Where is the fine-structure constant in this list? so John Baez has this nice list of what he enumerates as the 26 Fundamental Universal Physical Constants and they're dimensionless, as they should be to be meaningfully fundamental.  one if his grad students worked out the numerical values of those of the 26 that are known.
my question is, where is the low-energy fine-structure constant $ \alpha = \frac{e^2}{(4 \pi \epsilon_0) \hbar c} $ in all of this?  i see where the high-energy $\alpha$ is related to the gauge coupling constants, $g_{U(1)}$ and $g_{SU(2)}$:
$$ \alpha(m_Z) = \frac{1}{4 \pi} \frac{g_{U(1)}^2 \cdot g_{SU(2)}^2}{g_{U(1)}^2 + g_{SU(2)}^2} = \frac{1}{128.16} $$
but that is not the familiar $ \alpha = \frac{e^2}{(4 \pi \epsilon_0) \hbar c} = (137.0359991)^{-1} $ value.  can someone please explain where $\alpha$ fits into this list of 26 fundamental constants of nature?  how does $\alpha(m_Z)$ get related to $\alpha$?
 A: The fine structure constant is the electromagnetic coupling constant.  In the article he says 

Instead of the electromagnetic coupling constant together with the
  masses of the W, Z, and Higgs, we could have used 4 other constants:
  the U(1) coupling constant, the SU(2) coupling constant, the mass of
  the Higgs, and the expectation value of the Higgs field.

and all four of those are in the list.  I don't see where he shows the relationship between the four he lists and $\alpha$.  The linked Black paper shows the relationship.
A: I think your problem starts here:

...yet we understand $\alpha$ to be a universal fundamental constant. 

This isn't actually true, we do not understand $\alpha=e^2/hc$ to be a fundamental constant, instead the coupling constants are the fundamental constants.
Baez says this about the use of the gauge coupling constants instead of the fine structure constant (all emphasis mine):

I should warn you here: there are different ways of slicing the pie. Instead of the electromagnetic coupling constant together with the masses of the W, Z, and Higgs, we could have used 4 other constants: the U(1) coupling constant, the SU(2) coupling constant, the mass of the Higgs, and the expectation value of the Higgs field. These are the numbers that actually show up in the fundamental equations of the Standard Model. The idea is that the photon, the W and the Z are described by an U(1) × SU(2) gauge theory, which involves two coupling constants. The beautiful symmetry of this theory is hidden by the way it interacts with the Higgs particle. The details of this involve two further constants - the Higgs mass and the expectation value of the Higgs field - for a total of 4. If we know these 4 numbers we can calculate the numbers that are easier to measure in experiments: the masses of the W and Z, the electromagnetic coupling constant, and the mass of the Higgs. In practice, we go back backwards and use the constants that are easy to measure to determine the theoretically more basic ones

So Baez chooses to use the gauge coupling constants, rather than the fine structure constant, because U(1)$\times$SU(2) appears to be more fundamental because the electric charge is not a natural quantity in the unified theories, it's calculated from the coupling constants:
$$
e=\frac{g_{U(1)}\cdot g_{SU(2)}}{\sqrt{g_{U(1)}^2+g_{SU(2)}^2}}
$$
which is in Black's document, on the last page.
Black's document also shows how to obtain $\alpha(m_z)$ from $g_{U(1)}$ and $g_{SU(2)}$, which you represent in your question. Note that here, $m_z\sim90$ GeV whereas the electromagnetic coupling is at an energy of  511 keV. In this post, Lubos notes that,

The fine-structure constant is essentially constant below the mass of the electron - the lightest charged particle - which is 511,000 eV or so. You are extrapolating the running of the electromagnetic fine-structure constant $\alpha=1/137.03599$ all the way up to the Planck scale, about 
  10,000,000,000,000,000,000,000,000,000 eV. I chose to avoid the scientific notation to make it more explicit how far you have extrapolate

While you aren't quite running it up that far, you are trying to extrapolate up to 91,000,000,000 eV from 511,000 eV, so the principle remains: the fine-structure isn't as fundamental as you are led to believe.
