From the perspective of Quantum mechanics, Planck's constant is a model of the photon energy necessary to eject an electron from a flat surface. It is wave-length independent, so the surface is not a model of an atom. Atomic physics models the atom itself, in which case differences in photon energies ("colors") are detectable in the hyperfine structure, in which photons only have spin polarization, but not "rest" masses of their own (That is, the mass is modeled as interactive spin polarization; equal and opposite via the null vector characterization, and so relevant to the Binomial Theorem representation).
The relativistic circle is the limit as the interaction -> 0 for a single photon, where the energy is modeled as the final state of particle creation for a single photon of invariant wavelength (or "period")). This photon energy can be multiplied for a model of non-interacting bosons, and the context for probability is Bose-Einstein statistics.
The fine structure constant (as opposed to the hyperfine) models the interaction between charged particles. This is modeled by orbital angular momentum, which changes as electrons change shells within the atom; the neutrino is equivalent to the binding energy or the electron within (e.g.) a hydrogen atom, where h is an approximation that ignores the curved geometry of the atom but does introduce the continuum as the limit of . This introduces the concept of charge (as independent of photon polarization), so the probabilities are modeled as Bessel functions via the Schrodinger equation to model the changes in state for different configurations of the nucleus (as opposed to n(n+1) from a flat surface), resulting in the Periodic Table and classical (non-relativistic) quantum mechanics. The inclusion of spin is then the interaction between the electron spin and the photon polarization.
Fermi-Dirac statistics model this with invariant electron charge/mass ratio. It gets complicated - that's why they got the big bucks...
Nevertheless, the relativistic unit circle and the binomial theorem is the foundation of any mathematical model that includes interactions, expressible by comparing metrics - including gravity. It there are no interactions, there are only individual "affine" particles (ok, integers) and it is impossible to apply any relationship (except imaginary) because there is no common origin for any metric (all that exists is "counting" mentally, which was Wittgenstein's point in "Remarks on the Foundations of Mathematics"; he just hadn't made the mathematical jump to relativity and the Binomial Theorem.
The application of probabilistic analysis inn classical quantum mechanics (i.e., no hyperfine splitting) removes the states of the photon from the context (via the conjugate transpose of the wave equation relative to h, so the only observable is the difference between the initial and final states independent of Planck's constant (and thus light energy))
There is more to this story (neutrons, gravity), but I don't have the space to write it here...