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This work presents Enchan(cosmic), a physics-based deterministic solver for large-scale Ising/Max-Cut systems, and uses it to test the computational stability and reproducibility of the Enchan Field relaxation principle.

The framework was originally developed in a cosmological context, where stable structures are modeled as emerging through deterministic geometric relaxation,

S = Fix(F),

rather than through probabilistic annealing. The solver examined here is a discrete implementation of that pre-existing relaxation principle, not an independently developed optimization algorithm later reinterpreted as cosmology.

The continuous Enchan Field relaxation structure is mapped to a discrete deterministic gradient-flow formulation and tested on fully connected Max-Cut instances. The numerical experiments scale to 10,000 nodes, corresponding to 49,995,000 pairwise interactions.

For the largest instance, the solver reaches a Cut Score of

25,000,000 = N²/4,

which is the exact theoretical maximum for the complete-graph benchmark. Repeated executions under fixed parameters produce invariant final field-state hashes, demonstrating deterministic convergence without stochastic annealing or injected thermal noise.

The purpose of the benchmark is deliberately limited. The study does not claim a reduction in computational complexity, and all pairwise interactions in the tested complete-graph systems are evaluated explicitly. Instead, the benchmark asks whether the deterministic Enchan relaxation dynamics remain numerically stable, reproducible, and non-divergent when the interaction count reaches tens of millions.

The paper further examines a mathematical correspondence between the computational relaxation structure and the scalar-field language used in the Enchan cosmological framework. In this interpretation, field gradients and defect-like geometric structures provide a calculable route for exploring dark-matter-like phenomenology, including possible connections to galaxy-scale acceleration regularities.

This correspondence is presented strictly as a proof-of-possibility for geometric dark-sector modeling. The work does not establish that dark matter has been physically disproven or replaced, nor does it claim a completed derivation of the observed dark sector. Particulate, geometric, and mixed explanations remain open.

The manuscript includes:

the Enchan Field relaxation equation and its mapping to a discrete computational model;

the continuous-to-discrete derivation;

the numerical stability condition and gradient-flow interpretation;

scaling tests from 500 to 10,000 nodes;

exact Max-Cut benchmark results;

deterministic state-hash validation;

and the proposed correspondence between computational relaxation and geometric dark-sector modeling.

The manuscript states that the datasets, graph configurations, and output verification hashes generated during the study are available upon request for reproduction purposes. The underlying theoretical framework and computational implementation are archived separately on Zenodo.

Related Materials

Theoretical Framework — Enchan Field Notes v0.4.2https://doi.org/10.5281/zenodo.17979665

Computational Implementation — Enchan(cosmic)https://doi.org/10.5281/zenodo.18104955

Open Review

An open review has been requested through PREreview on August 26, 2026.PREreview: https://prereview.org/preprints/doi-10.5281-zenodo.18139449

Record-Level Patent & Public Disclosure Notice

The following notice is provided as metadata for this Zenodo record and is not part of the manuscript text.

This record also documents public disclosure relevant to Enchan001 and Enchan002, filed by M. Kobayashi in Japan in 2025. The manuscript itself should be read independently of this record-level patent and provenance notice.