DRCC-Width-Zeta Method (DWZ-V2.2)

Abstract

This work is a reproducible case study in operationalizing an abstract reconstruction-width framework — DRCC (Dimensional Reduction via Controlled Combinatorics, Hesamiy) — on one well-understood numerical problem: computing the Riemann zeta function ζ(s) in the critical strip 0 < Re(s) < 1. The question is not how to compute ζ(s) efficiently (Riemann-Siegel already does that), but what it takes to turn an abstract stability condition, 0 < d_rec(X) ≤ d_frag(X) < ∞, into a checkable number on one specific example, with every limitation made explicit. The naive partial sum diverges in the critical strip as the truncation width W grows. A first-order Euler-Maclaurin correction — classical, not a contribution of this work — resolves this, yielding a convergent width-zeta function (order O(W^-3/2) on the critical line), the DWZ method. The simplest correction is used deliberately: traceability, not speed, is the goal. "Width" W means the numerical truncation depth only, not the structural CPR-Width of DRCC; a formal lemma proves the active reconstruction state stays constant (O(1)) under sequential accumulation regardless of truncation depth, separating runtime length from state width. The method is reproduced against high-precision references for 100 nontrivial zeros (deviation 1.5×10⁻⁶ to 1.3×10⁻⁵), two found blindly, 29 further hardened via a hybrid search; a resulting pattern hypothesis shows no significant relationship. A runtime comparison against Riemann-Siegel on all 100 zeros confirms the expected efficiency disadvantage: about 30× (single-shot) and 280–430× (repeated measurement, mean ± SD, 15–30 runs). Nine tested gaps between neighboring zeros showed no additional candidates — not a completeness proof. The results demonstrate the operationalization's validity, not a contribution to the Riemann Hypothesis, which is not assumed.

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