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Using our discrete relativistic combinatorial bit?]string theory of physics in the context of the hydrogen spectrum, we calculate our first two approximations for the fine?]structure constant as ??(1) = 1/137 and ??(2) = [1 − 1/(30 ?~ 127)]/137 = 1/137.035 967 4 ?c ; we can then derive the Sommerfeld formula.

We base our theory of physics and cosmology on the five principles of finiteness, discreteness, finite computability, absolute nonuniqueness, and strict construction. Our modeling methodology starts from the current practice of physics, constructs a self-consistent representation based on the ordering operator calculus and provides rules of correspondence that allow us to test the theory by experiment. We use program universe to construct a growing collection of bit strings whose initial portions (labels) provide the quantum numbers that are conserved in the events defined by the construction. The labels are followed by content strings, which are used to construct event-based finite aned discrete coordinates. On general grounds such a theory has a limiting velocity, and positions and velocities do not commute. We therefore reconcile quantum mechanics with relativity at an appropriately fundamental stage in the construction. We show that 1) events in different coordinate systems are connected by the appropriate finite and discrete version of the Lorentz transformation, 2) three-momentum is conserved in events, and 3) this conservation law is the same as the requirement that different paths can ?interfere? only when they differ by an integral number of de Broglie wavelengths. The labels are organized into the four levels of the combinatorial hierarchy characterized by the cumulative cardinals 3, 10, 137, 2¹²⁷ + 136 1.7 ? 10³⁸. We justify the identification of the last two cardinals as a first approximation to c/e² and c/Gm = (M_planck/m_p)² respectively. We show that the quantum numbers associated with the first three levels can be rigorously identified with the quantum numbers of the first generation of the standard model of quarks and leptons, with color confinement and a first approximation to weak-electromagnetic unification. Our cosmology provides an event horizon, a zero-velocity frame for the background radiation, a fireball time of about 3.5 ? 10⁶ years, about the right amount of visible matter, and 12.7 times as much dark matter. A preliminary calculation of the fine structure spectrum of hydrogen gives the Sommerfeld formula and a correction to our first approximation for the fine structure constant, which leads to 1/ = 137.035 967 4 ?. We can now justify the earlier results m_p/m_e = 1836.151 497 ? and m/m_e = 274. Our estimate of the weak angle is sin² _Weak = 1/4 and of the fermi constant G_F ? m = 1 / (256)². Our finite particle number rela- tivistic scattering theory should allow us to systematically extend these results. Eteris paribus, caveat lector