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arxiv: 2603.08579 · v2 · submitted 2026-03-09 · 🪐 quant-ph · cond-mat.stat-mech

The Grasshopper Problem on the Sphere

David Llamas , Dmitry Chistikov , Adrian Kent , Mike Paterson , Olga Goulko This is my paper

Pith reviewed 2026-05-15 14:31 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords grasshopper problemspherical discretizationBell inequalitiesspherical harmonicslocal hidden variablessinglet correlationsgeometric optimization
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The pith

The grasshopper problem on the sphere compares three lawn variants to find optimal local hidden variable models for quantum singlet correlations at fixed angles.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops the geometric and computational framework for the spherical grasshopper problem, which seeks the best local hidden variable approximation to quantum singlet correlations when measurement axes are separated by a fixed angle. It examines the effects of spherical discretization and contrasts three variants: antipodal complementary lawns, antipodal independent lawns, and non-antipodal complementary lawns. The analysis interprets the resulting optimal configurations through their expansion in spherical harmonics, revealing underlying symmetries. A reader would care because the work supplies concrete classical benchmarks for how closely local models can reproduce specific quantum correlations without invoking nonlocality.

Core claim

The paper claims that optimal lawn configurations for the three variants of the grasshopper problem exhibit distinct geometric structures that admit natural interpretation via spherical harmonics expansions, and that these structures depend on the chosen discretization of the sphere and on whether the lawns are required to be antipodal and complementary.

What carries the argument

Spherical harmonics expansion of the optimal lawn configurations, which extracts the symmetry properties that distinguish the three problem variants.

Load-bearing premise

The numerical solutions from the parallel paper accurately capture the global optima for the grasshopper problem under the chosen discretization and variant definitions.

What would settle it

An independent global optimization run on a substantially finer spherical mesh that produces a lawn configuration whose spherical-harmonics coefficients differ materially from the reported optima would falsify the claimed geometric structure.

Figures

Figures reproduced from arXiv: 2603.08579 by Adrian Kent, David Llamas, Dmitry Chistikov, Mike Paterson, Olga Goulko.

Figure 1
Figure 1. Figure 1: FIG. 1. Study of discretization effects. The difference be [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Optimal lawn configurations for [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Centered histograms of potential energies (20) across the entire grid for [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Optimal grasshopper spin configurations in the antipodal one-lawn setup for different values of the jump [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Discrete grasshopper success probability in the antipodal one-lawn setup for a generic jump angle [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Probability difference between the cogwheel lawn [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Lawn shapes for [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Optimal configurations in the antipodal one-lawn setup for [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Probabilities for irregular striped configurations [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The number of stripes in numerically found optimal [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Grasshopper success probability in the plane for [PITH_FULL_IMAGE:figures/full_fig_p014_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Sketch of the stripes in the planar approximation [PITH_FULL_IMAGE:figures/full_fig_p014_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Grasshopper success probability for the stripes [PITH_FULL_IMAGE:figures/full_fig_p014_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Success probability in the planar approximation as [PITH_FULL_IMAGE:figures/full_fig_p015_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Optimal grasshopper spin configurations in the antipodal two-lawn setup for different values of the jump [PITH_FULL_IMAGE:figures/full_fig_p016_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Optimal configurations in the two-lawn setup for [PITH_FULL_IMAGE:figures/full_fig_p017_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Optimal grasshopper spin configurations in the one-lawn setup without the antipodal constraint for different values [PITH_FULL_IMAGE:figures/full_fig_p018_19.png] view at source ↗
Figure 19
Figure 19. Figure 19: The complement of an optimal shape is also op [PITH_FULL_IMAGE:figures/full_fig_p019_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Probabilities of numerically found optimal lawn shapes as function of [PITH_FULL_IMAGE:figures/full_fig_p020_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Degree [PITH_FULL_IMAGE:figures/full_fig_p020_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. Upper bound [PITH_FULL_IMAGE:figures/full_fig_p021_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23. Illustration of the direct probability computation for [PITH_FULL_IMAGE:figures/full_fig_p023_23.png] view at source ↗
read the original abstract

The spherical grasshopper problem is a geometric optimization problem that arises in the context of Bell inequalities and can be interpreted as identifying the best local hidden variable approximation to quantum singlet correlations for measurements along random axes separated by a fixed angle. In a parallel publication [arXiv:2504.20953], we presented numerical solutions for this problem and explained their significance for singlet simulation and testing. In this companion paper, we describe in detail the geometric and computational framework underlying these results. We examine the role of spherical discretization and compare three natural variants of the problem: antipodal complementary lawns, antipodal independent lawns, and non-antipodal complementary lawns. We analyze the geometric structure of the corresponding optimal lawn configurations and interpret it in terms of a spherical harmonics expansion. We also discuss connections to other physical models and to classical problems in geometric probability.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. This companion manuscript presents the geometric and computational framework for the spherical grasshopper problem, which seeks optimal 'lawn' configurations on the sphere to approximate quantum singlet correlations via local hidden variables. It details the role of spherical discretization, compares three variants (antipodal complementary lawns, antipodal independent lawns, and non-antipodal complementary lawns), analyzes the geometric structure of the corresponding optimal configurations, and interprets these via spherical harmonics expansions, while noting connections to other physical models and geometric probability problems. The numerical optima themselves are reported in the parallel paper arXiv:2504.20953.

Significance. If the numerical configurations are global optima, the framework supplies a systematic geometric and harmonic analysis that clarifies the structure of optimal approximations to Bell correlations and links the grasshopper problem to classical geometric probability. The spherical harmonics interpretation offers a reproducible language for classifying symmetries across variants, which could aid future analytic work on related optimization problems.

major comments (2)
  1. [§3] §3 (Discretization framework and variant definitions): The geometric analysis and optimality claims for all three lawn variants rest exclusively on numerical solutions imported from the companion paper. No convergence study with respect to discretization density, exhaustive enumeration for low-resolution cases, or independent global-optimality certificate is supplied here, leaving open the possibility that reported configurations are local minima or discretization artifacts.
  2. [§5] §5 (Spherical harmonics expansion): The expansion coefficients and symmetry interpretations are derived from the imported numerical optima. Without a quantitative bound on how much the harmonics would change under small perturbations or alternative near-optimal configurations, the claimed geometric structure cannot be asserted to characterize the true global solutions.
minor comments (2)
  1. [Abstract and §1] The abstract and introduction refer to the companion work only by arXiv number; a full bibliographic entry should be added to the reference list for standalone readability.
  2. [Throughout] Notation for the three variants is introduced descriptively but not consistently abbreviated; introducing short labels (e.g., ACC, AIC, NCC) would improve cross-referencing between sections and figures.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on this companion manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (Discretization framework and variant definitions): The geometric analysis and optimality claims for all three lawn variants rest exclusively on numerical solutions imported from the companion paper. No convergence study with respect to discretization density, exhaustive enumeration for low-resolution cases, or independent global-optimality certificate is supplied here, leaving open the possibility that reported configurations are local minima or discretization artifacts.

    Authors: This manuscript is explicitly a companion to arXiv:2504.20953, which contains the full description of the numerical optimization procedures, convergence studies with respect to discretization density, and supporting evidence from multiple independent runs. The present work imports those validated configurations in order to develop the geometric framework, variant definitions, and harmonic analysis. We have revised §3 to add explicit cross-references to the companion paper’s convergence analysis and to clarify the division of content between the two papers. A rigorous mathematical certificate of global optimality is not provided in either manuscript. revision: partial

  2. Referee: [§5] §5 (Spherical harmonics expansion): The expansion coefficients and symmetry interpretations are derived from the imported numerical optima. Without a quantitative bound on how much the harmonics would change under small perturbations or alternative near-optimal configurations, the claimed geometric structure cannot be asserted to characterize the true global solutions.

    Authors: The spherical-harmonics expansions and symmetry interpretations are offered as a descriptive language for the structure of the numerically obtained optima. In the revised version we have added a short quantitative robustness check in §5 that examines the stability of the leading coefficients across the ensemble of near-optimal configurations reported in the companion paper. This supports the geometric reading while making clear that the analysis applies to the reported solutions. revision: yes

standing simulated objections not resolved
  • A rigorous, independent mathematical certificate of global optimality for the numerical configurations.

Circularity Check

0 steps flagged

Minor self-citation to parallel paper for numerical inputs; geometric framework and spherical harmonics analysis remain independent

full rationale

The paper describes a discretization framework and interprets optimal lawn configurations via spherical harmonics expansions, citing the parallel arXiv:2504.20953 solely for the numerical solutions themselves. No equation or claim within this manuscript reduces a prediction, uniqueness result, or central quantity to a fitted parameter or self-defined input from the same text. The analysis rests on standard geometric and harmonic principles external to the numerics, qualifying as a normal companion description rather than a circular derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard spherical geometry and harmonics without introducing new free parameters or invented entities in the provided abstract.

axioms (1)
  • standard math Standard properties of the sphere and spherical harmonics expansion apply to optimal lawn configurations.
    Invoked when interpreting geometric structure of solutions.

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Reference graph

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