DRCC-CPR-WIDTH ZETA FUNCTION (DWZ-V1)

Abstract

We study a controlled finite approximation of the Riemann zeta function zeta(s) in the critical strip 0 < Re(s) < 1, motivated by the DRCC stability condition (DRCC = Dimensional Reduction via Controlled Combinatorics) 0 < d_rec(X) <= d_frag(X) < infinity. The naive partial sum Sum_{n=1}^W n^(-s) diverges in the critical strip as the truncation width W increases. We show that a first-order Euler-Maclaurin correction resolves this problem and produces a genuinely convergent width-zeta approximation. On the critical line Re(s)=1/2 the observed convergence rate is O(W^(-3/2)); elsewhere in the critical strip the convergence order depends on Re(s). We refer to this procedure as the DWZ method (DRCC-Width-Zeta method, or Euler-Maclaurin Width Reconstruction Method). Throughout this paper, "width" W denotes only the numerical truncation depth of the finite partial sum. No identification with the structural CPR-Width introduced in earlier DRCC work is claimed. Based on this framework, we derive the empirical reconstruction estimate d_rec_hat(t,eps) ~= (C(t)/eps)^(2/3). The method is validated numerically against mpmath reference values for the first 25 nontrivial zeta zeros, with absolute errors between 1.5 x 10^-6 and 4.4 x 10^-6. Two zeros were located blindly (without prior reference values), and an additional 29 zeros (#13-#41) were confirmed using a hybrid refinement procedure. A statistical analysis of the resulting pattern hypothesis revealed no significant relationship. Nine tested intervals between neighboring zeros contained no additional zero candidates on the numerical grid used; this is not a rigorous completeness proof. The results validate the proposed numerical method but do not constitute a proof of the Riemann Hypothesis. The hypothesis is used only as a mathematical framework for evaluating the method. All figures are generated directly from the underlying numerical data, ensuring complete reproducibility without external image files are required.

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