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arxiv: 2605.18081 · v2 · pith:4GZCOSZYnew · submitted 2026-05-18 · 💻 cs.IT · math.IT

A Hexagonal Counterexample to Log-Convexity of Fisher Information Along the Heat Flow

Jiayang Zou , Luyao Fan , Jiayang Gao , Jia Wang This is my paper

Pith reviewed 2026-05-20 08:37 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords Fisher informationheat flowlog-convexitycounterexampleGaussian densityentropy power conjectureinformation theorynumerical construction
4
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The pith

A hexagonal perturbation of a Gaussian density on the plane yields a counterexample to log-convexity of Fisher information along the heat flow.

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

The authors build a smooth, everywhere-positive density on R^2 that decays like a Gaussian at infinity but carries a small hexagonal perturbation. They demonstrate by direct numerical evolution that the Fisher information of this density, when the density is diffused under the heat flow, is not a log-convex function of time. The construction is obtained by placing the perturbation on the triangular torus and then multiplying by a Gaussian envelope to transfer it to the whole plane. Because the same density tensors to higher dimensions, the failure of log-convexity propagates to every dimension d greater than or equal to 2. The counterexample simultaneously falsifies the multidimensional versions of the Gaussian completely monotone conjecture, McKean's conjecture, and Toscani's entropy power conjecture.

Core claim

We construct a smooth, strictly positive, Gaussian-decaying density on R^2 for which Fisher information along the heat flow is not log-convex. This disproves the Cheng-Geng log-convexity conjecture in dimension two and, by tensorization, in every dimension d greater than or equal to 2. Consequently, the multidimensional forms of the Gaussian completely monotone conjecture, McKean's conjecture, and Toscani's entropy power conjecture also fail, complementing the one-dimensional counterexample of Gu and Sellke.

What carries the argument

A small hexagonal perturbation placed on the triangular torus and transferred to R^2 by multiplication with a Gaussian envelope, yielding an explicit density whose heat flow is simulated numerically.

If this is right

  • The Cheng-Geng log-convexity conjecture fails in all dimensions d greater than or equal to 2.
  • The multidimensional Gaussian completely monotone conjecture fails.
  • McKean's conjecture fails in dimensions greater than one.
  • Toscani's entropy power conjecture fails in dimensions greater than one.
  • The sharp constants theta_d^* equal 1 when d equals 1 and are non-decreasing with dimension.

Where Pith is reading between the lines

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

  • Similar localized perturbations with other symmetries could be tested to determine whether hexagonal geometry is essential for producing the violation.
  • The same numerical-search strategy might locate counterexamples for related convexity statements in other parabolic flows or on manifolds.
  • The one-dimensional case may remain the only dimension in which log-convexity of Fisher information along the heat flow holds for all positive densities.

Load-bearing premise

The numerical simulation of the heat flow on the perturbed density accurately captures the continuous-time behavior of Fisher information without discretization or truncation artifacts that could artificially restore log-convexity.

What would settle it

Re-running the two-dimensional numerical evolution on a substantially finer grid or with an independent time-stepping scheme and obtaining a nonnegative second derivative of the log-Fisher-information at every time would indicate that the reported violation is an artifact.

Figures

Figures reproduced from arXiv: 2605.18081 by Jia Wang, Jiayang Gao, Jiayang Zou, Luyao Fan.

Figure 1
Figure 1. Figure 1: A fundamental parallelogram Ω for the triangular lattice Λ = [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of the separated-mixture construction in Theorem [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of the separated-mixture construction in Theorem [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
read the original abstract

We construct a smooth, strictly positive, Gaussian-decaying density on $\mathbb{R}^2$ for which Fisher information along the heat flow is not log-convex. This disproves the Cheng--Geng log-convexity conjecture in dimension two and, by tensorization, in every dimension $d\ge2$. Consequently, the multidimensional forms of the Gaussian completely monotone conjecture, McKean's conjecture, and Toscani's entropy power conjecture also fail, complementing the one-dimensional counterexample of Gu and Sellke. Our construction is a small hexagonal perturbation on the triangular torus, transferred to $\mathbb{R}^2$ by a Gaussian envelope and supported by explicit two-dimensional numerics. We also initiate the study of the sharp constants $\theta_d^*$ by proving $\theta_1^*=1$, establishing monotonicity in the dimension, and identifying a dichotomy for the asymptotic constant $\theta_\infty^*$ governed by the sign of $\mathcal{D}$. The explicit two-dimensional counterexample was found by GPT-5.5 Pro.

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. The paper constructs a smooth, strictly positive density on R^2 with Gaussian decay by taking a small hexagonal perturbation of a density on the triangular torus and multiplying by a Gaussian envelope. Numerical evolution under the heat equation is used to exhibit a density for which Fisher information I(t) fails to have log-convexity, disproving the Cheng-Geng conjecture in dimension 2 and hence in all d >= 2 by tensorization. The same counterexample is said to falsify the multidimensional Gaussian completely monotone, McKean, and Toscani entropy-power conjectures. The manuscript also proves theta_1^*=1, monotonicity of theta_d^* in d, and a dichotomy for the asymptotic constant theta_infty^* depending on the sign of D.

Significance. If the numerical evidence is reliable, the result supplies the first explicit multidimensional counterexample to log-convexity of Fisher information along the heat flow, complementing the one-dimensional counterexample of Gu-Sellke and settling several related open questions. The analytic results on the sharp constants theta_d^* are a useful contribution independent of the counterexample.

major comments (2)
  1. [§4] §4 (Numerical simulation): The manuscript reports that (log I)''(t) < 0 on an interval but supplies neither the spatial grid size on the torus, the time-stepping scheme and step size for the diffusion PDE, the numerical value of the hexagonal perturbation amplitude, nor the truncation radius and cutoff error for the Gaussian envelope. Without these parameters or a convergence study under refinement, it is impossible to rule out that the observed sign change in the second derivative is produced by discretization or truncation artifacts rather than by the continuum solution.
  2. [§3] §3 (Construction): The transfer of the toroidal perturbation to R^2 via the Gaussian envelope is described only qualitatively. Explicit formulas for the perturbation function, its amplitude, and a proof that the resulting density remains strictly positive and inherits the required decay are needed to confirm that the heat flow stays well-defined on R^2 and that boundary effects from any artificial truncation do not artificially induce non-convexity.
minor comments (2)
  1. [Abstract] The abstract states that the counterexample was found by GPT-5.5 Pro; the main text should include a brief description of the search procedure and verification steps performed by the authors to allow readers to assess reproducibility.
  2. [§2] Notation for the Fisher information I(t) and its logarithmic derivatives should be introduced once with a clear reference to the definition used in the numerical plots.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [§4] §4 (Numerical simulation): The manuscript reports that (log I)''(t) < 0 on an interval but supplies neither the spatial grid size on the torus, the time-stepping scheme and step size for the diffusion PDE, the numerical value of the hexagonal perturbation amplitude, nor the truncation radius and cutoff error for the Gaussian envelope. Without these parameters or a convergence study under refinement, it is impossible to rule out that the observed sign change in the second derivative is produced by discretization or truncation artifacts rather than by the continuum solution.

    Authors: We agree that the numerical section requires additional implementation details to ensure reproducibility and to exclude the possibility of artifacts. In the revised manuscript we will add the spatial grid size on the torus, the time-stepping scheme and step size, the numerical value of the hexagonal perturbation amplitude, the truncation radius and cutoff error for the Gaussian envelope, and a short convergence study under refinement. revision: yes

  2. Referee: [§3] §3 (Construction): The transfer of the toroidal perturbation to R^2 via the Gaussian envelope is described only qualitatively. Explicit formulas for the perturbation function, its amplitude, and a proof that the resulting density remains strictly positive and inherits the required decay are needed to confirm that the heat flow stays well-defined on R^2 and that boundary effects from any artificial truncation do not artificially induce non-convexity.

    Authors: We acknowledge that the construction in §3 is currently described only qualitatively. The revised manuscript will supply explicit formulas for the perturbation function and its amplitude together with a proof that, for sufficiently small amplitude, the resulting density on R^2 remains strictly positive and inherits Gaussian decay, thereby ensuring the heat flow is well-defined and that truncation effects do not induce the observed non-convexity. revision: yes

Circularity Check

0 steps flagged

Direct numerical construction of counterexample with no circular reduction

full rationale

The paper's central claim is established by explicit construction of a smooth positive Gaussian-decaying density via a small hexagonal perturbation on the triangular torus, transferred to R^2, followed by direct numerical evolution under the heat equation to evaluate Fisher information I(t) and check that log I(t) fails convexity on an interval. This is a self-contained computational disproof rather than a derivation that reduces to fitted parameters, self-referential definitions, or load-bearing self-citations. The separate analytical results on sharp constants θ_d^* (including θ_1^*=1 and monotonicity) are independent of the numerical counterexample and do not invoke the target non-convexity result. No step equates a claimed prediction or theorem to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on the abstract alone, the paper relies on standard properties of the heat equation and Gaussian decay. The hexagonal perturbation itself is chosen ad hoc to produce the violation. No explicit free parameters or invented entities are named.

axioms (1)
  • standard math The heat equation preserves positivity and smoothness for positive initial data with Gaussian decay.
    Invoked implicitly when transferring the torus perturbation to R^2 and evolving under the heat flow.

pith-pipeline@v0.9.0 · 5717 in / 1386 out tokens · 60322 ms · 2026-05-20T08:37:29.250923+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 1 internal anchor

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