E8–Yarotsky hypothesis map

Status: structural match established; quantitative theorem application blocked Machine authority: hypothesis map

The projected E8 chain fits the architecture of Yarotsky's weakly interacting lattice systems: a finite local Hilbert space, a translation-invariant non-interacting Hamiltonian with a unique product ground state and gap, and a bounded nearest-neighbour perturbation. After an irrelevant optimal scalar shift, the bond perturbation has norm 0.0281165, or 0.128757 in units of the local gap.

The lowest odd onsite excitation is nondegenerate at 0.2183695 and lies below the two-particle threshold 0.436739. This is the candidate one-particle eigenvalue. The subtlety is the next global onsite eigenvalue: an even state at 0.2288234, only 0.0104539 away. The centered perturbation norm is 2.69 times that margin. Parity forbids their mixing and moves the next odd onsite state to 0.3141245, giving a much healthier 0.0957550 margin, but the cited theorem is not already written as the required parity-resolved extension.

Numerically, the seven-flavour projected first band spans 0.1816600–0.1872266 and remains separated from the second band, which begins at 0.1975589; the global projected band separation is 0.0103323. This supports the carrier interpretation but does not replace the missing infinite-volume estimate.

The decisive blocker is therefore explicit: the theorems state that the bounded perturbation must be “sufficiently small,” with constants depending on interaction range, but do not supply a directly evaluable threshold for this model. A valid proof must either extract explicit constants from the cluster expansion and incorporate parity, or use another quantitative stability theorem. The corpus must not turn structural agreement with a theorem into theorem certification.