Collective thermodynamic incompatibility of autocatalytic cores: minimally incompatible families of unbounded order

A potential autocatalytic core (PAC) is a minimal stoichiometric motif admitting a strictly productive reaction current. Kosc, Kuperberg, Rajon and Charlat (Proc. Natl. Acad. Sci. USA, 2025) proved that every isolated PAC can be realized by a positive activity vector under thermodynamically consistent mass action, and showed that families of interacting cores need not be jointly realizable; our predecessor paper quantified the two-core case exactly. Here we study the order of a collective conflict. Fix a reversible network with fixed transition-state factors and an activity box, call a family of selected cores compatible if one activity vector makes every selected core productive with its reactions running in the displayed directions, and call a family minimally incompatible if it is incompatible while every proper subfamily is compatible. We prove that minimally incompatible families of every order $n\ge 3$ exist among the sparsest conceivable cores: $n$ unit-factor two-reaction cores $X_s\rightleftharpoons X_t$, $X_t\rightleftharpoons 2X_s$ on $n$ species, arranged as a directed path with one shortcut. The full family has no common productive state anywhere in the positive orthant, every proper subfamily has one in $[9/10,1]^n$, and the full family nevertheless passes both standard necessary relaxations, reaction-direction feasibility and independent-complex feasibility, with witnesses inside the same box. The obstruction is a single exact inequality $f(f(x))<g(x)$ between a two-step response and the shortcut response. Consequently no test based on subfamilies of bounded size, even combined with both global relaxations, decides compatibility. Complementing this, we give an exact univariate elimination theorem for arbitrary bounded fans of autocatalytic triangles sharing one reaction; a three-triangle fan that is minimally incompatible on an explicit open neighbourhood of all seven kinetic factors and ten box endpoints while passing both relaxations; and a polynomial-time constructive compatibility certificate for graded assemblies of two-reaction cores. All source-level statements are verified in Lean 4 against Mathlib, and the hand-proved complements are listed explicitly.