On Structural Criticality: Atemporal Topological Discrimination of the Riemann Hypothesis

The Riemann Hypothesis (RH) has traditionally been approached as a purely analytic problem in number theory, addressed through asymptotic methods and time-bound computation. Its persistent unresolved status suggests that the difficulty of RH may reflect not only technical obstacles, but also limitations inherent in time-dependent mathematical reasoning. Motivated by foundational questions in computational complexity, we adopt an alternative perspective based on the atemporal complexity class O(J) and the Changbal Jump Machine (CJM) paradigm, originally developed to study structural solvability beyond temporal enumeration. Within this framework, RH is repositioned from a purely formal conjecture to a physically discriminable hypothesis. Rather than pursuing a traditional proof, we examine whether the defining features of RH can be embedded into a non-temporal setting where solvability is evaluated through structural coherence, resonance, and admissibility criteria independent of dynamical evolution, allowing ordered behavior to persist without reliance on temporal relaxation. From this perspective, the critical line at one-half is interpreted as a global ridge of phase stability, along which oscillatory components of the zeta function align coherently. Nontrivial zeros emerge as collective resonance points reflecting an atemporal equilibrium. We introduce a minimal CJM-inspired discrimination architecture and demonstrate its feasibility through software simulations based on standard approximations of the zeta function evaluated along the critical line. While no formal resolution is claimed, this study reframes the Riemann Hypothesis as a hypothesis admitting structural discrimination in principle, offering a new structural lens on critical-line phenomena. ========================================================================================= The Master Manifesto. We propose a shift in addressing the Millennium Prize Problems from exclusively formal, time-bound proof toward structural and physically discriminable experimentation, reinterpreting mathematical conjectures not as statements requiring asymptotic derivation but as questions of realizability within a structured, non-temporal state space. This series originates from the P versus NP problem, reformulated through the Changbal Atemporal Equation, P≡NPᴶ, and evaluated by the Changbal Jump Machine (CJM). The term Changbal is derived from the Korean conceptual notion of 창발 and denotes a discontinuous structural transition beyond constraint boundaries, distinct from gradual emergence; within this framework, solvability is defined by structural admissibility rather than computational effort. The technical foundations of the O(J) state space, the Changbal Atemporal Equation, and the CJM architecture have been developed and analyzed in detail in prior work [Yoon, K. (2025). P ≡ NPᴶ: On the end of time. Zenodo. DOI: https://doi.org/10.5281/zenodo.18139629]; accordingly, these elements are treated here as established primitives, and the present paper focuses exclusively on their application to a specific conjecture rather than on re-deriving or extending the underlying formalism. =========================================================================================