The phase of the reduced function is still [ω t + φ0]
It is no such thing because you have shifted the phase. You are just making this up as you go. That is not how phase is defined and you are ignoring the initial conditions. The link you keep referencing does not say what you are saying either. Any good physics text or calculus text will show that. I have several in my library and can get you some references when I get back but you should be able to find these on your own. Besides, you should know better anyway but I understand that things can be forgotten over time. How about you crack open some of your old college texts and have a look?
Unless you that insist sign is an element of the definition of phase (which I acknowledge many erroneously have)...
Erroneous according to you. You have no mathematical basis for this theory of yours as you have just made it up. Neither the physics nor the math support what you are claiming.
...my math is not even particularly difficult, let alone "new."
The Trig is not new but what is new is your re-defining phase constants such that they can be different but equal.
Please forgive me, I thought I had already discussed your spring analogy with you before.
Yes, I do beleive they are in phase; I don't beleive they are in synchronysim. Certainly things that ae synchronized are in phase, but the reverse isn?t necessarily true. I find it intertesting that some of the debaters are willing to give up amplitude,('cause their scope and scaling factors says so) but insist polarity is a requirement of phase.
I find it interesting that you just make these things up with no factual or sound mathematical basis. I'm pretty sure I could use Euler's identities to extract any portion of the phase constant I want and declare that the revised number is now the real "phase constant". That is what you are claiming and it just is not so.
In considering the physical system:
You can have an oscillating source drive one large output to the right or to the left. You can also have it drive two smaller outputs to the right, two smaller outputs to the left, or one small output to the left plus one small output to the right.
The fact that the source parts may or may not be coupled together is not what I am focused on. For the single-phase transformer, the source parts are coupled together and the forces are in phase. My position is that the single coupled source can also provide a set of outputs that are opposite in phase. This would be the same outputs you can get from source parts that are physically opposite in phase but can also be coupled together to work in phase.
Looking at the outputs instead of the source configuration makes it obvious that the coupled source, whether it is made of 0? displaced parts or 180? displaced parts is capable of delivering either two outputs with a 0? displacement or two outputs with a 180? displacement. Whether or not you want to call these phase differences is a matter of definition preference. If you use the technically correct definition of phase you can have two phases present at the output and a load that requires two phases with a 180? displacement can be served by the single-phase center-tapped transformer.
To try to use the fact that the source is single-phase, and to try to redefine the physics or try to redefine the math to force a fit to a label that is limited in scope does not make sense. The voltages are what they are where they are.
A ?180? phase displacement doesn't result in different phase,(i.e.,wt is still the same) unless you insist sign (or polarity) is necessary to define phase.
The physics and the math both show you are wrong and further support the fact that you are just making this up.