A Geometric Identity for the Fine-Structure Constant: From the 4D Unit Ball Volume and its Cube-Packing Deficit (Paper 7)

This paper presents a self-consistent algebraic identity for the fine-structure constant α from two elementary 4-dimensional geometric quantities: (i) N (1) = 137, the count of integer-centred unit cubes fully contained in a 4D ball of radius 3 (re-derived by direct enumeration), and (ii) V₄ (1) = π²/2, the volume of the 4D unit ball. The identity α⁻¹ = N (1) + V₄ (1) ·α is equivalent to a self-consistent quadratic (π²/2) α² + 137α − 1 = 0 whose positive root predicts α = 0. 0072984, agreeing with the observed value 7. 2974×10⁻³ to a relative accuracy of 1. 4×10⁻⁴ (0. 02%). The identity admits a perturbative reading 1 = 137α + V₄ (1) α², interpreted as "tree-level coupling on the inside (137 packed cubes) plus self-energy correction on the outside (boundary gap, 66% of the R=3 ball volume) ". Decisive differences from naive Eddington-style integer-fitting are spelled out: self-consistent (not pure integer) ; both 137 and π²/2 derived independently from 4D geometry; QFT-perturbative analogue. Whether the identity reflects geometric necessity or numerical coincidence is left as an open problem. The 0. 03% residual and first-principles derivation of the correction term are flagged as principal open questions. Companion paper to the BH Thermodynamics Programme (Papers 1–6).