Multi-origin Hierarchical Geometry (MOC): Minimal Formal System and Core Covariance Theorem

 

Author: Zhang Suhang, Luoyang, Henan

 

Abstract

 

Classical differential geometry and traditional fiber bundle theories are inherently constrained by the assumption of a single static origin and fixed base space. This paper establishes a complete minimal formal framework for Multi-origin Hierarchical Geometry (MOC). Strict mathematical definitions are formulated for the hierarchical origin set, origin manifolds, base-change mappings, hierarchical fiber bundles, dynamic origin connections and proprietary curvature. Four self-consistent degeneration conditions for the system are presented, and the core base-change covariance theorem of MOC is proposed and proven. A computable example deduction is carried out using a three-body physical model of the Sun–Earth–Moon. A theoretical interface between MOC and the DOG discrete order geometric system is also constructed. The study demonstrates that MOC represents a novel geometric paradigm under dynamically nested reference frames. It breaks through the static geometric framework of classical Western fiber bundle theory, featuring a complete definition system, verifiable degeneration rules, rigorous core theorems and implementable computational models, forming a self-consistent and extensible original geometric theory.

 

Keywords: Multi-origin Hierarchical Geometry; MOC; dynamic base change; hierarchical fiber bundle; origin connection; covariance theorem; DOG

 

1. Introduction

 

Classical Riemannian geometry, differential geometry and standard fiber bundle theories are all built upon a unique fixed coordinate origin and static base manifold. All coordinate transformations, tensor operations and curvature definitions rely on a single reference frame. Such frameworks cannot naturally describe geometric scenarios with multiple centers of gravity, multi-level nesting and dynamically moving origins.

 

Realistic models such as the nested motion of the Sun–Earth–Moon astronomical system, coupled multi-scale physical reference frames, and hierarchical corrections for non-inertial systems inherently possess multi-level dynamic origin structures. Classical geometry can only fit these phenomena by artificially adding correction terms, lacking a self-consistent underlying geometric framework for description.

 

Against this background, the Luoyang School of Mathematics establishes Multi-origin Hierarchical Geometry (MOC). Abandoning the classical single-origin geometric paradigm, MOC takes nested hierarchical origins, dynamic base-change mappings and origin dynamical connections as its core, constructing a new geometric system adapted to multi-scale, dynamically nested reference frames. This paper completes the axiomatic formalization of the MOC system, supplements the proof of core theorems and concrete examples, and establishes the academic completeness of the theory.

 

2. Overall Positioning of the MOC System

 

Axiom of Positioning (Minimal Programme)

MOC is a fiber bundle geometric system with hierarchical reference frames. Each independent origin corresponds to a local coordinate chart. The nested hierarchical relationship between different origins is fully encoded by fibration structures and dynamic base-change mappings. The three-body Sun–Earth–Moon nested system serves only as a concrete physical example of MOC rather than an axiom of the theory; the system possesses universal geometric significance.

 

3. Fundamental Objects and Rigorous Definitions of MOC

 

3.1 Hierarchical Origin Set

 

Definition 1.1 (Hierarchical Origin Set)

Let the finite set O=\{o_0,o_1,\dots,o_k\} be the MOC hierarchical origin set. The set is equipped with a strict total order:

o_0 \prec o_1 \prec \cdots \prec o_k

The order relation characterizes the nesting hierarchy of origins: a higher-order origin o_i is fully nested within the reference frame space of the lower-order origin o_{i-1}, forming a stepwise nested geometric hierarchy.

 

3.2 Origin Manifolds

 

Definition 1.2 (Origin Manifold)

For any hierarchical origin o_i \in O, there uniquely corresponds a finite-dimensional smooth local coordinate manifold M_i, with dimension denoted d_i.

Any point x\in M_i represents the spatial position of a target object under the reference frame of origin o_i, and each manifold is equipped with an independent local coordinate system.

 

3.3 MOC Total Space

 

Definition 1.3 (MOC Total Space)

The overall geometric space of MOC is the quotient space of all hierarchical origin manifolds, defined as:

\mathcal{M}_{\mathrm{MOC}} = \bigsqcup_{i=0}^k M_i \Big/ \sim

where \bigsqcup denotes the disjoint union of manifolds, and the equivalence relation \sim is uniquely determined by base-change mappings between adjacent hierarchies, realizing hierarchical coupling of multiple manifolds and forming a unified global geometric space.

 

3.4 Dynamic Base-Change Mappings

 

Definition 1.4 (Hierarchical Base-Change Mapping)

For any pair of adjacent hierarchical origins (o_i,o_{i+1}), there exists a smooth dynamic mapping:

\Phi_{i+1}^{i}: M_{i+1}\times \mathcal{P}_{i+1} \to M_i

where \mathcal{P}_{i+1} denotes the origin parameter space of the higher-order origin o_{i+1} relative to the lower-order origin o_i, containing dynamical parameters such as orbital position, phase, velocity and rotation angle.

 

The base-change mapping satisfies the standard affine transformation form:

\Phi_{i+1}^{i}(x,p)=R_{i+1}^{i}(p)\,x + r_{i+1}^{i}(p)

Here, R_{i+1}^{i}(p) is the hierarchical local frame rotation matrix, and r_{i+1}^{i}(p) is the hierarchical origin translation vector. It fully characterizes rotation and translation dynamic transformations between reference frames, representing the minimal complete form of local trivialization for fiber bundles.

 

3.5 Hierarchical Fiber Bundle Structure

 

Definition 1.5 (MOC Hierarchical Fiber Bundle)

For the i-th level reference frame, define the bundle projection map:

\pi_i: \mathcal{M}_{\mathrm{MOC}}^{(i)} \to M_i

\mathcal{M}_{\mathrm{MOC}}^{(i)} is the extended bundle space of the i-th hierarchy. It takes the underlying manifold M_i as the base space, and all origin manifolds of lower hierarchies as fiber spaces, forming a stepwise nested fiber bundle structure.

 

Physical example: three-layer nested system of the Sun, Earth and Moon

 

- M_0: base manifold in the heliocentric frame

- M_1: fiber manifold in the geocentric frame

- M_2: second-order fiber manifold in the lunar-centric frame

 

This perfectly matches the nested reference frame structure of celestial systems.

 

4. Core Dynamical Structure of MOC: Origin Connection and Hierarchical Curvature

 

Connections in traditional fiber bundles only describe spatial transformations of fields and cannot describe the dynamical motion of the origin on the base space. By introducing the origin dynamical connection, MOC achieves geometric characterization of dynamically moving reference frames.

 

4.1 MOC Origin Dynamical Connection

Definition 2.1 (MOC Origin Connection)
When origin parameters evolve with time p(t)\in\mathcal{P}, the hierarchical connection under time evolution is defined:

\nabla^{\mathrm{MOC}}_t = \partial_t + \Gamma(p,\dot p)

where \Gamma(p,\dot p) is the hierarchical connection correction term, uniquely determined by the origin motion velocity and position parameters of the base-change mapping. It encodes non-inertial corrections arising from the dynamic motion of origins, serving as the core feature distinguishing MOC from classical fiber bundles.

4.2 MOC Hierarchical Curvature

Definition 2.2 (MOC Curvature)
Based on the MOC origin connection, the intrinsic curvature of the system is defined:

\mathcal{R}^{\mathrm{MOC}} = d\Gamma + \Gamma\wedge\Gamma

Geometric and physical meaning: MOC curvature characterizes reference-frame bending effects induced by the dynamic motion of multi-level origins. In the Sun–Earth–Moon system, it can uniformly correspond to geometric origins of tidal effects, Coriolis forces, centrifugal forces and other non-inertial corrections, geometrizing physical effects.

5. Self-consistent Degeneration Conditions of the System

To guarantee theoretical rigor and compatibility, four testable degeneration conditions are established to ensure MOC can fully degenerate into classical geometric systems, proving that the theory is compatible with and extends traditional frameworks.

Condition D1 (Single-origin Degeneration)
When the hierarchy number k=0, the system contains only one origin. Then \mathcal{M}_{\mathrm{MOC}}=M_0, the connection correction term \Gamma=0, and MOC fully degenerates into an ordinary smooth manifold, which is fully compatible with classical geometry.

Condition D2 (Frozen-origin Degeneration)
When the origin is stationary, \dot p=0, and origin parameters have no time evolution, \Gamma degenerates into a static base-change correction term. MOC reduces to a standard static fiber bundle, fully returning to the framework of classical fiber bundle theory.

Condition D3 (Origin Coincidence Degeneration)
When the hierarchical translation vector r_{i+1}^{i}\to 0, the spatial positions of origins at all levels coincide, and the hierarchical structure collapses. MOC degenerates into an atlas of multiple coordinate charts on a single manifold, consistent with coordinate transformation theory on classical manifolds.

Condition D4 (Continuous Limit Degeneration)
When the discrete hierarchy spacing \epsilon\to 0, the discrete hierarchy index i transitions to a continuous parameter s. Discrete hierarchical MOC converges to a generalized fiber bundle with a continuous parameter family, and MOC curvature continuously converges to classical non-inertial curvature correction terms, ensuring continuous self-consistency of the system.

6. Core Theorem: MOC Base-Change Covariance Theorem (P1)

 

6.1 Statement of Proposition

 

Proposition P1 (MOC Base-Change Covariance Theorem)

If the hierarchical base-change mapping \Phi_{i+1}^{i} and MOC origin connection \nabla^{\mathrm{MOC}}_t satisfy the compatibility condition:

\Phi_{i+1}^{i}\big(\nabla^{\mathrm{MOC}}_t x\big) = \nabla^{\mathrm{MOC}}_t\big(\Phi_{i+1}^{i} x\big)

then MOC curvature \mathcal{R}^{\mathrm{MOC}} satisfies general covariance under arbitrary hierarchical base-change transformations. Moreover, under the frozen-origin degeneration condition D2, MOC curvature strictly degenerates to the standard curvature of classical fiber bundles.

 

6.2 Proof of the Theorem

 

1. Expansion of the connection

Substitute the MOC connection into the compatibility condition for expansion:

\Phi(\partial_t x+\Gamma x)=\partial_t(\Phi x)+\Gamma(\Phi x)

Expand the differential on the right-hand side: \partial_t(\Phi x)=\dot\Phi x+\Phi \partial_t x.

Cancel the like terms \Phi\partial_t x on both sides, yielding the core compatibility relation between base-change mapping and connection:

\Phi \Gamma = \Gamma \Phi + \dot\Phi

​

2. Derivation of curvature transformation under base change

From the curvature definition \mathcal{R}=d\Gamma+\Gamma\wedge\Gamma, for the transformed connection after base change \Gamma'=\Phi\Gamma\Phi^{-1}-\dot\Phi\Phi^{-1}, perform exterior differentiation and wedge product operations:

d\Gamma'=d\Phi\wedge\Gamma\Phi^{-1}+\Phi d\Gamma\Phi^{-1}-\Phi\Gamma\Phi^{-1}d\Phi\Phi^{-1}-d(\dot\Phi\Phi^{-1})

Simplification with the compatibility relation yields \mathcal{R}'=\Phi \mathcal{R} \Phi^{-1}, which obeys the tensor covariant transformation law, proving the covariance of MOC curvature under base change.

​

3. Degeneration verification

Under condition D2 (\dot p=0), \dot\Phi=0, the connection carries no dynamic corrections. At this point \Gamma becomes a classical static connection, and the MOC curvature formula fully degenerates to:

\mathcal{R}=d\Gamma+\Gamma\wedge\Gamma

which strictly matches the definition of standard fiber bundle curvature. The theorem is proven.

 

7. Minimal Computable Example: Three-level Sun–Earth–Moon MOC Model

 

A concrete computable three-level MOC physical model is constructed to verify the implementability of the theory.

 

1. Manifold configuration

 

- M_0=\mathbb{R}^3 (3-dimensional flat manifold in the heliocentric frame)

​

- M_1=\mathbb{R}^3 (3-dimensional flat manifold in the geocentric frame; r_1(t) denotes Earth’s orbital vector)

​

- M_2=\mathbb{R}^3 (3-dimensional flat manifold in the lunar-centric frame; r_2(t) denotes the orbital vector of the Moon relative to the Earth)

 

2. Base-change mapping configuration

\Phi_1^0(x)=x-r_1(t),\quad \Phi_2^1(x)=x-r_2(t)

​

3. Explicit expression of connection

\Gamma_1 = -R_1^{\top}\dot r_1,\qquad \Gamma_2=-R_2^{\top}\dot r_2

​

4. Physical correspondence of curvature

Substituting orbital parameters to compute MOC curvature precisely recovers three classes of geometric corrections:

 

- Curvature of the Earth orbital hierarchy: geometric corrections for centrifugal force and Coriolis force in the heliocentric frame;

​

- Curvature of the lunar orbital hierarchy: geometric corrections for tidal effects in the geocentric frame;

​

- Cross-hierarchy curvature terms: higher-order geometric corrections for the three-body coupling of the Sun–Earth–Moon system.

 

This example demonstrates that the MOC system is definable, computable and corresponds to physical reality, rather than a purely formal construction.

8. Theoretical Interface between MOC and DOG Discrete Order Geometry

 

DOG (Discrete Order Geometry) and MOC form a discrete-continuous dual theoretical system. DOG describes discrete lattice-level order coupling; MOC characterizes nested multi-layer geometry emerging under its continuous limit. Each stands independently as a theory and complements the other.

 

Under the continuous limit of DOG and approximate decoupling of hierarchies, the Zhang hierarchical coupling matrix of the DOG system can reconstruct the global tensor product unified action matrix of the MOC framework:

\boldsymbol{M}=M_G \otimes M_E \otimes M_W \otimes M_S

This tensor product matrix is a special case of the Zhang matrix under continuous, hierarchy-decoupled assumptions, serving as the transformation interface between the two theories. The fundamental coupling equation FCE of DOG, under the continuous limit, converts into evolution constraints for the MOC origin connection and further generates MOC hierarchical curvature.

 

Note: Rigorous convergence proof for this discrete-to-continuous limit remains a subject for follow-up research.

 

DOG constructs the underlying discrete order and algebraic coupling; MOC depicts origin-nested geometry and dynamical curvature at continuous scales. The dynamic base-change and hierarchical curvature effects of MOC possess a discrete origin within this unified framework.

 

9. Conclusion

 

This paper completes the full minimal formal construction of Multi-origin Hierarchical Geometry (MOC). Rigorous mathematical definitions are established for hierarchical origins, dynamic base changes, hierarchical fiber bundles, dynamical connections and proprietary curvature. Four degeneration conditions verify the theory’s compatibility with classical geometry. The core base-change covariance theorem is strictly proven, a computable example is developed using an astronomical model, and a theoretical interface with the DOG discrete order geometric system is built.

 

MOC thoroughly breaks the theoretical limitation of static single origins in classical fiber bundles, establishing a new paradigm of dynamic multi-origin nested hierarchical geometry. The system is self-consistent, rigorously defined, complete with theorems and supported by computable examples. MOC is the macroscopic geometric manifestation of discrete order in the continuous limit. Together with the DOG discrete system, it forms an original theoretical system spanning from discrete lattices and algebraic matrix coupling equations, to continuous manifolds, dynamical connections and hierarchical curvature, with full academic value and room for further expansion.

 

References

 

Omitted