The Hadamard closure theorem (paper §5): the Hadamard-finite series are closed under addition, scalar multiplication, the Hadamard product, and right derivatives. Addition, scalar multiplication, and the Hadamard product follow by concatenating the witnessing tuples (the Hadamard algebra is the pointwise ring, so the evaluation is a $ℚ$-algebra homomorphism); the right derivative is handled by the coincidence together with the right-derivative automaton (the final weights pre-composed with the letter endomorphism).