Actually, in practice, Ryze's whole combo takes less than three seconds to cast, so Trinity Force procs can be added (noting the 2 second internal cooldown) to the rotation so that you're maintaining damage per second while actually doing something all teamfight long. It works, trust me. You have to get timing down really well and have a Swiss clock for a brain, but it's viable. More so than the well-I've-used-my-burst-time-to-get-in-a-corner-and-try-not-to-die Mana/AP Ryze.
And feel free to get E. I personally hate the spell: 12 MPen for 5 seconds (Rank 1) sort of sucks; it has **** base damage and no mana scaling. AD Ryze doesn't need MPen at level 5 all that much. Take it if you want, for his passive, but it's underwhelming (to say the least). At that level, a woven Sheen proc is more damaging than casting Rank 1 Flux (50 damage before MR?!), potentially even if it meant the difference between an extra Overload, and it would be idiotic to get a point more than one at low levels.

and to make use of your passive you have to ignore Trinity Procs, and to make use of Trinity Procs you have to ignore your passive :/

one thing I REALLY REALLY don't like: not picking up level 1 in E untill the end. It lowers MR, deals a bit of damage, lowers cooldowns of all your other spells when cast, takes waves easily with your ultimate...

What's not to like?

I haven't published this yet, so there will eventually be a guide attached to this. I'll also fine-tune the items, runes, etc. This is just the build I currently go when I'm testing it out.
Anyway, it does work! Ryze's scaling with AP is so poor that I've removed it from the equation entirely! A good AD/Mana Ryze player can mix attacks in between spell casts, which is useful for the Trinity Force when it's off internal cooldown and makes up for a lack of attack speed. Between a Banshee's Veil, Frozen Heart and Manamune, you have plenty of mana. (More than some non-AD builds, actually; what mana does Rabadon's and a Void Staff give?)
Anyway, I'll stop here. If you like it, tell me what went right so I can add it to the guide.

Thank you!
Pranzatelli

At first I thought this looked terrible, but then I thought about it and nothing looks too bad about it. Ill have to try it out when I have time. Everything actually synergizes very well. The hex tech gunblade, even though it doesn't give mana, could give you the life steal and spell vamp you need to win a dragged out fight.
Ryze is a carry that tries to take things down In one or two bursts near the end of the match. You might have a little more less mana (which is converted to magic damage bla bla bla...) but it seems like the ad may make up for that. Plus all the added slow perks may be a large help.
keep tryin' to think of something bad to critic this build with but I cant. Usually the sixth item in ryzes build is hard to pick anyways. he doesn't seem to lose much besides small amounts of magic damage.
I guess an ad build of ryze would make his ultimate a little bit less useful, as you only gain spell vamp, Not life steal, and his ult snyergizes with all his spells more and gives nothing for any kind of ad. Also, not having the damage from the Rod of ages and rabadons
kinda sucks.
however well ad/mana ryze hybrid works I think you made this nearly perfectly, I couldn't improve it.
Edit: I actually think Your Runes and masteries could do some work, doesnt make sense to full out use armor pen runes when your a hybrid, the enemy team isnt going to focus as much armor because they won't suspect it.