E8 projected two-cell coupling

Date: 26 October 2026

Status: projected two-cell gap promoted; full two-cell and width gaps open

Evidence: two-cell ledger Certificate: two-cell verifier

The first explicit spatial extension uses two identical compact cells and

Each cell is projected onto its four lowest resolved states. Within that manifold the interaction is constructed from signed matrices, not transition weights:

where \(C_i\) represents \(\cos\theta_i\) and the real antisymmetric \(A_i\) is the \(i\sin\theta_i\) proxy. Their symmetry identities hold numerically to better than \(3\times10^{-14}\).

At every sampled \(J=0,0.01,0.025,0.05,0.1\), the projected first gap and, for \(J>0\), the first-doublet splitting pass two successive sub-one-percent shell increments from cutoffs 6 to 8 to 10. At \(J=0.1\), the projected gap is 0.1822513372 and the splitting is 0.0051948553. The lower one-excitation state is swap-even and therefore collective; the upper is swap-odd and therefore a relative-cell mode. Both remain inversion-odd.

This is the first dynamical meaning of the cell index in the nonlinear rotor lane: coupling does not merely shift the local line, it resolves motion in phase and out of phase between cells. The result is nevertheless a projected two-cell certificate. The local-root audit promotes the gap through eight roots but withholds the quantitative splitting coefficient. The full product charge basis was not diagonalized, and no width-independent or thermodynamic gap follows. Ten roots or a controlled downfolding remainder is next, followed by three- and four-cell chains.