Duality of Geometric Tests for Forward-Flatness

The paper proves P_k = E_{k-1}^⊥ for all k by an inductive argument that uses adapted (submersion) coordinates, an explicit normalization of bases, and two technical propositions that compute (i) the largest projectable subdistribution and (ii) the smallest codistribution invariant along the f-fibers, then closes the induction with dimension counting and annihilation checks . The candidate solution gives a clean alternative proof: it also works in adapted coordinates, but replaces the paper’s matrix-based normalization with two linear-algebra lemmas about projections/annihilators and a fiberwise span/intersection argument, then uses the pushforward/pullback maps to conclude P_{k+1} = E_k^⊥. The two approaches establish the same statement via different routes; we found no logical gaps in either, aside from a minor notation slip in the candidate solution (using f_* instead of f^* for covectors).