TL;DR
I booted Debian Linux on a 4-bit intel microprocessor from 1971 - the first microprocessor in the world - the 4004. It is not fast, but it is a real Linux kernel with a Debian rootfs on a real board whose only CPU is a real intel 4004 from the 1970s. The video is sped up at variable rates to demonstrate this without boring you. The clock and calendar in the video are accurate. A constant-rate video is linked below.
If it’s the first one when why the fuck is it called the 4004 and not just the 1?!
How is it “full Linux” when Linux has always required an integrated MMU (which is why it’s never supported anything less than a 386)? I mean, yes, you can modify it to run on more primitive chips at the expense of having proper virtual memory support, but if you do that I don’t think it counts as “full Linux” anymore!
If you read the article, it is indeed full Linux because the 4004 is running a MIPS emulator that provides the necessary memory management features. Pretty much all of the “run Linux on some old chip incapable of running Linux” projects achieve it via emulating a more featured architecture that Linux supports, not by somehow compiling Linux to natively run on a 4 bit, MMU-less architecture.
But does it run Doom?
If it is “full Linux” then the answer is clearly yes. Not very fast I suppose.
At first I was not very impressed. Then I read that that 4004 does not have any AND, OR, or XOR instructions. Yikes.