DRCC-WIDTH-ZETA Method (DWZ-V2.1)

Abstract

This work presents DRCC-Width-Zeta (DWZ), a reproducible numerical case study that operationalizes the abstract reconstruction-width ideas of Dimensional Reduction via Controlled Combinatorics (DRCC) and Controlled Partial Reconstruction Width (CPR-Width) for the Riemann zeta function in the critical strip. The study does not propose a faster method for computing ζ(s), nor does it contribute to a proof of the Riemann Hypothesis. Its purpose is methodological: to determine how an abstract stability condition can be converted into a concrete, testable, and accuracy-dependent numerical quantity on a nontrivial problem. The naive width model, defined by the truncated Dirichlet series, is shown to diverge for 0 < Re(s) < 1. A first-order Euler–Maclaurin correction is therefore adopted as the minimal convergent reconstruction model. This classical correction yields the DWZ approximation and supports the definition of a stable truncation threshold W_stab^(M1)(t,ε), interpreted as the smallest width beyond which the reconstructed zero remains within a prescribed tolerance ε. Numerical experiments indicate an error law close to W^(-3/2) on the critical line, consistent with the first omitted Euler–Maclaurin term under the stated local assumptions. From this behavior, an empirical estimator of the form W_hat_stab^(M1)(t,ε) ≈ (C(t)/ε)^(2/3) is derived, with C(t) fitted from the first fifty zeros. The method is tested on one hundred known nontrivial zeros. Ninety-eight zeros are locally refined from reference starting values, two are found through blind scans, and twenty-nine are additionally checked with a hybrid hardening procedure. The resulting absolute deviations lie approximately between 1.5 × 10^(-6) and 1.3 × 10^(-5). A statistical test of the proposed hardening pattern finds no significant association. Gap scans over nine neighboring-zero intervals reveal no additional candidates on the chosen numerical grids, but do not constitute a completeness proof. Comparisons with higher-order Euler–Maclaurin models and the Riemann–Siegel formula show that the first-order DWZ model is substantially slower and less accurate.This disadvantage is reported explicitly and is consistent with the case-study objective. The principal contribution is therefore not computational superiority, but a transparent operationalization of reconstruction depth, model dependence, state-width separation, and reproducible numerical verification within a DRCC/CPR-inspired framework.

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