$m_0$= mass when $v=0$ is rest mass, when the motion or velocity is 0, when the temperature is 0K, or absolute 0.
So, for the hydrogen atom, we have at absolute 0K, both the electron and proton, with no phononic energy (no temperature) at all.
Remember what the original question was? What is matter? We are attempting to find out what matter is from known elements, and then to find the fundamental constituents of those elements.
So, we focus on the hydrogen atom (because from all experience and all knowledge, it is the simplest element). Presently, there is no need to examine isotopes of hydrogen, as we are working on establishing a reference (focusing on protium) and then testing the theory against experiment. So far, I am not talking about dynamics, so this in no way really challenges quantum mechanics, we are simply working on defining the terms and masses. Especially since it appears these fundamental masses (proton and electron mass) and constants (c, h, R_H, etc) are not defined in mainstream physics - they are determined by measurement (well, defined by relationships to other other masses and constants). We intend to define them theoretically then verify via experiment.
The more clearly the question is defined not only the easier and more clearly the problem is solved, sometimes this allows the solution to present itself. It also avoids the confusion with mainstream quantum mechanics that already deals with the dynamics, like QED (electron and photon dynamics).
So, protium at 0K, a single hydrogen atom, a single proton and a single electron is the starting point.
or is it?
From our senses, from all experience, ALL we actually know exists (in this small world of our investigation) is this thing we call protium, and is has been or can be proposed that this is then composed of a proton and electron, however, what is an electron? what is a proton? This is why we have to proceed very carefully. The purpose is not to redefine physics, but to clearly, from first principles and fundamental constants, frame the problem so clearly that it demonstrates the proposed conjecture that the roots of this 8D multidimensional polynomial are indeed the fundamental masses and constants in Nature or is this simple the results of this model of forcing this viewpoint or structure?
So, while the focus of this blog cannot be determined by the Google algorithms, the focus being brought now to the question should clarify greatly the investigation point of this blog, because it is a very finely, precisely focused, quest: What is matter? And what are these masses and constants of physics? And we are using standard physics parameters, masses and constants, and focusing on the mass ratio of what appears to be two pieces of what constitute ordinary matter.
Also, have to consider that perhaps it is an "ionized" version of protium we are dealing with and an isolated electron, and it's "rest mass" behavior. This is part of the reason of going into such detail on the question because there are possibly finer details or ways of looking at things even before we get to dynamics, which we know form the success, that the dynamics are well known in QED.
We will proceed with looking at protium as a proton and an electron, however, a viewpoint that we are taking is that the proton is a packed stable vortex/dual vortex/magnetic_dipole in the super fluid aether and the electron is simply the inverse - an unpacked vortex. These ideas will need to be developed further, however, the direction will quickly go to looking at protium as a proton and an electron and save a full new analysis for either later or for simply developing the tools and equations to be able to resolve the problem. Since looking at the problem wholly as we are doing, we are able to talk about protium being a proton and an electron and simply provide more detail as to exactly what the proton and electron are rather than a model. Or a precise model of what they are, not an approximation. This is one way to recheck fundamental physics.
The concern is that the equation just happens to converge but it's simply a model that doesn't match Nature. After all, it's basically changing all the values of the constants starting at around the 4-6 decimal place, and -4% reduction in proton radius and likely proton magnetic moment.
More Later, (especially after I look at a cold matter physics or condensed matter/plasmas?)
(The answer very likely has been presented already, however, the question not so clearly.)
The Surfer, OM-IV
(part of the motivation to do this is that a -4% error on proton radius implies there may be some major reworked needed to mainstream approach)
(and because I think the proton IS a fundamental particle too)