The CFE Dirac Tensor: Resolving Hydrogen Fine-Structure Splitting and Extracting Quark Scattering Frequencies

**Preprint | Continuum Field Entropy Microscopic Validation Series** We present a deterministic field-dynamic resolution to the Hydrogen fine-structure variance using the Continuum Field Entropy (CFE) framework. Building upon the macroscopic scalar derivation of the Hydrogen atom, which successfully recovers the classical Rydberg formula, we establish a strict Correspondence Principle for the residual ~ 0.001% spectral variance by introducing the fully factorized CFE Dirac Tensor. By defining the vacuum as a non-Newtonian Cosserat continuum, we demonstrate that standard intrinsic spin and relativistic mass corrections natively emerge from physical field mechanics: isotropic wave-speed retardation and anisotropic spin-tension coupling. Applying these tensor mechanics naturally matches the fine-structure splitting of the Lyman and Balmer series to within 0.0008% of empirical NIST observations after correcting for standard atmospheric refraction indices. Furthermore, we bridge the macroscopic electron orbitals directly to the internal topology of the nucleus. By deploying a Global Harmonic Boundary Solver (Differential Evolution) constrained by the geometric Bohr nodes and the 938.27 MeV mass-energy limit, we organically extract the fundamental internal quark wavenumbers (k₁ = 1.90, k₂ = 3.80, k₃ = 5.70 fm⁻¹) without prior coordinate anchoring. These frequencies perfectly align with the 0.14 - 1.26 GeV² momentum transfer envelope observed in Deep Inelastic Scattering (DIS) experiments, proving that the atomic structure of Hydrogen is unified by a single, continuous thermodynamic field. **Project Integration:**This document is a standalone validation report. The underlying universal field equations, foundational axioms, and the complete multi-disciplinary validation framework can be found in the primary master manuscript (DOI: 10.5281/zenodo.20631794).