DRCC-Width-Zeta (DWZ-V2.2), A Methodological Demonstration of Reconstruction-Width Operationalization Using the Riemann Zeta Function

Abstract

This manuscript presents a methodological demonstration of how an abstract reconstruction-width framework can be translated into a transparent, auditable, and reproducible numerical workflow. The Riemann zeta function in the critical strip is used as a well-understood case study, not as a setting in which a new or competitive zeta-function algorithm is claimed. The central objective is to determine which definitions, finite surrogates, reconstruction models, error criteria, validation procedures, and limitations are required to operationalize the DRCC condition 0<d_{\mathrm{rec}}^{\mathrm{DRCC}}(X)\le d_{\mathrm{frag}}(X)<\infty in a concrete computational setting. The study follows a six-stage procedure: declaration of the abstract quantity and computational task; selection of an observable finite fragment; specification of an admissible reconstruction model; definition of measurable stability and error criteria; validation against independent references and established methods; and explicit documentation of scope, resource separation, and unresolved limitations. For the zeta-function instance, the naive partial sum diverges in the critical strip, whereas a first-order Euler–Maclaurin correction provides a convergent width-dependent approximation. The Euler–Maclaurin formula is classical and is used only as a minimal reconstruction model for examining the operationalization process. An empirical stability threshold W_{\mathrm{stab}}^{(M1)}(t,\varepsilon) is estimated using a power-law model calibrated on the first 50 reference zeros and evaluated out of sample on zeros 51–100. The aggregate trend transfers, but individual predictive accuracy remains weak, with R_{\mathrm{oos}}^{2}\approx0.08; the estimator is therefore interpreted only as an order-of-magnitude guide. Across 100 reference zeros, absolute position deviations range approximately from 1.5\times10^{-6} to 1.3\times10^{-5}. Statistical tests provide no defensible evidence for the proposed spacing relationship in the available n=20 hardening sample. Benchmarking further shows that the first-order realization has no algorithmic advantage over higher-order Euler–Maclaurin or Riemann–Siegel methods. The contribution is therefore a reproducible operationalization recipe, one fully documented numerical case study, and one exact finite combinatorial example—not a new zeta method, a universal DRCC advantage, or a contribution to the Riemann Hypothesis.

Keywords