Preliminary Abstract
The Hodge Conjecture addresses a foundational question in algebraic geometry: whether abstract topological cohomology classes necessarily admit concrete algebraic realization. While it is traditionally formulated as a problem of constructing algebraic cycles through sophisticated analytic, geometric, and cohomological techniques, its persistent resistance suggests that the obstacle may not lie solely in technical complexity. Rather, the conjecture appears to probe a deeper boundary concerning whether geometric structures themselves are intrinsically admissible within algebraic embodiment, independent of any particular constructive pathway.
From a structural perspective, this tension reflects a fundamental discontinuity between continuous geometric degrees of freedom and discrete algebraic generators. Topological classes inhabit a smooth and flexible configuration space, whereas algebraic cycles impose rigid integrality constraints. The conjecture therefore implicitly asks whether global geometric coherence can survive the collapse into discrete symbolic representation without loss of structural integrity. This gap between continuity and discreteness motivates a shift away from sequential derivation toward direct discrimination of structural admissibility.
In this work, we adopt an alternative framework grounded in the atemporal complexity class \( O(J) \) and the Changbal Jump Machine (CJM) paradigm, originally developed to evaluate structural coherence beyond time-ordered computation. Instead of attempting to explicitly construct algebraic representatives, we interpret the Hodge Conjecture as a problem of whether continuous geometric configurations can stabilize into admissible discrete embodiments when evaluated under non-temporal resonance constraints, coherence stability, and topological closure criteria.
Within this interpretation, realizability becomes a property of global structural alignment rather than local symbolic generation. Hodge classes correspond to latent structural modes whose admissibility depends on whether their internal degrees of freedom converge toward stable attractor configurations within an atemporal evaluation manifold. Algebraic cycles emerge as fixed configurations that satisfy resonance compatibility and closure conditions, while non-realizable classes persist as structurally unstable modes unable to achieve coherent consolidation across scales.
To operationalize this perspective, we introduce a minimal CJM-inspired discrimination architecture that encodes representative geometric proxies into a resonance-based evaluation space. Software-based experiments demonstrate the feasibility of probing structural stability and admissibility using computable invariants and spectral surrogates, providing a concrete pathway toward empirical discrimination without reliance on exhaustive symbolic construction or asymptotic convergence arguments.
Although no formal proof is claimed, this formulation reframes the Hodge Conjecture as a question of structural permission rather than explicit derivation. By shifting emphasis from construction to admissibility, the conjecture becomes accessible to physically grounded evaluation strategies that integrate topology, algebra, and non-temporal coherence. This perspective opens a broader program for investigating the conditions under which abstract structure may legitimately manifest as realizable form.
Keywords: Atemporal Computation; Changbal Jump Machine (CJM); O(J); Structural Equilibrium; Hodge 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.
