DOG Discrete Order Geometry: A Minimal Mathematical Paradigm for Unifying the Four Fundamental Interactions

Author: Zhang Suhang, Luoyang

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I. Fundamental Unified Substrate

Let the set of all discrete spacetime lattice points in the universe be

\mathcal{L} = \{\mathcal{L}_1, \mathcal{L}_2, \dots, \mathcal{L}_n\}

\]  

· Space ontology: Lattice adjacency coupling matrix \boldsymbol{A} 

· Time evolution: Lattice fiber unitary oscillation operator U(\tau) 

· Universal unified evolution equation:

\boldsymbol{\Psi}_{\tau+1} = \boldsymbol{M} \cdot \boldsymbol{\Psi}_\tau

\]  

Here \boldsymbol{M} is the total universal interaction matrix, satisfying the tensor product decomposition:

\boldsymbol{M} = \boldsymbol{M}_G \otimes \boldsymbol{M}_E \otimes \boldsymbol{M}_W \otimes \boldsymbol{M}_S

\]  

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II. Mathematical Definitions of the Four Interactions

1. Gravitation \boldsymbol{M}_G (Lattice Matrix Deformation)

Assign a mass weight m_i to each lattice point, modifying the adjacency matrix elements:

A_{ij} \to A_{ij} \cdot f(m_i, m_j)

\]  

Gravitation is equivalent to the gradient deformation of the discrete lattice matrix, abandoning continuous curved spacetime:

\boldsymbol{M}_G = \nabla \boldsymbol{A}

\]  

Macroscopic gravity is the large-scale topological distortion effect of the lattice adjacency matrix.

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2. Electromagnetism \boldsymbol{M}_E (U(1) Phase Fiber)

Homogeneous phase oscillation with a single degree of freedom:

U_1(\tau) = e^{i\alpha(\tau)}, \quad \boldsymbol{M}_E \in U(1)

\]  

Long-range phase synchronization propagation is achieved through weak matrix coupling, matching the global transmission characteristic of the electromagnetic interaction.

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3. Weak Interaction \boldsymbol{M}_W (SU(2) Chiral-Broken Fiber)

The twofold symmetric fiber oscillation belongs to the SU(2) group:

U_2(\tau) \in SU(2)

\]  

Introduce a chiral projection operator to break the inherent symmetry:

\boldsymbol{M}_W = P_{\text{chiral}} \cdot U_2

\]  

This naturally yields parity violation and matches the decay and short-range characteristics of the weak force.

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4. Strong Interaction \boldsymbol{M}_S (SU(3) High-Density Tightly Coupled Fiber)

High-dimensional color-space fiber transformation:

U_3(\tau) \in SU(3)

\]  

Introduce a strong coupling density coefficient \rho \gg 1 :

\boldsymbol{M}_S = \rho \cdot U_3

\]  

The extremely strong binding relationship of the matrix realizes color confinement, corresponding to the short-range, high-binding property of the strong force.

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III. Ultimate Unified Mathematical Expression

\boldsymbol{\Psi}_{\tau+1} = \Big( \nabla \boldsymbol{A} \;\otimes\; U_1 \;\otimes\; P_{\text{chiral}} U_2 \;\otimes\; \rho U_3 \Big) \boldsymbol{\Psi}_\tau

\]  

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IV. Core Mathematical-Physical Conclusions

1. Spatial configuration is uniquely determined by the coupling matrix, and the passage of time is defined by the unitary oscillation of the fibers.

2. The four fundamental interactions do not require independent field equations; they are merely tensor product combinations of matrix deformations and unitary transformations of different Lie group fibers.

3. The entire framework is rooted in the DOG discrete lattice ontology, compatible with Hilbert space vector evolution and the probabilistic norm-squared interpretation. The system is fully self-consistent, achieving a complete mathematical unification of fundamental physics.