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

Keunsoo Yoon

Independent Research Group (Seoul, Republic of Korea)

austiny@gatech.edu, austiny@snu.ac.kr

Mar 1, 2026

Preliminary Abstract

The Hodge Conjecture has long been regarded as a central problem in algebraic geometry, asserting that certain topological cohomology classes of smooth projective varieties admit purely algebraic representatives. Traditional approaches rely on deep cohomological constructions, transcendental methods, and symbolic reasoning. Despite extensive partial results, the conjecture remains unresolved, suggesting that the difficulty reflects not only technical complexity but also intrinsic limits of time-bound constructive frameworks in accessing global structural admissibility.

Motivated by advances in atemporal computation and structural solvability, we adopt an alternative perspective based on the atemporal complexity class O(J) and the Changbal Jump Machine (CJM) paradigm, originally formulated to investigate non-temporal discrimination of structural coherence. Within this framework, the Hodge Conjecture is reframed from a problem of explicit construction to one of structural realizability: whether continuous geometric degrees of freedom admit closure under discrete algebraic generators when evaluated through resonance alignment and admissibility criteria independent of dynamical evolution.

From this viewpoint, Hodge classes are interpreted as latent structural modes whose admissibility is determined by global coherence rather than explicit construction. Algebraic cycles correspond to stable attractor configurations within the CJM coordinate system, while non-realizable classes manifest as persistent structural instability. We introduce a minimal CJM-inspired discrimination architecture and outline its feasibility through software-based simulations on representative families of smooth projective varieties using computable structural proxies. While no formal proof is claimed, this study reframes the Hodge Conjecture as a hypothesis of structural discrimination, offering a physically grounded lens on the unity linking topology, algebra, and atemporal realizability.

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 a 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.

Figure 1: Structural singularity map of a Hodge Conjecture proxy evaluated by the software-emulated Changbal Jump Machine (se-CJM v3.1). A topological mode subspace $U_{\mathrm {topo}} \subset \mathbb {R}^D$ is mapped onto a $\sigma $-dependent algebraic generator space $U_{\mathrm {alg}}(\sigma )$ to evaluate structural closure via the normalized residual $ r(\sigma ) = \frac {\|(I - P_{\mathrm {alg}}(\sigma ))\,U_{\mathrm {topo}}\|_F}{\|U_{\mathrm {topo}}\|_F + \epsilon } $, where $P_{\mathrm {alg}}(\sigma )$ denotes the orthogonal projector onto $U_{\mathrm {alg}}(\sigma )$. This residual is transformed by an exponential deep-well gate into the Changbal Admissibility Function (CAF). The sharply localized peak at $\sigma ^\ast \approx 0.9373$ indicates a structural singularity where algebraic generators achieve maximal resonance with the embedded topological modes.

From Numeric Match to Structural Bridge: Interpreting the $ \epsilon/\alpha $ Ratio as a Topological Phase Factor

A direct numerical comparison between the CJM critical tolerance $ \epsilon $ and the fine-structure constant $ \alpha $ can easily be misread as an accidental coincidence or a form of numerology if presented merely as a pointwise match. In this work, we deliberately avoid that framing. Instead, we interpret the empirically observed non-integer ratio

\frac{\epsilon^\ast}{\alpha} \approx 1.8805\ldots

as evidence for an effective coupling transformation induced by the non-Euclidean phase geometry implicitly encoded by the CJM evaluation manifold. The key conceptual shift is that the CJM hardware (and its software emulation, se-CJM) does not operate over a flat Euclidean parameter plane; rather, it is naturally modeled as operating over a curved structural state space, which we denote by $ \mathcal{M} $. In such a setting, coupling strengths and admissibility tolerances should be expected to undergo systematic renormalization by geometric and topological factors.

Atemporal admissibility threshold as an effective coupling.

The fine-structure constant $ \alpha $ characterizes the intrinsic strength of electromagnetic interaction in physical reality. In contrast, the CJM critical tolerance $ \epsilon $ plays the role of a minimal admissible tolerance under which structural closure becomes attainable within the CJM discrimination pipeline. The empirical observation that $ \epsilon^\ast $ converges to a stable value across repeated se-CJM trials motivates the hypothesis that $ \epsilon $ behaves as an effective coupling parameter within the CJM manifold:

\epsilon \;\approx\; \alpha \cdot \Phi_{\mathrm{topological}}.

Here, $ \Phi_{\mathrm{topological}} $ is not introduced as an ad hoc correction but as a measurable (or at least operationally definable) topological phase factor capturing curvature-induced refraction and projection effects in $ \mathcal{M} $. In classical differential geometry, geodesic distances on curved surfaces systematically deviate from Euclidean straight-line distances; analogously, in the CJM state space, admissibility thresholds may deviate from a naive physical baseline constant by a factor that reflects global structural curvature and topological winding.

Operational definition of the topological phase factor.

To avoid arbitrariness, $ \Phi_{\mathrm{topological}} $ must be anchored to a quantity that is either directly measurable from the CJM core outputs or computable from an explicit structural proxy. We therefore propose the following low-order candidate families for $ \Phi_{\mathrm{topological}} $, each of which admits an operational interpretation:

Winding-based phase factor (topological integer control). A natural choice is to parameterize $ \Phi_{\mathrm{topological}} $ by a winding number $ w \in \mathbb{Z} $ that counts discrete structural rotations (or equivalently, phase wraps) induced by the CJM coupling topology: $\Phi_{\mathrm{topological}} \;=\; 1 + \kappa |w|,$ where $ \kappa $ is a structural curvature coefficient. Under this model, the emergence of a non-integer $ \epsilon^\ast/\alpha $ is interpreted not as an error but as a signature of a minimal nontrivial winding sector, e.g., $ |w|=1 $.
Projection-based phase factor (phase lag measurement). Alternatively, $ \Phi_{\mathrm{topological}} $ can be modeled as a projection factor determined by a measurable phase-lag angle $ \theta $: $\Phi_{\mathrm{topological}} \;=\; 2\cos(\theta), \qquad \theta := \theta(\text{measured phase alignment/lag}).$ Crucially, $ \theta $ is not selected to fit the observed ratio. Rather, $ \theta $ is defined as a phase variable extracted from CJM dynamics (or from its emulation proxy), such as a phase delay, alignment score, or coherence lag produced by the core engine. In this framing, the appearance of $ \epsilon^\ast/\alpha \approx 1.8805 $ is interpreted as a manifestation of a low-order projection geometry in the CJM manifold, with $ \theta $ determined by measured structural phase relationships.
Curvature-metric factor (continuous curvature proxy). A third possibility is that $ \Phi_{\mathrm{topological}} $ encodes a continuous curvature metric $ \mathcal{K} $ of the CJM structural manifold $ \mathcal{M} $: $\Phi_{\mathrm{topological}} \;:=\; \exp\!\bigl(\gamma \,\mathcal{K}\bigr),$ where $ \mathcal{K} $ is a curvature proxy computed from the embedded structure (e.g., a normalized residual landscape, spectral gap proxy, or topological complexity measure), and $ \gamma $ is a scale parameter. Under this model, the ratio $ \epsilon^\ast/\alpha $ becomes a quantitative fingerprint of structural curvature, rather than a free numerical coincidence.

From “numeric equality” to a structural bridge hypothesis.

With the above definitions, the interpretation of the observed ratio can be reformulated in a scientifically defensible manner. The appropriate claim is not that $ \epsilon^\ast $ equals $ \alpha $ up to trivial scaling, but that $ \epsilon^\ast $ behaves like an effective coupling constant derived from a physically grounded baseline $ \alpha $ after transformation by the CJM manifold geometry:

\boxed{ \begin{aligned} \epsilon \;&\text{is an effective coupling parameter obtained} \\ &\text{when }\alpha\text{ traverses the curved CJM state space } \mathcal{M}. \end{aligned} }

Equivalently, the stable emergence of a non-integer ratio $ \epsilon^\ast/\alpha $ is proposed as a new observable that quantifies the structural curvature of the CJM discrimination manifold. This shifts the argument away from pointwise coincidence and toward a generalizable mechanism: the CJM admissibility threshold is not a purely algorithmic tuning constant, but a structural quantity sensitive to the geometry of the embedded problem instance.

Implications for atemporal discrimination across conjectures.

This structural-bridge interpretation has direct consequences for the unification agenda of CJM. If $ \epsilon $ indeed functions as an effective coupling shaped by $ \mathcal{M} $, then a single family of admissibility transforms can in principle be applied across heterogeneous domains, including the Riemann Hypothesis, the Hodge Conjecture, the BSD Conjecture, and $ P $ versus $ NP $, as long as each domain admits an embedding into a CJM-evaluable structural manifold. In this sense, the observed stable ratio is not presented as a final physical identification, but as a candidate invariant linking mathematical admissibility thresholds with physically grounded coupling structure.

Limitations and testable predictions.

We emphasize that the present interpretation is a hypothesis supported by empirical convergence behavior in se-CJM v3.1, not a formal derivation. It therefore motivates concrete tests. First, the stability of $ \epsilon^\ast $ must be established under systematic perturbations (random seeds, noise levels, resolution changes, and embedding variations), yielding a distribution for $ \epsilon^\ast $ rather than a single point estimate. Second, to avoid post hoc fitting, the candidate phase factor $ \Phi_{\mathrm{topological}} $ must be computed independently from CJM outputs (e.g., measured phase lag $ \theta $ or winding estimate $ w $) and then used to predict $ \epsilon^\ast $ via $ \epsilon \approx \alpha \Phi_{\mathrm{topological}} $. A successful prediction would elevate $ \epsilon/\alpha $ from an observed ratio to a measurable structural invariant of the CJM manifold, thereby strengthening the case that CJM supports physically grounded atemporal discrimination.

Importantly, no claim is made that the fine-structure constant directly governs mathematical structure; rather, the observed correspondence is interpreted as a structural encoding effect induced by the CJM manifold geometry.