One Qubit, Two Classical States: A Certified Memory Gap

A quantum model can beat every classical model we train without establishing a memory advantage: a better classical model may remain undiscovered. We develop certificates that rule out this possibility for finite binary event distributions. They prove when a single qubit achieves lower total-variation error than every homogeneous two-state edge-emitting hidden Markov model. The key is shared dynamics: one physically feasible recurrence must explain every temporal context. Covering its two-dimensional coefficient region yields classical error lower bounds checked with exact arithmetic. These constraints detect structure missed by rank tests, although satisfying them need not yield a valid classical model. Across six designed quantum constructions, the method certifies candidates left unresolved by the specified clock, rank and complete-model search policies on every construction. On five, it brackets the best classical error within an interval of width 0.025. The comparison concerns persistent-state memory; the qubit remains inexpensive to simulate classically.