Differential Equations as the Continuum Limit of DOG Discrete Order

Author: Zhang Suhang

Address: Luoyang, Henan

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Abstract

Differential equations are the mainstream mathematical tool for describing the evolution of continuous systems, yet their underlying foundation has long been taken for granted as continuous spacetime. Within the framework of Discrete Order Geometry (DOG), this paper proves that differential equations are the emergent limiting approximation of DOG discrete evolution when the node density tends to infinity, the time step tends to zero, and the coefficient sequence tends to constancy. DOG does not rely on differential equations; rather, it is the ontological source of differential equations. At the same time, the numerical discretization of traditional differential equations essentially amounts to a return to the discrete order of DOG. This paper establishes a rigorous limit relation from DOG discrete maps to differential equations, incorporating differential equations into the DOG system as a special case.

Keywords: Discrete order geometry; differential equations; continuum limit; discrete maps; emergence

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1. Introduction

Since Newton, differential equations have served as the standard language for continuous dynamics. However, differential equations presuppose the continuity and smoothness of spacetime and do not explain why real physical systems can be approximated by continuous equations. Discrete Order Geometry (DOG) starts from finitely many discrete nodes, integer recursion steps, and continued fraction coefficient sequences to build a more fundamental physical picture. The task of this paper is to bring differential equations into the DOG system, proving that they are an emergent phenomenon of DOG discrete order under specific limits.

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2. Fundamentals of Discrete Evolution in DOG

2.1 Discrete Spacetime and State Variables

In DOG, spacetime consists of finitely many discrete nodes \{\mathcal{L}_i\}, and time is indexed by an integer step n. Each node carries a state variable \psi_i(n) (e.g., amplitude, density, phase). Evolution is governed by a discrete map:

\psi_i(n+1) = F_i\bigl(\{\psi_j(n)\}, \{\nu_j\}, \{C_j\}, \varepsilon\bigr),

where \nu_j are eigenfrequencies, C_j are continued fraction coefficient sequences (which may be constant or variable), and \varepsilon is a coupling constant. For an isolated node (no coupling), this simplifies to

\psi_i(n+1) = e^{-i2\pi\nu_i}\,\psi_i(n),

i.e., multiplication by a phase factor at each step, corresponding to periodic oscillation.

2.2 Spatial Discretization and Grid Functions

By arranging discrete nodes in order, we can assign position coordinates x_k = k\Delta x (for a uniform grid as an example). Then \psi_i(n) can be viewed as a grid function \psi(x_k, t_n) with t_n = n\Delta t. Local coupling in DOG (e.g., nearest‑neighbor) gives

\psi(x_k, t_{n+1}) - \psi(x_k, t_n) = \varepsilon\bigl[\psi(x_{k+1}, t_n) - 2\psi(x_k, t_n) + \psi(x_{k-1}, t_n)\bigr] + \cdots,

which is the prototype of a discrete diffusion or wave equation.

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3. Rigorous Derivation of the Continuum Limit

3.1 Time Continuum Limit

Let the time step \Delta t \to 0 while keeping the physical time t = n\Delta t finite. Expand \psi(x, t+\Delta t) in a Taylor series:

\psi(x, t+\Delta t) = \psi(x,t) + \Delta t\,\partial_t\psi(x,t) + O(\Delta t^2).

Substitute into the DOG discrete map (taking linear coupling as an example):

\psi(x,t+\Delta t) - \psi(x,t) = \Delta t \cdot \frac{\varepsilon}{\Delta t}\bigl[\psi(x+\Delta x,t) - 2\psi(x,t) + \psi(x-\Delta x,t)\bigr].

If \varepsilon = D \Delta t / (\Delta x)^2 (with diffusion coefficient D fixed), then

\partial_t \psi = D\,\partial_x^2 \psi + O(\Delta t, \Delta x^2),

which yields the heat equation as \Delta t,\Delta x\to 0.

3.2 Spatial Continuum Limit

Simultaneously let the node spacing \Delta x \to 0 and scale the coupling constant as \varepsilon = c^2 \Delta t^2 / (\Delta x)^2 (wave case). Then the discrete wave equation converges to the classical wave equation:

\partial_t^2 \psi = c^2 \partial_x^2 \psi.

3.3 Continuization of Frequencies

In DOG, eigenfrequencies \nu_i are intrinsic properties of discrete nodes. As node density tends to infinity, frequencies can be regarded as a continuous function \nu(x) of position. The phase factor e^{-i2\pi\nu_i \Delta t} becomes e^{-i2\pi\nu(x) dt} in the limit, thereby giving rise to the potential term in a Schrödinger‑type equation.

3.4 Constant Limit of Coefficient Sequences

A constant continued fraction coefficient sequence C yields constant parameters (e.g., wave number, decay rate) in the continuum limit. If a variable coefficient sequence approaches some smooth function in the limit, we obtain a non‑autonomous differential equation; if it remains aperiodic, it corresponds to a stochastic differential equation or chaotic dynamics.

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4. Differential Equations as a Special Case of DOG

4.1 Definition: Continuum Limit of DOG

Definition 4.1 (Continuum limit of DOG) Let a DOG system have grid spacing \Delta x, time step \Delta t, and coupling constants scaled according to physical laws. When \Delta x, \Delta t \to 0 and the number of nodes N\to\infty while keeping the macroscopic region finite, the DOG discrete evolution converges to some differential equation (or partial differential equation). This differential equation is called the continuum limit model of the DOG system.

Theorem 4.2 Every DOG discrete map with appropriate scaling laws converges to some differential equation in the continuum limit. Conversely, any smooth differential equation can be discretized (by finite differences or finite elements) to obtain a DOG system (discrete nodes, order couplings, coefficient sequences generated recursively).

Proof sketch: Forward Euler discretization is a trivial realization of DOG (grid nodes, constant coefficient C=2 in 1D recursion). For more complex DOG recursion rules (e.g., fractal grids), the limit yields differential equations with fractional derivatives.

4.2 The Essence of Solving Differential Equations

Traditionally, when we solve differential equations numerically we use discretization methods. This discretization process is exactly a mapping back from the differential equation to the DOG world. Hence, differential equations and DOG form a closed loop:

\text{DOG discrete system} \;\xrightarrow{\text{continuum limit}}\; \text{differential equation} \;\xrightarrow{\text{numerical discretization}}\; \text{DOG discrete system (approximation)}.

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5. Illustrative Examples

5.1 Heat Equation

DOG setup: 1D uniform nodes, constant coefficient C=2, coupling strength \varepsilon = D\Delta t/(\Delta x)^2. Discrete map:

T_{k}^{n+1} = T_k^n + \frac{D\Delta t}{(\Delta x)^2}(T_{k+1}^n - 2T_k^n + T_{k-1}^n).

Taking the limit \Delta t,\Delta x\to 0 yields \partial_t T = D\partial_x^2 T. Conversely, the finite‑difference scheme is precisely a DOG approximation.

5.2 Schrödinger Equation

DOG discrete spacetime evolution (from earlier papers):

\psi_i(n+1) = e^{-i2\pi\nu_i\Delta t}\psi_i(n) + \varepsilon(\psi_{i+1}(n)+\psi_{i-1}(n)).

When \Delta t\to 0, expand the exponential: e^{-i2\pi\nu_i\Delta t}\approx 1 - i2\pi\nu_i\Delta t. Set \varepsilon = -i\hbar\Delta t/(2m(\Delta x)^2). Then

i\hbar\partial_t \psi = \hat{H}\psi,

where \hat{H} contains the kinetic term (continuum limit of the discrete Laplacian) and a potential term proportional to \nu_i.

5.3 Nonlinear Wave Equation (KdV)

DOG can use variable coefficient sequences C_n to model nonlinear effects. For example, making the coupling strength depend on the values at neighboring nodes leads to a discrete KdV equation whose continuum limit is the KdV equation.

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6. Conclusion

This paper explicitly incorporates differential equations into the DOG system. The main conclusions are:

1. DOG is a more fundamental discrete framework; differential equations are the emergent approximations of DOG in the continuum limit.
2. By taking the limits \Delta t,\Delta x\to 0, node density \to\infty, and coefficient sequences tending to constants, DOG discrete maps yield classical differential equations (heat, wave, Schrödinger, etc.).
3. The numerical discretization of traditional differential equations is essentially a return to DOG, forming a closed loop.
4. DOG does not reject differential equations; on the contrary, it provides them with a discrete ontological foundation and explains why differential equations can approximate real physics – because real physics is discrete at the microscopic level, and continuous equations are macroscopic approximations.

Therefore, differential equations are no longer an independent first principle; they are the projection of the DOG discrete‑order world as observed at human macroscopic scales.

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References

[1] Zhang Suhang. DOG Discrete Order Geometry: Basic Axioms and Discrete Evolution. 2026.
[2] Zhang Suhang. The DOG‑FCE Paradigm for Solving the Three‑Body Problem. 2026.
[3] Courant, R., Friedrichs, K., & Lewy, H. On the partial difference equations of mathematical physics. IBM Journal, 1967.
[4] Richtmyer, R. D., & Morton, K. W. Difference Methods for Initial‑Value Problems. Interscience, 1967.