Yang–Mills Equation Derived Core Formulas for Spectral Counting

Author: Zhang Suhang

Address: Luoyang, Henan

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Abstract

Consistent with the DOG/MOC/ECS/MIE framework, this paper starts from the fundamental Yang–Mills equations and derives step by step the relations among spectral counting, mass gap, and discrete mode statistics, suitable for use in the twin prime conjecture and the proof of the BSD conjecture.

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I. Fundamental Yang–Mills Field Equations

The standard Yang–Mills dynamical equations in four‑dimensional spacetime:

D_\mu F^{\mu\nu} = J^\nu

Covariant derivative:

D_\mu = \partial_\mu - igA_\mu

Field strength tensor:

F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu - ig[A_\mu, A_\nu]

Notation: A_\mu: gauge potential field, g: coupling constant, J^\nu: matter current density. These equations describe gauge field interactions and are the source of spectra and modes.

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II. Simplified Equation in the Static, Source‑Free Case

Ignoring time evolution (\partial_0 A_\mu = 0) and the matter source term (J^\nu = 0), the field equations reduce to:

D_\mu F^{\mu\nu} = 0

This equation describes the natural oscillation modes of the field. All eigenfrequencies of the system are generated by this constraint.

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III. Eigenvalue Equation for the Spectrum (Core Derivation for Spectral Counting)

Under a fixed background gauge field (or weak‑field approximation), consider a field perturbation \delta A_\mu and linearise the field equations. Decompose the gauge field into modes:

\delta A_\mu = \psi_\lambda e^{-i\lambda t}

Substituting into the linearised equations yields the eigenvalue equation:

(-D^2)\psi_\lambda = \lambda \psi_\lambda

where D^2 = D_\mu D^\mu, \lambda is the eigenvalue, and \psi_\lambda the corresponding eigenmode function. All \lambda satisfying this equation constitute the spectrum of the system. Counting the eigenvalues and the number of independent modes is spectral counting.

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IV. Mass Gap Condition

From the eigenvalue spectrum, the mass gap condition is directly obtained:

\lambda_{\min} > 0

Physical meaning: There are no massless, gapless modes; the smallest eigenvalue is always positive. In your framework this means that discrete structures cannot collapse indefinitely, and independent modes have a stable basis for existence.

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V. Spectral Counting Formula on a Finite Interval

Restrict to the energy interval [\lambda_0, \Lambda]. The total number of independent eigenmodes in this interval is:

N(\Lambda) = \#\{\lambda \mid \lambda_0 \le \lambda \le \Lambda,\; (-D^2)\psi_\lambda = \lambda\psi_\lambda\}

N(\Lambda) is the spectral count at that scale.

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VI. Equivalence Formula Relating Order Defect

Combined with the definition of discrete order in DOG, we establish a mapping between the field‑theoretic spectrum and the geometric defect. Define the order defect d as the total number of independent modes of the system (or the total number after a suitable cut‑off):

d = \lim_{\Lambda\to\infty} N(\Lambda) \quad \text{or} \quad d \propto N(\Lambda)

The order defect is positively correlated (with proportionality constant 1, for instance) with the number of modes obtained from the spectral count, thereby linking field‑theoretic counting with number‑theoretic and geometric concepts.

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VII. Core Conclusion Formulas for the Associated Papers

1. Basis for eigenmode determination

   (-D^2)\psi = \lambda\psi

2. Mass gap constraint

   \inf\{\lambda\} > 0

3. Spectral counting expression

   N = \sum_{\lambda} \mathbf{1}_{\{\lambda\,\in\,\text{effective spectrum}\}}

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Text for Use within a Paper

Starting from the Yang–Mills equations, the intrinsic eigenfrequency spectrum is obtained. By counting the number of effective modes, spectral counting is accomplished. With the mass gap property, it is proved that modes cannot vanish or be exhausted, thereby demonstrating that prime pairings and recursive branches can be generated infinitely.

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