math.DS — Dynamical Systems
Unistationarity of Type II_l autocatalytic cores
We prove unistationarity of reversible Type II_l autocatalytic cores with l at least three under strictly positive degradation. The source-minimal wiring forces source exhaustion and unit return tails, allowing comparison of positive stationary states through current ratios, chain reduction and a cyclic determinant argument. The proof includes separated stems and the exceptional coincident three-fork branch, with all internal chain species retained. Exact six- and seven-species reversible mass-action examples show that the minimality assumptions cannot be dropped: both admit two distinct positive stationary states with strictly positive rates and losses. The principal uniqueness theorem has a complete registered Lean formalization. The manuscript distinguishes that theorem, broader historical formal evidence and conventional exact calculations; the supplied claim map records the boundaries of each.