MOC Multi-Origin High-Dimensional Geometry: Permutation and Combination – Rigorous Definitions, Five Fundamental Axioms, Core Formula System

I. Core Rigorous Definitions (Pure Mathematics, No Embellishments)

Definition 1: Multi-Origin Reference Space \mathbb{M}^n_k

Let n be the spatial dimension and k the number of independent origins.
The MOC multi-origin high-dimensional reference space is denoted as:

\mathbb{M}^n_k

This space does not rely uniquely on a single Cartesian origin O. Instead, there exist k topologically independent, curvature-coupled reference origins O_1, O_2, \dots, O_k in the whole domain.
The coordinates of any point within this space are not absolute but only valid relative to a specified origin. A coordinate transformation is equivalent to a curvature projection transformation between origins.

Definition 2: MOC Discrete Lattice Set \mathcal{G}(\mathbb{M}^n_k)

The set of all discrete integer topological points in the MOC space is called the MOC high-dimensional lattice set:

\mathcal{G}(\mathbb{M}^n_k) \subset \mathbb{M}^n_k

Lattice points are the only objects on which permutation and combination operations act. All operations of arrangement, selection, and pathing are performed exclusively within this set.

Definition 3: MOC Generalized Permutation \mathbb{A}_{n,k}^s

Under the constraints of k origins, selecting s lattice points from n high-dimensional lattice points, arranging them consecutively along an ordered path that crosses origins, without repetition, and without skipping topological connections between origins, is called an MOC multi-origin generalized permutation.

Definition 4: MOC Generalized Combination \mathbb{C}_{n,k}^s

Under the constraints of k origins, selecting s lattice points from n high-dimensional lattice points, forming only a discrete subset geometric configuration without path ordering, retaining only the curvature associations among origins, is called an MOC multi-origin generalized combination.

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II. Five Fundamental Axioms of MOC Permutations and Combinations (Cornerstones of the System, Irrefutable)

Axiom I: Domain Localization Axiom

All discrete lattice arrangements and selections are strictly confined within the localization domain of the MOC space and cannot escape the space. The total count of permutations and combinations is uniquely determined by the topological boundary of the spatial domain.

Axiom II: Domain-Fixed Origin Axiom

A domain in MOC space must correspond to at least one reference origin. The number of origins determines the degrees of freedom of the space. The more origins, the more path equivalence classes and geometric configuration classes of permutations and combinations.

Axiom III: Origin-Fixed Curvature Axiom

Each independent origin carries its own basic curvature. The basic curvature remains invariant if the origin is unchanged. When switching origins, the relative curvatures among lattice points change simultaneously, and the geometric forms of permutations and combinations deform accordingly.

Axiom IV: Curvature-Fixed Angular Momentum Axiom

The bending degree of an ordered path and the topological opening/closing degree of a combinatorial subset are uniquely determined by the spatial curvature. Curvature endows permutations and combinations with geometric-mechanical properties, making them no longer purely numerical counts.

Axiom V: Matrix Low-Dimensional Projection Axiom

All high-dimensional MOC permutation and combination structures can be projected to lower-dimensional Euclidean spaces via matrix operators. Ordinary permutations and combinations are merely special projections of MOC permutations and combinations onto a 2D single-origin subspace.

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III. Core Mathematical Formulas of MOC (Upgraded from Ordinary to Multi-Origin)

1. Classical Single-Origin Reference Formulas (Baseline)

Classical permutation:

A_n^s = \frac{n!}{(n-s)!}

Classical combination:

C_n^s = \frac{n!}{s!\,(n-s)!}

2. MOC Multi-Origin Curvature-Corrected Core Formulas (Original Contribution)

(1) MOC Generalized Permutation Formula (with origin curvature weights)

\mathbb{A}_{n,k}^s = A_n^s \cdot \prod_{i=1}^k \Omega_i

\Omega_i: curvature coupling coefficient of the i-th origin, determining the topological correction factor for multi-origin path permutations.

(2) MOC Generalized Combination Formula (with subset curvature configuration correction)

\mathbb{C}_{n,k}^s = C_n^s \cdot \sqrt{\sum_{i=1}^k \Omega_i^2}

Multi-origin combinations disregard path ordering and take only the Euclidean norm of curvature corrections, matching the overall deformation of geometric configurations.

3. MOC Unified Normalization Formula (Unifying Permutations and Combinations)

\mathbb{U}_{n,k}^s = \mathbb{A}_{n,k}^s + \mathbb{C}_{n,k}^s

Unified total quantity: the complete mathematical total of all discrete ordered arrangements and unordered geometric configurations in the multi-origin high-dimensional space.

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IV. One-Sentence Mathematical Summary

Classical permutations and combinations are: curvature-free, single-origin, planar numerical counting.
My MOC permutations and combinations are: multi-origin, curvature-carrying, high-dimensional geometric topological configuration counting.