Well, AP isn't useless on ryze, it's just less useful than most casters. My opinion is that you're seriously undervaluing AP in your calculations.

Let's take a typical combo from ryze as the standard for his damage:

Q --> W --> Q --> E --> Q

assume the E hits twice, since it bounces. (Maybe it hits more than twice, maybe it hits once... two hits seems like a workable average and in any case it's not a huge part of his damage so this assumption won't affect the numbers too much)

Each point of ability power will increase the damage of this combo by:

.2 * 3 (Q) + .6 (W) + .35 * 2 (E), or 1.9 damage per ap.

in comparison, each point of mana will increase it by:

(.08 * 3 (Q) + .05 (W) ) * 1.05 (expanded mind), or .3045 per point.

take 1.9 and divide it by .3045...

you can see that AP adds roughly the same damage as 6.24 mana. That means that the 80 AP from rod of ages or whatever else you buy is worth roughly the same, damage wise, as 500 points of mana. Mana is generally worth 2g per point, so 80 ap is worth 1000g, even for ryze. Less than most casters but nothing to sneeze at.

(AP also adds the additional benefit of increasing ryze's damage to buildings. And mana adds some health regen if you have that talent in defensive, and is also helpful in the extremely unlikely even that ryze runs out of mana late-game.)

After he's built his core items (Frozen heart, banshee's, tear, boots), and is level 18, he should have the following amount of mana:

1465 base (including passive bonus from ult)

+500 from frozen heart

+350 from tear

+375 from banshee's

up to 1000 from tear stacks, but really it's probably not higher than 600 since this takes so long to stack up. So lets just say 600 from tear stacks.

That gives him 3290 mana.

Let's say he upgraded the tear to an archangel's staff. That would give him 50 more mana, + 45 flat ap, + 3% of his mana as ap... which comes out to 100 AP (3340 *.03)

That gives him 145 ap + 50 mana for 1860 gold (the cost to upgrade tear to archangel's ). since one point of ap is worth 6.24 mana, that's worth 904.8 mana worth of ap, and 50 mana worth of mana. If we value mana at 2g per point, and mana regen at nothing, archangel's upgrade is worth 1909.6g, which is reasonably efficient, and slightly more efficient than stacking sapphire crystals. It will only get more efficient as you get more mana, so IMO you should upgrade to archangel's after your 4th or 5th item, rather than building it after your 6th as most guides recommend, unless you absolutely need a tank item at that point.

Compare to rod of ages... assume 1 health point = 1 mana point (may be bad assumption, lol), and you get the equivalent of 1349 mana at 0 minutes, and 1854.2 mana at 10 minutes, for a gold efficiency of 2.249g per mana at 0 min, and 1.637g per mana at 10 min.

They're pretty similar. Seems like if you want to build more DPS, you should upgrade to Archangel's as your next item, but if you want more beefyness you should get rod of ages.

Next up: abyssal scepter.

if you value magic resist at 15g per point, the mr on abyssal is worth 855g points.

the 70 ap is worth 436.8 mana, or 873.6g. Since the cost is 2650, that means the aura needs to be worth about 921.4g...

But what is the aura worth?

Consider:

If ryze has 3290 *1.05 mana and 0 ap, Ryze's combo, Q -> W -> Q -> E -> Q, discussed earlier, should be doing the following damage: (140 + 3290*1.05*.08 )*3 [Q] + (200 + 3290*1.05*.05) [W]+ 130*2 [E])

which comes out to 1881.8 damage by my calculations. as already discussed, each point of mana will increase this damage by .3045 before magic resist is factored in.

How much magic resist reduction will increase it is harder to say. Assume that the opponent has an average of 70 magic resist already accounting for sorc shoes, ryze's e, runes, etc (so probably upwards of 100 magic resist base):

Each point of mana will now add only .1791 damage (.3045 / 1.7)

and the damage from his combo is now 1106.94 damage. (1881.8 / 1.7)

but let's say the opponent had 50 magic resist instead (reduced by a further 20)

it comes out to 1254.53 damage (1881.8 / 1.5)

So the -20 magic resist aura increased his damage by 147.59

To increase his damage by that much through mana, he'd need 824 mana. if we value mana at 2g, the aura on abyssal scepter is worth 1648 gold (heavily dependent on how high the opponent's magic resist is... the lower the better).

That brings the total value of abyssal scepter to being worth way more than you payed for it, by my calculations. with 873.6g worth of ap, 855g worth of ap and 1648g worth of aura, or 3376g worth total, for 2650g.

Actually, abyssal scepter is a chronically undervalued item, so it doesn't surprise me that it's underrated on ryze also.

Heck, after doing that calculation i kindof want to buy haunting guise.

eyeballing it void staff is also a good option if they're stacking stupid amounts of magic resist but i'm too lazy to do any more math right now.

Honestly though, if you're getting abyssal scepter AND banshee's on the same build, with frozen heart, your armor+mr is going to be stupidly high with hardly any buff to your hp. So you'd probably want rod of ages as your 5th or 6th item regardless of any earlier math.. (i'm probably greatly undervaluing health in my above calculations, but i'm too lazy to do any more math right n ow)

....

So i'd say something like this would be a good ryze build:

CORE:

Tear

Sorc shoes

Frozen Heart

then in no particular order:

ROA (if you expect the game to go on a while)

Haunting guise (if you expect the game to end quickly.)

Banshees (overrated. It's worth 2250g + the bubble. pick it up if you need magic resist and want the bubble but i don't see how this is core and it's simpler to just buy a negatron or two then upgrade later if you just need MR.)

Abyssal (if nobody else took it)

Archangel's (pick up once you have like 4k+ mana or run out of inventory spots)

Get void staff if they're all stacking stupid amounts of magic resist (150+) which seems extremely unlikely to me.

As for will of the ancients, it would be nice if someone besides ryze would buy this, all you need is the aura...