Hyperuniformity of Random Measures on Euclidean and Hyperbolic Spaces

The candidate solution correctly proves the three asymptotic lower bounds (A)–(C) for the number variance on H^n by expressing NV as the L^2-mass of the spherical transform of ball windows and using standard Harish–Chandra asymptotics for spherical functions (principal and complementary series), including the σ=0 endpoint. This matches the core technical content of the paper’s approach: the paper derives an identity for χ̂_{B_R}(λ)=b_n sinh(R)^n ω^{(n+2)}_λ(R) and uses bounds/averages of |ω^{(n+2)}_λ(ar)|^2 to show (i) limsup_R NV/Vol(B_R)>0 when the principal diffraction is non-trivial and (ii) stronger growth when the complementary diffraction is present, with an R^2 factor at the endpoint λ=0 (Theorem 8.1 ⇒ Theorem 1.4) . However, as stated, Theorem 1.4 in the paper omits a necessary non-degeneracy assumption: for the degenerate invariant law that is almost surely a fixed multiple of the invariant volume, both diffraction measures vanish and NV≡0, contradicting limsup_R NV/Vol(B_R)>0; the proof step “Since |c_{n+2}(λ)|^2>0… the right-hand side is positive” implicitly assumes η̂^{(p)}_µ≠0 (or η̂^{(c)}_µ≠0) . The candidate explicitly flags this and restricts conclusions to non-degenerate invariant laws, so the model’s solution is correct under the standard, intended non-degeneracy assumption, while the paper’s theorem statement is incomplete on this point.