Algebraic enumeration of local density improvements for Thompson's group $F$

We study local deletions from the marked-forest sets of Belk and Brown in the standard Cayley graph of Thompson's group $F$. An interval selection rule is followed by disjoint three-vertex and two-vertex deletions whose eligibility depends on the split and merge operations at a tree root. We evaluate the construction by a nine-state table and elementary first-passage equations. In particular, root-sensitive context frequencies admit closed expressions without enumerating a large product automaton. The resulting finite subgraphs give \[ \operatorname{dens}(F;\{x_0,x_1\})>3.50074529. \] All quantities used in the proof are explicit elements of $\mathbb Q(\sqrt3,\sqrt{2\sqrt3-1})$. We also give the joint root-and-children law for the coarse forest categories and prove that the optimum over all retention rules on a fixed window is an exact weighted densest-subgraph problem. Its upper bounds have elementary edge-allocation certificates. One explicit certificate proves that a rule reading the marked category, its immediate right neighbour, and any fixed number of categories to the left cannot improve the limiting density $7/2$.