POLYNOMIAL EFFECTIVE DENSITY IN QUOTIENTS OF H3 AND H2 × H2

The paper’s main theorem exactly states the quantitative dichotomy for P-orbits in X = G/Γ (G = SL2(C) or SL2(R)×SL2(R)) with the parameters A, κ1, C1 and the alternative involving a nearby periodic H-orbit, and it gives a complete proof based on non-divergence, a projection/incidence step, spectral gap for the ambient space, and an effective closing lemma. By contrast, the model’s solution hinges on a global quantitative mean/pointwise ergodic theorem for the H-action along well-rounded P-averages with a polynomial rate uniform away from small periodic orbits; such a tool does not exist in the requisite generality here and is precisely the gap the paper fills by a different route. The model also conflates a K-thickening in G with sets lying inside H, and it normalizes exponents post hoc. Hence the paper’s argument is correct and novel, while the model’s proof relies on unproven assumptions.