Finite Gauss-Sum Modular Kernels: Scalar Gap and a Pure AdS$_3$ Gravity No-Go Theorem

We obtain closed-form expressions for the $ST^nS$ modular kernels of non-rational Virasoro CFTs and use them to construct fully analytic modular-bootstrap functionals. At rational width $\tau$, the Mordell integrals in these kernels reduce to finite quadratic Gauss sums of $\operatorname{sech}/\sec$ profiles with explicit Weil phases, furnishing a canonical finite-dimensional real basis for spectral kernels. From this basis we build finite-support "window" functionals with $\Phi(0)=1$ and $\Phi(p)>0$ on a prescribed low-momentum interval. Applied to the scalar channel of the $ST^1S$ kernel, these functionals yield a rigorous analytic bound on the spinless gap. As a second application we prove an analytic no-go theorem for pure AdS$_3$ gravity: no compact, unitary, Virasoro-only CFT$_2$ can have a primary gap above $\Delta_{\rm BTZ}=(c-1)/12$, because a strictly positive "Mordell surplus" in the odd-spin $ST$ kernel forces an odd-spin primary below $\Delta_{\rm BTZ}$.