I am going to calculate it, but this goes of base prices so it isn't 100 percent accurate(especially because 10 percent movement speed becomes way more viable if you have more movement speed.)
- for the ap:
amp tome gives 20 ap costing 435. 1 ap cost 21.75 gold.
ludens ekko has 100 ap, so an ap value of 2175 gold.
now for the 10 percent movement speed:
aether wisp is 5 percent movement speed + 30 ability power for 850 gold. 30 ap is 625.5 gold of ap. so 5 percent movement speeds is worth 225.5 gold.
ludens ekko has 451 gold in movement speed.
2175+
0451
2626 gold of stats alone.
Now the passive. The gives 100 base damage on your spell, which is almost impossible to calculate 100 percent how much it would be worth but, lets go with the following redenation. 1 ap item gives 4 spells a buff in damage, ludens gives 1 spell a buff. the bonus ap proc of ludens should have the value of around 1/4th of the damage of normal ap.
100/4 = 25 ap
25* 4 =543.75
this takes up the gold value to: 3169.75(lets say 3169 just to be save)
ludens ekko without ap has a gold value of 3169 by default. so you need some more ap to make it worth.
ap scales for 10 percent effectiveness ludens so it makes ap 10 times less viable.
so the calculation for the gold value that ludens gains per ap =
21.75/10/4=0.54375 gold extra value for ludens per one ap.
you need 31 gold to make the item viable.
31/0.54375=57.0114942529 ap.
so you need 57 ap to make ludens gold efficient according to my logic.