Doubtful with all the cutting fluid and all that in there. I'm not a smith so maybe someone in here will tell me I'm wrong but I don't think it would work. It'd be cool though!
If there was enough heat to melt that down, most of the impurities from cutting fluids and otherwise would burn off in the process.
I saw a guy take the filings, sanded debris, saw kerf material, and other "lost" material and make a bloom from that in a homemade foundry stack.
From there he took the bloom and essentially forge welded it and made a small ingot from it. He wouldn't trust it to make something for a customer, but he was happy he could achieve it. It now sits on his mantle.
As far as the shavings above, they are
Fairly pure/clean, so I would like to think it would be possible. I have not looked into the temps for forging 3V, or if forge welding is even suggested.
How the end result would end up, I am not entirely sure, but I wouldn't sell it. Though it would make for a nice conversation piece, or a small knife made for "dinners and special occasions" for Nathan himself to use.
I do not know what Nathan has for forging, but he would need access to a few hammers (2-7 lbs), an anvil, a nice forge, and tongs at the very least. A powered hydraulic press would work once he is through the hand work.
All of that big said, maybe he may come back to this idea once he has a little free time. He seems to be about as busy as he can stand making himself.
Lol