Finite Energy Barriers for Ramanujan Partial Sums

This preprint develops a finite-dimensional energy framework for front interpolation by partial sums of Ramanujan sums in a weighted discrete Hilbert space. For a prescribed initial segment and a finite detector budget, it introduces a visibility kernel whose inverse quadratic form gives the minimum Hilbert-space energy required to impose constant front data. The paper proves an exact Schur-complement innovation identity that separates the energy cost of adding a new front point into a prediction error and a local capacity term. An explicit divisorial cardinal correction yields a quantitative lower bound for the Schur capacity when the detector budget is at least twice the front length. The paper also develops a dyadic renormalization framework comparing vertical front growth with horizontal detector enrichment, reducing uniform energy boundedness to explicit prediction, transfer, or scalar-retention 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.