@shamathana

Of course you can contrive scenarios where certain entry/exit-patterns get more; payments do have a random component and so some patterns will result in more or less in the short run. (Sometimes betting on red at the roulette wheel results in gains over the next few spins.)

But there are an equal prevalence of scenarios where those same entry/exit-patterns, using information available at the moment of participate/don't-participate decisionmaking, get less.

The 'proof' you request is trivial: PPLNS payments are completely determined by a formula whose only inputs are future unknowable events uncorrelated with the recent past. There is thus no way to look at the history of a pool and decide, 'now if I enter(leave) I will get more(less)'. At every moment, the prospective value of participating is equal. Hence, no hopping by any rational miner.

Even a miner who is able to predict with certainty an upcoming inrush of new participants, or outrush, doesn't gain any net gain in expected-return. More co-miners mean a block will be hit quicker (by wall-clock time), but also old shares will expire quicker (by wall-clock time). Knowing an influx or outflux is imminent just means variance over time will fall or rise respectively, as is the case with any pool expanding or contracting. Not that expected return will rise or fall.