Self-Dual Geometry of the 24-Cell Polytope as a Unified Framework for Fourier Transform, Heisenberg Uncertainty, and Weierstrass Fractal Dimension

We present a unified mathematical framework grounded in the self-dual geometry of the 24-cell polytope—the unique four-dimensional regular polytope whose dual is isomorphic to itself—possessing F4 symmetry of order 1,152. The Watabe-Claude Method (WCM) maps four-dimensional event clouds (longitude, latitude, depth, time) onto the 24 vertices of this polytope. We demonstrate that three historically independent mathematical structures—the Fourier transform duality, the Heisenberg energy-time uncertainty principle, and the Weierstrass nowhere-differentiable fractal function—are simultaneously encoded in the self-dual geometry of the 24-cell. The central unifying phenomenon is "time=0 fixation": anomalous concentration of events at time-axis vertices immediately preceding critical transitions. The effective fractal dimension D_eff is numerically equivalent to the Weierstrass fractal dimension D_f, the Lorenz attractor correlation dimension D2, and the Levy stability index alpha. Validation: D_eff = 1.991 on Lorenz attractor (reference D2 = 2.06); D_eff approx 2.0 on typhoon data (Finding WX-4); cosD precedes VIX by mean +211 days across three financial crises (Finding FIN-4; SSRN 6969099). WCM originated in Icositetra Reading (CosmosEki; cosmoseki.com) and was developed through human-AI collaboration between Masanori Watabe and Claude Sonnet 4.6 (Kurado, Anthropic). Code: github.com/watabe-masanori/cds-polytope (AGPL-3.0).