The reality is that you have a very high gain DC amplifier with an unstable reference voltage.
I don't know where you got that "unstable reference voltage".
a more understandable comparison might well be that you have a really big car (The Regulator) moving across an area (output Voltage) without a driver (The Reference Voltage) High gain because the reference term is a current (Typ 100uA) controlling a regulator capable of an Ampere of current and it doesn't take a genius to figure out that 100 uA is a tenth of a milliamp and as the name implies a milliamp is 1/1000th of an ampere so we have a gain of 10 X 1000 and no stable input reference. The typical result for an NPN pass type (AKA 78XX series) is oscillation and for the PNP Pass types like the AMS1117 the reference term is a part of the load current... So there is a minimum current that must be drawn from the regulator for the regulator to be stable and a processor unfortunately doesn't fit that condition... again the result is typically oscillation.
Let me be the first one to say that I have absolutely no clue what the above is talking about and I doubt you do.
Those regulators work not by amplifying some kind of current. The current" is needed to run a voltage reference / error amplifier. In the case of LM117/317 for example, that current runs the bandgap voltage reference. That current itself has nothing to do with the output current, otherwise.
The minimum voltage drop you observed is due to Vce of the regulator. In the case of a regulator regulator (like LM317 or AMS1117), it is two Vbe voltage drop as the regulator works as a follower. In the case of a true LDO, that's the Vce of a common collector amplifier.
[/quote]I first ran into this issue ...[/quote]
I have no idea what you are talking about and I doubt you do either.
Now what can that mean? What kind of POWER DOES IT PASS
The kind of power that powers your arduino.
My intent here is to point out that
My intent here is to point out that just because the in-output voltage differential goes below the figure specified in the datasheet doesn't mean you cannot power your arduino with it.
If you would like to discuss more on technical terms, I am happy to.