, Sang-un (2025). Comparative Analysis of Riemann Hypothesis Proof Structures and Logical Hierarchy. figshare. 

https://doi.org/10.6084/m9.figshare.30509540.v1
Comparative Analysis of Riemann Hypothesis Proof Structures and Logical HierarchyThis document aims to comparatively analyze the proof structures for the Riemann Hypothesis (RH) and establish the ultimate logical hierarchy based on five core research papers provided. The analysis focuses on two central proof approaches: * Proof Structure 1 (Auxiliary Assumption-Dependent Type): The initial proof system relies on powerful external mathematical assumptions for the core link (Bridge) of the proof. Specifically, it assumes the relationship between analytic zeros and the quantum spectrum as an 'Spectral Equivalence Axiom (Axiom 4.1)' and cites the unproven 'Schanuel's Conjecture' to control the complexity of the exponential terms in the Zeta function. This structure takes the form of a conditional proof, complete only if both assumptions are true. * Proof Structure 2 (Internally Self-Contained Type): The subsequent proof system eliminates this external dependency, achieving logical self-sufficiency. 'Spectral Equivalence' is no longer an axiom but is elevated to a 'Deterministically Proven Theorem' through the analysis of geometric invariants of the 'RBSA (Rotation-Based Search Algorithm)'. Furthermore, the necessity of 'Schanuel's Conjecture' is replaced by the analysis of the dynamic properties of 'Analytic Energy Flow' and the 'Quantum Jump' mechanism, proving, via internal logic, that the system converges to the energy minimum (i.e., the critical line). This document clearly illuminates the evolution of the logical architecture by comparing these two approaches in parallel. Ultimately, the proof of the Riemann Hypothesis is presented as a complete logical hierarchy, consisting of a foundation (lower stratum) of 'Ontological Establishment of the Spectrum via RBSA' and a conclusion (upper stratum) of the '1/2-Line Alignment of the Spectrum via Energy Minimization Dynamics'.doi.org

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