# E8 Cartan-kernel compact-cell gap

**Date:** 26 October 2026

**Status:** provisional numerical compact-cell gap promoted; bulk/E8 response open

**Evidence:** [gap ledger](../../../evidence/data/translation-free-e8-cartan-compact-gap-v1.json)  
**Certificate:** [promotion verifier](../../../verification/verify/verify_translation_free_e8_cartan_compact_gap.mjs)

For the selected unit-coupling Hamiltonian

\[H={1\over2}n^TK_{E_8}n+\sum_i(1-\cos\theta_i),\]

complete E8 shells 8, 10 and 12 give first gaps (0.1023201434),
(0.1020184010) and (0.1019267219). The last two relative increments are
0.295% and 0.0899%, both below the frozen one-percent threshold. At shell 12
(1,604,401 states), the maximum low-level boundary probability is
(2.93\times10^{-5}) and the maximum residual is (9.28\times10^{-11}).

The promoted numerical compact-cell value is therefore

\[\Delta_{\rm cell}\simeq0.1019267219\quad(U=u=1),\]

with the last absolute increment (9.17\times10^{-5}) retained as a practical
truncation diagnostic, not a rigorous error bar.

Parity fixes detection: the first excitation is odd, cosine-dark and
sine-bright; the second is even and cosine-bright. Phase-quadrature drive is
therefore required to measure the actual gap.

This does not establish a rigorous lower bound, thermodynamic coupled-wire
gap, (c_-=8), the 240-current shell or a material realization. The next join
must sweep coupling and perturbations, then retain this cell spectrum while
increasing wire width and measuring the edge/current records.