n_X ≡ log₂(X/X_Planck): A Dimensionless Logarithmic Framework and Closed Geometric Derivation of α⁻¹ = 137.0359990838831 from T²/Z₈ QACavity

Abstract

The fine-structure constant inverse, α⁻¹ = 137.035999084(21) [CODATA 2022], remains one of the most precisely measured yet theoretically elusive constants in physics. Historical and recent attempts via T²/Z_N orbifold models have left geometric parameters fitted or suffered from dimensional inconsistencies.This Letter presents a closed-form, dimensionless, and falsifiable derivation from first principles on the double torus QACavity T²/Z₈. By introducing a formal logarithmic Planck normalization framework n_X ≡ log₂(X/X_Planck), we resolve dimensional errors and establish two exact results: the gravitational coupling gradient n_{α_G}=2n_M and holographic entropy scaling n_S=2n_M+log₂(4π).From T-duality self-duality conditions and Fibonacci anyon quantum dimensions, the twisted sector volume is derived as Vol=Φ²=2.618033988749895 (Φ golden ratio). Integrating this volume into the heat kernel expansion with Donnelly-Seeley boundary terms and performing a full Dyson bubble resummation, we obtain the master equation α⁻¹=N/Vol (1-b₁a-b₂a²), where N=360, b₁=Vol/√π=1.4770675058266744, b₂=-Vol/(32π+Vol)=-0.0253810896526106, a=α/π.This converges to α⁻¹=137.0359990838831, yielding relative error -0.000853 ppb inside CODATA 2022 bounds with zero free parameters. Explicit Lagrangian, heat kernel derivation, and numerical verification are provided. Full non-abbreviated version.