Homogeneity of arithmetic quantum limits for hyperbolic 4-manifolds

The paper’s Theorem 2 states exactly what the candidate claims: if Hecke eigenfunction probability measures on X = Γ\SO(1,4) converge to an A-invariant limit μ, then μ is proportional to a convex combination of homogeneous measures; each non-uniform ergodic component is supported on an orbit of a subgroup isomorphic to SO(1,3) with the stated R∩Q̄-structure (y ∈ G(R∩Q̄), m ∈ M, and mH̃m^{-1} defined over R∩Q̄) . The paper verifies the three dynamical inputs—(I) recurrence (Section 4, Lemmas 18–19) , (II) positive entropy (Section 5, via Theorem 14 and smallness of AM) , and (III) few exceptional returns (Section 6) —and then applies the restated Einsiedler–Lindenstrauss classification (Theorem 6) to deduce homogeneity and real rank-one structure of the stabilizer , concluding in Section 7 with the SO(1,3) alternatives . The candidate’s outline follows the same architecture: adelic model of X, verification of (I)–(III) via Hecke amplification/Marshall’s small-subgroup technology, application of the E–L classification, and elimination down to G or H̃ ≅ SO(1,3), together with the adelic R∩Q̄ refinements (Margulis–Tomanov). Minor inaccuracies in the candidate’s write-up (e.g., an imprecise description of M, and not explicitly normalizing a possibly sub-probability limit to apply Theorem 6) do not affect the main conclusion and are addressed in the paper’s precise statements (notably the ‘proportional to’ phrasing to handle possible escape of mass) , .