An exact computer-assisted lower bound for Fisher-information convexity along the two-dimensional heat flow
We study the sharp constant in the differential inequality \[ I(t)I''(t)\geq c\,[I'(t)]^2 \] for Fisher information along the probability-normalized heat flow on $\mathbb R^2$. 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 \[ c\geq \frac{231691}{250000}=0.926764. \] Together with an independent exact Fourier/Gaussian-envelope construction, this gives \[ 0.926764\leq c_*<0.94176665012062, \] where $c_*$ is the sharp constant for positive-time Gaussian convolutions of arbitrary probability measures on $\mathbb R^2$. 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.