A Tower Series for 137: Dimensional Closure from the Leibniz Sum

Copy We present a new series representation of the fine structure constant inverse: α⁻¹ ≈ T(3) = π + π² + 4π³ = 137.0363. The series T(D) = Σ(n=1 to D) 2^(2^n) × (π/4)^n is constructed in three layers: the Grandi oscillation (1 − 1 + 1 − 1 + ⋯ = 1/2), the Leibniz sum (1 − 1/3 + 1/5 − ⋯ = π/4), and a Tower of iterated powers through D spatial dimensions. The coefficients (1, 1, 4) are not fitted but follow from the closed form 2^(2^n − 2n), which measures the gap between exponential growth (2^n) and linear growth (2n) in the exponent. The series terminates uniquely at D = 3 — the n = 4 term would contribute 65536(π/4)⁴ ≈ 24937, causing divergence. We establish that the Tower shares its convergence mechanism…