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Can an equal emergence occur again, and as a unavoidable fact of infinites result in endless occurrences, given infinite chance in an eternal cosmos?

I say no. If the (falsely inferred as) 1/infinitey chance happens once it is not really unusual. The fault is in thinking this is like any probability solving problem. There is a 100% chance that a thing you see came to exist as itself in everyway. Just as its 100% chance that the cowboys did beat the bills in 2 past bowls. It is guaranteed sure that the presents exact qualities are what they are now and were emerging as at all times in result of any numbers of events and infinitive potentialities.

Now a second equal event that happens after or in parallel with the original is a statistical non event. The chance of it never being is the % 99.999... Repeating 9 decimal. Which is equal to 100% not possible.

That's easy to show as equal by taking 1 divide by three for .3333 repeating 3 decimal. Multiply by 3 to get .9999...etc three times a third is a whole.

So even in an infinite universe there is originality in structures and as a consequence nothing can be non unique in some minuet way.