I think what you want in checking -- if your circumstances allow for this kind of money-shuffling, at the least -- is more along the lines of 4 weeks of expenses, not 4 weeks of income, which would (hopefully!) be considerably higher. I'm not sure I'd call it a "large buffer," but even if you're not in the "shove absolutely everything you can into index funds and live like a pauper despite the fact that you're making $250K+ a year so you can retire at 35" camp, the more money you have in higher-return places, the better, so you probably don't want it in the checking account. (And probably don't want a lot in a low-interest savings account.)
This seems like a silly thing.