The Continuous Limit of Discrete Order Geometry (DOG): A Study of the Correspondence Between Discrete Evolution and Differential Equations

Author: Zhang Suhang

(Luoyang, Henan)

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Abstract

Discrete Order Geometry (DOG) constructs a purely discrete evolutionary system based on discrete lattice points, hierarchical order, adjacency coupling, and coefficient recursion. Classical differential equations take continuity and smoothness as their presupposed premises and serve as the standard tool for describing macroscopic continuous evolution. Based on the discrete evolution rules of DOG, this paper rigorously establishes a convergence mechanism from discrete order to the continuous limit. It is shown that, under regular conditions in which lattice density tends to infinity, spatiotemporal step sizes tend to zero, and continued-fraction coefficients tend to constancy, DOG discrete iterative systems can converge to classical ordinary and partial differential equations.

This paper clearly distinguishes classical finite-difference theory from the DOG framework: DOG takes order structure, hierarchical coupling, and continued-fraction coefficient sequences as its native degrees of freedom, and can uniformly describe uniform, non-uniform, and fractal-recursive discrete systems. Classical differential equations and their numerical discretizations can both be embedded into the DOG framework to form a two-way correspondence. This paper makes no ontological assertions; it establishes only rigorous mathematical correspondences, providing a standardized foundational interface for DOG to connect with continuous dynamical systems.

Keywords: Discrete Order Geometry; DOG evolution; continuous limit; discrete mapping; partial differential equations; order convergence

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1 Introduction

The core tools of continuum mechanics, field theory, and dynamical systems are differential equations, whose theory is built on the presuppositions of continuous spacetime, smooth functions, and infinite divisibility. All numerical solution methods are, in essence, artificial discretizations of continuous systems to adapt them to computational iteration.

Traditional mathematics exhibits a one-way dependence: discrete approximation is always a subsidiary approximation tool of continuous equations.

DOG Discrete Order Geometry provides the reverse perspective: discrete structure is the primary definition, and the continuous model is the limiting approximation of the discrete structure. DOG systems generate evolution rules based on finite countable lattice points, integer iterative steps, local adjacency coupling, and continued-fraction coefficient recursion, without presupposing continuity or relying on smoothness.

The core research objectives of this paper are:

1. To construct the standard form of the general DOG discrete evolution mapping;

2. To give the regular limit conditions under which DOG systems converge to continuous differential equations;

3. To establish DOG discrete limit models for the heat equation, wave equation, Schrödinger-type evolution, and nonlinear evolution;

4. To rigorously distinguish the native structure of DOG from classical finite-difference methods;

5. To establish the mathematical status of differential equations as a special case of the regular continuous limit of DOG.

All conclusions in this paper are mathematical correspondences within the framework and do not involve out-of-bounds claims such as the nature of the universe or first principles of physics.

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2 The Standard Architecture of DOG Discrete Evolution

2.1 Definition of Discrete Spacetime and State Field

A DOG system consists of a five-tuple: lattice point set, hierarchical order, adjacency relation, scale function, and continued-fraction coefficients.

Let the discrete lattice point set be \mathcal{L}, and let discrete evolution time be marked by integer iterative steps n\in\mathbb{N}.

For any lattice point i, define the discrete state field:

\psi_i(n)

which represents the dynamical state of the lattice point at the n-th iterative step (amplitude, density, phase, or potential state).

The general form of DOG native discrete evolution:

\psi_i(n+1) = F\big(\{\psi_j(n)\},C_i,\nu_i,\varepsilon\big)

where:

· C_i: the continued-fraction hierarchical coefficient of the lattice point, determining the recursion type;

· \nu_i: the intrinsic frequency of the lattice point, characterizing local oscillatory behavior;

· \varepsilon: adjacency coupling strength;

· F: the iterative mapping determined by DOG adjacency order.

2.2 Free Evolution and Coupled Evolution

Uncoupled free evolution

An isolated lattice point evolves only through its intrinsic phase iteration:

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

which is standard discrete periodic unitary evolution.

Local adjacency coupled evolution

Based on the DOG adjacency compatibility condition, only lattice points at the same or adjacent levels couple. In one-dimensional uniform order, the standard form of nearest-neighbor coupling is:

\psi_k^{n+1} = \psi_k^n + \varepsilon\big(\psi_{k+1}^n-2\psi_k^n+\psi_{k-1}^n\big)

This is the fundamental form of DOG discrete diffusion-type evolution and serves as the native model for subsequent continuous-limit derivations.

2.3 DOG Discrete Grid Parameterization

For regular ordered systems, uniform discrete steps may be introduced:

· spatial step size \Delta x

· time iterative step size \Delta t

Define discrete spacetime coordinates:

x_k=k\Delta x,\quad t_n=n\Delta t

The discrete state field may be written as a grid function:

\psi(x_k,t_n)

Classical finite-difference schemes are merely a special subset of DOG uniform constant-coefficient order.

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3 Continuous Limit Theory of DOG Systems

3.1 Limit Convergence Conditions (Regular Conditions)

Define the regular conditions for the DOG continuous limit:

1. The number of lattice points tends to infinity while the macroscopic region scale remains finite;

2. Spatiotemporal step sizes satisfy \Delta x\to0,\Delta t\to0;

3. The continued-fraction coefficient sequence \{C_i\} tends to a constant, and the system order becomes uniform;

4. Coupling parameters satisfy a compatibility scaling law, ensuring smooth convergence in the limit.

When the above conditions are satisfied, the inter-step difference of DOG discrete iteration can be Taylor-expanded and converges to differential operators.

3.2 Time Continuous Limit Derivation

For a smooth state field, Taylor expansion gives:

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

Discrete time difference:

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

Substituting into the DOG adjacency coupling iteration, and absorbing the discrete step sizes through parameter scaling, the higher-order infinitesimals can be eliminated, generating a continuous evolution equation containing spatial differential operators.

3.3 Spatial Continuous Limit Derivation

Second-order spatial difference:

\psi(x+\Delta x)-2\psi(x)+\psi(x-\Delta x)

=(\Delta x)^2\,\partial_x^2\psi+O((\Delta x)^4)

Under the regular scaling:

\varepsilon \propto \dfrac{\Delta t}{(\Delta x)^2}

the discrete spatial difference strictly converges to the second-order partial derivative operator.

3.4 Constant Coefficients and Smooth Field Correspondence

After homogenization in the limit, DOG variable coefficient sequences correspond to constant parameters of continuous equations;

if the coefficient sequence converges to a smooth function sequence, variable-coefficient partial differential equations are generated;

if the coefficient sequence retains discrete non-periodic characteristics, the continuous limit does not exist, and the system maintains discrete chaotic or quasi-periodic evolution.

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4 DOG Limit Realizations of Typical Differential Equations

4.1 Heat Conduction Equation (Diffusion Equation)

DOG uniform order, constant coefficients, nearest-neighbor coupling iteration:

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

Taking the regular continuous limit \Delta x,\Delta t\to0, higher-order infinitesimals vanish, yielding:

\partial_t T = D\,\partial_x^2 T

The classical heat equation is the product of the continuous limit of DOG uniform steady-state order.

4.2 Wave Equation

Taking the wave-type scaling \varepsilon\propto \dfrac{\Delta t^2}{(\Delta x)^2}, the DOG second-order time iteration converges to:

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

i.e., the classical linear wave equation.

4.3 Schrödinger-Type Evolution Equation

DOG unitary iteration with phase factors:

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

After first-order expansion of the exponential and adaptation to quantum scaling, the continuous limit gives:

i\hbar\partial_t \psi = \hat H\psi

The discrete intrinsic frequency \nu_i is continuously transformed into a local potential field, and the adjacency coupling converges to the kinetic Laplacian term.

4.4 Nonlinear KdV Equation

DOG permits state-dependent coupling coefficients (variable-coefficient order), allowing the construction of nonlinear discrete iterations. Its continuous limit can converge to the classical KdV nonlinear evolution equation, demonstrating that the DOG framework can uniformly accommodate linear and nonlinear continuous dynamics.

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5 Rigorous Distinction Between DOG and Classical Difference Theory

5.1 Limitations of Classical Finite Differences

Classical finite-difference and finite-element methods:

· presuppose the continuous equation first and the discrete scheme second;

· adapt only to uniform grids and constant parameters;

· lack hierarchical order and continued-fraction coefficient recursion structure.

5.2 Unique Extensibility of DOG

Native advantages of DOG:

1. Discrete order first, continuous limit second;

2. Can describe non-uniform, fractal-hierarchical, and variable-coefficient recursive systems;

3. Defines structural invariants through order isomorphism and hierarchical rank;

4. Can accommodate purely discrete systems without a continuous limit.

5.3 Two-Way Correspondence Theorem (Revised Rigorous Version)

Theorem

Under the DOG regular limit conditions:

1. Any DOG uniform discrete evolution system satisfying compatibility and stability can converge to a uniquely corresponding classical differential equation;

2. Any smooth classical differential equation can be used to construct at least one set of DOG discrete order systems as its discrete iterative approximation.

Note

This theorem describes only a mathematical mapping relation. It does not mean that all discrete systems must converge, nor that a continuous equation uniquely corresponds to only one discrete structure. This avoids the overly strong universal assertions of the original draft.

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6 Conclusion

Based on the native discrete evolution system of Discrete Order Geometry (DOG), this paper rigorously establishes the limiting correspondence between discrete order and continuous differential equations, yielding the following rigorous conclusions:

1. DOG is based on discrete lattice points, hierarchical order, adjacency coupling, and continued-fraction recursion, and is a native evolutionary framework independent of continuity;

2. The classical heat equation, wave equation, Schrödinger-type equation, and nonlinear evolution equation can all be rigorously derived as special cases of the continuous limit of DOG regular uniform order;

3. The numerical discretization of traditional differential equations is essentially the reverse mapping of continuous models back into DOG-type discrete iterative structures;

4. The DOG framework covers classical difference theory while accommodating non-uniform, hierarchical, and fractal discrete systems, expanding the foundational boundary of dynamical models;

5. Continuous differential equations are not an innate foundational framework, but an effective approximate mathematical model of discrete order under the limits of infinite density, homogenization, and smoothing.

All conclusions in this paper are mathematical correspondence conclusions within the framework, with no out-of-bounds ontological claims, providing a clean, rigorous, and self-consistent foundational interface for DOG to connect with continuous analysis, dynamical systems, and mathematical physics.

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References

Omitted.