Prime Structure in Multi-Origin High-Dimensional Geometry (MOC): Irreducible Origins and Projection Density

Abstract: Classical number theory is founded on the set of integers and divisibility properties. Within the framework of Multi-Origin High-Dimensional Geometry (MOC), this paper offers a novel geometric interpretation for integer systems and prime distribution. From the MOC perspective, integers correspond to discrete origins in space, and multiplication represents the composite projection of origins onto a low-dimensional interface. Primes are defined as irreducible origins that cannot be synthesised from other non-trivial origins, while composite numbers correspond to composite origins. This paper explores how this geometric viewpoint remains compatible with the Fundamental Theorem of Arithmetic and the Prime Number Theorem in classical number theory.

Keywords: multi-origin geometry; prime numbers; irreducible origin; low-dimensional projection; Fundamental Theorem of Arithmetic

I. Introduction

In classical number theory, primes (2, 3, 5, 7, …) serve as the multiplicative “atoms” of integers. Every integer greater than 1 admits a unique factorisation into a product of primes, which is the Fundamental Theorem of Arithmetic.

This indecomposability is often treated as a purely number-theoretic property. Within the MOC framework, however, it admits a direct geometric interpretation: integers are discrete origins in space, multiplication is the synthesis of origins, and primes are geometrically indivisible singular nodes.

II. Multi-Origin Embedding of the Integer System

From the MOC viewpoint, we construct a multi-dimensional “network of origins”.

1. Integers and discrete origins

Each integer n\in\mathbb{Z} corresponds to a discrete origin P_n in space. These origins do not lie along a flat straight line, but are distributed within a high-dimensional topological network.

2. Multiplication as low-dimensional synthesis of origins

In conventional arithmetic, a\times b = c. In MOC, this corresponds to the “synthesis” of two origins P_a and P_b on a low-dimensional projection plane:

P_a \oplus P_b \longrightarrow P_c

This means that if an origin P_c can be disassembled into the structures of two simpler origins, it is composite.

III. Geometric Definition of Primes: Irreducible Origins

Definition 3.1 (Irreducible Origin)

In the integer origin network of MOC, an origin P_p (p>1) is called an irreducible origin (i.e., a prime) if it cannot be expressed as the synthetic projection of any two non-trivial origins P_a and P_b with a,b>1.

Geometric intuition:

- A prime p is an indivisible fundamental geometric node with an intrinsic complete structure.

- A composite number c is a “composite origin” assembled from multiple prime-factor nodes.

MOC interpretation of the Fundamental Theorem of Arithmetic:

Every integer n>1 can be uniquely factorised into a product of primes.

Within MOC, this is equivalent to a topological geometric proposition: in a flat discrete space, any finite composite origin has exactly one decomposition into irreducible origins. Uniqueness guarantees the irreversibility of the synthesis path.

IV. Projection Density of Prime Distribution

The Prime Number Theorem in classical number theory states that the prime-counting function satisfies \pi(x)\sim x/\ln x. From the MOC perspective, this density decay receives a geometric explanation.

1. Projection from high-dimensional origins to the one-dimensional number line

While irreducible origins are distributed across the high-dimensional network, their observed “density” is constrained by projection rules when projected by an observer onto the one-dimensional natural-number axis.

2. Density decay and topological confinement

As x grows, namely as the observation range extends, cumulative effects of mutual occlusion and interference emerge during the projection of high-dimensional origins. Owing to their most compact structure, irreducible origins exhibit a logarithmic sparsification trend in projected spacing as x increases, matching the macroscopic distribution \pi(x)\sim x/\ln x.

V. Flat Limit: Recovery of Classical Number Theory

In the MOC space, when the distances between origins are contracted to zero and all origins converge onto a single flat coordinate axis (reverting to the conventional set of integers):

1. The “synthesis” of origins reduces strictly to classical multiplication.

2. Irreducible origins become fully equivalent to conventional primes.

3. The occlusion effect of high-dimensional projection vanishes, and the theory is naturally consistent with the foundational axioms of classical number theory.

VI. Conclusion

This paper reinterprets the structures of integers, primes and composites from the perspective of multi-origin geometry. Primes are irreducible geometric origins, composites are their synthesised products, and the Prime Number Theorem describes the density behaviour of the origin system when projected onto a one-dimensional axis.

This geometric viewpoint does not alter the computations of classical number theory. Instead, it supplies a spatial topological account for the questions “why primes are indecomposable” and “why primes follow such a distribution”. Further mathematisation of this multi-origin projection mechanism to derive more precise formulae for prime distribution constitutes the goal of future work.