Pressure-Mediated Gravity v6: Rotation curves, lensing and cosmology in a covariant superfluid vacuum framework — code and preprint

This repository contains the preprint manuscript and Python figure-generation code for the companion paper to Pressure-Mediated Gravity v6 (PMG v6; Jerrow 2026). PMG is an emergent-gravity framework in which gravitational attraction arises from pressure gradients in a superfluid vacuum medium governed by a logarithmic Gross–Pitaevskii Lagrangian. The present paper provides the first phenomenological application of the v6 framework to three observational regimes: (1) Galactic rotation curves and the Baryonic Tully–Fisher Relation (BTFR). The v6 interpolation function μ (x) = x / √ (1 + x²), derived from the AQUAL variational principle applied to the log–GP free energy, is used throughout. The corrected BTFR prediction v⁴flat = G Mb a₀ is demonstrated across five decades of baryonic mass. An earlier erroneous velocity claim (v4: 220 km/s for Mb = 6×10¹⁰ M☉) is explicitly corrected to the honest value of ~170 km/s. (2) Gravitational lensing in the weak-field limit. Vacuum pressure contributions to the lensing potential, arising from the covariant P (X) k-essence action of PMG v6, enhance deflection angles beyond the baryon-only Newtonian prediction. The Bullet Cluster remains an open challenge; the vacuum-memory argument is explicitly rejected on timescale grounds. (3) Background cosmological expansion. The Friedmann equation with the PMG vacuum as an effective dark energy fluid (w ≈ −1) is presented. The absence of a mechanism for the CMB acoustic peak structure is openly acknowledged as a priority for future work, with the Skordis–Zlosnik (2021) vector-field extension identified as the recommended path forward. The paper is framed explicitly as a research programme rather than a completed alternative to ΛCDM. All three major open challenges (Bullet Cluster, CMB, full SPARC re-fit) are enumerated transparently. The acceleration scale a₀ = cH₀/2π ≈ 1. 05 × 10⁻¹⁰ m s⁻² is derived from first principles (Jeans instability at the Hubble scale) and agrees with the empirical MOND value within ~12%. The covariant PMG action yields post-Newtonian parameters β = γ = 1 and α₁ = α₂ = 0, consistent with Cassini (Bertotti et al. 2003) and GW170817 (Abbott et al. 2017) constraints.