Ok. Then I'm afraid I don't find equations 12, 13, and 14 and the discussion surrounding them at all convincing. I just see curve fitting and handwaving.
Perhaps I'd better explain in more detail what I'm looking for by considering another idealized example: a static, spherically symmetric matter region of constant density surrounded by Schwarzschild vacuum. Of course this example is highly unrealistic because of the constant density assumption, but it's a common one used for pedagogy in textbooks because it has a known exact solution. The metric for this solution in Schwarzschild coordinates is
$$
ds^2 = - J(r) dt^2 + \frac{1}{1 - \frac{2m(r)}{r}} dr^2 + r^2 d\Omega^2
$$
where ##d\Omega^2## is the standard metric on a 2-sphere. There are two regions, the matter region and the vacuum region, with a boundary between them at ##r = R_0##. In the vacuum region, we have ##m(r) = M## and ##J(r) = 1 - \frac{2M}{r}##. In the matter region, we have
$$
m(r) = 4 \pi \rho \int_0^r r^2 dr = \frac{4}{3} \pi \rho r^3
$$
and
$$
J(r) = \left( \frac{3}{2} \sqrt{1 - \frac{2M}{R_0}} - \frac{1}{2} \sqrt{1 - \frac{2M r^2}{R_0^3}} \right)^2
$$
Note that the above equation for ##m(r)##, when we plug in ##r = R_0##, gives
$$
M = \frac{4}{3} \pi \rho R_0^3
$$
We can compute the orbital frequency ##\omega## of a free-fall circular orbit at radius ##r## using the methods in my Insights article on Fermi-Walker transport [1], which gives a simple way to compute the proper acceleration as a function of ##\omega## (and we can then set the proper acceleration to zero to find the free-fall orbit value of ##\omega##):
$$
A = \frac{1}{2} g^{rr} \left[ \left( \partial_r g_{tt} \right) u^t u^t + \left( \partial_r g_{\phi \phi} \right) u^\phi u^\phi \right] = \frac{1}{2} \left( 1 - \frac{2m(r)}{r} \right) \left( \frac{1}{J(r) - \omega^2 r^2} \partial_r J(r) + \frac{\omega^2}{J(r) - \omega^2 r^2} 2 r \right)
$$
The condition for ##A = 0## is then
$$
\partial_r J(r) = 2 \omega^2 r
$$
which gives
$$
\omega = \sqrt{\frac{1}{2r} \partial_r J(r)}
$$
For the Schwarzschild vacuum region, we have ##\partial_r J(r) = \frac{2 M}{r^2}##, so we get the familiar formula:
$$
\omega = \sqrt{\frac{M}{r^3}}
$$
For the interior matter region, we have
$$
\partial_r J(r) = 2 \left( \frac{3}{2} \sqrt{1 - \frac{2M}{R_0}} - \frac{1}{2} \sqrt{1 - \frac{2M r^2}{R_0^3}} \right) \frac{1}{4} \left( 1 - \frac{2Mr^2}{R_0^3} \right)^{-\frac{1}{2}} \frac{4Mr}{R_0^3}
$$
which gives
$$
\omega = \sqrt{ \left( \frac{3}{2} \sqrt{\frac{R_0^3 - 2M R_0^2}{R_0^3 - 2M r^2}} - \frac{1}{2} \right) \frac{M}{R_0^3} }
$$
Inverting this formula gives us a way to infer ##M## from observations of ##\omega## as a function of ##r## for objects in free-fall orbits in the interior matter region; in other words, it gives us a way to infer ##M_{RC}## from an observed rotation curve. And we will find that ##M_{RC}## inferred in this way is given by the formula for ##M## above, i.e.,
$$
M_{RC} = \frac{4}{3} \pi \rho R_0^3
$$
Now, suppose all the matter in the interior matter region is luminous, and obeys some known mass-luminosity relation. Then we can infer a mass ##M_L## from the observed luminosity. Since the mass-luminosity relation will be derived, as you note, from observations of stars in our own galaxy, we expect that ##M_L## for the interior matter region will be something like "integrate the density over the proper volume". But we know what that integral is:
$$
M_L = 4 \pi \rho \int_0^{R_0} r^2 \sqrt{g_{rr}} dr
$$
where the extra factor of ##\sqrt{g_{rr}}## is the correction to make the integral over the proper volume. We don't even need to evaluate this to see that, since ##g_{rr} > 1## for the entire range of integration, we must end up with ##M_L > M_{RC}##.
As I said, this model is obviously unrealistic; but just on heuristic grounds, I would expect a similar relationship ##M_L > M_{RC}## to hold for any stationary bound system (and a galaxy is such a system, certainly to a good enough approximation for our purposes here) in which all of the matter is luminous, for the simple reason that I've already given in a prior post, and which is obvious from comparing the integrals above: for any stationary bound system, the proper volume, which determines ##M_L##, will be larger than the "Euclidean volume" we infer based on the area of the system's boundary (which is what the ##r## coordinate in Schwarzschild coordinates is measuring), which determines ##M_{RC}##, and all of the other factors involved are the same. Therefore, if we see a stationary bound system, like a galaxy, where we have ##M_L < M_{RC}##, and by a large margin, we should infer that there is missing mass that is not luminous.
What I'm looking for is an argument from you, based on some kind of ansatz for a solution describing a galaxy (that will obviously be different from the ansatz I adopted above), for why the simple heuristic argument I gave above should
not apply to the actual galaxies we observe. Or, alternatively, you could make an argument that I'm somehow misinterpreting the ansatz I gave and how ##M_{RC}## and ##M_L## would be determined for that idealized case, and that when the correct method of determining them is used, we find ##M_L < M_{RC}##. Just saying "mass is contextual" won't do it. And just pointing to equations 12, 13, and 14 in the arxiv paper, and their surrounding discussions, won't do it, because I don't see any ansatz in there that is based on any kind of physical property that the actual galaxies we observe have (whereas my ansatz above is based on an obvious property they all have, that they are stationary bound systems); as I said at the start of this post, I just see curve fitting and handwaving.
[1]
https://www.physicsforums.com/insights/fermi-walker-transport-in-schwarzschild-spacetime/
