An Effective Closing Lemma for Unipotent Flows

The paper’s Theorem 1.2 is proved via a precise chain (effective non-divergence to stay in the thick part, a combinatorial covering using the λ-product property, construction of many small lattice generators and an “almost invariant Lie algebras” proposition, and finally a Diophantine/non-divergence dichotomy from [16]) that yields the exact alternatives and the e^{-k/A4} gain stated. The candidate outline glosses over essential steps, incorrectly attributes the e^{-k} gain to the λ-expansion rather than the Diophantine/non-divergence theorem, relies on an unnecessary and possibly false uniqueness/pigeonhole step for γ0, and does not reconstruct the crucial Section 3 argument ensuring the subgroup obtained is proper. Hence the outline is not a correct proof of the paper’s theorem.