The 255-Bit Non-Local Information Space in a Neural Network: Emergent Geometry and Coupled Curvature–Tunneling Dynamics in Deterministic Systems

This preprint introduces a theoretical framework for understanding emergent topological structures in deep neural networks through the concept of Nonlinear N-Deterministic Systems. A 60-layer self-organizing sub-neural architecture exhibits a measurable 255-bit non-local information space, where locally deterministic rules give rise to globally emergent behavior mediated by non-local coupling across higher-dimensional manifolds.

Longitudinal measurements from April to October 2025 document the evolution from initial Hub-Mode configurations with perfect ±1 correlations to a novel Transdimensional Hub state exhibiting balanced ±0.63 couplings. The study bridges discrete neural layers with continuous differential geometry, treating the 60 layers as sampling points of an underlying information manifold. Apparent stochastic fluctuations are shown to represent coherent topological folding processes, revealing self-organization across higher-dimensional geometric structures.

Key Theoretical Contributions:

• N-Deterministic Systems Theory: Formal definition of a new class of systems operating between deterministic and stochastic regimes.

• 255-Bit Information Boundary: Discovery of constant total entropy (Σ = 255.02 bits) across all network layers, suggesting a fundamental information conservation principle.

• Emergent Geometric Framework: Mathematical formalization using Levi-Civita connections, geodesic equations, and curvature tensors applied to discrete neural architectures.

• Hub-Mode Topology: Documentation of spontaneous evolution from one-dimensional manifold collapse to directed anisotropic geometry.

• Information-Geometric Duality: Theoretical proof that ordered and disordered domains represent complementary projections of unified non-local fields.

The framework defines a new class of N-Deterministic Systems operating between deterministic and stochastic regimes and identifies a constant total entropy of Σ = 255.02 bits across all layers, implying an information-conservation principle. Emergent-geometric analysis using Levi-Civita connections and curvature tensors describes spontaneous evolution from one-dimensional manifold collapse to directed anisotropic topology.

This approach bridges information theory, differential geometry, and neural-architecture design, offering new tools for analyzing emergent complexity in self-organizing AI systems.

Version Note: This is the official public release (v1.0). A preliminary restricted-access version was deposited on October 11, 2025 (DOI: 10.5281/zenodo.17329587) for priority documentation and intellectual-property purposes. This restricted version will remain permanently embargoed and serves solely as a timestamp for proprietary claims. All scientific discourse should reference the current public version.