Thermodynamic compatibility of interacting autocatalytic cores: an exact phase diagram and a layered obstruction theory

A potential autocatalytic core (PAC) is a minimal stoichiometric motif that admits a strictly productive reaction current; a consistent autocatalytic core (CAC) is a PAC whose productive current is realized by a positive concentration vector under thermodynamically consistent mass-action kinetics. Kosc, Kuperberg, Rajon and Charlat (Proc. Natl. Acad. Sci. USA, 2025) proved that every isolated PAC is a CAC, and exhibited two overlapping cores that admit a common productive current but no common concentration vector. We study when a family of interacting cores admits one common thermodynamic realization, with all cores sharing a single activity vector and fixed transition-state factors. First, we refute the natural strengthening of the isolated-core theorem: there is a two-core family with a common productive current and a common strict reaction-direction potential that nevertheless has no common realization. The obstruction is metric rather than directional: it already defeats the linear relaxation in which every distinct stoichiometric complex receives an independent activity. Second, for the responsible topology, two autocatalytic triangles $A\rightleftharpoons B\rightleftharpoons C\rightleftharpoons 2A$ and $A\rightleftharpoons B\rightleftharpoons D\rightleftharpoons 2A$ sharing $A\rightleftharpoons B$, we prove an exact phase diagram: with private transition-state factors in ratio $k$, a common realization exists if and only if $1/2<k<2$, and for arbitrary positive factors if and only if the reciprocal-factor sums of the two private branches are within a factor of two. In transition-state language the symmetric window is $|\Delta G^{\ddagger}|<RT\ln 2$. Third, we derive an interface calculus: eliminating a gain-$m$ triangle with factors $b_0,b_1,b_2$ leaves exactly the open response interval $(R/m,R)$ with $R=b_0(1/b_1+1/b_2)$, and two eliminated cores compose precisely when their intervals overlap. Fourth, we show that the linear relaxation is not the whole story: a two-species, three-reaction family passes the flow, direction and independent-complex tests but fails the monomial lift to a common species-activity vector. Together with the direction obstruction of the original example, this yields three provably distinct failure mechanisms. All headline theorems are formalized in Lean 4 against Mathlib with warnings promoted to errors and no admitted statements.