The Hydrogen Atom Derived: The Speed of Light, Planck's Constant, the 360° Circle, the Rydberg Formula, and Integer Quantisation from One Geometric Construction

This paper presents five geometric derivations from a single construction on the hydrogen atom. First, the speed of light is shown to be a distance, with the value c = 2.99846604 × 10⁸ metres per hydrogen second. The derivation defines the per-degree arc length rc as the arclength subtending one degree of central angle at radius c × 10⁻¹², then independently derives rc from the framework's atomic primitives via a concentric-circle construction; solving for c gives the value above. Second, Planck's constant is derived from the same apparatus by combining frequency-from-geometry with the Einstein-Planck relation E = hf, giving h = 6.6248736 × 10⁻³⁴ J·s per hydrogen second. The derivation exposes the reciprocal symmetry of c and h — where c carries the second in the denominator, h carries it in the numerator — and the 0.018% offset from the SI values of c and h is located by elimination in the SI second's Babylonian inheritance. Third, the hydrogen atom's geometric structure forces the angular unit count n = 360 in any internally consistent application of the foundational equation: 360° is a physical fact about hydrogen, not a Babylonian convention. Fourth, the closure theorem: a wave on a circle of circumference 2π satisfying smooth-periodicity conditions must have an integer wavelength count. Fifth, the empirical Rydberg formula (1888) is derived as a theorem from the preceding results: λ(n→m) = Ly × m²n² / (n² − m²), where the integers come from closure and the length Ly is the Lyman boundary wavelength. The speed of light, treated as a velocity since Rømer's 1676 measurement, becomes a distance. Planck's constant, introduced reluctantly in 1900 and unexplained for one hundred and twenty-five years, becomes a derived quantity. The angular unit 360°, treated as a Babylonian convention for four thousand years, becomes a physical fact. The integer rule of quantum mechanics, postulated by Planck, Bohr, and Schrödinger between 1900 and 1926, becomes a geometric necessity. The Rydberg formula, treated as an empirical pattern-fit since 1888, becomes a theorem. We discuss extensions to fine structure, the wavefunction, and multi-electron atoms, and identify the open questions.