Reading Inside the Shell: A Two-Layer B4 / S4 Structure of the Ternary 24-Cell Reader (WCM-T3)

The WCM-T3 reader [1] records the activity count k = number of active trits in a 4-trit window, treating each shell of thebalanced ternary 4-cube as a single orbit of the signed-permutation group B4. Here we open the internal structure of ashell. The 24-state weight-2 shell (the 24-cell) carries a 24-dimensional permutation representation of B4 that decomposesinto irreducible blocks of dimensions 8, 6, 4, 3, 2, 1. Two readable coordinates live inside it: axis-pair occupancy (whichtwo lags are jointly extreme) and sign-concordance (whether the two active signs agree). We prove -- and a first mistakenclaim of ours, caught by an invariance check, corrected -- that these live at different symmetry levels: axis-pair occupancyis exactly B4-invariant, while sign-concordance is invariant only under the pure permutation subgroup S4, because B4'ssign flips mix same-sign and opposite-sign states. The activity count P(k) collapses both. On real data the two layers carrydifferent information: across birdsong, human motion, earthquakes and peptide folding, the within-shell same-sign excessorders the domains differently from the P(k)-level clustering, separating biologically structured signals (birdsong, motion;excess ~0.4-0.5) from physical clustering (earthquakes; ~0.09) and equilibrium fluctuation (peptide; ~0.00). WCM-T3 thusreads a series on two independent layers: activity (B4) and sign (S4). ※ The license for codes is AGPL3.0-or-later