$K$-spherical horospherical averages on the Nagao quotient: tree combinatorics and exact discrepancy

Let $q$ be a prime power, $F=\mathbb F_q(\!(t^{-1})\!)$, $G=\mathrm{SL}_2(F)$, $Γ=\mathrm{SL}_2(\mathbb F_q[t])$, and $K=\mathrm{SL}_2(\mathbb F_q[\![t^{-1}]\!])$, and let $U<G$ be the upper unipotent subgroup. We study right $K$-spherical averages along $U$ on $X=Γ\backslash G$. Expanding translate