The Closure Is Load-Bearing: Gravity's Spatial Curvature and Gravity Probe B Require All Three Volumes

The bending of light, the Shapiro delay and the precession of orbiting gyroscopes depend on the spatial curvature that a mass produces, measured by the post-Newtonian parameter γ. In the framework of The Volumes: Geometry of Electromagnetism the value γ = 1 needs one ingredient from each volume: the rate of the particle's clock, carried by g₀₀ (Volume II); the boost relation, which ties the time, space and time–space parts of the metric together (Volume I); and the closure of the particle's motion (Volume III). Volume III finds that a particle's whole mass is a motion at c around a closed cycle of one Compton wavelength. Motion at c carries stress, and the closure is what fixes γ. If the stress were left unbalanced, a body would gravitate with twice its energy, give γ = 1/3, bend light by two thirds of the observed amount per unit of measured mass, and resist acceleration with 4/3 of its energy. Von Laue's theorem says that every stable, self-contained system has zero net stress, and with zero net stress all four discrepancies vanish at once. In Gravity Probe B's orbit the geodetic precession splits as 1/2 from the clock rate and γ from the closure; the alternatives γ = 0 and 1/2 miss the measured drift by 240 and 120 standard deviations. The Cassini time delay bounds any net stress to below about 2×10⁻⁵ of the rest energy, and the MICROSCOPE test limits an inertia–weight difference of the electron to about 10⁻¹⁰. These are the values of general relativity, a premise of Volume III. What is shown is that a mass made of motion at c meets them only if its motion is closed, and that the framework needs all three volumes to reach them.