E8 joined-rotor nonlinear reconnaissance
Date: 26 October 2026
Status: 117,361-state projected spectra resolved; continuum gaps not converged
Parent: adaptive-shell solver
Evidence: cutoff reconnaissance ledger
Solver: matrix-free shell Lanczos
Certificate: recomputation and cutoff audit
The nonlinear models are already distinguishable
The common E8-shell compiler was executed for both conditional step graphs. At every cutoff the Hamiltonian is applied without constructing a dense matrix, and a fully reorthogonalized Lanczos projection resolves four low levels. Complete shells through kinetic energy six contain 117,361 states.
| cutoff | states | (K)-kernel ground | (K)-kernel gap | (K^{-1})-kernel ground | (K^{-1})-kernel gap |
|---|---|---|---|---|---|
| 1 | 241 | 6.276774 | 0.429336 | 7.532843 | 0.501231 |
| 2 | 2,401 | 5.468592 | 0.241911 | 7.273991 | 0.590525 |
| 3 | 9,121 | 5.082400 | 0.135459 | 7.057355 | 0.474016 |
| 4 | 26,641 | 4.865702 | 0.112768 | 6.898491 | 0.468880 |
| 6 | 117,361 | 4.710110 | 0.105608 | 6.685580 | 0.368244 |
The two branches are not related by merely relabelling the low nonlinear spectrum. The shared kinetic multiplicities coexist with different potential graphs, and those graphs produce separated ground energies and gaps.
Why this is not yet the claimed gap
The maximum projected Ritz residual at cutoff six is below (4\times10^{-14}). That certifies the numerical eigensolve inside the projected basis. It does not certify that omitted shells have stopped shifting the result.
From cutoff four to six, the (K)-kernel first gap changes by 6.35% and the (K^{-1})-kernel gap changes by 21.46%. Both fail the preregistered one-percent promotion threshold. The result is therefore a successful nonlinear reconnaissance and a failed continuum-gap promotion. Reporting either last number as "the E8 gap" would conflate solver residual with cutoff error.
Engineering consequence
Full reorthogonalization is excellent for an auditable reconnaissance but stores every Lanczos vector. The next complete shells contain 340,321, 794,161 and 1,604,401 states. Reaching them safely requires a restarted block Lanczos or Davidson implementation that retains a small Ritz subspace, plus a boundary-leakage estimator and drive-transition weights. The sparse Hamiltonian itself remains cheap: no row has more than sixteen neighbors.