Zombie Compositions in Assembly Algebras and an Upper Bound on the Size of Chemical Space

Abstract

In this paper we present construction systems -- tuples $(X, BB, \oplus, ν)$ comprising objects, building blocks, an assembly operation, and a joining multiplicity -- as a general algebraic framework for studying how complex objects are built from simpler parts. To each construction system we associate a toric ideal, a toric variety, and a matroid, obtaining analytical bounds on the growth function $N(a)$ (the number of objects of construction complexity $\leq a$) purely from the design signature $(m, ν, n_0)$. For systems equipped with a type system and valence bounds, we define the composition polytope $P_{\mathrm{val}} \subset \mathbb{R}^m$, whose integer points count the feasible compositions. Compositions outside $P_{\mathrm{val}}$ -- termed zombies -- are combinatorially valid but physically unrealisable. We prove that the zombie classification is sound (zero false positives) and conservative: the true infeasibility rate is at least as high as the polytope predicts. Specialising to the molecular graph assembly system of Morales Parra et al. ($m = 19$ bond types, $5$ atom types with valences $1$--$4$), we identify a composition polytope whose lattice points capture the physically realisable compositions, and show that the resulting growth exponent tightens from $0.73$ to the exact value $\log 2 \approx 0.693$. While the difference of $0.037$ appears small, in the doubly-exponential regime it corresponds to a tightening of the bound by a factor exceeding $10^{198}$ at assembly index $10$. The framework also recovers the bioorthogonal click-chemistry system of the author's prior work as a second instance, with $m = 8$ and a partition matroid.