Flux Selects Charge-Sector Ground States without Gapping a Motzkin Ring

Local frustration-free constraints can transport amplitudes consistently on every bond yet fail around a ring. We solve this obstruction for an interacting periodic spin-one Motzkin Hamiltonian with arbitrary nonzero complex ordered-bond parameters. The complete finite-size ground-space degeneracy and surviving-sector pattern depend only on the complex product of the bond parameters: for every ring size N ≥ 2, the dimension is 2N + 1 at trivial holonomy and exactly 3 otherwise. On the unit circle, the principal total boundary flux removes every nonextreme charged exact ground state and shifts the two charge-conjugate one-flat bands oppositely. Because those bands are invariant sectors of the full Hamiltonian, a fixed nonzero total flux gives a rigorous many-body gap upper bound of order N⁻². Moreover, every lifted charged sector contains a positive level at or below the squared principal flux divided by 4N, yielding at least 2N − 2 soft eigenvalues, counting multiplicity. Boundary flux therefore collapses the growing exact ground manifold without opening a thermodynamic gap. No gap lower bound, bulk transition, topology, or quantized transport is claimed. This v1 record contains the eight-page combined preprint and its minimal four-file LaTeX source archive. This is a preprint and has not undergone journal peer review.