E8 projected-chain boundary comparison

Date: 26 October 2026

Status: periodic finite-width plateau promoted; thermodynamic bound open

Evidence: boundary ledger Certificate: boundary verifier

At cutoff eight, eight local roots and \(J=0.05\), closing the projected chain gives periodic first gaps through width six

The successive changes are \(4.1\times10^{-6}\) and \(6.5\times10^{-8}\), followed by \(1.17\times10^{-9}\). The first excitation is inversion-odd and reflection-even at every width. Periodic momentum-like excited pairs appear above it, while the minimum itself remains uniform. This is consistent with a bulk \(k=0\)-like collective band minimum rather than an end mode.

The corresponding open-chain gaps approach the same scale. Their relative differences from the periodic values are 1.65%, 0.843% and 0.499% at widths three, four and five. Thus boundary sensitivity is already below one percent at the last two widths.

The width-six calculation contains 262,144 projected states and has first-state residual below \(8\times10^{-15}\). This is still a finite boundary-condition certificate, not a thermodynamic lower bound. It also does not identify an E8 edge anomaly: the first mode is a bulk collective excitation of the projected chain. Larger widths should reuse a serialized local operator dictionary so spatial scaling no longer repeats the expensive charge-shell reconstruction.