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,

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