Finite-Window Operator Euler Calculus for the Weil Prime Form: Exact Dirichlet Transport, Sharp Euler Leakage, and Boundary-Innovation Geometry

This preprint develops a finite-window operator calculus for the prime-translation part of the localized Weil quadratic form. Zero-extended translations on L^2(-a,a) give a faithful finite representation of truncated Dirichlet convolution, yielding exact operator analogues of the Euler product, Möbius inversion, the von Mangoldt logarithmic derivative, and logarithmic Möbius tapering. The nonconstant arithmetic ideal is identified with the Jacobson radical and its Loewy filtration is computed exactly through the arithmetic depth \Omega(n), showing that ordinary eigenvalues are blind to nonconstant arithmetic data even though the operator representation itself is faithful. After adjoining adjoints, complete prime-power rays admit positive Cayley factorizations with Kac–Murdock–Szegő fibers. Fourier transformation identifies the finite-window boundary defect with Paley–Wiener Hankel leakage. For each Euler factor, an exact retained-plus-leaked energy decomposition is proved, and the normalized leakage is shown to be a scalar multiple of an orthogonal projection. On every KMS fiber the leakage has rank one, corresponding to a single boundary-innovation mode. The single-prime leakage is then transported exactly into Suzuki’s K_a=(-\Delta_N)^{-1} denominator metric, while normalized cross-prime leakage separates into an explicit arithmetic amplitude and a contraction measuring boundary-innovation alignment. The diagonal critical-weight leakage budget grows only as 2a+O(1). The work does not establish positivity of the full Weil operator or prove the Riemann hypothesis; the remaining problem is the collective cross-prime alignment together with the centered prime term and pole/archimedean contribution in the relative K_a geometry.