A series is recognisable by a polynomial automaton iff its reversal is Hadamard-finite (paper appendix). Only-if: the component series $g_i(w) = π_i(q_I · w^R)$ generate a Hadamard algebra closed under left derivatives containing $f^R = F(g_1, …, g_k)$. If: from a left-derivative- closed Hadamard algebra $ℚ{g_1, …, g_k}$ containing $f^R = F(g_1, …, g_k)$, rebuild the polynomial automaton with initial state $(g_1(ε), …, g_k(ε))$ and transitions given by the left-derivative polynomials; an induction on the word shows $g_i(w) = π_i(q_I · w^R)$, hence $f = ⟦B⟧$.