Supplement I: Explicit Construction of \Omega_i (Curvature Coupling Coefficient)

The coefficient \Omega_i was undefined in the original formulas; we now provide a minimal viable definition intrinsically linked to the geometry of \mathbb{M}^n_k.

Let \mathbf{R}_i be the curvature tensor of the i-th origin O_i in \mathbb{M}^n_k, and let \theta_i(x,y) be the geodesic deviation angle between any two adjacent lattice points x,y in the lattice set \mathcal{G}_{n,k} with respect to O_i. Define:

\Omega_i \;:=\; \exp\!\left(-\frac{1}{|\mathcal{G}|}\sum_{(x,y)\in\mathcal{E}} \bigl(1 - \cos\theta_i(x,y)\bigr)\right)

where \mathcal{E} denotes the set of adjacent edges of the lattice set (topological connections).

Properties: \Omega_i \in (0,1]. In flat space (\theta_i\equiv 0), \Omega_i=1; the larger the curvature, the smaller \Omega_i — curvature suppresses the effective number of high-dimensional paths.

For an introductory simplified version, we may use normalized scalar curvature R_i:

\Omega_i = \frac{1}{1 + \alpha \|R_i\|}

where \alpha>0 is a coupling constant to be determined by experiments or symmetry constraints.

 

Supplement II: Derivability Relations Between Axioms and Formulas

(Showing Self-Consistency of the System)

- By Axiom III (Origin Determines Curvature): Each O_i contributes independently to \Omega_i, so the correction terms in the permutation and combination formulas take the form \prod_i \Omega_i or \sqrt{\sum \Omega_i^2}.
- By Axiom IV (Curvature Determines Angular Momentum): The “total geometric action” of permutation paths is proportional to \prod \Omega_i (product of path-independent curvature factors); the “total configuration angle” of combinatorial configurations is proportional to \sqrt{\sum \Omega_i^2} (norm superposition).
- By Axiom V (Matrix Projection to Lower Dimensions): When k=1 and \Omega_1=1 (flat single-origin space),

\mathbb{A}_{n,1}^s = A_n^s,\quad \mathbb{C}_{n,1}^s = C_n^s,

and the classical formulas are naturally recovered.

 

Supplement III: Generalized Normalization Identity for MOC Permutations and Combinations (Advanced, Optional)

Define the MOC generating function:

G_{n,k}(x,y) = \sum_{s=0}^n \left( \mathbb{A}_{n,k}^s \cdot x^s + \mathbb{C}_{n,k}^s \cdot y^s \right).

From the core formulas, we directly obtain:

G_{n,k}(x,y) = \sum_{s=0}^n A_n^s \bigl(\prod\Omega_i\bigr) x^s \;+\; \sum_{s=0}^n C_n^s \bigl(\sqrt{\sum\Omega_i^2}\bigr) y^s,

or equivalently,

G_{n,k}(x,y) = \bigl(\prod\Omega_i\bigr) \cdot {}_n\!P_s(x) \;+\; \bigl(\sqrt{\sum\Omega_i^2}\bigr) \cdot {}_n\!C_s(y),

where {}_n\!P_s(x) and {}_n\!C_s(y) are the generating functions of classical permutations and combinations, respectively.

When \Omega_i \equiv 1 and x=y, the expression reduces to identities related to the binomial theorem.

 

Complete Entry:

Core Formula System for MOC Permutations and Combinations

(Including Explicit Definition of Curvature Coefficients)

\boxed{
\begin{aligned}
&\Omega_i = \exp\!\left(-\frac{1}{|\mathcal{G}|}\sum_{(x,y)\in\mathcal{E}}(1-\cos\theta_i(x,y))\right), \\[4pt]
&\mathbb{A}_{n,k}^s = \frac{n!}{(n-s)!}\prod_{i=1}^k\Omega_i, \\[4pt]
&\mathbb{C}_{n,k}^s = \frac{n!}{s!(n-s)!}\sqrt{\sum_{i=1}^k\Omega_i^2}, \\[4pt]
&\mathbb{U}_{n,k}^s = \mathbb{A}_{n,k}^s + \mathbb{C}_{n,k}^s.
\end{aligned}
}

When k=1 and \Omega_1=1 (flat space), the system reduces to classical permutations and combinations.

 

Suggestion: Two Sets of Notation

- Keep the current set (\mathbb{A}_{n,k}^s, \mathbb{C}_{n,k}^s, \Omega_i), as it directly shows the structure “classical quantity × curvature correction” for intuitive understanding.
- Introduce a more compact set for dense formula derivations:

Original Notation Compact Notation (Recommended) Description
    Number of origins as subscript, dimension as superscript
    Lattice set
      for Permutation
      retained, in script form
    or     suggests curvature weight
      for Total

The final normalization formula can be written in compact form:

\mathcal{T}_{n,k}^{\,s} = \underbrace{A_n^s \prod_{i=1}^k \kappa_i}_{\mathcal{P}_{n,k}^{\,s}} \;+\; \underbrace{C_n^s \sqrt{\sum_{i=1}^k \kappa_i^2}}_{\mathcal{C}_{n,k}^{\,s}},

where

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