I see... So, how does the LM317T avoid the Ouroboros effect resulting in exponential change? What's fed into ADJ affects Vout which is then changed by R1 & R2 and fed back into ADJ.

The program current is set by the resistance between the Vout and Vadj terminals. It is designed to be a stable reference/current.

There is no feedback from changes in the value of R2 as long as Vout stays within the range required for the regulator to function. What the regulator is doing is maintaining a constant 1.25V between Vout and Vadj.

Where'd you get that schematic from? I can't find it in the one I have from Mouser.

Did Mouser offer datasheets from multiple manufacturers? I prefer to use the one from National Semiconductor. It always contained more information than datasheets from other manufacturers. A quarter of TI's version is dedicated to package information. You can get the Nat Semi version here.

Why can I ignore the effect? "Delta Iadj"?

If you choose a resistor that will give a program current of 10mA, the effect of Iadj (100uA max) can be neglected because it can be compensated for by the pot and the effect of Delta Iadj (5uA max) will be insignificant (5uA*1kΩ = 5mV).

What's with that second Vout after R2 in your schematic? Am I supposed to join the two Vouts?

There is only one Vout. The "second" one shows where circuit ground and the output are taken.

 

There is no feedback from changes in the value of R2 as long as Vout stays within the range required for the regulator to function. What the regulator is doing is maintaining a constant 1.25V between Vout and Vadj.

And the light bulb turns on... So, by altering R2 the regulator detects that difference between Vout and Vadj is not 1.25V and adjusts as long as Vout within Io(max) on the datasheet. I'm assuming the speed at which it adjusts is dependent on some type of frequency, timing, or clocking mechanism that I have yet to learn.

Did Mouser offer datasheets from multiple manufacturers? I prefer to use the one from National Semiconductor. It always contained more information than datasheets from other manufacturers. A quarter of TI's version is dedicated to package information. You can get the Nat Semi version here.

Yep, they did. I was using the one from ST Microelectronics. Another point to experience in knowing to use National Semiconductor.

The "second" one shows where circuit ground and the output are taken.

So the second one is figurative; i.e. I don't have to jump it to Vout? If I don't jump it, I'm assuming that it goes to ground.

 

So, by altering R2 the regulator detects that difference between Vout and Vadj is not 1.25V and adjusts as long as Vout within Io(max) on the datasheet.

Not quite.

The regulator will maintain a constant voltage between Vout and Vadj. That will give a constant current for I1. Vout will be 1.25V + I1*R2.

I'm assuming the speed at which it adjusts is dependent on some type of frequency, timing, or clocking mechanism that I have yet to learn.

LM317 is a linear regulator, so there is no frequency or clock. It will continually adjust it's output voltage to be Vref + VR2. It will do that as long as Vin is sufficiently high and the regulator self protection limits aren't exceeded.

So the second one is figurative; i.e. I don't have to jump it to Vout? If I don't jump it, I'm assuming that it goes to ground.

No, the two voltages with the arrows are simply annotations.