BCT Letter 3: The Fine Structure Constant from Pure Geometry — alpha = alpha_0(1 - 2*alpha_0) with alpha_0 = (sqrt(2)-1)(sqrt(6)-2)/8pi

The electromagnetic fine structure constant α ≈ 1/137.036 is the most precisely measured fundamental constant with no theoretical explanation for its value. We derive it from the interstitial void geometry of a body-centred tetragonal (BCT) lattice with axial ratio c/a = √ 2. The octahedral and tetrahedral void radii are roct = (√2 − 1)/2 = 0.20711 and rtet = (√6 − 2)/4 = 0.11237, fixed by sphere-packing geometry. The BCT crystal coupling α0 = roct rtet/π = (√ 2 − 1)(√6 − 2)/8π = 0.0074081 yields the fine structure constant via one-loop vacuum polarisation: α = α0(1 − 2α0) = 1/137.018. This differs from the measured value 1/137.036 by −0.013%, or 1.8 parts in 104. The formula contains no measured inputs, no fitted parameters, and no arbitrary choices—only square roots, integers, and π. We discuss the physical origin of each factor, the one-loop correction mechanism, and the implications for the longstanding question of whether α is calculable from first principles.