Is there an O(1) solution to find the kth-smallest element in an implicit min-heap? I know this would be an O(k log n) operation on a traditional heap, and I know there are ways to maintain Kth-smallest over a stream of inserts/deletes for constant-time access...

My question though is, when we have an implicit array representation of a min-heap, e.g.:

min_heap = [1, 2, 4, 3, 6, 5, 54, 12, 12, 34]


In this case, is it possible to infer the Kth-smallest value's index from the value of K, so you can use the array index to get the value in O(1) time?
This is not possible. If this is the case then we could extract a sorted list in linear time; $O(1)$ for each $k$th smallest element. We can also create a min heap in linear time thus we could sort in linear time! We know comparison sorting has an $\Omega(n \log n)$ lower bound worst case, so this is not possible.