Open Families of High-Schmidt-Number PPT Channels with Interior Entanglement-Breaking Triple Compositions

We construct, in every local dimension d≥6, nonempty open families of triples of CPTP channels whose Choi matrices and partial transposes are strictly positive, each with Schmidt number at least d−floor(sqrt(2d−1)) and individually non-entanglement-breaking, but whose triple compositions lie in the interior of the entanglement-breaking cone. The proof combines Johnston–Lovitz high-Schmidt-number PPT states with an explicit separable-order sandwich lemma. We also give a conservative exact rational interval for robustness of a fixed 10→6→10 PPT-channel pair, with independently auditable integer Kraus matrices and numerical checks. This is an existence and sufficient-condition result, not a solution of the general PPT-cubed conjecture or a universal triple-composition theorem. Historical priority and expert peer review remain pending.