Karthus has one of, if not, fastest jungle clear in the game. Because karthus is heavily realiant on farming in the early game, you will not be ganking or have first drag priority most of the time, or scuttle priority. Most matchups are terrible, counterjungling is a big issue playing karthus jungle. Approximately 2% of people are taking first strike on karthus compared to dark harvest. While dark harvest helps later, first strike lets you ramp your scaling in the early to mid game. You won't be proccing first strike at all besides this game is a rush to lvl 6. You want to full clear your camps every game and watch intently for invades. Once you get lvl 6, even if nobody is killable, continue to spam it. Whenever it's off cooldown, you should look to use it for the amount of gold you get. People don't understand that will liandries burn will allow you to get assists easier, even if it's not low, because the damage has lingered for so long. I'm not done adding to this guide. I don't even know if I can edit it ater publishing, but I hope you enjoy this. Axiom arc is an essential item that differentiates this build from typical builds, as well as first strike being very unpopular. The item is indeed trash because we're not getting any stats we want besides cdr and the axiom arc ability. We're actually paying for wasted stats. Nevertheless, i've had lots of success and look to update this build in future with more information.

Okay okay, let's be real here. Axiom arc is complete garbage. You are paying an expensive 3000g for the wasted stats (lethality, ad). You are buying this item for the sole purpose of its ability haste and passive. 98% of the time, a different ap item will help you scale better by making clears faster, ultilizing every stat, doesn't throw you off your build path. With faster clears, you use your time to focus on positioning for ganks and teamfights. If laners can repeatedly solo kill enemies, and you face no risk of jungle invades, then it's trash. Essentially any other item will provide more damage in skirmishes.

But... will anyone play devi's advocate for the item?

I believe this playstyle barely suits the use of the item. For one, you have first strike. You want to be able to proc first strike as much as possible, which is why you need as much ability haste as possible, even if it means losing ap in the process. You need to using the item's passive to the best of the ability.