Hyperuniformity in Regular Trees

The paper proves a non-geometric hyperuniformity theorem (Theorem 1.3) for all invariant locally square-integrable point processes on (q+1)-regular trees: either complementary-series mass forces super-volume growth, or principal-series mass forces a positive Cesàro lower bound, which implies limsup NV*μ(r)/|Br| > 0. This is established via Proposition 5.1 and the spherical transform of balls, and the proof explicitly splits by spectral type, covering all nontrivial cases . The candidate reproduces correct kernel identities and the two sharp lower bounds for atoms at 0, τ/2 and on the imaginary arcs, and a valid Cesàro argument for interior real spectrum, but incorrectly flags as a remaining possibility the case where the diffraction measure is supported entirely at the two removed boundary atoms {i/2, τ/2+i/2} (which would make NV* vanish). The paper’s argument shows one necessarily has nonzero principal or complementary diffraction in the nontrivial cases considered, and then concludes the desired positivity of limsup; see the NV* definition and spectral decomposition, the main theorem statement, and its proof line selecting ε with σ(p)μ((ε, τ/2−ε)) > 0 . Hence the model’s “incomplete” conclusion is unwarranted, while its partial calculations match the paper’s bounds.