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

What the math gives to ML

The paper provides an explicit coarse-graining of horospherical dynamics: after restriction to right-K-invariant observables, the quotient dynamics collapse to a one-dimensional height process on an asymmetric rooted ray with q inward descendants and one outward cusp edge. Its transferable asset is the exact even-sector stationary height law and finite-shadow discrepancy structure, which can be used to design tree-routed neural modules with analytically controlled expert-depth or feature-scale occupancy. The most practical neural use is a q-ary multiscale architecture whose routing distribution is regularized toward the geometric height law and monitored through shell-discrepancy decay, testing whether balanced coverage and stable long-depth propagation emerge.