A Dual Geometric Test for Forward-Flatness

The paper rigorously defines the codistribution sequence (Algorithm 11), proves well-posedness via Proposition 6 and Corollary 10, establishes nestedness (Proposition 13), integrability (Proposition 14), and the main equivalence (Theorem 17) through Theorem 15 and Lemma 16, all self-contained and consistent with the shift-operator framework, including the exact backward-shift formula for 1-forms δ−1(ωi(f) dfi) = ωi(x) dxi (equation (3)) . By contrast, the candidate solution relies on an incorrect identity Tf[v,w] = [Tf v,Tf w] to claim involutivity of W = ker Tf; this identity is not valid for a general submersion. The paper correctly handles involutivity by passing to adapted coordinates (θ = f(x,u), ξ = h(x,u)), where span{df}⊥ = span{∂ξ} is manifestly involutive . In addition, the model’s integrability argument is only a sketch, whereas the paper supplies a complete wedge-form proof for both the ω- and ρ-blocks (equations (14)–(15)) . The paper’s equivalence proof is also sharper, linking dim-drop to a triangular decomposition (Theorem 15) and then iterating (Lemma 16) to obtain Theorem 17 (forward-flatness iff the sequence terminates at P̄ = 0) .