Building the most accurate DIY CNC lathe in the world [video]

For folks who are not familiar w/ machine shops, the lathe is a fundamental tool in a shop, and is the only tool in a shop which can replicate itself --- there is even a book series which uses this conceit, the "Gingery Books":

https://gingerybookstore.com/

where Vol. 1 has one setting up an aluminum casting foundry in one's backyard, and Vol. 2 has one using it to make a lathe which is then used to either improve itself or make a better lathe, then one uses it to make the balance of the tools in a machine shop.

A lathe can't actually replicate itself completely. Specifically, a lathe can only make ways smaller than its own cross slide's stroke. It would also be impossible to make a typical lathe bed on a lathe, though you theoretically could design an unconventional lathe bed that is possible to make on a lathe, even if grossly impractical.

The real starting point for machine precision is rubbing 3 granite plates together.

Lathes can certainly make cylinders, and a tube-based lathe bed is not a stretch.

A lathe can't replicate its own assembly, of course. It can't seat the spindle in the constraint bearings, for instance.

A CNC (without the word lathe) can make most of itself, and possible all. Nope: certainly all, if two of its dimensions fit within its work volume.

Lathes can make cylinders, but not of unlimited length in one setup so they lose some accuracy making cylinders longer than their carriage travel. And their beds are by necessity longer than their carriage travel, since the carriage rides along the bed and isn't infinitely thin. They also can't make things like motor stator laminations, and you definitely need a motor for a replica of a motorized lathe. So lathes can't replicate themselves exactly.

Milling machines are also just lathes with a different orientation, an extra travel axis, and a motor optimized for higher speeds & lower torques, it's possible (and reasonably common) to use a mill like a lathe or a lathe like a mill in many cases. So "only machine" part is also a stretch.