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As far as I can tell, a sorcerer in padded armor has to apply a check penalty to attack rolls and all DEX and STR based skill checks… but padded armor has check penalty 0. Does that mean that a sorcerer in that armor doesn't suffer any other penalties, even if he is not proficient with it? Just a 5% failure check on spells?

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So, a couple things.

An arcane spellcaster suffers from the listed spell failure chance while casting spells - but only those that have somatic components; see Arcane Spells and Armor in the SRD.

Anyone wearing armor takes the Armor Check Penalty to all STR and DEX based skill rolls. However, nonproficient characters (like your sorcerer) also take that penalty to attack rolls and all skill checks involving moving, see Armor Check Penalty and Light Armor Proficiency in the SRD. But yes, in this case the penalty is 0. You could go up to leather with an ACP of 0, though you do go to 10% spell failure.

Of course, if you choose a sorcerer build that has a lot of non-somatic spells and spells that don't require hit rolls, you're good with even heavier armor. This is likely to be a very low level workaround as that armor won't stack with Mage Armor, Bracers of Armor, or all the other tricks most mage types use to have a decent AC.

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