Does Arcane Surge + Lifetap work?

Path of Building is saying I would get more dps with my spell, but I'm using Lifetap. Arcane Surge says "Each supported spell will track how much mana you spend on it" but I'm spending life...

So is PoB wrong or would the wording of Arcane Surge change to "Life" because of Lifetap?

POB assumes you can trigger things even if you can't.

you must spend mana to trigger arcane surge.