Finite Weil Windows: Euler Boundary Geometry, Hyperbolic Parity, and Smoothed Prime Errors

We investigate the coupling between finite Euler shifts, hyperbolic parity, archimedean gamma contributions, and logarithmically smoothed prime errors in finite-window Weil quadratic forms. An exact positive Euler boundary-leakage decomposition yields a spatial operator normal form. A sharper endpoint analysis identifies two asymptotically localized directions with distinct leakage scales, 1 and 2a − γ_E. In the associated two-dimensional compression, the smoothed Chebyshev error appears as an off-diagonal boundary coupling rather than a diagonal leakage cost. Using the classical explicit formula, we relate this coupling directly to the nontrivial zeros of the Riemann zeta function. Its exponential growth envelope determines the supremum of their real parts. Conditional on the Riemann Hypothesis (RH), its Bohr–Fourier coefficients recover individual zero ordinates and multiplicities. The corresponding compressed spectral response provides a complementary quantitative characterization of hypothetical RH violations. We also derive a refined unconditional hyperbolic-parity transfer formula in which the leading quadratic-in-window arithmetic and archimedean corrections cancel exactly. One-sided Mellin–Landau arguments and a distinct two-sided Mellin-holomorphy argument give RH-equivalent characterizations. The results provide a unified operator-arithmetic interpretation of boundary localization, prime-number fluctuations, and zeta-zero information. They do not prove RH, establish global Weil positivity, or certify a new positive window. Analytical limitations, numerical regression checks, and literature-priority qualifications are explicitly documented.