The Bound-Electron g Factor in the Clock Picture: Electromagnetism Adds, Gravity Multiplies

An electron bound in a hydrogen-like ion of nuclear charge Z has a slightly smaller g factor than a free one; Breit's result from the Dirac equation is g = 2 − ⅔(Zα)² + O((Zα)⁴) for the ground state. This note computes the same quantity from the ingredients of the framework of The Volumes: Geometry of Electromagnetism. The question the framework must answer is how an electric potential acts on the particle's clock. In the forced metric of Volume III a change of gauge is a shift of the particle's own time, so a potential adds qΦ to the energy of every state of the charge and advances all of its phases equally. Spin precession is a frequency difference between two spin states, so the potential cancels from it, and the moment stays the one set by the mass, with g = 2. Relativistic kinematics then give −(Zα)² + ⅓(Zα)² = −⅔(Zα)², Breit's value. The binding shifts of ¹²C⁵⁺, ¹⁶O⁷⁺ and ²⁸Si¹³⁺ agree to 0.02–0.2%, the remainder being the size of the known higher-order terms. The alternatives fail: treating the potential as gravity is treated, scaling all of the charge's energies by the clock rate 1 + Σ_EM, gives −(8/3)(Zα)², and letting the potential energy change the moment gives +(4/3)(Zα)². The electromagnetic clock effect is additive, set by a fixed charge; gravity's, set by energy, multiplies. For the ground state the same two mechanisms, evaluated on the circular orbit that the Hydrogen Test uses for the exact Dirac levels, give Dirac's result to all orders in Zα, g = 2[⅓ + ⅔(E/mc²)], including the heavy ions where the leading term alone fails. The orbit was fixed by the energy levels, not by the g factor, so this is an independent check. In de Broglie–Bohm theory the orbit is the equatorial trajectory of Dirac's own ground state: every trajectory circles the spin axis, on the equator at exactly Zαc, and the orbit's potential energy is Dirac's mean. Averaging the same mechanisms over all the trajectories, or over the probability density, instead fails, and for excited s states the orbit result is close to Dirac's but not exact. The result is Breit's.