Gapped Vacuum Polarization: A Pure Thermodynamic Derivation of the Effective Fine-Structure Constant

This paper (Pillar 12 of a unified framework) presents a rigorous, purely thermodynamic derivation of the effective fine-structure constant (alpha_eff) and the modified vacuum polarization in Quantum Electrodynamics (QED). Building upon the UV/IR Dephasing Duality established in the derivation of emergent gravity (Pillar 11), we model the QED vacuum as a gapped open quantum system. We compute the vacuum polarization tensor and rigorously verify the Ward identity, proving that gauge invariance is perfectly preserved and the photon remains exactly massless (m_gamma = 0). Despite the exact conservation of gauge symmetry, the dielectric function of the vacuum is modified by the saturation ratio eta(s) of the gapped bath. This yields the master equation for the effective electromagnetic coupling: alpha_eff = alpha / eta(s). For a hard gap (s=0), this theory predicts a strict and parameter-free 11.1% enhancement of the measured fine-structure constant (alpha_eff = 1.111 alpha). This framework provides a novel, non-perturbative explanation for the 4.2-sigma Muon g-2 anomaly and predicts specific, falsifiable modifications to the Uehling potential and the Lamb shift. The derivation relies exclusively on established quantum field theory theorems, containing no fitting, no new beyond-Standard-Model particles, and no circular reasoning. A fully validated Python script (4/4 tests passing) is provided in the appendix.