Can two quantum chains look identical in small tests, yet behave differently when they become long? This research candidate gives a precise example—and a classification explaining why. The spectral gap is the energy needed to leave the ground-state space. A uniformly gapped chain keeps a positive energy threshold as its length grows. In a gapless chain, that threshold can shrink towards zero. The paper studies a restricted but natural class: neighbouring sites are constrained by one forbidden quantum state, and that state has exactly two equal nonzero Schmidt probabilities. These probabilities describe its entanglement across the pair. The classification turns on how two support planes meet. If they share a line, a special geometric condition decides whether a low-energy disturbance can propagate. On the gapless side, the paper gives an exact formula for every finite chain length. On the other side, it proves a positive lower bound that does not vanish as the chain grows. The most revealing family has the same entanglement probabilities, the same complete two-site and three-site spectra, and the same number of ground states at every length. Nevertheless, its endpoint is gapless while all earlier points are gapped. Those particular summaries therefore do not determine the long-chain gap. Four overlapping sites reveal information the shorter spectral lists miss. The package contains analytic arguments, exact finite certificates, numerical diagnostics and an exact-input classifier. A supplied review identified a symbolic-parameter bug; the revision rejects unspecialised parameters and tests the exceptional case explicitly. This remains an unrefereed candidate, not a formally verified theorem or an authenticated independent reproduction. It does not classify arbitrary interactions, periodic chains or every meaning of a quantum phase. This is Evidence Press, flat Schmidt-rank-two chains, released on the twenty-eighth of September twenty twenty-six. The paper and evidence are linked on this page. This synthetic AI voice is an explanation, not additional scientific evidence.