The electrons have not changed their relative direction of flow relative to the conductor(s) in the transformer
Not the point. The point is the difference in time can be represented by the phase-opposed voltages and unless we were there at the time of creation, it is impossible to tell the difference.
nor have you altered the chemical reaction in a battery by using a sharpie marker to re-identify the terminals.
Some batteries can be charged and discharged but that doesn't mean you have altered physics. Besides, to try to make the battery analogy work in the other post, the batteries were being flipped around in regular cycles to emulated the AC cycle (so we could always have a single direction of current from the battery cell but a cycling direction of current coming out of the battery box).
The posters have chosen a relative reference point which is contradictory to the physical devices.
Absolutely incorrect. See above. To me, it is not about how the voltages were created (a single transformer, two transformers, a phase-shifting function generator circuit, etc) but what kind of sources can the output voltages represent. The single-phase transformer is a single-phase transformer and the single flux through the single core makes that a given. The voltage waveforms are what I am discussing and they could have been created with a single transformer or multiple transformers.
At some point in the analysis, a negative sign must be injected to align the equations with the physical devices. Failure to do so will eventually lead to an analysis where the result opposes the physical movement of the electron flow.
Incorrect. It is about the movement of the electrons based on a relative point in time. The electrons from wave A started their life moving "forward" at time a. The electrons from wave B started their life moving "forward" at time a + 1/2 cycle. In other words, the B electrons were moving forward when the A electrons were moving back-wards. To get them to move in synchronization, we did indeed make a polarity change and tied the two strings together in reverse.
You are arguing the case (case #1) that all electrons in a single source were originally moving forward at the same time. To get the end result of them moving in opposite directions, we must break the source and align the halves in the opposite directions. I agree that is a valid case.
For a two-source scenario, you would arguing the case (case #2) that all electrons in both source were originally moving forward at the same time. To get the end result of them moving in opposite directions, we align the sources in the opposite directions. I agree that is a valid case.
I say there is also a case that exists (case #3) where the electrons in one source were originally moving in the opposite direction from the electrons in the other source at the same time. To get the end result of them moving in the same direction, we align them in opposite directions. You fail to recognize this as a valid case.
I do not say there is a case (case #4) where a single source has electrons moving in two different directions.
I do say that the electrons moving forward in half A and the electrons moving forward in half B can represent the two sources of case #2 aligned in the same direction, or the two sources of case #3 aligned in opposite directions and that it would be impossible for us to look at the steady-state output and tell the difference.
These same posters have concealed this negative sign in many forms during previous discussions
On the contrary, I am not trying to conceal anything, but rather show the nature of the physics of the system from a general standpoint rather than by its conventional name. I have tried my best to use different analogies and examples to clarify my statements, not conceal any part of it.
in an effort to "force" their relative reference point to "appear" as though it were absolute.
There is no absolute reference point. To define a voltage, one must either pick a reference point or accept one that someone else has already picked. I have argued that the selection of a reference point is a choice, not an absolute. X1 and X2 are BOTH valid choices.