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An exact computer-assisted lower bound for Fisher-information convexity along the two-dimensional heat flow

This paper establishes a machine-verifiable, exact lower bound of approximately 0.9268 for the sharp constant in the Fisher-information convexity inequality along the two-dimensional heat flow by constructing a sum-of-squares certificate with eleven positive rational Gram matrices, thereby significantly improving upon the previously known universal value of 0.5 without requiring symmetry or log-concavity assumptions.

Original authors: Fayez Abu-Ajamieh, Xiaoyan Huo

Published 2026-10-16

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CC BY 4.0

Original authors: Fayez Abu-Ajamieh, Xiaoyan Huo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of probability and physics, there is a fundamental question about how information changes as a system evolves over time. Imagine a drop of ink spreading out in a glass of water. As the ink diffuses, the pattern becomes smoother and more predictable, a process governed by the heat equation. Scientists measure the "sharpness" or concentration of this spreading pattern using a concept called Fisher information. Think of this as a gauge for how much we can learn about the original position of the ink from its current, blurred state. A key property mathematicians have long studied is how this gauge behaves as time passes. Specifically, they wanted to know if the rate at which the information changes follows a strict, predictable curve. For a long time, it was believed that in a two-dimensional space, this curve was perfectly smooth and predictable in a specific way, a property known as log-convexity. However, recent work has shown that this perfect smoothness does not hold true in two dimensions, leaving a gap in our understanding of exactly how the curve bends.

A team of researchers has now stepped into this gap to find the precise limit of how much the curve can bend. They set out to determine the sharpest possible mathematical rule that describes the relationship between the Fisher information and its rate of change for any random distribution spreading out in a two-dimensional plane. Their goal was to find a specific number, a constant, that acts as a floor. This number guarantees that the information curve will never bend more sharply than a certain amount, no matter how the initial ink drop was shaped or how it started. The team discovered that while the curve is not as smooth as once hoped, it is still remarkably well-behaved. They proved that the bending is constrained by a value of approximately 0.926764. This means the curve cannot be as sharp as the theoretical maximum of 1, but it is far from the previously known, looser limit of 0.5.

To reach this conclusion, the researchers did not rely on simple approximations or computer simulations that might contain rounding errors. Instead, they constructed a rigorous, computer-assisted proof that functions like a mathematical certificate. They built a complex algebraic structure involving polynomials and matrices, which they verified using exact rational arithmetic. This method ensures that every step of their logic is flawless, with no room for the tiny errors that often plague numerical calculations. The proof demonstrates that for any possible starting shape of the probability distribution, the relationship between the information and its changes will always stay above their calculated threshold. They also constructed a specific, highly complex example of a spreading pattern that comes very close to this limit, showing that their lower bound is tight and that the true limit is likely just slightly higher, but definitely below 0.94176665012062.

The significance of this work lies in its precision and its independence from special assumptions. Previous results often required the starting distribution to have certain symmetries or to be well-behaved in specific ways, such as having a finite variance. This new result holds true for absolutely any probability measure, even those that are singular or irregular. The researchers used a powerful combination of symbolic algebra and numerical optimization to find a candidate solution, and then they reconstructed that solution using exact fractions to verify it without any doubt. They checked hundreds of specific conditions within their proof to ensure that every part of the mathematical structure remained positive and valid. This level of scrutiny confirms that the bound is not just a likely guess, but a mathematically certain fact for the class of problems they studied.

The findings also shed light on a broader question regarding the "power" of the information curve. By establishing this new, tighter bound, the researchers have narrowed down the range of possible values for a related mathematical exponent that describes how the information behaves under different transformations. Their work rules out a wide range of previously considered possibilities and suggests that the true value lies in a very narrow window. While the exact sharp constant remains a mystery, the team has effectively boxed it in with unprecedented precision. This achievement represents a major step forward in understanding the fundamental geometry of information flow, providing a solid foundation for future investigations into how randomness and diffusion interact in the physical world. The work stands as a testament to the power of combining human insight with computational rigor to solve problems that were previously thought to be out of reach.

Technical Summary: An Exact Computer-Assisted Lower Bound for Fisher-Information Convexity Along the Two-Dimensional Heat Flow

Problem Statement
The paper investigates the sharp constant $c^*$ in the differential inequality governing the Fisher information $I(t)$ along the probability-normalized heat flow on $\mathbb{R}^2$ . Specifically, the authors seek the largest constant $c$ such that the inequality
$I(t)I''(t) \geq c [I'(t)]^2$
holds for all $t > 0$ and all initial probability measures $\mu$ .
While the unrestricted log-convexity value $c=1$ holds in one dimension (Ledoux, Nair, and Wang) but is false in two dimensions (disproved by a hexagonal construction), the known "square-root convexity" theorem establishes a universal lower bound of $c = 1/2$ . The precise value of the sharp constant $c^*$ in two dimensions remains an open problem, with previous work suggesting the interval $1/2 \leq c^* < 1$ .

Methodology
The authors employ a hybrid approach combining rigorous computer-assisted algebraic verification with explicit analytical construction.

Lower Bound via Sum-of-Squares (SOS) Certificate:
- Formulation: The problem is reformulated using explicit two-dimensional index notation. The authors introduce an auxiliary parameter $\tau$ to convert the global product inequality $ID \geq cQ^2$ (where $I, Q, D$ are expectations of local derivative invariants) into the non-negativity of a local polynomial expectation $F(\tau) = \langle d_3 - 2c\tau q + c\tau^2 r \rangle_t$ .
- Algebraic Identity: They construct a polynomial identity of the form:
  $H^3 (d_3 - 2c_0\tau q + c_0\tau^2 r) = S_0 + \tau S_1 + H D_i J_i - 2J_i \partial_i H$
  where $H$ is a strictly positive denominator polynomial, $S_0$ and $S_1$ are sum-of-squares polynomials represented by positive semidefinite Gram matrices, and the remaining terms form a weighted total derivative that vanishes upon integration.
- Verification: The existence of this identity is verified exactly over the rational numbers. The authors use semidefinite programming (SDP) only to discover a candidate solution. This candidate is then projected onto exact rational coefficients. The final certificate involves 11 rational Gram matrices and 491 leading principal minors, all strictly positive. The polynomial identity has zero residual across 15 variables and 925 linear coefficient equations.
- Robustness: The proof imposes no assumptions on the initial measure $\mu$ regarding symmetry, log-concavity, finite variance, or finite entropy.
Upper Bound via Explicit Construction:
- To establish an upper bound for $c^*$ , the authors construct a specific smooth, strictly positive probability density.
- Structure: The density is based on a periodic trigonometric polynomial $g(x)$ (a sum of 729 Fourier modes on a triangular lattice) raised to the power of 6, multiplied by a broad Gaussian envelope.
- Calculation: Using exact rational interval arithmetic, they compute the ratio $I(t)I''(t)/[I'(t)]^2$ for this specific density at $t=0$ (and by continuity for small $t>0$ ). The periodic nature allows for exact evaluation of the necessary moments via coefficient extraction.

Key Results
The paper establishes a rigorous, narrow bracket for the sharp constant $c^*$ :
$0.926764 \leq c^* < 0.94176665012062$
Specifically:

Lower Bound: $c \geq \frac{231691}{250000} = 0.926764$ . This is proven via the exact SOS certificate.
Upper Bound: $c^* < 0.94176665012062$ . This is derived from the explicit counterexample construction.

The authors also translate these bounds into the context of power convexity. Defining the sharp power threshold $\alpha^* = 1 - c^*$ , they show that $I(t)^\alpha$ is universally convex for $\alpha \geq 0.073236$ , while universal convexity fails for $\alpha \leq 0.05823334987938$ .

Significance and Claims
The paper claims to provide the first "exact, machine-verifiable" lower bound for this problem that significantly improves upon the known $c=1/2$ .

Exactness: The authors emphasize that the lower bound is not a floating-point approximation. The final certificate is reconstructed over the rationals and independently verified to have zero residual and strictly positive minors.
Generality: The lower bound holds for any probability measure on $\mathbb{R}^2$ without requiring regularity assumptions (e.g., finite entropy or variance) on the initial data, relying only on the properties of the heat flow for $t>0$ .
Openness: The authors explicitly state that the exact sharp value $c^*$ remains open, as their lower and upper bounds, while close, do not coincide. They suggest that the gap may be closed by improving the denominator $H$ , enlarging the divergence ansatz, or finding a different extremizing sequence.

The work relies on computer assistance for the discovery of the SOS representation and the exact verification of the polynomial identities, but the mathematical validity rests entirely on the exact rational arithmetic and the logical structure of the sum-of-squares decomposition.

Technical Summary: An Exact Computer-Assisted Lower Bound for Fisher-Information Convexity Along the Two-Dimensional Heat Flow

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