Hmm, so it is.

I still don't understand why \$$\\{ \mathbb{C},+,0, \le, ||z|| \\} \$$ doesn't count as a commutative monoidal poset.

To me, and I'm sorry if I'm skipping ahead, the modulus is a homomorphism from \$$\\{ \mathbb{C},+,0, \le, ||z|| \\} \$$ to \$$\\{ \mathbb{R}_+,+,0, \le, r \\} \$$, ie,

$$||z|| :: \mathbb{C} \rightarrow \mathbb{R}_+ .$$