Variational Derivation Program for the Fine-Structure Constant: Asymmetry, Closure Functional, and Charge Duality in QMU

This work develops a variational derivation program for the fine-structure constant within the Aether Physics Model (APM) using Quantum Measurement Units (QMU). A preceding paper established the Singularity as a pre-geometric bifurcation event in which first stable closure separates into boundary and torsional sectors. The present work extends that framework by asking whether the fine-structure constant \(\alpha\) can be derived as the normalized geometric residue of first stable asymmetry. The paper begins from a quadrupolar perturbation of the first admissible closure geometry,\[R(\theta)=R_0\left(1+\epsilon P_2(\cos\theta)\right),\]where \(\epsilon\) is a dimensionless asymmetry parameter. Since the fine-structure constant is dimensionless, the derivation program treats \(\alpha\) as a normalized function of this asymmetry. The charge duality relation is written as\[\frac{e^2}{{e_\mathrm{emax}}^2}=8\pi\alpha.\]The factor \(8\pi\) is interpreted as the product of full spherical angular closure, \(4\pi\), and a dual-sector factor, \(2\). The remaining factor is identified as the asymmetry residue of the bifurcated loxodromic closure. A concrete loxodromic candidate is proposed:\[\Phi_{\mathrm{lox}}(\epsilon)=\frac{\frac{4}{5}\epsilon^2}{1+\frac{4}{5}\epsilon^2}.\]This function arises by relating the normalized surface-area excess of a quadrupolar perturbation to the pitch fraction of a loxodromic traversal. In this model,\[\alpha = \Phi_{\mathrm{lox}}(\epsilon_\ast),\]where \(\epsilon_\ast\) is the stationary asymmetry selected by the closure functional. The closure functional is constructed from boundary, torsional, and volumetric contributions:\[\mathcal{S}[\epsilon]=\mathcal{S}_{\partial}[\epsilon]+\mathcal{S}_{\tau}[\epsilon]+\mathcal{S}_{V}[\epsilon].\]To leading order this gives a quartic stability form,\[\mathcal{S}[\epsilon]=\mathcal{S}_0+k_2\epsilon^2+k_4\epsilon^4+\cdots,\]with a nontrivial bifurcated solution when \(k_2