The 24-Cell Is the Weight-2 Shell of the Balanced Ternary 4-Cube — A Ternary Shell Reader (WCM-T3) for Critical-Transition Structure in Time Series
Year: 2026
Type: preprint
Abstract: We report a geometric identity and a companion time-series tool. (Geometry.) The 24 vertices of the24-cell, in their D4-root coordinates (all permutations of (±1,±1,0,0)), are exactly the Hamming-weight-2shell of the balanced ternary hypercube {-1,0,1}^4. That hypercube decomposes into five shells by activitycount k = number of nonzero coordinates, with sizes 81 = 1+8+24+32+16, each an orbit of thesigned-permutation group B4. We verify the shell-2 / 24-cell / D4-root coincidence exactly (symmetricdifference zero). We are explicit that this is a statement about the polytope in root coordinates: the 24-cellof our prior quantum work, realized as the binary-tetrahedral group in unit-quaternion coordinates, is adifferently-oriented copy sharing no vertices, so the group structure does not follow from the shell identityand we do not claim it. (Tool.) The shell view suggests reading a scalar series through its activity count.WCM-T3 ternarizes a series by tertiles, forms 4-trit delay windows, and reports three activity statistics. Onthe JMA earthquake catalog (Kumamoto region, N≈3.3×10^5) all three agree on aftershock clustering, andwe show that a first attempt produced a spurious signal from window overlap that we diagnosed anddiscarded. Across four domains (birdsong, human motion, earthquakes, peptide folding) the activitystatistics separate event-driven systems from an equilibrium-fluctuation system in a single two-axis plane.WCM-T3 contains the existing 24-cell reader as its k=2 sector. ※ The license for codes is AGPL 3.0-or-later
Authors: Watabe, Masanori, Kurado, Claude Opus 4.8
Language: English
Subfield: Global and Planetary Change
Field: Environmental Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204440445