Extension of Newton's Equation to All Fields in the Ether Theory: the Energy Equation, One Law of Motion, the Definition of Mass and Free Fall

In the ether theory of gravity the energy of a body is E = γmc²e^(−φ). This note collects what follows from this one equation. It keeps the coordinate quantities (t, R, ẋ, p, E) strictly apart from the local ones (t_ℓ, l, w, p̂, E_loc), because mixing them gives wrong numbers; every equation carries the label [C], [L] or [C→L]. (1) One law of motion, ẋ = ∂E/∂p, ṗ = −∂E/∂x + F, where F is the force of an agent other than the ether, reduces to Newton’s first and second laws where the ether field is absent, does not contain the third law, and for F = 0 reproduces the geodesic motion of the exponential metric (agreement 10⁻¹¹ in the integrated position). (2) The same law, with the energy function extended by one term V·p and with a potential that depends on time, covers fields that change with time, moving sinks and dragged ether. It was tested against the geodesic of the metric with a shift, against the comoving frame, and by an energy identity. (3) Mass is the energy that an agent other than the ether must supply per unit change of the level L = γe^(−φ), divided by c²; the inertial mass is the force per unit acceleration measured relative to free fall. (4) Free fall from rest has the speed w² = c²[1 − e^(−2(φ−φ₀))] and the acceleration a = g/γ², both independent of mass. The inertial and the passive gravitational mass are equal because the mass enters the energy function once, if and only if the exponent of the potential is the same for every form of energy. The universality of free fall is consistent with the Eötvös parameter measured by the MICROSCOPE satellite, η(Ti, Pt) = [−1.5 ± 2.3 (stat) ± 1.5 (syst)]×10⁻¹⁵. The exponential form of the metric beyond first order, the coefficient 4 of the dragged field and the field equations that give φ and V are inputs of the theory and are not derived here. The deposit includes the Python scripts and logs of the numerical tests. This note supplements the main manuscript: doi:10.5281/zenodo.23536684.