Unistationarity of Type IIℓ and Type V autocatalytic cores with nonnegative degradation

Nandan, Nghe and Unterberger classified the minimal autocatalytic cores of diluted 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 the exception of the Type $\mathrm{II}_\ell$ cores with $\ell>2$ forks, which were settled subsequently. They asked whether uniqueness survives mixed degradation, in which some species are degraded and others are not, and left this open for the Type $\mathrm{V}$ cores and the Type $\mathrm{II}_\ell$ cores with $\ell>2$. We answer both cases affirmatively. For the reversible mass-action extension of every source-minimal Type $\mathrm{II}_\ell$ core ($\ell\ge3$) and every source-minimal Type $\mathrm{V}$ core, with arbitrary positive forward and reverse rate constants and an arbitrary nonnegative degradation vector, there is at most one strictly positive stationary concentration vector; the conclusion includes every internal species of the unit paths permitted by the classification and does not assume that any degradation constant is positive. Combined with the published results for the remaining families, this gives unistationarity across the whole classification for all nonnegative degradation vectors. Two mechanisms are used. Exact elimination of passive reversible unit paths, which remains valid when internal losses vanish, reduces Type $\mathrm{V}$ to three polynomial equations whose coordinate ratios admit a strict extremum argument. For Type $\mathrm{II}_\ell$ a stationary-flow factorization of the Jacobian and an admissibility shift show that the stationary Jacobian is nonsingular at every positive stationary state, even on the boundary of the degradation orthant, by transporting the positive-degradation current criterion through the positive diagonal orthant with a multiaffine determinant argument; the implicit-function theorem then transfers uniqueness from positive degradation to its boundary. We also derive a quantitative lower bound $\det(-H)\ge(\det N)^2\prod_i\min(p_i,q_i)$ for the scaled stationary Jacobian, local smooth continuation of the positive state through vanishing loss coefficients, and existence and local asymptotic stability of the unique positive state for all sufficiently small nonnegative degradation. The two family uniqueness theorems and the algebraic kernel criteria are verified in Lean 4 against Mathlib with warnings treated as errors and no unproved declarations.