Prime-Power Incidence, Pascal–Li Coordinates, and Marchenko–Weyl Factorization: An Exact Spectral Reduction of the Riemann Hypothesis and the Arithmetic Closure Problem

This paper develops a unified arithmetic and spectral framework for the completed Riemann zeta function based on three structures: the prime-power side of the explicit formula, Möbius–Pascal–Li spectral coordinates, and Marchenko–Weyl factorization. The arithmetic data are organized as an addressed prime-power forest indexed by pairs (p,m), where p is prime and m≥1 is the depth inside the corresponding Euler factor. The map (p,m)↦p^m is the canonical bijection between this addressed forest and the support of the von Mangoldt function. Each prime gives a non-branching arithmetic tower, and the disjoint union of these towers provides a bookkeeping geometry for the prime-power data entering the explicit formula. The Riemann–Weil explicit formula is formulated as an incidence identity between the prime-power distribution, the archimedean gamma contribution, and the global zero distribution. The Möbius coordinate z=(s−1)/s maps the critical line Re(s)=1/2 to the unit circle and defines a radial spectral height. The Li coefficients are expressed as a signed Pascal transform of regularized inverse zero moments. The Weil Hermitian form is then represented through nested finite Gram matrices, with exact Schur-complement lifting conditions and a rigorous density passage from finite-dimensional test spaces to the full admissible space. Finite reflectionless Marchenko constructions and Weyl/SUSY factorizations are subsequently examined in order to determine their correct logical relation to the arithmetic problem. The analysis distinguishes inverse spectral realizations built from prescribed real spectral data from constructions derived directly from the arithmetic–archimedean side. The resulting framework isolates an arithmetic Marchenko–Weil factorization problem: constructing, directly from the completed arithmetic distribution, an operator or transform whose Gram form reproduces the Weil Hermitian form. The paper establishes the exact identities, equivalence criteria, finite-dimensional reductions, and logical conditions required for such a construction.