Einstein Dynamics from First Principles: Rigidity-Weighted History Selection, Universal Pullback, and Covariant Conservation

This paper derives the Einstein field equation from a completion-locked bundle/history datum. It does not begin with a metric manifold, an assumed Einstein–Hilbert action, or an independently chosen Newton constant. The datum consists of admissible histories equipped with a nonnegative rigidity functional, Gibbs weights, boundary projection data, and tautological connection data. The equilibrium measure concentrates on minimal-rigidity sectors of the history space. A completion lock fixes the admissible history class, equivalence relation, boundary channel, projection data, selection rule, connection-bundle data, and realization maps before any Lorentzian chart, metric representative, action coefficient, or field equation is extracted. The derivation uses the Lorentzian causal chart certified by the causal-chart recovery theorem from the same locked datum. On that recovered chart, the rigidity-weighted selection law selects the curved equilibrium branch. An admissible boundary-channel realization supplies a selected preimage section of the total history bundle. The compatible connection-bundle section realizes the universal pullback through the tautological connection, producing the pulled-back connection and curvature of the selected branch. Thus the recovered causal chart, selected branch, and Regge curvature representative are tied to one pulled-back connection and curvature. Curvature enters first as curvature of the selected connection; scalar curvature and the Einstein tensor enter only after the spin-foam/Regge realization and the four-dimensional second-order uniqueness theorem are applied. The remaining terms in the field equation are fixed by the same locked datum. The completion-locked vacuum extraction determines the cosmological term. Locked matter-response functionals determine stress-energy. The curvature coefficient is the spin-2 stiffness of the selected curved branch, reported in Einstein units as the corresponding Newton coupling. Covariant conservation follows from the descended diffeomorphism Noether identity. Stationarity of the locked effective action yields the Einstein field equation with cosmological and matter terms. Under the stated closure hypotheses, the Einstein field equation is obtained from one completion-locked bundle/history datum: the causal chart is first certified, the curved branch is selected, connection and curvature are pulled back, the vacuum and matter-response terms are fixed, and the field equation follows as the covariantly conserved variational law on the recovered chart. License note: Distributed under CC BY-NC-ND 4.0.