Discrete Order Geometry (DOG) Fundamental Coupling Equation: Generative Construction from Zhang Matrix to FCE

Author: Zhang Suhang

(Luoyang, Henan)

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Abstract

Discrete Order Geometry (DOG) takes discrete lattice points, hierarchical order, adjacency coupling, and continued-fraction coefficients as its foundational architecture. After the geometric layer is defined, DOG requires a native equation layer matching it. Within the DOG framework, this paper first defines the Zhang matrix (coupling matrix), formed by arranging order-coupling coefficients, and then constructs the Fundamental Coupling Equation (FCE) from the Zhang matrix. The 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 the 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 matrix; coupling matrix; Fundamental Coupling Equation; FCE; continued fractions

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1 Introduction

The basic claim of DOG (Discrete Order Geometry) is that a geometric system need not rely on spatial connectivity, but instead takes hierarchical nesting, order isomorphism, and continued-fraction scales as the core criteria for structural determination.

DOG emerged from fractal geometry and uses continued fractions to realize recursive hierarchical generation. Gauss’s research on the intrinsic connection between continued fractions and analytic functions provides an intellectual origin for DOG’s extension from discrete recursive structures to the function level.

In previous work, DOG has completed:

· systematic definitions, morphisms, and order isomorphism (geometric layer);
· field-evolution correspondence in the continuous limit (limit layer);
· the Zhang group–Lie group emergence relation (group-theoretic layer).

However, the equation layer has not yet been established.

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

1. to define the Zhang matrix (coupling matrix);
2. to construct the Fundamental Coupling Equation (FCE) from the Zhang matrix;
3. to give the steady-state, logarithmic, and dynamic forms of the FCE;
4. to explain the position of the FCE within the DOG system.

This paper does not involve specific physical applications and does not assert ontological conclusions; it only completes the generative construction from the geometric layer to the equation layer.

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2 Basic DOG Setup

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. lattice points have an adjacency relation \mathcal{R}, and only lattice points at the same or nearby hierarchical 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 generation sequence.

Notes:

· lattice points are discrete and non-connected;
· lattice points are connected by order;
· the form of order is a continued fraction, a ratio, or a function;
· the interior of a single lattice point may be regarded as continuous (discreteness at one point is a special case).

Regarding \omega_p:

\omega_p is the intrinsic order quantity on a lattice point. Its value is not preliminarily restricted; it 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 another representation, \omega_p takes the corresponding value.

The physical labels of \omega_p are only classification entries and do not constitute the mathematical specification of the FCE.

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3 Zhang Matrix

Definition 3.1 (Zhang 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 elements C_{pq} are determined by a continued-fraction generation sequence;
3. Hierarchical structure: nonzero elements are arranged in blocks according to the hierarchical order \preceq.

Call \mathbf{C} the Zhang matrix (also called the coupling matrix).

Notes:

· The Zhang matrix is an algebraic arrangement of DOG order coupling, distinct from an ordinary graph-theoretic adjacency matrix;
· its nonzero structure determines the order neighborhood of lattice points;
· its hierarchical structure corresponds to DOG hierarchical nesting.

Definition 3.2 (Order neighborhood)

For a lattice point p, define its order neighborhood as

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

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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 the Zhang matrix. For each lattice point p\in\mathcal{L}, define the Fundamental Coupling Equation as

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

where:

· \lambda_p: the native generation coefficient of the lattice point;
· \omega_p: the intrinsic order quantity of the lattice point, whose value is determined by the Zhang group representation;
· C_{pq}: the coupling coefficient of the Zhang matrix;
· \operatorname{Neigh}(p): the order neighborhood determined by the positions of nonzero elements of the Zhang matrix.

Notes:

· The left-hand side is the product of the lattice point’s own order-quantity term and the neighborhood coupling product;
· the right-hand side is 1, corresponding to the DOG lattice order-conservation condition;
· the product structure comes from the product recursion property of DOG continued fractions.

Why 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. Algebraically, this nesting naturally corresponds to product combination rather than linear superposition. Therefore, the order coupling among lattice points uses a product constraint.

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 FCE form remains unchanged; the label determines which formula is used.

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5 Equivalent Forms of the FCE

5.1 Logarithmic Form

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

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

After logarithmic transformation, the product equation becomes a sum 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\operatorname{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 respect to the order iteration step, and this product constraint continues to hold at any order time.

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6 Position of the 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 matrix established in this paper
Equation layer FCE established in this paper
Limit layer FCE \to continuous equations related work exists

The FCE is DOG’s native expression at the equation layer.

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7 Relationship to Existing Work

· With graph-theoretic adjacency matrices: the Zhang matrix is an adjacency matrix with hierarchical structure and continued-fraction generation;
· With coupled-oscillator equations: the FCE is product-type coupling, unlike linear-superposition coupling;
· With continuous-limit work: the continuous limit of the FCE points to differential equations, and related work exists.

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

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8 Conclusion

Within the DOG framework, this paper completes:

1. the definition of the Zhang matrix (coupling matrix): an algebraic arrangement of order-coupling coefficients;
2. the construction of the FCE from the Zhang matrix: a product-type Fundamental Coupling Equation;
3. the steady-state, logarithmic, and dynamic forms of the FCE;
4. clarification of the position of the FCE within the DOG system.

The FCE is the first step of DOG from the geometric layer toward the equation layer. Its continuous limit, group-theoretic correspondence, and physical interpretation are left for future work.

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References

Omitted.

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Note: This paper is a generative construction of the equation layer within the DOG framework. All definitions are exploratory attempts and do not constitute quantitative conclusions. No specific physical applications are involved.