Area B complete. This level is about six seconds faster by my
extremely rough measurements, but three seconds slower on the boss.
The previous run did this level second-to-last (and all of them in a strange order). For some reason, the punching power was doubled by that point, and so the boss was defeated faster.
(later...)
I have figured out how to upgrade punching strength. Ducking down on the right (P) platforms with 100 gold will give you a more powerful punch. I'm not sure how many times this can be upgraded, but it means I must redo the first level to see if it's faster, which I'm almost positive it is.
If anyone knows how many times you can upgrade your punch, that'd be cool.
Edit: It's one. Lame. Also, two hits and you lose your upgrade (indicated by your vest color, just like in real life). So it may not save time as much as I thought.