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

Let $$F={\mathbb{F}}_q(\text{\hspace{0.17em}⁣}(t^{-1})\text{\hspace{0.17em}⁣}),G={\mathrm{S}\mathrm{L}}_2(F),\mathrm{\Gamma }={\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_q[t]),X=\mathrm{\Gamma }\mathrm{\backslash }G,$$ and let $K={\mathrm{S}\mathrm{L}}_2(\mathcal{O})$, where $\mathcal{O}={\mathbb{F}}_q[\text{\hspace{0.17em}⁣}[t^{-1}]\text{\hspace{0.17em}⁣}]$. We study right $K$-spherical averages along the upper unipotent subgroup, the horospherical subgroup associated with the standard cusp, on the Nagao lattice quotient. The basic observation is that the $K$-spherical projection converts two natural dynamical families - expanding translates of compact unipotent orbits and cusp-adapted truncations of dense unipotent orbits - into the same rooted descendant problem on the Bruhat–Tits tree. In the even bipartite sector the limiting height law is the explicit probability measure $${\rho }^{\mathrm{e}\mathrm{v}}(0)=\frac{q-1}{q},{\rho }^{\mathrm{e}\mathrm{v}}(2m)=(q^2-1)q^{-2m-1}(m\ge 1)\mathrm{.}$$ We prove an exact discrepancy formula: in the backward state the error is a pure top-shell term minus a missing tail, while in the forward state the error is a first-turn weighted sum of backward errors. These formulas give quantitative $K$-spherical equidistribution for expanding translates of compact $U$-orbits and for dense-orbit truncations. For compactly supported $K$-spherical observables in the expanding translates of compact orbits, the discrepancy is eventually exactly zero. In the dense case the rate is controlled by the continued-fraction expansion of the boundary point attached to the orbit.