One Clock Rule, Two Forces: the Second-Order Clock Term for Electromagnetism and Gravity

The identity h = E Δt of Volume II of The Volumes: Geometry of Electromagnetism says that one completed cycle of a particle's phase carries action h, so a cycle of a clock of energy E lasts Δt = h/E (at rest, the Compton period T_C = h/mc²) and the clock's rate is proportional to its energy. Applied to a clock at rest in a potential, with the potential energy counted in E, the rule fixes the second-order term of g₀₀ that a first-order metric leaves open. For electromagnetism, which couples to a fixed charge, the clock rate is linear in the potential and g₀₀ = −(1 + Σ_EM)², with Σ_EM = qΦ/mc²; this is the rest-frame component of the metric that Volume III derives, and it reproduces the hydrogen Hamiltonian through α⁴. For gravity, which couples to energy, each step of potential acts on energy already reduced by the steps before, so the effect compounds: −g₀₀ = e^(−2U) = 1 − 2U + 2U² + …, with U = GM/rc². This gives the post-Newtonian parameter β = 1 and Mercury's perihelion advance of 43.0″ per century, where the non-compounding rule would give 50.1″. One rule therefore supplies the correct second-order term in both sectors and accounts for why they differ. With Volume I's boost relation, gravity's source taken as energy, momentum and stress together, and the zero net stress of a closed mass (Volume III), the spatial part follows: γ = 1, a frame-dragging coefficient of 4 and the full deflection of light, confirmed here by direct ray tracing. Gravity's existence and strength are inputs, and beyond second order the framework uses general relativity's field equations; the result is a consistency check across the three volumes, not a derivation of gravity.