Now emulate the 4004 logic design in redstone on the 4004

That’s a really good question and I had to know, the answer is kinda neat:

Intel's chip-naming scheme at that time used a four-digit number for each component. The first digit indicated the process technology used, the second digit indicated the generic function, and the last two digits specified the sequential number in the development of that component type. Using this convention, the chips would have been known as the 1302, 1105, 1507, and 1202. Faggin felt this would obscure the fact that they formed a coherent set, and decided to name them as the "4000 family".[24] The four chips were the following:
the Intel 4001, a 256-byte 4-bit ROM;
the Intel 4002, DRAM with four 20-nibble registers (total size 40 bytes);
the Intel 4003, an I/O chip comprising a 10-bit static shift register with serial and parallel outputs; and
the Intel 4004 CPU.

Why they jumped to 4000 from 1000 tho, I didn’t see.

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.

If it is “full Linux” then the answer is clearly yes. Not very fast I suppose.