An Operator Theoretic Route to the Riemann Hypothesis via the Capacity Barrier

This preprint presents a deterministic proof strategy for the Riemann Hypothesis based on a rigid “energy conflict” between (i) the localized footprint that any hypothetical off–critical-line zero would impose on a smoothed detector and (ii) a structural upper bound coming from a near-diagonal bilinear form controlled by a bounded operator norm. The method introduces a sliding scanner: instead of testing at a fixed real part, the detector position adapts to the location of a hypothetical zero. This eliminates classical “coverage” issues (zeros far from the critical line cannot hide outside the detection window) and strengthens the forcing mechanism in regimes where arithmetic noise is smaller. The argument proceeds in three stages: Footprint / forcing: a hypothetical off-line zero generates a localized spike of size proportional to the inverse smoothing scale on a short interval near its ordinate, including the “very close” regime. Gate-P reduction: the detector is related to a truncated Dirichlet model plus a remainder term. The remainder bound is stated and used uniformly with respect to the smoothing scale, avoiding hidden derivative losses. Capacity barrier / contradiction: the Dirichlet model is bounded above by an arithmetic “energy budget” (a second-moment quantity) and a near-diagonal operator ceiling. This yields an upper bound too small to accommodate the footprint spike, producing a contradiction once a single fixed parameter is chosen large enough. The exposition emphasizes logical closure and removal of common hidden dependencies: no finite-height numerical verification is required, no “sufficiently large” thresholds remain in the final statements, and the key remainder estimate is organized to be uniform in the smoothing parameter. The result is a streamlined, self-contained chain of lemmas leading to the claimed elimination of all off-critical-line zeros.