# E8 projected-chain boundary comparison

**Date:** 26 October 2026

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

**Evidence:** [boundary ledger](../../../evidence/data/translation-free-e8-projected-chain-boundary-comparison-v1.json)
**Certificate:** [boundary verifier](../../../verification/verify/verify_translation_free_e8_projected_chain_boundary_comparison.mjs)

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

\[
\Delta_{3,4,5,6}=(0.1826163504,0.1826156104,0.1826155985,0.1826155983).
\]

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.