Prolate Leakage and Near-Zero Spectral Clusters for the Semilocal Weil Operator

We study quantitative spectral localization for the semilocal Weil operator $\widetilde A_\lambda$ in logarithmic coordinates through arithmetic sums of prolate spheroidal wave functions. For each fixed integer $m\ge1$ and all sufficiently large real $\lambda$, we obtain at least $m$ eigenvalues, counted with multiplicity, in a symmetric interval about zero of radius $K_m\lambda^3\sqrt{1-\chi_{4m+4}(\lambda)}$, where $\chi_j(\lambda)$ are the time-frequency concentration eigenvalues. An exact exterior action identity and a uniform lower norm bound for arithmetic summation transfer source leakage to the whole trial space. We also study the canonical zero-integral combination of modes $0$ and $4$. For its normalized even arithmetic image $b_\lambda$, a WKB phase-fold analysis yields $\|\widetilde A_\lambda b_\lambda\|_2\le C\lambda^{29/6}e^{-2\pi\lambda^2}$, without an additional logarithmic factor. Natural-scale tests give complementary lower bounds for the residual centered at the Rayleigh quotient of $b_\lambda$. For this residual norm $R_\lambda$, the bounds imply $\log R_\lambda=-2\pi\lambda^2+O(\log\lambda)$ as $\lambda\to\infty$ through real values. The lower-bound argument uses fixed smooth test shapes at a growing family of arithmetic nodes and an explicit Bessel profile.