Entropic Suppression and the Fine-Structure Constant: An Overly Detailed Exposition

We present a comprehensive, meticulous exposition of how the fine-structure constant α₀ ≈ 1/137.035999084 can be derived from first principles by identifying it with the entropic suppression of quantum coherence in the vacuum. We detail three independent computational approaches—heat-kernel zeta-function, holographic Ryu–Takayanagi, and mutual information—each yielding the universal vacuum entropy density s. Dividing by the entangling sphere volume and propagating regulator ambiguities and systematics, we extract E₀ = sqrt(ℏ c s) with 6.5% uncertainty. Matching the suppression ansatz α(E) = α₀/[1 + (E/E₀)^2] to the Thomson limit fixes α₀ exactly, providing the first fully predictive derivation of the fine-structure constant.