A VP Cosmological Model of the Hubble Tension, Structure Growth, Dark Matter, and CLASS/CAMB Falsification: From the Constitutive Field to BAO, Lensing, Halo Phenomenology, and MCMC Closure

This paper presents the "Universe in Terms of Planck Pressure" (VP) cosmological model, a unified field theory designed to resolve the Hubble (H0) and structure growth (S8) tensions, while explaining dark matter phenomenology. Based on a complex order parameter Psi whose amplitude locks the metric clock field and pressure, the VP model introduces a covariantly averaged energy density, restoring dimensional balance in the Friedmann equations. An early VP energy component locally reduces the sound horizon, increasing the CMB-inferred H0. Concurrently, a late-time suppression window reduces structure growth, directly addressing the S8 tension through a "tug-of-war" mechanism with cold dark matter. The perturbation sector is derived fundamentally: anisotropic stress arises from spatial Korteweg gradients and Israel-Stewart relaxation. Dark matter is described through a hybrid approach: standard CDM is retained in the background, while galactic scales are dominated by VP medium polarization, explaining MOND-like rotation curves and closing lensing via a covariant Weyl potential. The provided C/Fortran modules enable direct CLASS/CAMB testing, ensuring strict falsifiability.