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An Explicit Family of Log-Concave Counterexamples to the Gaussian Completely Monotone Conjecture
Jiayang Zou, Luyao Fan, Jiayang Gao, Jia Wang
TL;DR
The paper gives a direct analytic construction of smooth, strictly log-concave counterexamples to the Gaussian completely monotone conjecture. It reduces the sign calculation to a two-frequency entropy expansion, transfers it to the real line, and extends the construction to every dimension.
Problem
Earlier work established a log-concave counterexample through heat regularization and backward propagation, while this paper seeks a direct analytic construction in the log-concave class.
Method
The proof analyzes a two-frequency factor on the circle, identifies its dominant spectral level, and transfers the asymptotic to the real line using exact heat evolution of Gaussian-windowed Fourier modes.
Results
For every sufficiently large m, the explicit family has a negative signed mth entropy derivative, and the failure persists for sufficiently small positive times; Gaussian tensorization yields counterexamples in every dimension.
Takeaways & Limitations
The Gaussian completely monotone conjecture fails for smooth strictly log-concave densities on R and on R^d for every d≥2.
Abstract
from arXiv · showhide
We construct smooth, strictly log-concave counterexamples to the Gaussian completely monotone conjecture in every dimension. In one dimension, they form an explicit family $f_m$ whose signed $m$th entropy derivative at time zero is negative for every sufficiently large $m$; the inequality persists for all sufficiently small positive times. Tensorization with a broad Gaussian factor gives the higher-dimensional examples. The argument is analytic and self-contained. It reduces the sign to a two-frequency entropy calculation on the circle and transfers the resulting asymptotic to the real line through an exact heat-flow formula for Gaussian-windowed Fourier modes. The proof was developed by GPT-5.6 Sol Pro under the authors' guidance.
I. Introduction
The paper gives an explicit analytic family of smooth, strictly log-concave densities that violates the Gaussian completely monotone conjecture in one dimension, with tensorized extensions to every dimension. Its proof reduces the sign calculation to two Fourier frequencies on the circle and transfers it to the real line through exact heat-flow formulas.
- Main result: For every sufficiently large m, the signed mth entropy derivative at time zero is negative, and the failure persists for sufficiently small positive times.Thus the conjectured inequality fails for smooth strictly log-concave densities on R.
- Motivation: The Gaussian completely monotone conjecture concerns signed entropy derivatives along Gaussian convolution and belongs to a hierarchy of Gaussian entropy inequalities.Prior work established several low-order cases, including through order five for log-concave inputs in every dimension.
- Context: The conjecture would imply Fisher-information log-convexity, but one-dimensional counterexamples are not detected by that weaker consequence.This distinguishes the higher-order obstruction from known log-convexity failures in dimensions at least two.
- Related work: A previous construction produced the smallest possible unrestricted one-dimensional failure order, while a log-concave counterexample was known only existentially by heat regularization.The earlier higher-order obstruction used a symmetric measure supported on seventeen points and an exact numerical certificate.
- Proof strategy: The proof isolates a dominant level in a two-frequency periodic entropy expansion, then restores the Gaussian envelope using exact heat evolution with uniformly controlled errors.The Gaussian curvature enforces strict log-concavity, analyticity preserves the sign, and Gaussian tensorization yields higher-dimensional counterexamples.
II. The Periodic Calculation
The periodic calculation expands the entropy of a two-frequency factor into spectral levels. The unique dominant level is j=3, whose two terms produce a negative leading constant, while all remaining levels are uniformly controlled.
- Entropy expansion: The periodic factor remains uniformly small enough for a convergent entropy expansion for all s≥0 when m≥5.The bound ||u_m(s,·)||∞<1/2 permits termwise expansion around 1+u_m.
- Spectral grouping: Symmetry eliminates terms with an odd number of first-harmonic factors and organizes the surviving monomials by level j.Writing p=2h gives j=h+2q.
- Remainder control: All other spectral levels contribute a uniformly bounded remainder, while the absolute expansion sum is O(β_m).The finite low levels away from j=3 are O(1), and the tail is uniformly bounded for m≥10.
III. Gaussian Localization and Completion of the Proof
The proof restores the Gaussian envelope and transfers the periodic entropy asymptotics to the real line using exact Gaussian-windowed Fourier evolution. These estimates establish the negative derivative sign while preserving strict log-concavity, then extend the counterexamples to every higher dimension by Gaussian tensorization.
- Gaussian localization: Exact heat evolution of Gaussian-windowed Fourier modes reduces the real-line calculation to the periodic one, with Fourier-filtering and time-change errors that are o(β_m).The Gaussian window interacts exactly with the heat flow, enabling the localization from the circle to the real line.
- Fourier filtering: The Fourier series is uniformly controlled on a complex strip, with nonzero coefficients bounded by C e^−|n| uniformly on the time disk.Analyticity and normal convergence support termwise integration and derivative estimates.
- Error control: The periodic entropy contribution dominates the Gaussian entropy and time-change errors, yielding the claimed asymptotic sign for sufficiently large m.The Gaussian entropy term is o(β_m), while the time-change relative error is o(1); combining the estimates proves the target relation.
- Completion in one dimension: The constructed one-dimensional densities are smooth, positive, Gaussian-decaying, and strictly log-concave, yet violate the Gaussian completely monotone conjecture.The negative sign is established analytically rather than through a separate numerical certificate.
- Higher-dimensional extension: Tensorizing with a broad Gaussian preserves the required regularity and strict log-concavity while producing counterexamples on R^d for every d≥2.Heat-semigroup tensorization and entropy additivity transfer the negative sign to higher dimensions.