The only information that any given weighing (of equal number of coins) reveals to us:
- If pans are equal, either both or none of the pans have fake coins.
- If pans are unequal, either one or both of the pans have fake coins.
We also notice that finding actual weights of coins in terms of other coins is pointless, since we might end up with equations like $Fake\ coin\ 1 \geq 200 \times Fake\ coin\ 2$. The only that seems of practical importance is sorting the coins relatively (in ascending/descending order).
Let's make the assumption that equal pans always means no fake coins in the given scenario. (There exist no two sets of fake coins with equal weights.) However, we will not know this. So even if we stumble upon two equal pans, we will repeatedly use different weighings till we conclude that there is no fake coin. This can take a long time. This also means that the fact as to which pan is greater is of no importance, once we know we have unequal pans.
According to worst case, we will never get equal pans unless this is logically implied from previous facts (in which case, weighing them is pointless).
Hence we can conclude that we will never get equal pans in our perfect strategy under the worst case.
If pans are unequal, we gain no new information.
Hence, we conclude that it most be possible to achieve a state where it is logically implied that a particular coin is genuine, without us ever getting equal pans in our weighings.
This only seems possible by doing $70$ weighings, as others have said.