A Deterministic Hydrogen Validation of the Scretching Quantum Chain and the Maxwell–Scretching Electromagnetic Chain III: The Balmer Series (n = 3…7)

This work presents a deterministic, first-principles validation of the Scretching Quantum Chain (SQC) and its macroscopic electromagnetic continuation, the Maxwell–Scretching Chain (MSC), using analytically solvable hydrogen Balmer-series transitions (2→n, n = 3…7) as an independent reference-grade testbed. Whereas Part I and Part II established deterministic closure using the Lyman series, the Balmer series provides an orthogonal validation spanning visible wavelengths, a different lower-state manifold, and distinct statistical-weight structure. Hydrogen is uniquely suited for this validation because its electronic structure is exactly solvable and its oscillator strengths and Einstein coefficients follow directly from quantum electrodynamics rather than empirical fitting. Using five Balmer transitions (H-α through H-ε), this study demonstrates a strict linear invariant linking oscillator strength f, transition frequency ν, statistical-weight ratio (g₁/g₂), and the Einstein spontaneous emission coefficient A₂₁. The proportionality constant is shown to be fixed entirely by fundamental electromagnetic constants, yielding agreement at the ≲2.7×10⁻⁴ relative level, with deviations attributable solely to tabulated rounding and convention differences. The Balmer-series validation confirms that the SQC invariant is independent of spectral region, principal quantum number, or lower-state manifold, and that deterministic closure persists across transitions spanning more than an order of magnitude in wavelength and transition probability. Apparent discrepancies among tabulated Balmer A-values are shown to arise from differences in statistical-weight averaging and fine-structure conventions rather than physical inconsistency. The work further demonstrates that the Maxwell–Scretching Chain (MSC) is the unique Maxwellian projection of the same quantum electromagnetic coupling encoded in oscillator strength. Through the MSC, f deterministically fixes absorption cross-section, absorption coefficient, extinction coefficient, dielectric loss, and macroscopic absorbance without introducing empirical parameters. Together, the SQC and MSC form a continuous quantum→electromagnetic chain linking atomic transition physics to bulk optical response. This paper establishes the hydrogen Balmer series as a second canonical anchor—complementary to the Lyman series—for validating deterministic quantum–electromagnetic closure. The results provide a reference-grade diagnostic framework: any deviation from linearity in more complex atomic, molecular, or condensed-matter systems must originate from unit errors, convention mismatches, or experimental systematics rather than breakdown of the SQC–MSC structure. Future work will extend this validation to additional hydrogen manifolds (Paschen, Brackett) and to multielectron atoms, enabling systematic testing of relativistic, fine-structure, and correlation effects within the same unified framework. KeywordsHydrogen Balmer series; Oscillator strength; Einstein coefficients; Quantum electrodynamics; Maxwell’s equations; Optical spectroscopy; Deterministic modeling; Electromagnetic closure