Algebraic Varieties and Ideal Theory in Combinatorial Click-Reaction Design

Abstract

We study compatibility-constrained combinatorial chemical assembly through the lens of commutative algebra. Given a finite set $F$ of chemical families, a finite set $H$ of handle types, and a compatibility relation $Pairs(f) \subseteq H \times H$ for each $f \in F$, we construct an assembly ideal $I = J_{bool} + J_{sel} + K_{compat}$ in a polynomial ring $R = k[F,H,H']$ whose variety $V(I) \subseteq \{0,1\}^n$ encodes the set of feasible reaction triples. We prove that $I$ is zero-dimensional and radical, whence $R/I \cong k^{|V(I)|}$. Elimination ideals characterise handle diagnosticity (whether a handle determines its family), the toric ideal of the log-linear model on $V(I)$ measures redundancy in the compatibility relation, and a multi-step ideal $I^{(k)}$ encodes orthogonality constraints among simultaneous assembly plans; the clique number $\omega(G_\perp)$ of the associated orthogonality graph gives the maximum number of mutually compatible plans. We derive a necessary and sufficient criterion for a new family to raise $\omega$. The framework is instantiated on the bioorthogonal click-chemistry landscape ($|F|=8$, $|H|=17$), yielding $|V(I)|=30$, a toric ideal with 2 generators, ML degree 1, and $\omega(G_\perp)=4$. All computations are verified over $\mathbb{Q}$ in SymPy.