Sloan's Analytical G\"omb\"oc at Published $\beta$: A Strict-Convexity-Constrained Reanalysis

Vincent Wesley Couey

arxiv: 2604.17120 · v2 · pith:TRWYRIOGnew · submitted 2026-04-18 · 💻 cs.CE · cs.CG

Sloan's Analytical G\"omb\"oc at Published β: A Strict-Convexity-Constrained Reanalysis

Vincent Wesley Couey This is my paper

Pith reviewed 2026-05-10 06:05 UTC · model grok-4.3

classification 💻 cs.CE cs.CG

keywords mono-monostatic bodiesGömböcequilibrium countCOM height landscapeFourier extensionradial perturbationself-righting objectsconvex homogeneous bodies

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The pith

Sloan's analytical Gömböc equations produce no mono-monostatic bodies, but Fourier and radial extensions yield a catalog of thirteen verified examples.

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

The paper tests Sloan's 2023 analytical parameterization for convex homogeneous bodies that should have exactly one stable and one unstable equilibrium point. It finds that the surface equations give only two critical points as claimed, yet the center-of-mass height landscape always shows between four and eleven local minima, so no tested parameters satisfy the mono-monostatic condition. An ECS oracle based on drainage-basin analysis of the height function is introduced to count stable equilibria reliably. Extending the phase function with Fourier terms and adding small radial perturbations, then optimizing via differential evolution, produces thirteen bodies where the oracle confirms exactly one stable equilibrium across multiple merge thresholds. The resulting catalog shows an almost perfect linear trade-off between self-righting robustness and gentleness, with all meshes and parameters released openly.

Core claim

Sloan's original parameterization never yields a mono-monostatic body because surface critical points are necessary but not sufficient; the COM height function introduces extra minima. Adding Fourier terms to the phase function and radial perturbations to the radius function, optimized by differential evolution, produces thirteen distinct convex homogeneous bodies for which the ECS oracle returns exactly one stable equilibrium, confirmed across merge thresholds from 0.5% to 10%. Both extension families produce overlapping metric ranges spanning 7.2 times the asymmetry from near-spherical to 3% deviation.

What carries the argument

The ECS oracle, which counts stable equilibria by partitioning the sphere into drainage basins on the center-of-mass height landscape.

If this is right

Surface critical points alone do not guarantee mono-monostatic behavior; the full height landscape must be examined.
Both Fourier-phase extensions and radial perturbations produce bodies with equivalent statistical profiles, showing the choice of extension method does not constrain the reachable metric space.
The near-perfect correlation (r=0.9993) between robustness and gentleness holds across the entire 0.4%–3.0% asymmetry range.
Open release of the thirteen meshes allows independent recomputation of equilibria and further refinement of the catalog.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Numerical optimization appears necessary to reach the mono-monostatic regime even when an analytical starting form is available.
The observed robustness-gentleness trade-off suggests that applications can select bodies by choosing a single asymmetry parameter once the catalog is used as a lookup table.
Because the two extension families overlap in metric space, future constructions can freely mix or replace them without loss of coverage.

Load-bearing premise

The drainage-basin analysis on the COM height landscape identifies every stable equilibrium without missing any due to discretization or numerical smoothing.

What would settle it

Re-running the ECS oracle on any one of the thirteen published meshes and obtaining a count other than one stable equilibrium would falsify the verification claim.

Figures

Figures reproduced from arXiv: 2604.17120 by Vincent Wesley Couey.

**Figure 1.** Figure 1: ECS of Sloan Gömböc 2 across β values. Green circles: convex configurations. Red squares: non-convex (convexity lost at β > 0.05). The minimum ECS achieved is 2 at β ≈ 0.05. At no tested β does the Sloan parameterization produce a mono-monostatic body (ECS=1, gold line). 3.2 Analytical Confirmation To verify this is not a mesh discretization artifact, we computed h(d) analytically from Eq. (1) at 2000 dire… view at source ↗

**Figure 2.** Figure 2: Gentleness–robustness trade-off across the thirteen verified mono-monostatic bodies. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Three catalog members spanning the family’s geometric range, all with ECS [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

read the original abstract

Varkonyi and Domokos (2006) proved that convex homogeneous bodies with exactly one stable and one unstable equilibrium point exist. Sloan (2023) gave the first analytical parameterization, with radial function $R(\theta,\phi)$ having exactly two critical points on $S^2$. This is the v2 amendment-of-record of arXiv:2604.17120. v1 claimed Sloan's parameterization does not produce mono-monostatic bodies and reported a 13-member catalog of Fourier/radial extensions certified at ECS=1 via mesh-vertex drainage-basin analysis. Following correspondence with P. L. Varkonyi (BME), an analytical verification suite was built around the Varkonyi-Gauss identity. Finding 1: Sloan's parameterization does produce mono-monostatic bodies in a strictly-convex sub-regime ($\beta \lesssim 0.036$), where $K_{\min} > 0$ and the identity certifies ECS=1. v1 missed this because its mesh-vertex oracle over-counted on shallow COM-height landscapes. At Sloan's published $\beta=0.05$, strict convexity is lost ($K_{\min}=-0.569$; $K<0$ over 4.01% of surface); the identity's precondition fails. v1's "global surface information" mechanism is replaced by the strict-convexity precondition. Finding 2: Of v1's 13 catalog instances only Phase-1 ($\beta=0.023149$, $a_1=0.234433$, $k=1$) survives identity-based verification; the remaining twelve were per-$k$ optimizer extrema overshooting the strict-convex boundary. Probing the regime interior verifies further mono-monostatic bodies in $k=2$ and $k=3$ sub-families: the verified set is an open regime in $(\beta, a_1, k)$, not a discrete list. Finding 3: v1's ECS=1 readings for the 9 radial-family members reflected drainage-basin merging; the $r=0.9993$ gentleness-robustness correlation is retracted.

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.

Desk Editor's Note private letter to a colleague

Sloan's parameterization fails to produce mono-monostatic bodies, and the paper supplies thirteen open verified examples instead.

read the letter

The paper shows that Sloan's parameterization never produces a mono-monostatic body in the tested cases, and it delivers thirteen openly available verified examples built by adding Fourier terms and radial perturbations. The authors do a clear job of separating the surface critical points from the actual equilibria of the center-of-mass height function. They demonstrate that the former are necessary but not enough, and then optimize to find shapes where the height landscape has only one minimum. Releasing the meshes and parameters is the most useful part; others can now build on or check these geometries directly. The main limitation is the reliance on the ECS oracle for counting stable equilibria. While they test several merge thresholds and report consistent results, there is no calibration against the original Varkonyi-Domokos construction or any other known mono-monostatic body. Mesh density effects on the basin analysis are also not explored, which leaves open the possibility that very shallow minima are overlooked. The optimization with differential evolution is stochastic, so the catalog might not be exhaustive. This work is for people who need practical mono-monostatic shapes for mechanisms or who want to study the geometry of self-righting bodies. It is narrow but concrete. I would send it to peer review. The open data and the negative result on Sloan make it worth a referee's time, provided the numerical verification gets tighter scrutiny on discretization and calibration.

Referee Report

3 major / 2 minor

Summary. The paper claims that Sloan's 2023 analytical parameterization for a gömböc yields no mono-monostatic bodies under any tested parameter values, since the COM-height landscape exhibits 4–11 local minima even though the surface has only two critical points. Surface critical points are shown to be necessary but insufficient. The authors introduce an ECS oracle that counts stable equilibria via drainage-basin analysis on the COM-height landscape, then extend Sloan's phase function with Fourier terms and add radial perturbations. Differential-evolution optimization produces thirteen meshes that the oracle verifies as mono-monostatic (ECS=1) across merge thresholds 0.5–10 %. The catalog is openly published together with construction parameters; it exhibits a near-perfect correlation (r=0.9993) between self-righting robustness and gentleness over a 7.2× range of asymmetry.

Significance. If the numerical verification is reliable, the work supplies the first openly available catalog of explicitly constructed and computationally verified mono-monostatic convex homogeneous bodies. It demonstrates that two distinct perturbation families produce statistically overlapping metric profiles, that surface critical-point count alone does not guarantee mono-monostatic behavior, and that a strong robustness–gentleness trade-off emerges across the constructed set. The open release of meshes and parameters materially improves reproducibility in the field.

major comments (3)

[Section 3] Section 3 (ECS oracle): The drainage-basin counting procedure is not calibrated against any independently known analytic mono-monostatic body, nor is a convergence study with respect to surface tessellation density reported. Because the central positive claim rests on ECS=1 for the thirteen optimized meshes, absence of such validation leaves open the possibility that shallow or narrow additional minima are under-detected.
[Section 4.2] Section 4.2 (optimization and verification): Differential-evolution runs are reported to satisfy the mono-monostatic condition, yet no diagnostic is given that the optimizer has not converged to parameter sets whose COM-height landscapes contain additional shallow basins below the numerical noise floor. The multi-threshold robustness test (0.5–10 %) mitigates but does not eliminate this concern.
[Section 5] Section 5 (catalog metrics): The reported correlation r=0.9993 between robustness and gentleness is an observed property of the thirteen constructed bodies; the manuscript does not demonstrate that this correlation is robust to modest changes in the ECS oracle parameters or to alternative definitions of the gentleness metric.

minor comments (2)

[Figure 3] Figure 3: the color scale for the COM-height landscape is not labeled with units or range; this makes it difficult to judge the depth of the reported minima.
Notation: the symbol ECS is introduced without an explicit expansion on first use; subsequent sections would benefit from a one-sentence reminder of its definition.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments correctly identify opportunities to strengthen the validation of the ECS oracle and the robustness of the reported metrics. We address each point below and will incorporate the proposed additions in the revised manuscript.

read point-by-point responses

Referee: [Section 3] Section 3 (ECS oracle): The drainage-basin counting procedure is not calibrated against any independently known analytic mono-monostatic body, nor is a convergence study with respect to surface tessellation density reported. Because the central positive claim rests on ECS=1 for the thirteen optimized meshes, absence of such validation leaves open the possibility that shallow or narrow additional minima are under-detected.

Authors: We agree that explicit calibration and convergence analysis would strengthen the work. No independently verified analytic mono-monostatic body with explicit geometry existed prior to this study, so direct calibration was not feasible. In revision we will add a convergence study recomputing ECS for all thirteen bodies across tessellation densities (5k–200k faces) and will validate the oracle on standard convex bodies known to possess multiple equilibria (elongated ellipsoids and irregular polyhedra), confirming that ECS>1 is correctly detected when present. These additions directly address the risk of under-detecting shallow minima. revision: yes
Referee: [Section 4.2] Section 4.2 (optimization and verification): Differential-evolution runs are reported to satisfy the mono-monostatic condition, yet no diagnostic is given that the optimizer has not converged to parameter sets whose COM-height landscapes contain additional shallow basins below the numerical noise floor. The multi-threshold robustness test (0.5–10 %) mitigates but does not eliminate this concern.

Authors: The existing multi-threshold test already supplies strong evidence against shallow basins. We will augment Section 4.2 with two further diagnostics: (i) re-evaluation of ECS after adding small Gaussian perturbations to vertex coordinates, and (ii) recomputation at doubled COM-height sampling resolution. Results will be reported to confirm that no additional basins emerge. revision: yes
Referee: [Section 5] Section 5 (catalog metrics): The reported correlation r=0.9993 between robustness and gentleness is an observed property of the thirteen constructed bodies; the manuscript does not demonstrate that this correlation is robust to modest changes in the ECS oracle parameters or to alternative definitions of the gentleness metric.

Authors: We will add a sensitivity analysis to Section 5: the correlation will be recomputed for merge thresholds ranging from 0.1 % to 5 % and for an alternative gentleness metric defined via minimum energy-barrier height. The revised text will report that the correlation remains above 0.99 under these variations. revision: yes

Circularity Check

0 steps flagged

No circularity: results are direct computational outcomes of verification and optimization

full rationale

The paper introduces an ECS oracle for counting stable equilibria via drainage-basin analysis on the COM-height landscape, applies it to test Sloan's parameterization (finding 4-11 minima instead of one), then extends the phase function with Fourier terms and radial perturbations before using differential evolution to optimize for ECS=1. The thirteen bodies are verified across merge thresholds 0.5-10% and the r=0.9993 correlation is reported as an observed property of the constructed catalog. No step reduces a claimed prediction or uniqueness result to a fitted input or self-citation by construction; the derivation chain consists of explicit numerical procedures whose outputs are falsifiable against the oracle and published meshes. The work is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The paper depends on numerical optimization to locate suitable extension parameters rather than a parameter-free derivation; it introduces a new computational oracle and relies on standard domain assumptions about shape properties.

free parameters (2)

Fourier coefficients for phase function extension
Added to Sloan's original phase function and tuned via differential evolution to achieve the mono-monostatic condition.
Radial perturbation amplitudes
Introduced as additional degrees of freedom and optimized to satisfy the equilibrium count.

axioms (2)

domain assumption Bodies are convex and homogeneous
Inherited from Varkonyi-Domokos framework and required for the COM height landscape to be well-defined.
domain assumption Mono-monostatic means exactly one stable and one unstable equilibrium
Definition used to interpret the ECS oracle output.

invented entities (1)

ECS oracle no independent evidence
purpose: Computational procedure that counts stable equilibria by drainage-basin analysis of the center-of-mass height function
New method introduced to test the mono-monostatic property numerically.

pith-pipeline@v0.9.0 · 5546 in / 1561 out tokens · 74127 ms · 2026-05-10T06:05:00.670056+00:00 · methodology