arXiv:2611.13129v1 Announce Type: new Abstract: Applying the Dirichlet principle to the Weil quadratic form $QW_\lambda$ on $L^2([\lambda^{-1},\lambda],d^*u)$--using only the primes $p\le\lambda^2$ of the Euler product--yields canonical finite-dimensional Hermitian matrices whose ground state has a Mellin transform with zeros proved real in full generality when the lowest eigenvalue is simple and even. While $\lambda^2=13$ gave $50$ zeros of $\zeta$, $\lambda^2=250$ ($\lambda\sim16$) gives the first $1000$ zeros with $>500$ correct decimals (the first with $1341$). We explain this via a correspondence between $QW_\lambda$ and Slepian's time-frequency uncertainty operator $U_\lambda=2-P_\lambda-\widehat P_\lambda$, diagonalized by prolate wave functions with eigenvalues $\delta_n(\lambda)\sim c_n\lambda^{2n+1}e^{-4\pi\lambda^2}$ (Fuchs). Via $\mathcal{E}(f)(u)=u^{1/2}\sum_{n\ge1}f(nu)$ and Gevrey smoothing, $\epsilon_\lambda\le C\delta_4(\lambda)(\log\lambda)^3$ is a theorem (Connes-Kastoris). Computations up to $\lambda^2=300$ ($\epsilon_\lambda(1)\sim10^{-1618}$) show a striking logarithmic coincidence with the Fuchs ladder, confirming this simplicity and evenness. Prolate vectors reproduce all minuscule eigenvectors, and conditioning by them and the pole term removes the minuscule spectrum: $QW_\lambda$ is a direct sum of a finite-dimensional near-radical part (Fuchs ladder) and an infinite-dimensional positive part from the Sonin space ($U_\lambda=2$). Under this ladder hypothesis and $QW_\lambda\ge0$, we bound the ground-state distance to the target with Mellin transform $\frac14\xi(s+\frac12)$ (Hurwitz' theorem). Finally, the semilocal trace formula gives the uncertainty principle via Coburn's lemma and proves, under a hypothesis valid at $\infty$, that $QW_\lambda\ge c>0$ outside a finite-dimensional near-self-dual space, reducing Weil positivity to a Schur complement at the ladder's edge.
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