The Hadamard anti-derivative closure (paper §5): over a finite alphabet, if $g$ is a left anti-derivative of a tuple $f$ of Hadamard-finite series ($leftDeriv a g = f a$ for all $a$), then $g$ is Hadamard-finite. The witnessing tuple for $g$ is $g$ itself followed by the concatenation of the witnessing tuples of the $f a$'s: $g$ is trivially a Hadamard polynomial in a tuple containing it (its own variable), and the tuple is closed under left derivatives because $leftDeriv a g = f a$ is a Hadamard polynomial in the (closed) witnessing tuple for $f a$, and each $f a$'s witnessing tuple is itself closed. The finiteness of the alphabet is what makes the combined tuple finite.