Model Structures and Bousfield Reachability on Finite Horizontal Sums

Martínez Alberga and Roitzheim [6] classified model structures on the five-point pentagon and proved their Bousfield reachability. We describe every model on \( H(p,q) \) by two weak-block partitions and two acyclic-fibration threshold vectors. Two endpoint conditions complete the classification and give explicit counts. We extend the classification to every nonempty finite horizontal sum \( H(p_1,...,p_k) \) by imposing the endpoint conditions on each pair of branches. Every classified model is reachable. Two cuts remove the interactions at the shared endpoints. The remaining model is constructed branch by branch; at most two further set-localizations restore the target. Classification and reachability therefore give complementary descriptions: which models exist, and how to construct them.