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Native Derivation of Yang-Mills Equations within the DOG Framework (Under the Unified Four-Force Paradigm)

Author: Zhang Suhang
Affiliation: Independent Researcher, Luoyang, Henan

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Abstract

Traditional Yang-Mills theory is a field-theoretic construction based on artificially postulated gauge symmetry groups. It is an empirical system of field equations that cannot incorporate gravity and remains confined to the three-force framework of the Standard Model. This paper, based on the DOG (Discrete Order Geometry) spacetime system that has already achieved unification of the four forces, and relying on the underlying geometric structure of the spatial order matrix and the temporal fiber bundle, natively derives the Yang-Mills equations without additional axioms or external assumptions.

This paper demonstrates that the Yang-Mills equations are not fundamental physical axioms but rather necessary geometric corollaries of DOG discrete spacetime under the continuous low-energy approximation and non-gravitational gauge field domain. The gauge field structures of electromagnetic, weak, and strong interactions are all embedded within the node coupling and fiber oscillation system of DOG spacetime, thereby fundamentally resolving the core defects of traditional gauge field theory: its incompatibility with gravity and its artificial imposition of symmetry.

Keywords: DOG Discrete Order Geometry; unification of four forces; temporal fiber bundle; spatial coupling matrix; gauge field; Yang-Mills equations

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I. Introduction

The Yang-Mills gauge field theory, established in 1954, is a cornerstone of modern particle physics. Its core logic, "symmetry dictates interaction," employs SU(2), SU(3), and U(1) gauge groups to construct non-Abelian gauge field equations that accurately describe the three fundamental interactions: electromagnetic, weak, and strong. However, this theory suffers from essential paradigmatic defects:

1. Gauge symmetry groups are artificial mathematical assumptions without support from an underlying spacetime geometric origin.
2. The system naturally excludes gravity, failing to achieve unification of the four forces and resulting in a fragmented physical picture.
3. The field equations are empirically variational constructions rather than natural consequences of spacetime structure.

In previous research, the author established DOG (Discrete Order Geometry), reconstructing the ontology of spacetime: space as an ordered coupling matrix of discrete lattice points, time as the unitary oscillation of lattice fibers. Within this framework, a complete unification of the four forces has been achieved, along with the geometric origination of probability formulas, spacetime evolution, and interaction mechanisms.

With the global closure of the DOG system taking shape, all traditional gauge field equations can be naturally derived from the underlying DOG structure. The core task of this paper is to natively derive the standard Yang-Mills field equations from the original DOG spacetime structure, establishing their true physical position as high-order geometric approximations.

II. Core Prerequisite Geometric Structures of DOG (Unified Four-Force Paradigm)

DOG spacetime abandons the classical continuous manifold hypothesis, constructing a dual unified structure of a discrete spatial matrix and local temporal fiber bundles. All physical interactions, field equations, and statistical laws originate from the coupling interplay between these two. The core prerequisite definitions are as follows:

2.1 Discrete Spatial Order Matrix

The fundamental basis of spacetime is a finite set of discrete lattice points \{\mathcal{L}_i\} without preset coordinates. There exist intrinsic order couplings between lattice points, characterized by the adjacency matrix A_{ij} : the matrix elements represent the strength of spatial order coupling between two lattice points, serving as the most fundamental geometric reference scale of spacetime.

After normalization, the coupling strength reference is set to unity, corresponding to the origin of the constant term in the previous probability formula P = \dfrac{1}{1+(\Delta\nu)^2} . The reference scale for all fields, coupling strengths, and connection bases is uniquely determined by the spatial matrix.

2.2 Unitary Oscillation of the Temporal Fiber Bundle

Each spatial discrete lattice point \mathcal{L}_i is locally attached with an independent temporal fiber space \mathcal{F}_i . The fiber undergoes unitary periodic oscillation with discrete evolution steps. The oscillation rate is defined as the eigenfrequency \nu_i of the lattice point, and the single-step evolution operator is e^{-i2\pi\nu_i} .

The frequency difference \Delta\nu between different lattice points corresponds to the fiber phase difference and rhythm difference, serving as the core source of spacetime excitation, field curvature, and interaction strength, corresponding to the dynamical term (\Delta\nu)^2 in the probability formula.

2.3 Core Logic of DOG Four-Force Unification

Electromagnetism, weak force, strong force, and gravity are essentially coupling modes of spatial coupling and temporal fiber oscillation at different dimensions and symmetry orders:

· Low-order Abelian symmetric mode → Electromagnetic interaction (U(1))
· Second-order non-Abelian symmetric mode → Weak interaction (SU(2))
· Third-order non-Abelian symmetric mode → Strong interaction (SU(3))
· Global discrete topological coupling mode → Gravitational geometric effect

The three gauge symmetries of the traditional Standard Model are merely a low-energy special subset within the infinite symmetry system of DOG.

III. Geometric Derivation of Yang-Mills Equations within the DOG Framework

The standard geometric form of the Yang-Mills equations is the covariant curvature conservation equation:

d_A * F = 0

where A is the gauge connection, F is the gauge field curvature, d_A is the covariant exterior derivative, and * is the Hodge dual operator.

This paper, based on the original DOG structure, establishes geometric correspondences and natively derives the equations step by step, without any artificial assumptions or external field theory implantation.

3.1 DOG Origin of the Gauge Connection A: Upgraded Spatial Matrix Coupling

The traditional gauge connection is an artificially defined field quantity. In the DOG system, the gauge connection is precisely the matrix order coupling of discrete lattice points.

Extending from basic scalar coupling between two nodes to multi-node multi-symmetry dimensions, the coupling strength is upgraded to a matrix-valued connection field:

A_{ij} \to A_\mu(x)

This field satisfies local symmetry and corresponds to the generator structure of the SU(N) gauge group. In essence, all gauge connections of the Standard Model are continuous-field and symmetry approximations of the DOG spatial coupling matrix.

3.2 DOG Origin of the Gauge Curvature F: Temporal Fiber Frequency-Difference Phase Curvature

The traditional curvature is defined as F = dA + A \land A , lacking an underlying physical origin. In the DOG system, curvature possesses a clear geometric ontology:

Gauge field curvature = Topological accumulation of frequency differences and phase differences of temporal fiber oscillations across multiple lattice points

The discrete fiber frequency difference \Delta\nu , in the continuous limit, transforms into the phase gradient of the field; the accumulation of phase deviations from non-commutative coupling across multiple nodes naturally generates the non-Abelian curvature term A \land A . Thus, DOG natively generates the curvature definition:

F = \text{gradient}(\Delta\nu) + [A, A]

which is completely consistent with the standard Yang-Mills curvature formula.

3.3 Derivation of the Field Equation via the Extremum Principle

The core steady-state criterion of DOG spacetime is: all physical systems tend toward a balanced equilibrium state between spatial coupling and temporal rhythm — i.e., the geometric curvature takes a minimum value.

This criterion corresponds to the action minimization principle in field theory. Constructing the DOG geometric action (steady state of pure curvature squared):

S = \int \text{Tr}(F \land *F)

Solving for the variational extremum with respect to the connection A , i.e., imposing the steady-state geometric constraint, directly yields:

\delta S = 0 \implies d_A * F = 0

This is precisely the complete standard Yang-Mills equation.

IV. Paradigmatic Reconstruction of Yang-Mills Equations from the DOG Perspective

The equations derived in this paper are formally identical to the traditional Yang-Mills equations, but their physical essence is radically innovated, achieving a paradigmatic reduction and subversion:

4.1 Symmetry Transforms from "Assumption" to "Geometric Necessity"

· Traditional: SU(2) and SU(3) symmetries are artificial mathematical assumptions without physical origin.
· DOG: Gauge symmetries are natural symmetric modes of discrete lattice coupling and fiber oscillation, inherent properties of spacetime geometry.

4.2 Field Equations Transform from "Empirical Construction" to "Structural Corollary"

· Traditional: The Yang-Mills equations are artificial variational fits to experimental laws.
· DOG: The equations are necessary consequences of spacetime steady states, requiring no experimental fitting and no axiomatic presuppositions.

4.3 Fundamental Resolution of the Four-Force Incompatibility Problem

· Traditional: Yang-Mills fields cannot accommodate gravity; the Standard Model can never achieve grand unification.
· DOG-derived version: Naturally compatible with gravity: gravity, as a global topological coupling mode, is unified with the three gauge fields within the same discrete spacetime framework, truly realizing a grand unified field theory of the four forces.

V. Conclusion

1. The standard Yang-Mills equations are not fundamental physical axioms but rather precise corollaries of DOG discrete order geometry under the continuous approximation, low-energy steady state, and gauge field domain.
2. The gauge connection originates from the order coupling of the DOG spatial matrix; the gauge curvature originates from the frequency and phase differences of DOG temporal fibers. All core concepts of gauge fields possess a clear geometric ontology.
3. The DOG four-force unification framework completely encompasses and surpasses traditional gauge field theory, definitively ending the century-old paradigmatic defect of the Standard Model: three forces standing apart, excluding gravity.
4. This paper demonstrates that all classical field equations can be natively derived from the underlying discrete spacetime structure of DOG. DOG is the ultimate unified geometric paradigm covering both gravity and gauge fields.

References

[1] Zhang Suhang. A New View of Spacetime in Discrete Order Geometry (DOG): Spatial Matrix and Temporal Fiber Bundle. 2026.
[2] Zhang Suhang. Frequency as the Origin of Probability: From Discrete Order Geometry to an Endogenous Quantitative Theory of Probability. 2026.
[3] Yang, C. N., & Mills, R. L. Conservation of Isotopic Spin and Isotopic Gauge Invariance. Physical Review, 1954, 96(1), 191-195.
[4] Zhang Suhang. The Complete Geometric Closed-Loop Theory of Four-Force Unification within the DOG Framework. 2026.

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