DRCC-Width-Zeta (DWZ): A Methodological Demonstration of Reconstruction-Width Operationalization Using the Riemann Zeta Function

Abstract

his manuscript presents a methodological demonstration of how an abstract reconstruction-width framework can be translated into a transparent, checkable, and reproducible numerical workflow. The Riemann zeta function in the critical strip is used as a deliberately well-understood case study, not as a setting in which a new or competitive zeta-function algorithm is claimed. The central question is not how to compute \zeta(s) most efficiently, but which definitions, numerical surrogates, reconstruction models, stability criteria, validation procedures, resource distinctions, and limitations are required to operationalize the DRCC condition 0 < d_{\mathrm{rec}}^{\mathrm{DRCC}}(X) \leq d_{\mathrm{frag}}(X) < \infty in a concrete computational setting. The study develops a six-stage workflow: declaration of the abstract quantity and computational task; selection of an observable finite fragment; specification of an admissible reconstruction model; definition of measurable error and stability criteria; validation against independent references and established methods; and explicit documentation of scope, resource separation, and unresolved limitations. For the zeta-function case, the naive Dirichlet partial sum fails in the critical strip, whereas a first-order Euler–Maclaurin correction yields a convergent width-dependent approximation. The classical correction is used only as a minimal reconstruction model. An empirical threshold W_{\mathrm{stab}}^{(M1)}(t,\varepsilon) is calibrated on the first 50 reference zeros and tested out of sample on zeros 51–100. The aggregate trend transfers, but individual prediction remains weak. Across 100 zeros, absolute position deviations are approximately 1.5\times10^{-6} to 1.3\times10^{-5}; two zeros are located by blind scans, while 98 use known starting values. Benchmarking shows no algorithmic advantage over higher-order Euler–Maclaurin or Riemann–Siegel calculations in practice. The contribution is therefore a reproducible operationalization recipe and one fully documented numerical case study, not a new zeta-function method, not a proof of universal DRCC advantage, and not a contribution to the Riemann Hypothesis.

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