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.
@ future unknowable events uncorrelated with the recent past
yes, and true for prop too. the deciding factor is in hopping prop/pplns, that if the round is better than average u get the reward, but if it's below average u still make more than the normalo. i can go into more details if u wish, but first i will eat.
No, pure proportional payments are not strictly a function of future unknowable events. In pure proportional there is also one important factor which affects payouts and is already known: how long ago the last block was found. Nothing before that block is paid. Thus a hopper knows when an extra-valuable participation period is happening (early), or when an less-valuable participation period is happening (late).
This is very different from PPLNS. There is no 'round' that resets when random generations happen, affecting how future shares will be paid. Every single share is in its own 'round' that last lasts exactly N shares into the future. There are some periods that in retrospect turn out to be better, but it is impossible to predict those periods in advance.
The next N shares to be submitted from all miners will completely determine what a share will be paid. Those next N shares are all of unknowable (but equal on average) value. Their value has no relation to the timing of past generation fees. Hence, there is a consistent equal average return to participating.