Finite Interpolation and Minimum-Norm Estimates for Ramanujan Partial Sums

This preprint develops a finite-dimensional interpolation and minimum-norm framework for partial sums of Ramanujan sums in a weighted discrete Hilbert space. The paper first proves that, for every finite initial segment ({1,\ldots,L}), the consecutive family (S_2,\ldots,S_{L+1}) interpolates arbitrary complex data. The proof uses divisorial coordinates and an explicit determinant identity for a shifted Ramanujan-sum matrix. An explicit Möbius formula is then derived for the unique interpolant of the constant data on a finite initial segment. The quantitative part formulates interpolation as a constrained minimum-norm problem. For a finite generating family, the effective evaluation kernel[K_{L,N}=A_{L,N}G_N^{-1}A_{L,N}^*]gives the minimum Hilbert norm required to impose constant values on the first (L) points. A Schur-complement identity separates the cost of extending the interpolation by one point into a prediction error and a local correction capacity. An explicit divisorial cardinal function yields an unconditional lower bound of order (L^2/\log L) for the corresponding Schur complement when (N\ge 2L). The paper also studies restricted generating families, establishes a composite-channel obstruction for prime and prime-power indices, proves a rigidity result for nonnegative annihilating charges, and introduces dyadic projection criteria that reduce uniform minimum-norm bounds to explicit finite prediction and correlation defects. All proved results are finite-dimensional and unconditional. The paper does not claim a global closure theorem, an equivalence with a Nyman--Beurling criterion, or a proof of the Riemann hypothesis.