Moment exponents for Gaussian directed polymers in three dimensions

For the directed polymer based on simple random walk in $\Z^3$ with iid standard Gaussian disorder, let $p^*(\beta)$ be the supremum of the orders for which the normalized partition function has moments bounded uniformly in the polymer length. We prove that $\beta\mapsto p^*(\beta)$ is a continuous strictly decreasing bijection from $(0,\beta_c]$ onto $[5/3,\infty)$, where $\beta_c$ is the weak-disorder critical point. The main step is local stability of every strictly subcritical moment above order two under an increase of $\beta$. A concave power of a terminal polymer moment has a one-step deficit proportional to the endpoint overlap; Gaussian differentiation bounds the perturbation by the same overlap. This yields a moment bound for all lengths. Gaussian hypercontractivity and the established critical and one-sided continuity results then give the full statement.