Functions under Multi-Origin High-Dimensional Geometry (MOC)

Abstract: In classical mathematics, a function is generally defined as a mapping rule between two sets, for instance f: X \to Y. Based on the Multi-Origin High-Dimensional Geometry (MOC) framework, this paper investigates the ontological geometric status of functions. It demonstrates that within MOC, a function is not an abstract rule detached from space, but a path description of origins undergoing multidimensional displacement, projection and mutual interaction in space. When an origin unfolds continuously along a specified direction, it forms a continuous function in the classical sense; when an origin switches between different hierarchies, the behaviour manifests as composition and multi-valued mappings. This paper provides an ontological geometric interpretation for the concept of a function.

Keywords: multi-origin geometry; function; origin trajectory; dimensional-reduction projection; spatial mapping

I. Introduction

Since Leibniz and Euler, functions have gradually been abstracted into pure correspondence rules: given an input, an output is produced. In modern mathematics, functions are further grounded in set theory. This abstraction has greatly advanced mathematics, yet it decouples the concept of a function from concrete geometric intuition.

The MOC (Multi-Origin High-Dimensional Geometry) framework seeks to restore geometric substance to functions. The core thesis of this paper is that a function is fundamentally not a “rule”, but a geometric trajectory generated when origins move, project and switch within a spatial structure.

II. A Function as a Continuous Motion Trajectory of an Origin

1. Limitations of the conventional viewpoint

The conventional function y = f(x) describes a curve in one-dimensional or low-dimensional space. This description presupposes an absolutely stationary coordinate system as the background.

2. Reconstruction within the MOC viewpoint

In the MOC system, there exists no absolutely stationary global background. The so-called independent variable x is in fact the state of a local origin moving continuously through space; the dependent variable y = f(x) is the projected trace left by this moving origin on another reference interface (or the reference origin for dependent quantities).

- A function is not a “mapping”, but a process of origin motion.

- The graph of a function is the imprint of an origin’s motion trajectory on a low-dimensional observation surface.

III. Multi-Origin Switching and Composite Functions

Composite functions g(f(x)) in conventional mathematics receive a highly intuitive geometric interpretation under MOC.

1. Switching of reference frames

Suppose f(x) represents the trajectory of origin P projected from origin A to origin B, and g(y) represents the projection trajectory from origin B to origin C.

Then the composite function g(f(x)) essentially corresponds to origin P undergoing a continuous transfer across multiple reference frames: A \to B \to C.

2. Path dependence

Within a complex topological structure with multiple origins, there may be more than one transfer path from A to C. If the space contains structural distortions, different paths (i.e., different orders of composition) yield distinct final projection results. This accounts for why the order of composition fails to commute under certain algebraic structures.

IV. Hierarchical Nesting of Functions and Multi-Valued Behaviour

In high-dimensional multi-origin space, the form of a function is determined by the dimension of the observation interface.

1. Multi-valuedness arising from dimensional reduction

When the motion trajectory of an origin in high-dimensional space is projected onto a low-dimensional observation surface, trajectories that do not intersect in the higher dimension may overlap on the projection plane. Classically, this phenomenon is described as a “multi-valued function” (e.g., the circle equation x^2+y^2=r^2). Within MOC, this is merely the sectional manifestation of a high-dimensional geometric structure on a low-dimensional interface.

2. Hierarchical nesting

The same origin exhibits different functional characteristics in different hierarchies (large-scale macroscopic hierarchies and small-scale local hierarchies). A function with violent oscillations at small scales (high-frequency perturbations) may appear as a smooth continuous function (low-frequency envelope) under large-scale projection.

V. Flat Limit: Recovery of Classical Functions

Within the MOC framework, suppose all origins coalesce into a single origin, and the reference interface for origin motion remains flat (no hierarchy switching and no dimensional collapse):

1. Continuous motion of the origin degenerates into a curve in a rectilinear coordinate system.

2. Multi-valued sections reduce to single-valued mappings.

3. The function fully reverts to the “mapping rule” defined in classical set theory.

A classical function is therefore a degenerate special case of the multi-origin geometric structure restricted to a flat, single-origin interface.

VI. Conclusion

This paper reinterprets functions as motion and projection trajectories of origins across different reference frames from the perspective of multi-origin high-dimensional geometry. A function is no longer a suspended mapping rule, but a manifestation of the dynamic evolution of spatial structure itself.

This interpretation does not alter the computational rules for functions. Instead, it seeks to establish an ontological geometric footing for the concept. Subsequent work will undertake the topological reformulation of more sophisticated notions in calculus, such as limits and continuity, within the multi-origin framework.

Note: This paper presents a foundational conceptual discussion within the MOC theoretical system and does not introduce changes to specific computational methods. All newly proposed geometric interpretations can revert completely to the definition of a function in classical mathematics under the limit of a single origin and a flat interface.