Multi-Origin Curvature (MOC) Framework: Basic Definitions and Geometric Conservation Law

Author: Zhang Suhang, Luoyang

Abstract

This paper establishes the basic geometric definitions of the Multi-Origin Curvature (MOC) framework. Three core concepts are defined: MOC (Multi-Origin Curvature), MIE (Maximum Information Efficiency), and ECS (Elliptic Coupled Conservation System). Curvature is proposed as a geometric property of a higher-dimensional vector, with its magnitude and direction explicitly specified. The geometric conservation law under the MOC framework is given, and its formal correspondence with the conservation of angular momentum in classical mechanics is pointed out. This paper is a purely geometric framework; it does not involve physical conclusions and does not replace any physical theory. The mathematical structures of MIE and ECS are left for subsequent work.

Keywords: multi-origin curvature; curvature vector; geometric conservation law; elliptic family

I. Introduction

The purpose of this paper is to establish a geometric framework for describing curvature relations under multiple reference origins.

This paper does not revise, extend, or replace any physical theory. This paper does only one thing: define geometric objects and propose geometric propositions.

Specific applications such as the three-body problem, many-body problems, and celestial orbits are left for subsequent articles.

II. Definitions of the Three Core Frameworks

2.1 MOC (Multi-Origin Curvature)

Definition 2.1 (MOC): Suppose there exist multiple reference origins O_i in a system, and each origin corresponds to a curvature vector K_i. The geometric meaning of this vector is: the degree and direction of bending of a trajectory when the trajectory of a certain reference point is observed from origin O_i.

The core proposition of the MOC framework is: each K_i remains constant in the absence of external perturbations.

2.2 MIE (Maximum Information Efficiency)

Definition 2.2 (MIE): Refers to the minimization of the information entropy of a system's geometric configuration while maintaining curvature conservation.

MIE is the principle for selecting the simplest geometric representation under the MOC framework. Its mathematical structure (the expression of information entropy, extremal conditions) is left for subsequent work.

2.3 ECS (Elliptic Coupled Conservation System)

Definition 2.3 (ECS): A system satisfying the following conditions is called an ECS:

· The trajectories of all participants can be mapped to ellipses (or degenerate ellipses);

· Different ellipses are coupled through rules such as sharing foci and addition of curvature vectors;

· The total amount of curvature is conserved, and each local curvature vector is also conserved with respect to its own origin.

ECS is the geometric framework of MOC for handling multi-origin problems. The mathematical form of its coupling rules is left for subsequent work.

III. Geometric Properties of the Curvature Vector

Postulate 3.1 (Vector Nature of Curvature): In the MOC framework, curvature K is a higher-dimensional vector. It satisfies:

· Magnitude |K|: describes the intensity of trajectory bending;

· Direction K-hat: describes the normal direction of the plane in which bending occurs or the orientation of the orbital plane.

The curvature vector carries both bending strength and bending orientation.

Postulate 3.2 (Independence of Multiple Origins): For different reference origins O_i ≠ O_j in a system, the corresponding curvature vectors K_i and K_j are mutually independent, each satisfying its own conservation law. They are related to one another through geometric relations (such as translation and rotation).

IV. Formal Correspondence Between the Curvature Vector and Angular Momentum

Proposition 4.1 (Formal Correspondence): In a central force field, the angular momentum of a particle L = r × mv is consistent with the orbital curvature vector in direction, and there exists a functional relationship between their magnitudes.

Note: This correspondence is formal and does not constitute a proof of physical equivalence. The MOC framework does not claim that the curvature vector "equals" angular momentum; it only points out that in the case of a central force field the two have the same direction, and that there may be some functional relationship between their magnitudes.

Definition 4.1 (MOC Curvature Vector): In the MOC framework, for a particle moving about an origin, its curvature vector may be defined as K := cL, where L is the angular momentum and c is a unit-system constant. This definition makes the MOC curvature vector formally correspond mathematically to classical angular momentum.

Note: This definition is a choice of the MOC framework, not a theorem of physical equivalence.

V. Reformulation of Classical Mechanics Concepts in MOC Geometric Language

This section demonstrates how several concepts in classical mechanics can be reformulated in MOC geometric language. This is not a replacement; it is a translation.

5.1 Orbital Motion

Classical description: position r(t), velocity v(t), angular momentum L = r × v.

MOC formulation: The orbit is described using the curvature vector K and its corresponding elliptic geometric parameters (major axis, eccentricity, focus). When K is a constant vector, the orbit is a conic section.

5.2 Rotation and Spin

Classical description: moment of inertia tensor, angular velocity, spin angular momentum.

MOC formulation: A rigid body is regarded as a collection of many particles, and the sum of the curvature vectors of each particle relative to the center of mass gives the total spin curvature vector. The direction of the spin axis is consistent with the direction of the total spin curvature vector.

5.3 Interactions

This section is not expanded. How "interactions" are expressed in the MOC framework through the geometric compatibility of curvature vectors is left for subsequent work.

5.4 Conservation Laws

Classical Mechanics MOC

Conservation of angular momentum Conservation of curvature vector

Conservation of energy Left for subsequent work

Conservation of momentum Left for subsequent work

VI. Statement of Framework Positioning

This paper does not claim that MOC has overturned any physical theory. MOC is a reformulation of mechanical quantities at the geometric level. All correct conclusions of classical mechanics can be reformulated under the MOC framework, but this does not mean that MOC replaces classical mechanics.

The positioning of MOC is: to provide a geometric language for describing curvature relations under multiple origins. Whether this language has physical value is left to be tested by subsequent work.

VII. Conclusion

This paper has completed the following work:

· Defined the three core concepts MOC, MIE, and ECS;

· Established the geometric properties of the curvature vector;

· Pointed out the formal correspondence between the curvature vector and angular momentum;

· Gave a reformulation of classical mechanics concepts in MOC geometry.

This paper is a purely geometric framework. The mathematical structures of MIE and ECS, as well as the correspondence between MOC and physical theories, are left for subsequent work.