A rare-event obstruction to the uniform exponential GEPP tail

For every $n\ge2$ and $C\in\{0,I_n\}$, a matrix $G$ with independent standard normal entries satisfies $\Pr\{\rho_{\rm PP}(C+G)>\tfrac12(3/2)^{n-1}\}\ge2^{-n^2(n^2+n+5)}$, where $\rho_{\rm PP}$ is the growth factor of Gaussian elimination with partial pivoting in exact arithmetic. Consequently no positive constants $c_1,c_2$ give $\Pr\{\rho_{\rm PP}(\bar A+\sigma G)>x(n/\sigma)^{c_1}\}\le2^{-c_2x}$ for all $n$, all $\|\bar A\|_2\le1$, $0<\sigma\le1$ and $x\ge1$; the center $\bar A=I_n$ with $\sigma=1$ already violates it. The proof uses an explicit matrix with unit diagonal, strict lower entries $-1/2$ and last column of ones, on which partial pivoting never exchanges rows and the growth is $(3/2)^{n-1}$. Every matrix in the entrywise box of radius $2^{-(n^2+n+1)}$ around it is nonsingular, keeps the same strict pivots and has growth above half that value, and the Gaussian probability of the box gives the bound.