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, ⊕, 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 ≤ a) purely from the design signature (m, ν, n0). For systems equipped with a type system and valence bounds, we define the composition polytope Pval ⊂ ℝm, whose integer points count the feasible compositions. Compositions outside Pval -- 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 ≈ 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 10198 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.
Related
BibTeX
@misc{ribas2026zombie,
title = {{Zombie Compositions in Assembly Algebras and an Upper Bound on the Size of Chemical Space}},
author = {Vicent Ribas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.21363},
eprint = {2606.21363},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.21363},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Zombie Compositions in Assembly Algebras and an Upper Bound on the Size of Chemical Space
AU - Vicent Ribas
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2606.21363
UR - https://arxiv.org/abs/2606.21363
AB - In this paper we present construction systems -- tuples (X, BB, ⊕, 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 ≤ 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_val ⊂ ℝ^m, whose integer points count the feasible compositions. Compositions outside P_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 ≈ 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.
ER -