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Have you seen this stated officially somewhere or did you do your own testing? Just curious.
Cost = BaseCost × (1.4 - 0.012 × min(Skill, 100))
Magnitude determines damage. Higher mag costs more, so higher destruction skill offsets that cost.
For weapons it increases per point by 1.5%.
Where have you found it documented that the damage output of a spell scales with the skill level of the relevant school of magic? I don't think that is accurate.
It's not accurate.
This is how it has always been since 2006 with the original.
For example, the simple flare spell I started with (on Xbox) was initially like 20 something magicka. I've maxed out destruction so now it's like 4.
Destruction magick is awesome in this game because of spell-crating and easily makes it the most damaging method of gameplay possible. The real trick is not making spells so powerful you don't have the magicka to cast them.
Destruction skill does not affect the damage of a destruction spell.
https://en.uesp.net/wiki/Oblivion:Destruction
It never has, you have never seen it do so. You are misremembering or misunderstanding what you looked at.
Skill level itself does not increase damage of magic you already have. Nothing does. Only wearing armor may decrease magic efficiency, which is explained in game.
Simple as that.
Weapon skills increase damage with given weapon. Just as heavy/light armor skill level increases armor of heavy/light armor.