Covariant Chu-Kovasznay Decomposition: Resolving Thermodynamic Ambiguity in Compressible Flows

We establish the Covariant Chu--Kovasznay Decomposition (CCKD), a geometric framework that resolves thermodynamic ambiguity in compressible mode content by formulating the decomposition on the effective acoustic spacetime. Enforcing orthogonality in the covariant Chu energy norm, we show that shock--turbulence interaction, often treated as a scattering source, is, in the idealized linear, inviscid setting, a near-unitary (Chu-isometric) scattering map constrained by conservation of covariant Chu-energy flux. In the canonical Shu-Osher problem, CCKD characterizes the shock as a thermo-acoustic lens, mathematically demonstrating that the transfer of entropy fluctuations into sound follows a geometric blue-shift ($k_{\mathrm{out}}=\Lambda k_{\mathrm{in}}$) analogous to gravitational blue-shift. Thus, while the mean flow produces entropy across the shock, the fluctuation mapping is information-preserving on the retained subspace; practical information loss arises from noise, truncation, and model mismatch, not shock physics.