Quantum Mechanics from a 360◦ Circle: Quantisation as the Count of Whole Turns, and the Bound-State

The integers that organise the bound-state spectrum of quantum mechanics — Planck’sn, Bohr’s n, Schrödinger’s n — are read off the geometry of a turning circle rather thanpostulated. Planck’s constant is one square degree of the hydrogen circle’s energy, partedwith once per turn: h = (π/180)2 (1 + α2π)−2 10−12 ELy. Energy therefore leaves in wholepackets, and the integer is a count of whole turns. The size of the packet is settled geometry;that the count is a whole number is reduced to one open property of a closed circle, not proved— a sharper thing to leave open than the bare integer Planck, Bohr, and Schrödinger eachassumed. From this reading the hydrogen spectrum is one length in rational disguises, and thetwo-particle bound-state catalogue of quantum electrodynamics becomes rational arithmeticon a single energy unit, α2ELy = 4πELy ¯λc/λLy ≈ 724.5 μeV: hydrogen fine structure,the Darwin term (1 × α2ELy/n3, exact to nine figures), the 21-centimetre line (from 360,λLy, α, with the proton’s magnetic moment absent), positronium hyperfine ( 76 α2ELy), andSchwinger’s anomaly — which is the framework’s electron-extent factor (1 + α2π) minus one.The wavefunction is read as the circle itself. The claim is structural, not a new dynamics:every measured number is unchanged. One consequence is falsifiable — because the 21-centimetre identity contains only geometric primitives, the rest-frame frequency does notdrift with cosmological time, a prediction testable by high-redshift 21-cm cosmology. Theopen frontier is named to the digit: the two-loop coefficient of the electron anomaly resolvesinto sums over the integer ladder (ζ(2) and ζ(3) supplied, the alternating ln 2 tower open),and the Lamb-shift coefficient waits on a geometry of the unbound continuum