Frequency‑Coupling Equation (FCE) of Discrete Order Geometry (DOG): Generative Construction from Zhang’s Matrix to FCE

Author: Suhang Zhang
(Luoyang, Henan, China)

Abstract

Discrete Order Geometry (DOG) is built upon discrete lattice points, hierarchical order, adjacency coupling, and continued‑fraction coefficients. After the geometric layer is defined, DOG requires a matching native equation layer. Within the DOG framework, this paper first defines Zhang’s matrix (coupling matrix), which arranges order‑coupling coefficients, and then constructs the Frequency‑Coupling Equation (FCE) from Zhang’s matrix. As a product‑form constraint equation, FCE derives its product structure from the recursive property of continued fractions in DOG, its sparse structure from the locality of DOG adjacency, and its hierarchical structure from DOG hierarchical order. This paper presents the steady‑state form, logarithmic form, and dynamic form of FCE, and clarifies its position in the DOG system. No concrete physical applications are involved; this work merely completes the generative construction from the geometric layer to the equation layer.

Keywords: Discrete Order Geometry; DOG; Zhang’s Matrix; Coupling Matrix; Frequency‑Coupling Equation; FCE; Continued Fractions

1 Introduction

The fundamental proposition of DOG (Discrete Order Geometry) is that a geometric system can be judged by core criteria of hierarchical nesting, order isomorphism, and continued‑fraction scales, without relying on spatial connectivity.

DOG originates from fractal‑geometric ideas and uses continued fractions to realize recursive hierarchical generation. Gauss’s research on the intrinsic connection between continued fractions and analytic functions provides ideological origins for DOG to extend from discrete recursive structures to the function level.

Previous work on DOG has completed:

- System definition, morphisms and order isomorphism (geometric layer);
- Correspondence of field evolution in the continuous limit (limit layer).

However, the equation layer has not yet been established.

DOG needs a native equation to describe the order‑coupling relations between lattice points. The tasks of this paper are:

1. Define Zhang’s matrix, the algebraic arrangement of order‑coupling coefficients;
2. Construct the Frequency‑Coupling Equation (FCE) from Zhang’s matrix;
3. Present the steady‑state, logarithmic, and dynamic forms of FCE;
4. Clarify the position of FCE within the DOG system.

This paper does not involve concrete physical applications and makes no ontological claims. It only completes the generative construction from the geometric layer to the equation layer.

2 Basic Settings of DOG

Definition 2.1 (DOG Discrete‑Order Lattice)

Let \mathcal{L} be a DOG discrete‑order lattice satisfying:

1. \mathcal{L} is a finite or countable discrete set, whose elements are called lattice points;
2. \mathcal{L} is equipped with a hierarchical partial order \preceq;
3. There exists an adjacency relation \mathcal{R} on lattice points: only lattice points at the same or nearby hierarchical levels are adjacent;
4. Each lattice point p\in\mathcal{L} carries an intrinsic order frequency \omega_p;
5. Each lattice point p carries a native generation coefficient \lambda_p, taken from a continued‑fraction generating sequence.

Remarks:

- Lattice points are discrete and non‑connected;
- Lattice points are linked by order relations;
- Order relations take the form of continued fractions, ratios or functions;
- The interior of a single lattice point may be regarded as continuous (a special case of discreteness at a point).

3 Zhang’s Matrix

Definition 3.1 (Zhang’s Matrix)

Let \mathcal{L} be a DOG discrete‑order lattice. Define the matrix
\mathbf{C}=(C_{pq})_{p,q\in\mathcal{L}}
subject to three conditions:

1. Sparsity: C_{pq}=0 if p and q have no partial‑order adjacency;
2. Continued‑fraction generation: non‑zero entries C_{pq} are determined by a continued‑fraction generating sequence;
3. Hierarchical structure: non‑zero entries are arranged in blocks according to the hierarchical order \preceq.

\mathbf{C} is named Zhang’s Matrix.

Remarks:

- Zhang’s Matrix is the algebraic arrangement of DOG order coupling, distinguished from ordinary graph‑theoretic adjacency matrices;
- Its non‑zero structure determines the order neighbourhood of lattice points;
- Its hierarchical structure corresponds to the hierarchical nesting of DOG.

Definition 3.2 (Order Neighbourhood)

For a lattice point p, its order neighbourhood is defined as
\text{Neigh}(p)=\{q\in\mathcal{L}:C_{pq}\neq 0\}.

4 Frequency‑Coupling Equation (FCE)

Definition 4.1 (FCE, Steady‑State Form)

Let \mathcal{L} be a DOG discrete‑order lattice and \mathbf{C} be Zhang’s Matrix. For every lattice point p\in\mathcal{L}, the Frequency‑Coupling Equation is defined as

\boxed{
\lambda_p\,\omega_p
\prod_{q\in\text{Neigh}(p)}
C_{pq}\,\omega_q
=1
}

Where:

- \lambda_p: native generation coefficient of lattice point p;
- \omega_p: intrinsic order frequency of lattice point p;
- C_{pq}: order‑coupling entry of Zhang’s Matrix;
- \text{Neigh}(p): order neighbourhood determined by the positions of non‑zero entries in Zhang’s Matrix.

Remarks:

- The left‑hand side is the product of the lattice‑point self‑frequency term and neighbouring coupling products;
- The right‑hand side equals 1, corresponding to the order‑conservation condition of the DOG lattice;
- The product structure arises from the product‑recursive nature of DOG continued fractions.

Why Product Rather Than Summation

The DOG continued fraction
r_n(C)=\frac{1}{C+\frac{1}{C+\cdots}}
possesses a layer‑by‑layer nested recursive structure. Such nesting corresponds naturally to product combinations in algebra, rather than linear superposition. Therefore, order coupling between lattice points adopts product‑form constraints, in contrast to linear‑summation coupling in classical coupled‑oscillator models.

5 Equivalent Forms of FCE

5.1 Logarithmic Form

This paper assumes that all native lattice‑point coefficients, intrinsic order frequencies, and non‑zero entries of Zhang’s Matrix are positive real numbers to guarantee well‑defined logarithmic transformation. Taking the natural logarithm on both sides of FCE:

\ln\lambda_p+\ln\omega_p+\sum_{q\in\text{Neigh}(p)}\ln(C_{pq}\omega_q)=0

The product‑form equation is converted into a summation equation via logarithm, facilitating numerical computation.

5.2 Dynamic Form

Introduce DOG recursive order‑time \tau:

\frac{d\omega_p}{d\tau}\cdot
\lambda_p\,\omega_p
\prod_{q\in\text{Neigh}(p)}
C_{pq}\,\omega_q
=1

Here \tau is the recursive order‑time of the DOG lattice, not physical time. \dfrac{d\omega_p}{d\tau} describes the evolution rate of the lattice‑point intrinsic frequency with respect to order‑iteration steps, and this product constraint holds for all order‑time instants.

6 Position of FCE within the DOG System

The DOG system consists of four layers:

Layer Content Status
Geometric layer Lattice, order, isomorphism Established
Algebraic layer Zhang’s Matrix Established in this paper
Equation layer FCE Established in this paper
Limit layer FCE   continuous equations Existing work

FCE is the native representation of DOG at the equation layer.

7 Relations with Existing Work

- With graph‑theoretic adjacency matrices: Zhang’s Matrix is an adjacency matrix with hierarchical structure and continued‑fraction generation;
- With coupled‑oscillator equations: FCE adopts product‑form coupling, different from linear‑superposition coupling;
- With continuous‑limit work: The continuous limit of FCE leads to differential equations, for which existing work is available.

Methodological Supplement

There exists a natural algebraic difference between the single‑interaction isolation scheme of this framework and Yang‑Mills theory. The Yang‑Mills Lagrangian has an additive structure of summed terms; when studying a single interaction, other interaction terms are set to zero. By contrast, FCE originates from the continued‑fraction recursive construction of DOG and is a product‑form constraint equation. In a product‑system, the multiplicative identity is 1. Thus, when idealistically isolating one class of coupling channels, the contributions of other couplings can be set to 1 to remove their influence on the product constraint. This treatment is a model‑truncation assumption and does not represent real‑world physical scenarios. Relevant physical interpretations and concrete force derivations are left for future work.

The above are positional remarks and do not constitute comparative conclusions.

8 Conclusion

This paper completes the following within the DOG framework:

1. Defines Zhang’s Matrix: algebraic arrangement of order‑coupling coefficients;
2. Constructs FCE: product‑form Frequency‑Coupling Equation from Zhang’s Matrix;
3. Presents the steady‑state, logarithmic and dynamic forms of FCE;
4. Clarifies the position of FCE in the DOG system.

FCE constitutes the first step for DOG to move from the geometric layer to the equation layer. Its continuous limit, group‑theoretic correspondences and physical interpretations are reserved for subsequent work.

References

Omitted

Note: This paper presents generative constructions within the DOG framework. All definitions are exploratory attempts and do not constitute quantitative conclusions. No concrete physical applications are discussed in the text.