Scale-Invariant Phase Rebound: A Missing Algebraic Structure for Residual Recovery in Mathematical Records of Natural Systems

We formalise Scale-Invariant Phase Rebound (SIPR), a structural mechanism that restores recoverable content at critical compression thresholds in mathematical records of natural systems. Under a strengthened recoverability axiom—requiring that ledger content remain recoverable at thresholds—an inverse, compression-weighted, ledger-preserving transfer is necessary for consistency. The mechanism is realised in eight continuum and kinetic settings, shown to be available wherever a compressive variable, a threshold and a ledger coexist, and supported by counterfactual tests in gravitational collapse, focusing nonlinear Schr¨odinger dynamics and chemical reaction networks. The residual tipping point is of observed order 1–2 % and is pressure-tunable.