Hi Malcolm,
The first approximation of the fine-structure constant is primary and has several confirmations from the same geometry of the Cosmological Circle (from previous work), and is perhaps more significant to the classical geometry of the Pythagorean unity-duality-harmony progression. The next two approximations are more related to the regular heptagon, which also has approximations to the dodecahedron geometry. Still, all three are related to the unity-duality-harmony progression.
We probably would not have found these approximations without your initial discovery of quintessence from your geometry of momentum model, and our subsequent geometric approximation from the regular heptagon and its various connections with the dodecagon and dodecahedron of the Cosmological Circle. While the first approximation may be qualitatively "primary," the quantitatively "most intriguing" might be the fourth approximation included in the text at the bottom of page four, as it has both the silver constant and the power of twelve that is so prevalent in the other harmonic equations for fundamental physical constants.
Thanks, Michael