Do enlightenment orb games bank?

This time around, the enlightenment orbs have a set of 5 bars beneath them, apparently indicating the number of games you've played. When the "claim" button is grayed out (while waiting for a new orb), it says that we can play "up to 5 games" to increase the progress.

My question is: Can you over-play one day and have the extra games carry over to the next? So if I play 8 games in one day, and then claim my orb, are there 3 games already counted toward my next orb? And if this is the way it works, can this go indefinitely? Can I play 20 games, and have the next 4 orbs completely "charged"?