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 with the Borwein integrals: both constructions draw from odd reciprocals filling a unity container, and both converge because the Leibniz sum π/4 < 1. Borwein fills unity additively (6 = D! terms); the Tower fills multiplicatively (D = 3 dimensions). A basis change from π to e^(2/√3) yields the closed-form approximation α⁻¹ ≈ e⁵ − 6√3 − 1 + 1/66 = 137.03600, matching CODATA to 0.05 ppm, where the correction 1/66 is the first harmonic residual of the 2D-to-3D transformation. We provide historical context, comparing the Tower to previous attempts at deriving α by Eddington (1929), Wyler (1969), and Atiyah (2018). The Tower is distinguished by its derivable coefficients, elementary construction, connection to proven mathematics, natural dimensional termination, and — uniquely — its fertility: the Tower series was the ancestor of the Bootstrap Universe framework, from which 74 preprints have derived multiple independent physical constants including the muon-electron mass ratio, hydrogen binding energy, lepton g−2, nuclear magic numbers, and the cosmological constant. As a secondary result, we prove that the Leibniz partial sum deviations satisfy n(Lₙ − π/4) → (−1)ⁿ/4, with prime indices providing a clean convergent subsequence.