> Are you arguing something like, "there are X irreducible inputs to the Standard Model. For any particular input, a, you might be able to swap it out for a different one, g, so that you still have X irreducible inputs, but now they are a different set.

That's part of what I'm saying. Some trivial examples from the Standard Model are the choice of angles you use to parametrize the CKM and PMSN matrices, Weinberg angle vs electroweak gauge couplings and the scale at which you choose to fix those couplings.

Maybe it will help to call the prediction of values for one such set of parameters from the values of another such set of parameters a "horizontal prediction": you have one theory T, a value a_A of some parameter A, and you predict a value b_B of some other parameter B: B_b = T(A_a). It is "horizontal" because A is no more fundamental than B; you could equally well use T to predict A_a from B_b.

y = T(x) is of course the general form of any prediction of anything at all from theory T.

The reason you saw fit to "correct" walru1066 is that you implicitly expanded "prediction" to "prediction from a more fundamental theory". That's too long to write, so I'll call it a "vertical prediction": you have a more fundamental theory F with some set of parameters A and a less fundamental theory L with some set of parameters B, and you predict B from A using F: B = F(A). It is "vertical" because F is more fundamental than L.

How do we know that F is more fundamental than L, and not just an equivalent description of the same theory? That's easy: because the set A is smaller than the set B. :)

walrus1066 mentioned a prediction of the fine structure constant, and he was right; that's what's done in [1] (I'm pretty sure he was remembering that paper, but not the exact reference; who does?). It's a horizontal prediction. Like all proper predictions, it only works if the theory works, so it is a perfectly valid test of the theory (the topic of his post).

You saw "prediction" and expanded it to "vertical prediction", but that was never mentioned or intended.

> Do you take issue with the phrasing of the passages in this wikipedia article (https://en.wikipedia.org/wiki/Dimensionless_physical_constan...)?

I do not take issue with the full phrasing of it, which you snipped out. The complete sentence is

Other physicists do not recognize this usage, and reserve the use of the term fundamental physical constant solely for dimensionless physical constants that cannot be derived from any other source.

In other words, there is no consensus about whether dimensional quantities can be called "fundamental physical constant". The reason is obvious: once you've settled on a system of units (if you are doing fundamental physics, presumably natural units [2]), you can always turn any dimensional quantity into a dimensionless one combined with a fixed dimensional factor.

I can imagine a parallel to this thread in that context: Somebody posts "the mass of the electron is a fundamental constant of the Standard Model", you reply "no it's not, it's dimensional, so it's not fundamental", and I end up writing a long post explaining that you can factor it into a dimensionless Yukawa coupling and a dimensional Higgs expectation value, so it's really fine to call it fundamental even by your definition (i.e. we do not currently have a more fundamental theory which predicts the mass of the electron, unless you are happy with it being a random value).

Regarding this part of your question,

> Fundamental physical constants cannot be derived and have to be measured.

I have no problem with the first part of that sentence (can't be derived; that would require having a more fundamental theory) but the "have to be measured" is subject to interpretation. If you take it to mean directly measured, it's really too restrictive (just have a look at what really goes into determining the properties of short-lived elementary particles). If you allow for measuring some quantities and performing a bunch of calculations on the general form of a horizontal prediction (the only kind possible within the confines of a single theory) then fine.

As for "classification of the fine-structure constant as a fundamental physical constant", I have no problem with it (at the current state of knowledge).

[1] https://arxiv.org/abs/1205.5368

[2] https://en.wikipedia.org/wiki/Natural_units