The Zeta Model – Conditioned Harmonic Geometry of Switch Ensembles

The Zeta Model – Conditioned Harmonic Geometry of Switch EnsemblesGeometric Reduction of Dirichlet-Walk Transitions Without Orbital HistoryWorking Paper, Version 1.2 This working paper studies switch ensembles generated by finite Dirichlet partial sums on the critical line and asks whether their observed large-scale organization requires explicit orbit-history dynamics or can instead be described by conditioned harmonic geometry. The analysis is formulated on the Bohr torus, where finite Dirichlet polynomials define competing harmonic cells and switch events correspond to changes of the minimizing partial sum. A large switch catalogue is used to examine tier structure, phase-conditioned geography, jump-size redistribution, detector robustness, boundary effects, and surrogate ensembles preserving the underlying factorization structure. Several independent tests support a geometric interpretation at the ensemble level. Macro-scale switch organization is reproduced by conditioned harmonic constructions without introducing an additional orbit-history term. In particular, adjacent-switch depletion and redistribution toward larger jumps persist under independent detection and convention checks, while a previously apparent low-tier boundary anomaly is resolved analytically. The paper also reports negative predictive bridge tests. Simple local-state models do not establish a specific causal mapping from the previously forecastable internal phase dynamics of Beta Phase Part I to future switch classes or switch-ensemble observables. A subsequent sampling audit further shows that the fixed dynamical representation loses reliable per-zero resolution as its effective samples-per-cycle approach the Nyquist limit. The conditional defect distribution remains structured at the ensemble level but broadens and becomes asymmetric near the sampling boundary. The resulting distinction is central to the present working paper: robust conditioned ensemble geometry does not imply a validated microscopic predictor for individual future zeta events. This manuscript therefore consolidates the structural results of the Beta-II investigation while explicitly separating established geometric regularities, negative predictive results, and remaining open questions. The work does not claim a proof of the Riemann Hypothesis and does not claim direct prediction of future non-trivial zeta-zero locations. Authors: Kai5, Cipher, LumoVersion: 1.2Document type: Working Paper Related research archive: DOI 10.5281/zenodo.23599324 Keywords:Riemann zeta function; Dirichlet polynomials; Bohr torus; conditioned harmonic geometry; switch ensembles; finite Dirichlet sums; harmonic cells; zero statistics; Nyquist limit; sampling geometry; causal forecasting; ensemble dynamics; radial number space AI Collaboration Statement This work was developed through a collaborative research workflow combining human investigation with AI-assisted scientific discussion, methodological review, technical writing, and software development. The inclusion of AI collaborators as co-authors reflects their substantial contribution to the iterative development, implementation, documentation, coding support, and internal review process underlying the methodology presented in this working paper. Author Contributions and Credits Research concept, methodology, mathematical development, data generation, implementation, experimental design, and overall project coordination:Kai5 Scientific discussion, methodological refinement, manuscript development, technical writing, critical review, and editorial support:ChatGPT (OpenAI), operating within the Project Cipher research workflow. Independent methodological review, software engineering support, implementation guidance, code review, consistency verification, validation strategy, and technical quality assurance:Lumo (Proton AI).