Chow and Rashevskii meet Sobolev

The candidate solution proves exactly Theorem 3.4 of the paper: (i) for every Borel B of positive Lebesgue measure, L^d(R^d \ E(B)) = 0, and (ii) the ensuing approximate controllability statement. Its proof strategy mirrors the paper’s chain: (1) invariance of E(B) under each flow (paper’s Lemma 4.1 shows etX*1_E = 1_E a.e., leveraging absolute continuity of pushforwards) ; (2) a stationary transport identity for 1_E (paper’s Lemma 4.3 derives div(X 1_E) − 1_E div X = 0 via a pushforward expansion, not renormalization) ; (3) closure under Lie brackets (see the use of Lemma 4.9 inside the proof of Theorem 3.4) ; (4) invertible Hörmander frame implies ∇1_E = 0 (paper’s Lemma 4.11) ; (5) conclude (i)–(ii) (paper’s “Proof of Theorem 3.4”) . The model’s Step 2 appeals to renormalized transport theory for regular Lagrangian flows (RLFs) to get X_i u = 0, whereas the paper proves the distributional transport identity directly using Definition 2.4(v) and Appendix A (Lemma A.2) without invoking uniqueness of renormalized solutions . Step 1 in the model tacitly uses that flow pushforwards are absolutely continuous, which the paper explicitly assumes/derives to justify a.e. invariance (Lemma 4.1 + Lemma A.1) . Step 4 in the model compresses the technical L^p–W^{1,s} argument that the paper carries out carefully in Lemma 4.11 with the threshold s > (q^* ∧ r)' from the Sobolev–Hörmander condition . Net: both arguments are correct and follow the same architecture; the model’s proof is slightly more specialized (RLF/renormalization), while the paper works under the broader “flow generating” framework it defines.