On Structural Form: Atemporal Coherence of Hodge Classes in Algebraic Varieties

The Hodge Conjecture asks whether certain topological cohomology classes admit algebraic realization. Despite substantial progress, its continued resistance suggests that the central difficulty may lie not only in technical complexity, but also in the assumption that realizability must be established through sequential construction. This motivates a different perspective: realizability may be governed less by symbolic derivation than by structural admissibility.

In this paper, we reinterpret the Hodge Conjecture within the framework of the Changbal Jump Machine (CJM) as a problem of structural admissibility in the atemporal complexity class O(J). Under this view, configurations are mapped into a Hodge-closure representation, and admissibility is evaluated through global coherence, stability, and topological closure. Transformation therefore precedes computation: the Hodge structure is treated not primarily as an object to be sequentially constructed, but as a structural lens through which realizability can be discriminated.

On this basis, we introduce CJM--Hodge as a specialized admissibility filter within the broader CJM architecture. In this formulation, realizable states correspond to structurally stable attractors rather than to outcomes of symbolic derivation. No formal proof of the Hodge Conjecture is claimed. Rather, the aim is to propose a conceptual and structural reformulation in which algebraic realizability is interpreted as a question of closure, coherence, and stability. This perspective is intended to clarify how mathematical structure may be recast in terms of structural permission, while preserving the conjecture itself as an open problem.

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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.

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