E8 coupling operating corridor

Status: sampled finite projected corridor promoted through $J=0.1$; strong coupling diagnostic only Machine authority: coupling corridor evidence

At periodic width four and eight local roots, the lowest state remains even and the first pole remains odd and reflection-even across the sampled range $J=0.01$ to 1. The finite gap decreases from 0.2111, reaches roughly 0.0526 near $J=0.5$, and remains nonzero in the scan. The uniform lowest-odd drive retains 99.83% first-pole weight at $J=0.1$ and 92.21% even at $J=1$.

Only $0.01\leq J\leq0.1$ is promoted as the present operating corridor. That interval is supported by the earlier local-root audit as well as this width scan. Throughout it, the gap is at least 0.1494, the carrier retains at least 0.976 excess excitation units, and the uniform creator retains at least 99.83% of its normalized spectral weight in the first pole.

Above $J=0.1$, the data are diagnostic. The product-vacuum one-particle band crosses zero between $J=0.25$ and 0.3, while the exact finite many-body gap remains positive. This is a coordinate failure: the dressed vacuum has reorganized enough that excitation energy relative to the bare product vacuum is no longer physical. It is not evidence that a phase transition occurred, and it does not license a strong-coupling claim without renewed local-root and width convergence.

The corridor is sampled, not continuous. Its practical control variables are coupling $J$, odd-drive pole frequency, oscillator strength and excess excitation count. A device leaves the weak-dressing regime when the latter two begin to erode or the one-particle estimate ceases to track the pole, even if a finite-size gap remains visible.