This paper proves that spacetime recovery is completion closure, not a consequence of boundary invariance alone. Invariant boundary laws can admit many representatives; the theorem identifies the finite-dimensional closure conditions that eliminate representative retuning and certify the closed boundary law to carry a Lorentzian causal-chart representation. Those closure conditions require quotient selection, scale transport, refinement coherence, spectral-pairing admissibility, a shared heat/real-time generator, descended covariance, variational current identity, spin-2 response, and hyperbolic response-symbol certification to close in the same completion lock. The starting datum is not a background manifold, metric, diffeomorphism group, or causal cone. It is a completion-locked boundary ensemble: histories with rigidity weights, a locked boundary Hilbert space and observable algebra, admissible representative changes, boundary-channel realizations, and fixed comparison data. The theorem proves completion closure for the locked boundary ensemble: admissible moves and refinements preserve the boundary law without retuning, positive-gap quotient selection gives boundary convergence, and the scale, coherence, spectral-pairing, and shared heat/real-time residuals vanish exactly when the corresponding hidden-retuning channels are closed. The theorem then proves Lorentzian causal-chart certification. When the closed response contains a nondegenerate gapless spin-2 sector and a hyperbolically admissible principal symbol whose chart residuals vanish, the limiting boundary law admits a Lorentzian causal chart. The chart is not assumed, but recovered when boundary invariance, selection, scale, coherence, spectral, response, covariance, current, spin-2, and hyperbolic-symbol conditions close together. The present paper supplies the causal-chart certification. A companion paper then uses the same completion-locked datum to recover Einstein dynamics and the Einstein field equation by adding the curvature-action, coefficient, matter-response, and covariant-conservation closures. License note: Distributed under CC BY-NC-ND 4.0.
Salimah Meghani (Tue,) studied this question.