Let
Ys=sX+Z, where
Z is standard Gaussian and independent of the real random variable
X. We prove that, under the square-exponential moment condition
EeβX2<∞ for some
β>0, the scalar minimum mean-square error
mmseX(s) is analytic at zero signal-to-noise ratio if and only if
X is Gaussian, with constant random variables included as degenerate Gaussians. The proof converts estimation in the Gaussian channel into a backward heat flow acting on the moment-generating function
M(z)=EezX. Under the stated tail condition, every non-Gaussian input forces
M to have a nonzero complex zero. We show that each zero cluster produces a finite singularity in its localized Borel transform at the action
ξ=z02/2. After removing the action scale, the Borel coefficients have a nonzero
n−1/2 prefactor for a simple zero. A zero of multiplicity
m≥2 splits according to the roots of a Hermite polynomial and instead contributes a prefactor
n−m/2erm2n. A finite-disc localization and relative-cycle continuation argument then show that at least one such singularity survives in the full Borel transform. Thus, for every non-Gaussian input in the stated class, the formal zero-SNR expansion is Gevrey-1 but divergent. Rational-MMSE rigidity and the analogous analyticity criterion for mutual information follow as corollaries.