Type II Cores: Uniqueness and the Role of Minimality

Nandan, Nghe and Unterberger classified the minimal autocatalytic cores of diluted chemical reaction networks into five families and proved that, under strictly positive linear degradation of every species, every classified core has at most one positive stationary state, with one exception: the family of Type $II_\ell$ cores with $\ell>2$ cyclically ordered one-to-many forks. We prove the missing case. Starting from the source normal form of a Type $II_\ell$ core, source minimality yields a sharp dichotomy: either every weak stem is strictly separated, or the core is the fully coincident three-fork quotient. In the separated case minimality forces every stoichiometric multiplier on each return tail to equal one, and the tails contract to passive positive two-ports. Given two positive stationary states, their concentration ratio and one-way currents satisfy an exact linear current-kernel equation. A back-first divided difference of the product monomials gives every fork row a common sign pattern, Schur elimination of the passive stems produces a cyclic three-term matrix with a uniform diagonal margin and a strictly positive comparison vector supplied by the base stationary current, and a continuant and multiaffine-determinant argument shows that this cyclic matrix is nonsingular. Hence every ratio equals one. The coincident three-fork quotient is handled by a direct ratio-extremum argument. We also show that the minimality hypothesis cannot be dropped: an explicit six-species reversible mass-action network with the Type $II_3$ fork wiring, coefficient-one back products, strictly positive rates and strictly positive degradation has two distinct positive stationary states with exact rational data, and a seven-species weak-order network with six forks shows the same for coincident stems. Both networks violate exactly the minimality consequences used in the proof. The main theorem and all indispensable reductions have been compiled in Lean 4 against Mathlib with no unproved declarations, and both counterexample certificates are checked exactly.