Recovery of plane-wave profiles from clock-marked nested volumes: finite-family uniqueness, dimension dependence and certified reconstruction

Can integrated causal geometry retain enough information to recover a changing gravitational field? We study clock-marked prefix Alexandrov volumes in a four-dimensional plane wave with fixed polarisation. On a seven-dimensional even bump-polynomial family, quadratic volume coefficients at 28 prescribed proper-time depths determine every profile up to overall sign. Interval rank and rational sum-of-squares certificates place the lifted inverse margin between 2.24 × 10⁻⁹ and 2.45 × 10⁻⁹ in physical L² and RMS data norms. Certified margins decrease strictly through dimensions two to seven. A common-record construction supplies a quantitative resolution obstruction. We derive an exact volume representation, a uniform remainder, and a conditional clock/readout budget. Independent finite-amplitude forward tests support the quadratic expansion; additional tests examine amplitude scaling and departure from the chosen family. The result establishes mathematical identifiability with quantified conditional stability. For a profile-error target of 0.2, a conservative uniform remainder constant and an equal allocation to data error and fit residual give an optimistic calibration ceiling of 2.21 × 10⁻²² after optimisation over physical amplitude. Its one-standard-deviation Poisson comparison is about 2 × 10⁴³ elements per diamond; this scale depends on the remainder bound and residual allocation. Separately, a common coefficient record with error radius 1.76 × 10⁻⁹ forces a profile error greater than 0.712 for one of two admissible truths, independently of the recovery method. Feasible measurement and unrestricted history recovery remain open.