Pressure-Mediated Gravity v6: An Emergent Superfluid Vacuum Framework for Galactic Dynamics

═══════════════════════════════════════════════════════════════════Pressure-Mediated Gravity: An Emergent Superfluid Vacuum Framework for Galactic Dynamics═══════════════════════════════════════════════════════════════════ AUTHOR: Dr. Mo Jerrow, Independent ResearcherDATE: February 2026LICENSE: CC BY 4. 0VERSION: 6 (Current) ═══════════════════════════════════════════════════════════════════ We present PMG v6, a revised and self-consistent formulation of Pressure-Mediated Gravity, an emergent gravity framework modelling spacetime as a superfluid condensate governed by a logarithmic Gross-Pitaevskii Lagrangian. Key advances: (i) the interpolation function μ (x) =x/√ (1+x²) is derived from the AQUAL variational principle applied to the logarithmic GP free energy; (ii) the characteristic acceleration scale a₀=cH₀/2π≈1. 05×10⁻¹⁰ m s⁻² is derived from first principles; (iii) Lorentz covariance is resolved through a covariant superfluid construction in which the PMG action is a scalar under all general coordinate transformations. Open challenges (Bullet Cluster lensing offset, CMB acoustic peaks) are identified with precise falsifiability conditions. PMG is presented as a research programme making falsifiable predictions at galactic scales. ═══════════════════════════════════════════════════════════════════ What's New in Version 6 This version represents a substantive theoretical advance over v4: ✅ DERIVED: Interpolation function μ (x) = x/√ (1+x²) from AQUAL variational principle applied to logarithmic Gross-Pitaevskii free energy✅ IDENTIFIED: AQUAL free function ℱ (Q) = √Q·√ (1+Q) − arcsinh (√Q) analytically✅ RESOLVED: Lorentz covariance via covariant superfluid construction (Section VIII) ; gravitational waves propagate at c (consistent with GW170817) ✅ EXPANDED: Bullet Cluster analysis with sterile neutrino (~2 eV) hybrid proposal and precise falsifiability condition✅ MAPPED: CMB challenge to Skordis no mathematical rigor || v2 | Superseded | First field equations; undefined depletion term || v2. 1 | Superseded | Brans-Dicke scaling argument (now retracted) || v4 | Superseded | Analog gravity framing; documented limitations || v6 | ACTIVE | AQUAL derivation; covariant construction; falsifiability conditions | See the main manuscript for detailed theoretical development and Section VI for precise falsifiability conditions. ═══════════════════════════════════════════════════════════════════What PMG v6 Achieves═══════════════════════════════════════════════════════════════════ Despite remaining open challenges, this version successfully demonstrates: • Theoretical derivation: μ (x) = x/√ (1+x²) is no longer imposed by hand but derived from the AQUAL variational principle applied to the log-GP free energy, with ℱ (Q) identified analytically • Covariant foundation: The PMG action is manifestly Lorentz-covariant through promotion of the condensate phase to a scalar field φ with X = −g^μν ∂_μφ ∂_νφ; gravitational waves propagate at c • Empirical consistency: Asymptotically flat rotation curves and exact BTFR slope (v⁴ ∝ Mb) recovered with no free parameters beyond baryonic mass and cosmologically-derived a₀ • Falsifiability: Precise conditions enumerated for Bullet Cluster (numerical simulation must reproduce ~720 kpc lensing offset) and CMB (vector-extended action must match Planck TT spectrum) • Reproducibility: Complete Python implementation provided with AQUAL free function, interpolation function, and BTFR prediction ═══════════════════════════════════════════════════════════════════Critical Open Challenges (Documented in v6) ═══════════════════════════════════════════════════════════════════ 1. Bullet Cluster Lensing Offset: PMG in its current form predicts lensing should track baryonic gas, but observations show offset ~720 kpc toward galaxies. The sterile neutrino hybrid proposal (~2 eV) is identified as the most viable near-term resolution, but quantitative simulation is required. Falsifiability condition: numerical solution must reproduce observed offset within uncertainty. 2. CMB Acoustic Peaks: PMG has no current mechanism to reproduce the ~7 acoustic peaks in the CMB power spectrum. The Skordis & Zlosnik (2021) existence proof provides a blueprint, but implementation within the log-GP framework remains to be demonstrated. Falsifiability condition: vector-extended PMG must match Planck 2018 TT spectrum across 2 ≤ ℓ ≤ 2500. 3. Parameter Degeneracy: Gₑff = γκ/4πρ₀ involves three independent parameters. The healing length ξ is proposed as an observable target, but laboratory or sub-galactic constraints on deviations from 1/r² at scale ξ have not yet been obtained. 4. ~12% a₀ Discrepancy: The PMG prediction a₀ = cH₀/2π ≈ 1. 05×10⁻¹⁰ m s⁻² lies ~12% below the empirical MOND value. A rigorous treatment of vacuum acoustic modes near the Hubble horizon may modify the 2π factor, but this calculation is deferred to future work. ═══════════════════════════════════════════════════════════════════Invitation for Collaboration═══════════════════════════════════════════════════════════════════ This version is released to invite collaboration on the unresolved challenges above. Feedback from the community on: • Numerical implementation of covariant PMG for Bullet Cluster simulations• Vector field extension following Skordis & Zlosnik blueprint for CMB compatibility• Laboratory or sub-galactic tests constraining healing length ξ• Rigorous derivation of O (1) corrections to a₀ = cH₀/2π from vacuum mode structure. . . would be invaluable. The author actively welcomes constructive criticism, collaboration, and independent verification of the numerical implementation. ═══════════════════════════════════════════════════════════════════References & Context═══════════════════════════════════════════════════════════════════ This work builds on established emergent gravity and modified gravity literature: • Bekenstein & Milgrom (1984): AQUAL framework for MOND• Blanchet & Le Tiec (2009): Quasi-static limit procedure for connecting scalar field theories to MOND• Skordis & Zlosnik (2021): Relativistic MOND theory reproducing CMB peaks• Berezhiani & Khoury (2015): Superfluid dark matter framework• Zloshchastiev (2011), Avdeenkov & Zloshchastiev (2011): Logarithmic Gross-Pitaevskii potential for stable condensates See main manuscript for complete reference list. ═══════════════════════════════════════════════════════════════════Files Included═══════════════════════════════════════════════════════════════════ • PMGᵥ6₂4. 2. 26. pdf — Complete manuscript with all sections, figures, and Python code• rotationcurves. pdf — Figure 2: Galactic rotation curves for two baryonic masses• btfrᵣelation. pdf — Figure 3: Baryonic Tully-Fisher Relation recovery• pmgcodeᵥ6. py — Python implementation with AQUAL free function and interpolation ═══════════════════════════════════════════════════════════════════Keywords═══════════════════════════════════════════════════════════════════ emergent gravity, superfluid vacuum, MOND, AQUAL, Gross-Pitaevskii, BTFR, galaxy rotation curves, dark matter alternatives, covariant gravity, Bullet Cluster, CMB, Lorentz invariance ═══════════════════════════════════════════════════════════════════Author's Position Statement═══════════════════════════════════════════════════════════════════ "This work represents an ongoing exploration of how spacetime geometry and gravitational phenomenology might emerge from a superfluid vacuum condensate governed by a logarithmic Gross-Pitaevskii Lagrangian. The explicit documentation of limitations, open challenges, and precise falsifiability conditions is intentional—to demonstrate scholarly integrity, to guide future researchers toward productive directions, and to invite rigorous testing of the framework. Criticism, collaboration, and independent verification are actively welcomed. " — Dr Mo Jerrow, Independent Researcherdr. jerrow@gmail. com ═══════════════════════════════════════════════════════════════════Citation═══════════════════════════════════════════════════════════════════ Jerrow, M. (2026). Pressure-Mediated Gravity v6: An Emergent Superfluid Vacuum Framework for Galactic Dynamics Preprint. Zenodo. https: //doi. org/10. 5281/zenodo. DOIWILLAPPEAR ═══════════════════════════════════════════════════════════════════