P1. There is no possible world where the truthfulness of universals are falsified (i.e. no state of affairs where 2 plus 2 equals 9, or triangles have four sides)
A wordy way of saying that tautologies are necessary true. Which itself is a tautology.
P2. If universals are true across all possible worlds, they are not dependent on human cognition.
P3. If they are not dependent on human cognition, they are dependent on another cognition. Namely a universal cognition.
False. They are not dependent on any form of cognition.
P4. A universal cognition that apprehends the truth value of all necessary propositions (universals) can apprehend the truth value of all particular and contigent propositions.
Not necessarily true.
P5. Any mind that apprehends the truth value of all propositions is omniscient.
P6. An omniscient mind exists.
Since some of your premises fail, this does not follow. But even if we grant your premises, this does not follow. From P3 we can only deduce that, for any universal truth, it is dependent on some non-human cognition. You have not shown that all universal (or their contingent consequences) must be dependent on the same, singular cognition. Maybe they, collectively, are dependent on 2 non-human cognitions. Or 5, Or 326,875,923,465 cognitions. Collectively, they are omniscient, but individually they are not.
P7. Therefore God exists.
God has additional properties other than omniscience, so positing the existence of an omniscient being does not necessarily lead to God existing.