A Modular Route to the Riemann Hypothesis via Spectral Tails and Background Dominance

This work presents a modular “locked-chain” architecture aimed at deriving the Riemann Hypothesis from a sequence of verifiable interface statements. The central idea is to convert a quantitative spectral cancellation input from analytic number theory into strong tail control for a smoothed detector field, then upgrade that probabilistic tail control into a deterministic pointwise floor (“background dominance”) on each dyadic interval. An off-critical zero would force a macroscopic negative footprint in the same detector field; the pointwise floor forbids such footprints, yielding the final contradiction. The manuscript is organized as a set of black-box modules with explicit contracts and dependencies: ADDV (near-diagonal reduction package): restructures high moments into diagonal, far off-diagonal (negligible), and near-diagonal packages, producing a finite collection of bilinear forms in the arithmetic regime while tracking all combinatorial and dyadic losses. Stage-B (spectral engine): treats the near-diagonal bilinear forms using Kuznetsov/trace-formula technology and a spectral large sieve to export a single quantitative power-saving exponent. This exponent is the only genuinely new numerical margin in the chain; all downstream constants depend on it transparently. SDNP (spectral-to-tail transfer): projects the Stage-B saving through the moment method to obtain sub-Gaussian tail bounds for the relevant arithmetic fluctuations (in the normalized measure on each dyadic interval), with an explicit cutoff range for the moment order. Background Dominance: combines the tail bounds with a smoothing–thickness principle to upgrade “rare deep dips” into a deterministic prohibition of any deep negative excursion anywhere on the interval. TSK closure: tunes the smoothing scale so that any off-critical zero would generate a negative footprint deeper than the allowed floor, contradicting Background Dominance. This closes the chain and excludes zeros off the critical line. A key contribution of the paper is the “single-source-of-truth” bookkeeping: every loss (dyadic partitions, admissible weight transforms, capacity/multiplicity factors, and polylogarithmic overheads) is routed into declared consolidation factors, ensuring that no hidden dependencies leak across modules. The result is a proof architecture designed to be auditable: each module can be checked independently against its interface, and the final logical closure uses only pointwise statements.