Preliminary Abstract
The Birch and Swinnerton-Dyer (BSD) Conjecture links the analytic behavior of an elliptic curve at \( s=1 \) to the latent arithmetic capacity encoded in its rational points. While classical approaches pursue this correspondence through deep analytic continuation and arithmetic invariants, the persistence of the problem suggests that rank may reflect a structural property that resists purely sequential inference.
We reinterpret BSD as a problem of generative admissibility within the atemporal complexity class \( O(J) \) and the Changbal Jump Machine (CJM) paradigm. Rather than computing rank explicitly, the question becomes whether arithmetic degrees of freedom admit coherent structural activation under non-temporal resonance constraints.
In this formulation, the vanishing order of the \( L \)-function functions as a resonance signature of latent generative potential, while rational points correspond to stabilized activation modes within the CJM state space. A minimal discrimination architecture is outlined using computable spectral surrogates. No formal resolution is claimed; instead, BSD is positioned as an operationally discriminable hypothesis of arithmetic emergence.
Keywords: Atemporal Computation; Changbal Jump Machine (CJM); O(J); Birch and Swinnerton-Dyer; BSD conjecture; Trinity Resonance; P vs NP; NP Problem; Time Crystal; allthingsareP; 창발
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 ≡ NP\(^{J}\), 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 [?]; 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.
