Hammock coordinates for silting subcategories of derived-discrete algebras

We introduce hammock coordinates to study the combinatorial structure of partial silting configurations in the bounded derived category of a derived-discrete algebra $Λ(r,n,m)$. By extracting exact integer inequalities from the Bobiński--Geiß--Skowroński Hom-hammock formula, we establish a closed-form criterion for pairwise partial silting both within any Z-component and across distinct components. Using this coordinate framework, we establish three primary results. First, we prove the universal upper bound |S| ≤ n+m on the cardinality of any partial silting configuration S of Z-objects ---both globally via Bongartz completion in the silting-discrete setting, and combinatorially within a single component via a Σʳ-invariant staircase argument. Second, we prove that this bound is universally sharp by constructing an explicit L-shaped partial silting configuration of size $n+m$ for every $Λ(r,n,m)$, alongside specialized diagonal configurations. Third, we show that the configuration lattice carries a faithful action of the cyclic group Z/g where g = (r+m, n-r). We conclude by showing a structural obstruction to local determination: pairwise partial silting of consecutive objects in the lattice ordering does not imply global partial silting, due to the strict additivity of a coordinate defect along chains.