Finite Front Interpolation and Divisorial Structure of Ramanujan Partial Sums

This preprint develops a finite theory of Ramanujan partial sums. It studies the arithmetic and geometric structures generated by these functions on finite initial segments of the positive integers. The paper proves exact divisorial factorization, a divisorial basis for every finite front, periodic anchoring for finite approximation spaces, and exact front interpolation by Ramanujan partial sums. It also proves that prime-only and prime-power-only families omit an essential composite divisorial channel, and that finitely supported nonnegative charges invisible to finitely many detectors are confined to a least-common-multiple lattice. The central conclusion is that local interpolation on every fixed finite front is exact, while the remaining open issue is quantitative control of the weighted tail energy outside a growing front. The paper is self-contained, finite in scope, and does not assert an equivalence with another approximation criterion or a proof of the Riemann hypothesis.