Vacuum Condensate Tomography: Reconstructing Emergent Acoustic Geometry and EFT Parameters from Metrology

This work formulates vacuum condensate tomography as a well-posed inverse problem in which emergent spacetime is reconstructed operationally from signal-based metrology. The vacuum is treated as a condensate medium, and geometry arises as an effective description of excitation propagation rather than a fundamental input. A unified reconstruction pipeline links redshift, time-of-flight, and loop observables to an effective acoustic geometry, followed by inversion to background fields and global inference of effective field theory parameters. The gravitational sector emerges as a medium response governed by a Poisson-type equation with controlled higher-derivative corrections, providing a direct link between macroscopic observables and microscopic structure. A central result is the identification of a structural degeneracy between effective coupling and background energy in single-state data. This is resolved through multi-state protocols that enable consistent parameter separation. Independent ultraviolet calibration is incorporated via near-horizon scaling, allowing discrimination between genuine higher-order effects and reconstruction artefacts. The framework admits a microscopic closure in which condensate degrees of freedom act as effective moduli, with Wilson-line holonomy and radion-like modes controlling the state dependence of the coupling sector, providing a dynamical origin for variations in effective interaction strength. Reconstruction of density-dependent parameters is conditioned on access to proper-time-sensitive clock channels, while kinematic observables alone constrain only the conformal class of the geometry. Validation on synthetic datasets demonstrates robustness to noise, geometric perturbations, and model mismatch, and provides stability diagnostics that distinguish ultraviolet structure from operator-level and discretization artefacts. Overall, the work establishes emergent gravity as a quantitatively testable inverse problem, unifying analogue gravity, condensate effective field theory, and metrological reconstruction within a single operational framework.