A spectral-gap classification for flat Schmidt-rank-two chains

For homogeneous open nearest-neighbour chains with a normalised rank-one local projector whose forbidden vector has exactly two equal nonzero Schmidt probabilities, a written proof candidate classifies gappedness in every finite local dimension. Gaplessness is equivalent to a balanced-marker form and has exact finite-length gap 1−cos(π/N). All other interactions in the class have explicit uniform lower bounds. A complete qutrit short-spectrum fibre has fixed Schmidt probabilities, two- and three-site spectra and ground-space dimension at every length, yet only its endpoint is gapless. The proof distinguishes compatible propagation from an obstruction to four-site saturation.

Unrefereed candidate. Producer-side exact and numerical replay corroborates finite statements, not universal analytic proofs. Supplied referee implementation and review are not authenticated unaffiliated reproduction or peer review. No exhaustive priority, experimental or impact claim.

Version 1.0.1 repairs the symbolic classifier domain and adds precise antecedent locators and a named full Gram-decomposition lemma with explicit basis appendix. Original prose/data CC0-1.0; original code MIT; third-party exceptions in LICENSES.md.