Basic Coupling Equation of Discrete Order Geometry (DOG): Generative Construction from Zhang's Matrix to FCE

Author: Zhang Suhang

(Luoyang, Henan)

---

Abstract

Discrete Order Geometry (DOG) takes discrete lattice points, hierarchical order, adjacency coupling, and continued-fraction coefficients as its basic architecture. After the geometric layer has been defined, DOG requires a native equation layer that matches it. Within the DOG framework, this paper first defines Zhang's matrix (coupling matrix), formed by the arrangement of order-coupling coefficients, and then constructs the Fundamental Coupling Equation (FCE) from Zhang's matrix. FCE is a product-type constraint equation: its product structure comes from the recursive properties of DOG continued fractions, its sparse structure comes from the locality of DOG adjacency, and its hierarchical structure comes from DOG hierarchical order. This paper gives the steady-state form, logarithmic form, and dynamic form of FCE, and explains its position within the DOG system. This paper does not involve specific physical applications; it only completes the generative construction from the geometric layer to the equation layer.

Keywords: Discrete Order Geometry; DOG; Zhang's matrix; coupling matrix; Fundamental Coupling Equation; FCE; continued fractions

---

1 Introduction

The basic claim of DOG (Discrete Order Geometry) is that a geometric system can avoid dependence on spatial connectivity and instead take hierarchical nesting, order isomorphism, and continued-fraction scales as the core criteria for structural determination.

DOG originates from fractal geometric ideas and uses continued fractions to achieve recursive hierarchical generation. Gauss's research on the intrinsic connection between continued fractions and analytic functions provides an intellectual source for DOG's expansion from discrete recursive structures to the function level.

In previous work, DOG has completed:

· System definitions, morphisms, and order isomorphism (geometric layer);

· Field evolution correspondence in the continuous limit (limit layer);

· Zhang group–Lie group emergence relation (group-theoretic layer).

However, the equation layer has not yet been established.

DOG requires a native equation to describe the order-coupling relations among lattice points. The tasks of this paper are:

1. Define Zhang's matrix (coupling matrix);

2. Construct the Fundamental Coupling Equation (FCE) from Zhang's matrix;

3. Give the steady-state, logarithmic, and dynamic forms of FCE;

4. Explain the position of FCE within the DOG system.

This paper 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} carries a hierarchical partial order \preceq;

3. There is an adjacency relation \mathcal{R} among lattice points, and only lattice points of the same or nearby levels are adjacent;

4. Each lattice point p\in\mathcal{L} carries an intrinsic order quantity \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 connected by order;

· The form of order is continued fractions, ratios, or functions;

· The interior of a single lattice point may be regarded as continuous (discreteness at a point is a special case).

Remarks on \omega_p:

\omega_p is the intrinsic order quantity on a lattice point. Its value is not pre-limited, but is determined by the representation of the Zhang group G_Z.

· When the Zhang group takes a scalar representation, \omega_p is a scalar;

· When the Zhang group takes a Lie algebra representation, \omega_p is a Lie algebra value;

· When the Zhang group takes other representations, \omega_p takes the corresponding value.

The physical label of \omega_p is only a classification entry and does not constitute the mathematical specification of FCE.

---

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}}

satisfying the following three conditions:

1. Sparsity: if p,q have no partial-order adjacency, then C_{pq}=0;

2. Continued-fraction generation: the values of nonzero entries C_{pq} are determined by a continued-fraction generating sequence;

3. Hierarchical structure: nonzero entries are arranged in blocks according to the hierarchical order \preceq.

\mathbf{C} is called Zhang's matrix (also called the coupling matrix).

Remarks:

· Zhang's matrix is an algebraic arrangement of DOG order coupling, different from an ordinary graph-theoretic adjacency matrix;

· Its nonzero structure determines the order neighborhood of a lattice point;

· Its hierarchical structure corresponds to DOG hierarchical nesting.

Definition 3.2 (Order Neighborhood)

For a lattice point p, define its order neighborhood as

\text{Neigh}(p)=\{q\in\mathcal{L}:C_{pq}\neq 0\}.

---

4 Fundamental Coupling Equation (FCE)

Definition 4.1 (FCE, Steady-State Form)

Let \mathcal{L} be a DOG discrete order lattice, and let \mathbf{C} be Zhang's matrix. For each lattice point p\in\mathcal{L}, define the Fundamental Coupling Equation as

\lambda_p\,\omega_p

\prod_{q\in\text{Neigh}(p)}

C_{pq}\,\omega_q

=1

where:

· \lambda_p: native generation coefficient of the lattice point;

· \omega_p: intrinsic order quantity of the lattice point, whose value is determined by the Zhang group representation;

· C_{pq}: coupling coefficient of Zhang's matrix;

· \text{Neigh}(p): order neighborhood determined by the positions of nonzero entries in Zhang's matrix.

Remarks:

· The left-hand side of the equation is the product of the lattice point's own order quantity term and the product of neighborhood couplings;

· The right-hand side is 1, corresponding to the DOG lattice order conservation condition;

· The product structure comes from the multiplicative recursive property of DOG continued fractions.

Why It Is a Product, Not a Sum

The DOG continued fraction

r_n(C)=\frac{1}{C+\frac{1}{C+\cdots}}

has a layer-by-layer nested recursive structure. This nesting relation naturally corresponds algebraically to multiplicative combination rather than linear superposition. Therefore, the order coupling among lattice points uses product constraints.

On Physical Labels

\omega_p can carry different physical labels:

Label Corresponding Group Corresponding Formula

Electromagnetic U(1) U(1) coupling equation

Weak SU(2) SU(2) coupling equation

Strong SU(3) SU(3) coupling equation

Labels are classification entries, not values to be substituted. A label corresponds to a group, and the group corresponds to a formula. The form of FCE remains unchanged; the label determines which set of formulas to use.

---

5 Equivalent Forms of FCE

5.1 Logarithmic Form

When \omega_p,\lambda_p,C_{pq} take positive real values, take the natural logarithm of both sides of FCE:

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

After logarithmic transformation, the product equation becomes a summation equation, which is convenient for numerical solution.

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

where \tau is DOG lattice recursive order time, not physical time; \dfrac{d\omega_p}{d\tau} describes the evolution rate of the lattice point's intrinsic order quantity with order iteration steps, and this product constraint continues to hold at any order time.

---

6 Position of FCE in the DOG System

The DOG system is divided into 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 related work exists

FCE is the native expression 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 is product-type coupling, different from linear superposition-type coupling;

· With continuous limit work: the continuous limit of FCE points to differential equations, and related work exists.

The above relations are positioning statements and do not constitute comparative conclusions.

---

8 Conclusion

Within the DOG framework, this paper completes:

1. Defining Zhang's matrix (coupling matrix): the algebraic arrangement of order-coupling coefficients;

2. Constructing FCE from Zhang's matrix: a product-type fundamental coupling equation;

3. Giving the steady-state, logarithmic, and dynamic forms of FCE;

4. Clarifying the position of FCE within the DOG system.

FCE is the first step of DOG from the geometric layer toward the equation layer.

 

Future Outlook

The Zhang coupling matrix constructed in this paper adopts continued-fraction iteration rules, and the FCE takes the form of product-type coupling. This framework is extensible. Future work may explore other sequence generation rules, more types of coupling terms, and generalizations to multi-degree-of-freedom and many-body systems.

 

---

References

Omitted

---