Dynamics of the QCD Vacuum Condensate Amplitude in Dense Matter and Cosmology

Within the theoretical frameworks of Null-Vector Gravity (NVG) and the Vacuum Mass Fraction (VMF), the vacuum condensate is described by a complex scalar field Φ (x). Using a polar representation, the field is parameterized by a real radial mode W (x), which governs the in-medium effective mass of hadrons M*, and an angular mode θ (x) representing the Goldstone phase. This work provides a rigorous mathematical derivation of the classical equations of motion for the real amplitude field W (x) and its gauge-invariant coupling to the baryon current and vector meson fields in a dense nuclear medium. The coupling parameters are calibrated via QCD pion-nucleon ΣπN = 44 MeV and strange σₛN = 30 MeV sigma-terms. The energy-momentum tensor is derived, and its Friedmann-Lemaître-Robertson-Walker (FLRW) reduction is performed. We demonstrate that the melting of the vacuum condensate W → 0 under extreme compression leads to a violation of the Strong Energy Condition (SEC). This generates a quadratic density correction to the energy density, yielding a modified Friedmann equation that triggers a smooth cosmological bounce and eliminates the Big Bang singularity. The limits of the mean-field approximation are discussed using the Ginzburg criterion (Gi ≈ 10⁻³), local spatial fluctuations, and Coleman-Weinberg radiative corrections. The vacuum condensate melting transition occurs at a critical density nB ≈ 2. 05 n₀, which is far below the Planck scale, providing a macroscopic QCD-scale mechanism for singularity avoidance. Repository: https: //github. com/infosave2007/vmfZenodo: https: //zenodo. org/records/20214457