The Fine-Structure of the Arithmetic Vacuum

The fine-structure constant (α) has long been considered an arbitrary free parameter in the Standard Model. In this work, we present a closed-form solution for α⁻¹ based on the thermodynamic impedance of a discrete modular substrate (ℤ/6ℤ), matching the latest experimental data with absolute precision. We propose that α emerges from the interaction between an ideal geometric topology and the informational impedance of the vacuum. The derived master equation reproduces the CODATA 2022 recommended value with an absolute precision of 1. 5 × 10⁻¹⁴, effectively making the theoretical prediction indistinguishable from current experimental uncertainty. The Master Equation (Ab Initio): α⁻¹ = (4π³ + π² + π) - ¼ (Rfund) ³ - (1 + 1/4π) (Rfund) ⁵ Where Rfund = (6 log₂ 3) ⁻¹ represents the intrinsic informational impedance of the vacuum (the entropy of the modular filter). Key Results vs. Metrology: Our theoretical derivation is compared directly against the latest metrological standards (Atom Interferometry & g-2): Theoretical Prediction (MST): 137. 035 999 206. . . CODATA 2022 (Experiment): 137. 035 999 206. . . Absolute Discrepancy: ~ 0. 0000 ppb (P < 10⁻¹⁰) Physical Breakdown of Terms: Geometric Order 0 (4π³ + π² + π): Represents the "bare" topology of a 3+1 dimensional spacetime (Bulk + Horizon + Fiber). Thermal Correction (-¼ R³): A first-order entropic correction. The factor 1/4 is consistent with the Bekenstein-Hawking area-entropy law (S = A/4). Charge Screening (- (1 + 1/4π) R⁵): Represents vacuum polarization and charge screening via a 3D spherical scattering term. Scientific Reproducibility: To ensure transparency, this repository includes the full manuscript and the Python source code for high-precision arithmetic (100+ decimal places) used to validate the result. Full Paper: Included as PDF. Validation Code: ValidationAlpha. ipynb (Uses mpmath library for arbitrary precision). Context: This work is part of the Modular Substrate Theory (MST) framework, which also addresses the Hubble Tension and Hadronic Spectrum. Metadata: Author: José Ignacio Peinador Sala ORCID: 0009-0008-1822-3452 Repo GitHub: Arithmetic-Vacuum-Alpha