Strong disorder and critical-point shift for a Gaussian pinning model

We study renewal pinning with independent standard Gaussian disorder and gap law $K(n)=n^{-5/4}/\zeta(5/4)$. For every inverse temperature $\beta>0$, we prove that almost-sure extinction of the normalized free partition function at the annealed field $-\beta^2/2$ implies a strictly larger quenched critical field. The decreasing renewal hazard allows us to control future intersections through the polymer's age distribution. Under strong disorder, this gives a $1/t$ lower tail for the maximum constrained partition function. Testing favorable stretches along a planted renewal, while retaining the final renewal endpoint, then gives summable fractional moments and a positive change of field with zero free energy. A quantitative estimate of Alexander and Berger excludes extinction at and below the second-moment boundary.