A bell-shaped curve can look almost Gaussian and still fail a global shape constraint far out in its tails. This candidate studies symmetric stable probability densities, a family that includes both the Cauchy distribution and the Gaussian. Each density admits some negative power that turns it into a convex function. But how weak must that requirement become as the distribution approaches the Gaussian? The written proof gives a sharp answer. The best exponent deteriorates like minus one sixth times the logarithm of the reciprocal distance from the Gaussian index. At the Gaussian itself, the optimal exponent returns to zero. The proof tracks a moving region where the Gaussian and algebraic contributions to the first derivative balance. It also controls the whole remaining line, so this is not an inference from a finite plot. A second theorem rules out a previously proposed density exponent for every non-Gaussian symmetric stable law except the Cauchy case. This contradicts the density conjecture in the inspected Laha, Miao and Wellner preprint. It does not settle their weaker constraints on distribution and survival functions. A finite interval-arithmetic certificate gives an additional explicit witness. The general theorems rest on the written argument, not on that computation. This is an unrefereed Evidence Press candidate dated September eighth, twenty twenty-six. Internal checks do not establish external specialist validation or historical priority. The full paper, code and evidence are linked on the release page. This is an AI-generated voice summary, not additional mathematical evidence.