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But that may be unnecessarily confusing. I'll get some more input and then modify accordingly; thanks for the note!
So my initial thoughts are that Malignance is worth rushing as a first item. Also after my test game with AP Kog (end of the video), Rylais will likely return to the order. It's just so hard to land multiple hits on a row if you don't have it!
Or you could just swap out the resolve tree for the precision tree because adding a negligible amount of armor and HP doesn't actually help kog with being long range artillery, or even a mid-range mage. May as well have more mana and an execution damage amp instead.
1. I tested Phase Rush quite a lot prior, and found it underwhelming. I couldn't get it to activate when I needed it to (except like once or twice in the laning phase), and later on in the match when it was activating, it usually made no difference in the overall outcome of a fight. It wasn't useless, but generally wasn't making a meaningful impact.
2. Kog is only mana hungry if you exceed the barrage limit regularly. Otherwise, the amount of mana with biscuits and Manamune (which is now in the build order) is enough as long as the reserve bullets aren't dipped into regularly. Getting more mana options will only allow like, 1-3 extra shots later in the match because of the sheer amount of mana it takes to dip into the reserve.
3. Clarity is an option that can be picked from Unsealed Spellbook
4. I find the opposite to be true, that the amount of mana and damage added is largely negligible enough that I prefer having 15% more effective HP instead.
I didn't read your guide thoroughly because I am not planning to play this champion whatsoever, but again, I know your other guides as I have commented, so I know that this will be useful for my friends and probably will share it with them :>
Granted, the lack of access to chaining is a big issue, and sometimes it's better to go for rylai's earlier due to the giant's belt buildup, and you'll still want to grab it second. But I think ludens synergizes perfectly with the way the ult has been changed to work.
My full build comes down to usually Sorc shoes, Ludens, Rylais, Void Staff, Archangels, Liandry/Defense/Nashor's Tooth.
With Full penetration quints to make up for Liandry's, rather than MS, since Ludens handles that.
As for damage, while guise/liandrys/voidstaff gives more penetration and/or % HP damage, usually the targets you want to snipe with AP kog, especially in the midgame around the lvl 11 power spike, are pretty squishy. Ludens gives a flat boost to damage earlier on, so that synergizes with the spike. In addition, ADCs and stuff don't tend to build tremendous MR often, and certainly not early, thus negating the benefits of MPen, since with Sorc Shoes, masteries, and Pen runes you'll be hitting 30 Pen without any other items.
Most enemies run some kinds of MR glyphs, and Liandrys still does better gold/dmg with the ultimate even without having to rely much on its magic penetration; the burn is just that good according to the math.