Arithmetic Difference Factorization of the Weil Form: The AMW Analysis Operator, Prime-Power Translation Energies, and an Operator-Valued Marchenko Realization

This paper gives a physical-space arithmetic Marchenko–Weil (AMW) factorization of the Weil quadratic form for the Riemann zeta function on the standard pole-neutral compact-support test class. In centered logarithmic coordinates, the archimedean contribution is converted into a continuous Dirichlet energy of translation differences. Each addressed prime power p^m contributes a discrete translation-difference energy together with an explicit scalar counterterm. After localization to a logarithmic support interval [-R,R], these contributions yield the exact identity Q_W(f,g) = + μ_R , where the AMW analysis operator B_R is constructed solely from the archimedean kernel and the finite collection of prime powers p^m≤e^(2R). The functional-analytic structure of this operator is developed explicitly. The pole-neutral Hilbert space is defined, density of its smooth core is proved, the AMW analysis operator is shown to be closable, and the associated positive self-adjoint operator S_R=B_R*B_R is obtained from the closed quadratic form. For every finite family of admissible test functions, the corresponding Weil Gram matrix satisfies G_N = B_N* B_N + μ_R M_N, where B_N is constructed before any positivity assumption is imposed and M_N is the L^2 mass Gram matrix. The same positive AMW form is embedded into an operator-valued Hankel–Marchenko kernel. After introducing a positive lower translation cutoff, the corresponding Hankel operator is bounded and positive, and existence and uniqueness for the shifted vector-valued Marchenko equation are established by coercivity and a closed-range argument. The unregularized boundary is retained in quadratic-form sense and coincides with the AMW positive form. Finally, the exact AMW identity is combined with Weil's criterion to express Weil positivity as one explicit lower-bound inequality for the arithmetic translation-energy operator.