On the Homology of Dynamics, Information Theory and Neural Networks under a Multi-Origin High-Dimensional Geometric Framework

Based on the concept of multi-origin curvature space and high-dimensional projection, this paper explores the potential homologous structures of dynamics, information theory and neural networks. Existing theories remain valid within their respective domains. This paper does not replace classical approaches; it merely offers a geometric perspective for speculative reflection.

I. Foundation of the Framework: Multi-Origins, Curvature Space and High-Dimensional Projection

Mass points, information sources and neurons can all be regarded as independent origins.
Each origin possesses its dedicated curvature dimension, and curvature stiffness corresponds to mass, inertia or feature intensity.
Origins couple with one another via curvature gradient differences, forming a dynamic, non-flat high-dimensional space.
Dynamical evolution, information transmission and computational processes are external manifestations of this curvature space, rather than externally attached modules.

II. Dynamics: Motion Generated by Curvature Gradient Differences

Classical dynamics establishes a causal chain of force, acceleration and displacement within flat coordinates.
Within this framework, curvature gradient differences induce shifts in high-dimensional projection, which manifest as displacement, velocity and acceleration.

Forces and torques arise from the coupling of curvature gradients between origins. Momentum is the first-order flow quantity of curvature, and angular momentum the second-order circulation quantity of curvature. Kinetic and potential energy correspond to hierarchical differences in curvature energy across origins. Temporal evolution is the natural progression of iterative curvature changes among multiple origins.

Motion is not merely a computed result; it emerges naturally from the flow of curvature iteration.

III. Information Theory: Integral Volume as the Measure of Information

Conventional information theory is grounded in probability spaces, where entropy is defined by probabilistic expressions.
This framework quantifies information in terms of geometric measure, with entropy corresponding to the integral volume of a high-dimensional state space.

Information entropy, which describes system uncertainty, equals the high-dimensional integral volume. Mutual information is the measure of overlapping regions between curvature structures. Channel capacity denotes the maximum number of distinguishable integral regions. Coding compression is the controllable dimensional collapse of integral regions, and error correction relies on integral redundancy to restore states.

Information is no longer merely an abstraction of probability; it becomes a geometric fact of spatial measure.

IV. Neural Networks: High-Dimensional Curvature Conduction in Place of Flat Matrix Operations

Traditional neural networks rely on layer-wise matrix operations, featuring long information paths and dense parameters.
This framework adopts multi-origin curvature conduction, where information propagates along the shortest geometric geodesics.

Neurons act as independent origins carrying local features and curvature benchmarks. Weights represent high-dimensional association strengths between origins, namely curvature coupling coefficients across origins. Bias stands for the inherent curvature offset of a single origin. The activation function serves as a curvature threshold switch that controls dimensional projection. Forward propagation is directional conduction along geodesics, while backpropagation traces errors backward along curvature gradients for correction. The loss function measures the total geometric deviation of projections from the origin cluster. Gradient descent continuously adjusts coupling relationships between origins. Feature mapping refers to the projection of high-dimensional geometric structures into low-dimensional space.

Information processing and dynamical evolution share the same geometric language.

V. Intrinsic Logic of Homology

The three fields share one carrier: multi-origin curvature space.
They share one driving mechanism: curvature gradient differences, which drive motion, alter information volume, and update network coupling relations.
They share one constraint: inherent topological relations among origins, with no need for additional multipliers or regularization terms.
They share one output: high-dimensional curvature projected onto low-dimensional space, yielding observable quantities such as displacements, symbols and predictions.

Dynamics is the kinematic manifestation of curvature iteration.
Information theory is the measure-theoretic manifestation of curvature space.
Neural networks are the computational manifestation of curvature iteration.