Kinetic order is invisible to D-cores: a support-preserving mass-action lift and an exact counterexample

Unstable-core methods explain the instability of a reaction-network equilibrium through a small child selection: a square submatrix of the stoichiometric matrix, chosen by assigning to each of a few species one reaction in which it is a reactant, that is unstable under some positive diagonal scaling. A child selection is built from the stoichiometric matrix $S$ and the reactant-incidence relation only; it does not record how many copies of a reactant a reaction consumes. We prove that this forgotten coordinate, the kinetic order, is decisive under classical fixed-exponent mass-action kinetics. First, for every finite network, every strictly positive rational flux in the kernel of $S$ and every rational reactivity matrix that is positive exactly on reactant incidences, there is an ordinary integer-exponent mass-action network with the same stoichiometric matrix and the same reactant incidences whose linearization at an explicit positive equilibrium is exactly the prescribed Jacobian. The construction adds the same nonnegative integer matrix to both sides of every reaction (stoichiometrically silent catalytic padding), which leaves every child-selection matrix unchanged. Second, we exhibit a four-species, five-reaction network that is Hurwitz stable at every positive mass-action equilibrium, together with a padding of it that has a positive equilibrium whose Jacobian has the eigenvalue $\tfrac{1}{500}+\tfrac{51}{500}i$, although the common child lattice contains no D-unstable core. Hence the classical mass-action analogue of D-unstable-core necessity is false, already in dimension four, and any corrected localization theorem for mass action must see kinetic order. We also give a compact witness with maximal exponent $200$. The general lifting theorem, the universal stability of the unpadded network, the padded instability, the absence of D-unstable cores, and the resulting refutation are verified in Lean 4 with Mathlib, with warnings treated as errors and no unproved placeholders.