The comic uses an old story mentioned by Momiji about two woodcutters. The first dropped his ax into a spring, and when the spirit of the spring asked which one he had dropped, he got a gold ax and a silver ax for answering honestly. The second woodcutter, hearing this story, lied out of greed and so got nothing.

It's the fable of Mercury and the Woodman. We even have a tag for it, which is honest_axe because it's shorter, and because it's what tvtropes uses, I guess.

Thanks for pointing that out. I'd never have guessed there was such a tag.