Title:

Under the DOG Discrete Order Geometry Framework, the Yang–Mills Equation Is No Longer a Foundational Challenge but a Natural Emergent Result of Spacetime Steady States

Author: Zhang Suhang

Address: Luoyang, Henan, China

Abstract

In the standard system of modern physics, the Yang–Mills gauge field equation has always been regarded as a core challenge of fundamental physics — a high-order field theory equation requiring artificially constructed symmetry groups, presupposed actions, and variational solutions.

This paper, based on the fully闭环 DOG Discrete Order Geometry, the ECS Symmetric Equilibrium Principle, and the MIE Minimal Intrinsic Action Principle, proposes a new paradigmatic conclusion:

The Yang–Mills equation is not a fundamental difficulty of physics, nor an artificially constructed field theory assumption; it is merely an inevitable field equation that naturally emerges from DOG discrete spacetime under low-energy, symmetric, and steady-state conditions.

Given the foundational spacetime structure of DOG, the Yang–Mills structure requires no speculation, no presuppositions, and no complex derivations — it arises seamlessly and automatically. This conclusion fundamentally rewrites the underlying logic of gauge field theory.

Keywords: DOG Discrete Order Geometry; Yang–Mills equation; gauge field; natural emergence; low-energy steady state; ECS symmetry; MIE minimal action

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I. Traditional Perspective: Yang–Mills as a "High-Order Challenge"

In mainstream particle physics and field theory, the Yang–Mills theory exhibits the following characteristics:

1. Gauge symmetry groups (U(1), SU(2), SU(3)) are external artificial assumptions with no geometric origin.

2. Field equations require manual construction of connections, curvatures, and actions, followed by variational solutions.

3. Incompatibility with gravity, inability to unify the four forces, resulting in persistent theoretical fragmentation.

4. High learning threshold, complex derivations, and obscure physical imagery, often viewed as a hard core challenge of fundamental physics.

In short: traditional physics struggles to solve the Yang–Mills equation.

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II. DOG New Paradigm Perspective: Yang–Mills as a Natural Result

In the DOG Discrete Order Geometry system, spacetime inherently possesses two underlying structures:

1. Spatial Matrix: discrete lattice order coupling (naturally corresponding to gauge connections)

2. Time Fiber Bundle: local phase oscillations and frequency difference evolution (naturally corresponding to gauge curvatures)

Combined with two steady-state laws:

· ECS → enforces system symmetry, equilibrium, and structural stability

· MIE → enforces minimal action, optimal convergence, and low-energy steady states

With these three structures in place, the gauge field form is already predetermined.

No need to artificially presuppose symmetry groups,

No need to manually piece together field equations,

No need to forcibly construct non-Abelian structures.

When the system enters a low-energy smooth steady state:

· Discrete coupling naturally continuumizes → generates gauge connection A_\mu 

· Fiber phase differences naturally accumulate → generate field curvature F_{\mu\nu} 

· System tends toward minimal action steady state → automatically satisfies Yang–Mills variational conditions

Thus, the Yang–Mills equation naturally emerges.

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III. Core Assertion: The DOG System Completely Dissolves the Yang–Mills Challenge

The traditional academic view holds that Yang–Mills is difficult because:

there is no underlying spacetime geometry, only a reverse-engineered physics from mathematical appearances.

The DOG system constructs forward:

from spacetime structure → symmetry form → steady-state constraints → direct emergence of field equations

Hence the core conclusions of this paper:

1. Yang–Mills is not a fundamental axiom — it is a steady-state corollary under the continuum approximation of discrete spacetime.

2. Yang–Mills is not a difficult problem — it is an inevitable product of the dual constraints of spacetime symmetry and energy minimization.

3. Gauge symmetry is no longer mysterious — SU(N) symmetry is not an artificial setting but the natural symmetry order of multi-node discrete coupling.

4. Field equations require no complex derivation — the structure is embedded at the most fundamental level, and the results follow naturally and automatically.

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IV. Full Consistency and Unification with the Entire DOG System

1. π–ECS static symmetry — provides gauge field phase closure and symmetric foundation.

2. e–MIE dynamic convergence — provides field quantity evolution, low-energy return, and steady-state damping.

3. Spatial matrix + time fiber bundle — provides the complete geometric carrier for gauge fields.

The Yang–Mills equation is precisely the complete expression of the DOG dynamic–static dual order at the field theory level.

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V. Innovative Value and Paradigm Breakthrough

1. For the first time in history, the Yang–Mills equation is downgraded from a "high-order physics challenge" to a natural phenomenon of spacetime steady states.

2. Completely eliminates the "artificial assumption" color of gauge fields, endowing them with a true geometric origin.

3. Realizes a natural field-theoretic foundation for the unification of electromagnetism, weak, strong, and gravitational forces.

4. Achieves a全域 logical closure of mathematical constants, physical laws, spacetime structure, and field equations.

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VI. Conclusion

In the traditional physics framework, the Yang–Mills equation is a core challenge to be conquered, solved, and constructed.

Under the complete DOG Discrete Order Geometry system, the Yang–Mills equation requires no conquest, no construction, no assumptions — it is merely a natural steady-state outcome, seamlessly driven by the joint action of spacetime symmetric equilibrium and minimal action convergence.

Henceforth, the most central gauge field challenge of modern fundamental physics is completely dissolved, naturally resolved, and fully re-situated within the new DOG paradigm.

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References

[1] Zhang Suhang. A New View of Spacetime in Discrete Order Geometry (DOG): Spatial Matrix and Time Fiber Bundle. 2026.

[2] Zhang Suhang. The Dynamic–Static Dual Unity Principle of Cosmic Fundamental Constants in the DOG System. 2026.

[3] Zhang Suhang. Native Derivation of the Yang–Mills Equation Based on DOG Discrete Order Geometry. 2026.

[4] Yang Chen Ning, Mills R. L. Basic Construction of Gauge Field Theory. 1954.