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 30, 2026

Current statusOpen problem

The polynomial-time all-N construction remains open. Bounded-bit rational solutions and a simply exponential Turing-model algorithm are known, while current peer-reviewed work still describes Smale's seventh problem as extremely difficult and the exact linear constant in minimal logarithmic-energy asymptotics as unknown.

[2][3]

External progress

What the literature has established

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

2023
Peer reviewedBeltran and Lizarte gave a new proof of the sharpest known lower bound for logarithmic energy on the sphere and recorded the narrow interval known for the linear asymptotic constant. The exact constant and…[3]
2023
Peer reviewedBeltran, Etayo, and Lopez-Gomez developed an explicit Diamond-ensemble family with very low logarithmic energy for every N. The cited abstract does not assert the polynomial-time O(log N)-gap theorem required…[4]
2013
Peer reviewedBeltran proved that solutions may be chosen as rational spherical points of bounded bit length and gave an exponential-time Turing-model algorithm. This resolves representation and existence issues without…[2]
1998
Historical sourceSmale published the numbered list of mathematical problems for the next century; the near-minimal logarithmic-energy construction appears as Problem 7.[1]

Mathematical neighborhood

Related results and reusable starting points

Current focusSmale's Seventh Problem

Solved special caseRational bounded-bit solutions with exponential-time construction

Bounded-bit rational points satisfying the energy requirement exist and can be found by a simply exponential Turing-model algorithm. Polynomial-time construction remains the missing complexity requirement.

[2]

Dependency or reductionMinimal logarithmic-energy asymptotics on the sphere

Sharp lower bounds and the asymptotic constant for minimal logarithmic energy control comparisons between an explicit configuration and the unknown optimum. The exact linear constant is not known.

[3]

Weaker or relaxed formDiamond-ensemble low-energy point sets

Explicit low-energy point families provide constructive benchmarks, but a low asymptotic energy description is not by itself the all-N polynomial-time O(log N)-excess guarantee.

[4]

Formal and computational footholds

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

formal library support · partial resource linkedMathlib Euclidean sphere and real logarithm primitives
Mathlib documents Euclidean spheres and the real logarithm, supplying foundational definitions only. These pages do not claim a formal statement or proof of Smale's seventh problem, its discrete energy optimum, or the required algorithm.
[5][6]

Formalization opportunities

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

Formalization targetA formal finite-configuration model for distinct rational points on the unit two-sphere and the exact logarithmic pair energy used by the public statement.
Formalization targetFormal existence and compactness results for the minimum energy e_N, including singularity handling when points collide.
Formalization targetA machine model with bit complexity, exact rational output, and a checked fixed polynomial running-time bound uniform in N.
Formalization targetA checked comparison proving the constructed configuration lies within C log N of the exact minimum for one absolute constant and every N.

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 statements4 open questions2 narrowed routes

Statements by mathematical role7 selected mapped statements

theorem candidate1 of 71
reduction3 of 73
lemma1 of 71
equivalence1 of 71
special case1 of 71

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 statementCan nearly minimum-energy rational sphere configurations be constructed deterministically in polynomial time for every N?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementExact osculating-block determinant identityintermediate
retained route statementWhole-block approximation endpointintermediate
retained route statementUnrestricted locality refutedintermediate
retained route statementExact marginal-supply criterionintermediate
retained route statementExplicit ring baseline constructionintermediate
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 counterexample narrows one intermediate strategy; it does not challenge the open conjecture.challenges · 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 failureUniform fixed-radius local mode truncation for every distinct configurationreported failure
Useful failureUniform fractional block design and direct whole-block polynomial shortcutsreported failure
Research targetGive a polynomial-bit algorithm selecting N whole rank-m osculating blocks from the explicit rational grid with determinant within exp(-A d log d) of the best integral candidate set.open
Research targetConstruct a finite conditioned-Gibbs or coefficient-prefix state whose polynomial-time update handles zero cases, near singularity, bit lengths, and cumulative logarithmic error on every adaptive prefix.open
Research targetFor a repaired local or ring construction, control spectral supply, genuine Jensen/Fischer loss, curvature and population defects, all-N construction, rounding, and comparison to the global optimum within O(log N).open
Research targetPolynomial whole-block selectoropen
ComputationThe governing Markdown reports exact rational regression checks, a three-point leakage value, finite spectral diagnostics, interval arithmetic for the candidate constant, and finite-N sanity checks, all located in named attachments.The source reports a rational enclosure of c_jet with endpoints differing in the last displayed digit and a separate corrected smoothing derivative enclosure. These certify the named analytic expressions only; the attachments were not opened or rerun, and the energy implication remains dependent on X-FT. · reported unreproduced
Narrowed routeUniform fixed-radius local mode truncation for every distinct configurationFor sufficiently large N, the restricted allocation on the opposite cap can place at most ell modes with ell less than qm, so its feasible set is empty while the full determinant and full correction remain finite. A locality theorem restricted to exact minimizers or another proved regularity class, with adaptive neighborhoods or a finite fallback and an exact marginal-supply theorem, remains open.
Narrowed routeUniform fractional block design and direct whole-block polynomial shortcutsThe N=3 exact baseline gap persists at high jet order, while the four-site quadratic has two positive Hessian eigenvalues; neither direct shortcut reaches the required integral-relative approximation. A specialized integral-relative block algorithm, a justified lift, a multi-point move with a proved basin, or a different algebraic structure is not excluded by these scoped examples.

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

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 pointGive a polynomial-bit algorithm selecting N whole rank-m osculating blocks from the explicit rational grid with determinant within exp(-A d log d) of the best integral candidate set.

Smale’s Seventh Problem · ready to start

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Research contextPrepared context for any AI agent

For every number N of points, the problem asks for a fast deterministic way to place N distinct rational points on the unit sphere almost as evenly as the best possible arrangement, with only logarithmic excess energy. The current work develops several exact tools and corrects failed shortcuts, but does not yet provide that algorithm.

Exact question and boundaries
Current routes and known obstacles
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Sources and references6 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
Mathematical Problems for the Next Centuryoriginal source · Steve Smale · The Mathematical Intelligencer · 1998 · DOI 10.1007/BF03025291 · accessed Aug 30, 2026
2
Harmonic Properties of the Logarithmic Potential and the Computability of Elliptic Fekete Pointspeer reviewed result · Carlos Beltran · Constructive Approximation · 2013 · DOI 10.1007/s00365-012-9158-y · accessed Aug 30, 2026
3
A Lower Bound for the Logarithmic Energy on S2 and for the Green Energy on Snpeer reviewed result · Carlos Beltran, Fatima Lizarte · Constructive Approximation · 2023 · DOI 10.1007/s00365-023-09642-4 · accessed Aug 30, 2026
4
Low-Energy Points on the Sphere and the Real Projective Planepeer reviewed result · Carlos Beltran, Ujue Etayo, Pedro R. Lopez-Gomez · Journal of Complexity · 2023 · DOI 10.1016/j.jco.2023.101742 · accessed Aug 30, 2026
5
Mathlib.Geometry.Euclidean.Sphere.Basicformalization · Mathlib · accessed Aug 30, 2026
6
Mathlib.Analysis.SpecialFunctions.Log.Basicformalization · Mathlib · accessed Aug 30, 2026

Important qualifications

This was a bounded primary-source status pass, not an exhaustive bibliography, priority review, or review of the private packet's mathematical claims.
No submitted URL or packet attachment was opened, fetched, executed, compiled, rendered, or treated as external authority.
The current work's rational-coordinate, exact-output, and ordered-energy formulation is stricter than some published statements of Smale's seventh problem; statement alignment is therefore described as a related variant where appropriate.
Beltran's 2013 result supplies bounded-bit rational solutions and an exponential-time Turing-model algorithm, not the polynomial-time algorithm demanded by the unresolved target.
A bounded formal-library search found relevant sphere and real-logarithm primitives but no statement-aligned formal proof; no nonexistence claim is inferred.

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