The Vacuum Supersolid Hypothesis: A Complete Mathematical Framework for the Emergence of Spacetime, Gauge Fields, and Quantum Mechanics from a Single Elastic Order Parameter

We present and systematically develop a rigorous theoretical framework in which the physical vacuum is identified as a quantum supersolid medium with nonlinear elastic properties. Within this framework, the speed of light c is interpreted as the phase speed of transverse shear waves in the vacuum medium, dark energy is recog- nised as the elastic restoring pressure arising from the bulk modulus K under cosmic volumetric strain, and general relativity and quantum field theory emerge as low- energy approximations in different energy scales and excitation modes. We prove that the longitudinal wave speed vL defines the absolute upper bound of causality, that c is merely its transverse subset, and that superluminal travel does not generate closed timelike curves, thereby excluding the grandfather paradox mathe- matically. From first principles, we derive the complete elastic constitutive relations, dual-wave dispersion laws, a modified Friedmann equation, the threshold condition and cone-angle formula for vacuum Cherenkov radiation, and a modified prediction for the large-scale structure power spectrum. Systematic comparisons with ob- servational data from DESI, DES, Planck, LIGO/Virgo, and PVLAS demonstrate compatibility with all high-precision observations within error bounds, while pro- viding a natural physical explanation for the dynamical behaviour of dark energy. The framework introduces no free parameters beyond known physics, and all predictions are explicitly falsifiable. In addition, we extend the framework to ordinary condensed-matter media such as water and glass, establishing that their acoustic, viscoelastic, and structural relaxation phenomena are projections of the same elastic constitutive equations at finite cutoff parameters.