An exact computer-assisted lower bound for Fisher-information convexity along the two-dimensional heat flow
Abstract
We study the sharp constant in the differential inequality I(t)I''(t)≥ c [I'(t)]2 for Fisher information along the probability-normalized heat flow on ℝ2. The unrestricted log-convexity value c=1 is false in two dimensions, whereas the known square-root convexity theorem gives the universal value c=1/2. We construct an exact, machine-verifiable sum-of-squares certificate proving the stronger universal bound c≥ 231691/250000=0.926764. Together with an independent exact Fourier/Gaussian-envelope construction, this gives 0.926764≤ c*<0.94176665012062, where c* is the sharp constant for positive-time Gaussian convolutions of arbitrary probability measures on ℝ2. We formulate the proof in explicit two-dimensional index notation. The lower bound follows from an exact rational polynomial identity involving a strictly positive denominator, a weighted total derivative, and eleven positive rational Gram matrices. All $491$ recorded leading principal minors are strictly positive and the complete polynomial identity has zero residual. No symmetry, log-concavity, finite-variance, or finite-entropy assumption is imposed on the initial measure. The sharp value remains open.
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BibTeX
@misc{abuajamieh2026exact,
title = {{An exact computer-assisted lower bound for Fisher-information convexity along the two-dimensional heat flow}},
author = {Fayez Abu-Ajamieh and Xiaoyan Huo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20653},
eprint = {2610.20653},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20653},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An exact computer-assisted lower bound for Fisher-information convexity along the two-dimensional heat flow
AU - Fayez Abu-Ajamieh
AU - Xiaoyan Huo
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.20653
UR - https://arxiv.org/abs/2610.20653
AB - We study the sharp constant in the differential inequality I(t)I''(t)≥ c [I'(t)]^2 for Fisher information along the probability-normalized heat flow on ℝ^2. The unrestricted log-convexity value c=1 is false in two dimensions, whereas the known square-root convexity theorem gives the universal value c=1/2. We construct an exact, machine-verifiable sum-of-squares certificate proving the stronger universal bound c≥ 231691/250000=0.926764. Together with an independent exact Fourier/Gaussian-envelope construction, this gives 0.926764≤ c_*<0.94176665012062, where c_* is the sharp constant for positive-time Gaussian convolutions of arbitrary probability measures on ℝ^2. We formulate the proof in explicit two-dimensional index notation. The lower bound follows from an exact rational polynomial identity involving a strictly positive denominator, a weighted total derivative, and eleven positive rational Gram matrices. All $491$ recorded leading principal minors are strictly positive and the complete polynomial identity has zero residual. No symmetry, log-concavity, finite-variance, or finite-entropy assumption is imposed on the initial measure. The sharp value remains open.
ER -