Muzzer, try to understand something. I bought this 4th axis rotary chuck and it comes with a belt giving 1:6 reduction. It is NOT something which is a choice, the belt is a necessary thing in it, not an "expensive reduction gear" which I design. In fact, direct driving would be impossible because there is a hole through the rear, so unless you want to give that up, you MUST have a belt or some sort of arrangement to drive the chuck. Besides, if we take your own torque chart as example and my actual 1:6 gearing then you can see that at 2280rpm that motor should have approximately 10 oz-in (units make no difference in the calculation). Now because of the gearing, the torque is 6 x 10 = 60oz-in but the rpm is down to 380. So, yes, while the torque is not that much bigger, but still bigger, and is definitely 6 times the value read from the chart. Not much is gained compared to direct drive at low rpm, but never the less, there is a gain, even with your motor chart. I have also said that the gearing is mostly a help between zero and very low rpm, I am aware of that, but still, there is no way you can directly drive this chuck, except through putting in a 1:1 belt, but that would mean I lose even holding torque, which is the very reason these belt ratios are used the first place.

Now, to make something clear, this is the motor I am considering: https://www.jmc-motor.com/product/980.html
It says: "Nema 23 Integrated AC servo motor 180W" ...and it is not a printing error, because I am sure it is the same as any other AC servo, except that the driver is fed using maximum 50VDC, so the rectifier is outside the driver, as I said before. The motor has a 0.6Nm (85 oz-in) torque all the way to 3000rpm with a max torque of 1.5Nm. Now, by using the same 1:6 belt I'd get 3.6Nm holding torque AND all the way to my current maximum chuck speed, which is 380rpm. In other words, I gain at the speedy end, but lose on holding torque, since my current steppers have 1.8Nm x 6 = 10.8Nm, but my current stepper has 0.5Nm at 1500rpm, which is quite a bit from 2280rpm (my current speed). I have no torque table above 1500rpm. My guess is that the torque is very low at that high speed. With the servo I could go to 3000 rpm (or even more), so that would definitely be beneficial for the rotation, but not for holding torque.

Yes, I know that combining a 4th axis and use it as a lathe is not a good idea according to a professional, but to me, this is just for fun, I don't aim at competing with anyone except myself. I aim at getting experience, knowledge, having fun, enjoying my time, exercise a hobby and so on.

Correct me if I am wrong, but don't say that I "happy to redefine terminology", because I am not. I did not make any calculations before, but definitely not redefined anything and torque is definitely increasing when you add a reduction gear. I think you understand that very well, but if you don't understand what I am saying that's a different thing, and is perfectly possible that I am not that clear, but that does not justify such statement. I am not stupid, I can read charts, make calculations (in ANY sort of units) and so on. You are not the only one with engineering background and long and prosperous career.