You read that right, that beautiful good looking red axe that is with woodie in the constant is fake and I will show evidences that supports it.
You can see in walter's vídeo that he was looking for evidences about the werebeaver, and went to a cabin in the woods:
Starting here we can already see the real Lucy the Axe there, and figure out who the owner of the cabin is: woodie.
Why would Woodie leave his favourite axe alone, outside from the cabin ?
The explanation is very simple:
You can see in this pictures (and on the first one, with the outside view of the cabin) that there are several spider webs on the cabin, indicating that some time has passed since anyone would show up on the cabin. And even Lucy the Axe has some moss on it, showing that nobody, not even woodie, has used it for a long time.
Why? because Woodie was already taken by Them to the constant.
Woodie was taken alone to the constant, leaving the real Lucy the Axe in the real world.
He has beeing hanging out, chopping with a fake copy of his axe this whole time!
Cheating on his axe, with his axe's fake evil twin
Edit (more evidence)
Walter came to the constant after the moon breaking incident, and Woodie was in the constant before it, since he only discovered new powers after the moon broke and fell to the ocean. Proving that Woodie was taken before Walter, and proving that real Lucy the Axe is alone in the real world