A Geometric Proof of the Leibniz Formula for π via Gibbs Phasor Chain Convergence

A geometric proof of the Leibniz formula π/4 = 1 − 1/3 + 1/5 − 1/7 + ⋯ in which π emerges from the polygon-to-circle limit rather than from any prior analytic identity. The partial sums of a Fourier series are treated as a phasor chain in the complex plane whose discrete curvature is controlled by a single parameter. At one setting the chain folds into a regular (N+1)-gon; at another it collapses to the Leibniz alternating sum. The Dirichlet kernel is the intrinsic velocity of the deformation connecting these two states. In the limit N → ∞ the polygon becomes a circle, chords become arcs, and the integral evaluates to π/4 by the same mechanism as Archimedean exhaustion. The convergence rate decomposes into a fast O(1/N²) polygon-to-circle error and a slow O(1/N) Gibbs tail, explaining geometrically why the series converges so slowly.