A Discussion on the Fundamental Axioms of Multi-Origin Hierarchical Differential Geometry (MOC-DG)

Abstract: Traditional differential geometry is built upon a single background manifold and global coordinate systems. Based on the Multi-Origin High-Dimensional Geometry (MOC) framework, this paper presents an exploratory discussion of a hierarchical formulation of differential geometry. Within this framework, space is regarded as a nested network composed of origins at different scale levels. Each level carries its own local differential structure and curvature, and couples with adjacent levels via dimensional-reduction projection rules. Classical single-origin differential geometry emerges as a degenerate special case of this framework under the limit of a single level. This work aims to provide a foundational mathematical perspective for multi-scale physical and geometric systems.

Keywords: multi-origin geometry; hierarchical structure; differential geometry; dimensional-reduction projection; local curvature

I. Introduction

Classical differential geometry (e.g., Riemannian geometry) is generally founded on a single manifold, with all local coordinate charts and metric tensors defined over one shared global background. When addressing multi-scale physical scenarios where macroscopic astrophysical systems coexist with microscopic local systems, a single metric field encounters inherent difficulties in crossing vastly separated scales.

The MOC framework introduces a hierarchically nested geometric viewpoint: space is not a single flat or curved manifold, but a network nested by origins of multiple scales (galactic, stellar-system, planetary, for instance). Each level possesses its own local geometric structure, and levels are linked through specified projection rules.

II. Axioms for Hierarchical Multi-Origins

Axiom 1: Nested Origins across Hierarchies

Space consists of hierarchical origins on distinct scales. Origins at different levels (such as the galactic origin, solar-system origin, Earth origin) are geometrically independent and irreplaceable, each establishing its own local reference benchmark.

Axiom 2: Curvature Governance within Each Level

The origin of each hierarchy governs the local differential structure and curvature distribution of its own layer. Large-scale origins dominate the macroscopic geometric background, while small-scale origins govern local geometric perturbations. Geometric properties of different hierarchies cannot overwrite one another arbitrarily; instead, they nest through topological boundaries.

Axiom 3: Dimensional-Reduction Projection Principle

High-dimensional geometric structures from upper hierarchies may be projected onto lower-dimensional spaces at subordinate hierarchies. During projection, information of the high-dimensional structure is imprinted as local metrics and connections on the low-dimensional interface. When the hierarchical structure converges to a single origin, the entire MOC differential geometry system naturally degenerates into classical single-origin differential geometry.

III. Metrics and Connections from the Multi-Origin Viewpoint

Within the MOC-DG framework, core objects of classical differential geometry (metrics and connections) are extended hierarchically:

1. Hierarchical metric: The metric tensor g_{\mu\nu} is no longer a globally unique matrix, but a local mapping varying with origin hierarchy. Within the domain governed by each hierarchical origin, the metric is determined by projection parameters of that level.

2. Cross-hierarchy connection: Geometric transitions between hierarchies correspond to cross-hierarchy connections under the multi-origin framework. Projection paths from higher to lower dimensions endow local space with emergent curvature behaviours, such as effective gravitational fields or inertial forces.

3. Geodesic deviation: The trajectories of particles or fields follow the geodesics intrinsic to their current hierarchy. When crossing a hierarchical boundary, trajectories deflect due to switching of projection rules.

IV. Flat Limit and Recovery of Classical Theory

MOC-DG is constructed to maintain compatibility with classical frameworks. Consider the following degenerate conditions:

1. The hierarchical structure converges to a single origin (number of hierarchies N=1).

2. Cross-hierarchy projection rules reduce to identity maps.

3. Local metrics degenerate into the flat Euclidean metric.

Under this limit, hierarchical differential structures cease to operate, and MOC-DG automatically collapses into conventional single-origin differential geometry. All classical theorems, including the Gauss–Bonnet theorem and the Einstein field equations, remain rigorously valid.

V. Conclusion

Based on the MOC framework, this paper proposes a formulation for hierarchically nested multi-origin differential geometry. By decomposing space into hierarchical origins and dimensional-reduction projections, the single-origin limitation of traditional differential geometry is partially lifted.

In subsequent work, we will derive precise connection equations for cross-hierarchy projection and explore how this hierarchical structure can be introduced for local metric corrections in general relativity and field theory.

Note: This paper constitutes an exploratory discussion of foundational axioms for the MOC theoretical system. It does not involve concrete engineering applications or quantitative experimental verification. All newly constructed geometric structures can fully revert to classical differential geometry under the limit of a single origin and identity projection, ensuring the consistency and self-compatibility of the system.