Pressure-Mediated Gravity V: Covariant Lensing Formalism and Galaxy-Scale Tests

In Pressure-Mediated Gravity (PMG), spacetime emerges from a superfluid vacuum condensate governed by a covariant kk-essence action. This paper derives the weak-field lensing equation from first principles, finding ∇2Φlens=4πG(ρb+5ρv/2)∇2Φlens=4πG(ρb+5ρv/2), where the vacuum equation of state Pv=c2ρvPv=c2ρv enhances lensing by a factor of 5/25/2 relative to vacuum density alone. Four critical physics corrections relative to the previous preprint are incorporated: (i) an explicit derivation of the perfect-fluid approximation from the kk-essence stress-energy tensor; (ii) a corrected treatment of the gravitational slip Φ≠ΨΦ=Ψ, showing γPMG=1+3ρv/ρb>1γPMG=1+3ρv/ρb>1, with a deviation from unity of order 10−1110−11 at galaxy-halo scales (undetectable by current surveys); (iii) an honest derivation of the coupling constant κκ as a model parameter fixed by two independent conditions; and (iv) a corrected weak-lensing calculation using baryonic mass MbMb directly. The corrected calculation yields a tangential shear γt(50 kpc)≈0.002γt(50 kpc)≈0.002, a factor of ∼10∼10 below the CFHTLenS anchor of 0.021±0.0040.021±0.004—identified here as a genuine quantitative failure requiring additional physics in Paper VI. The predicted slope γt∝1/r2γt∝1/r2 (from constant MbMb) falls faster than the isothermal 1/r1/r profile. We further show that the Bullet Cluster centroid offset requires ∼2∼2 eV sterile neutrinos, as the vacuum fluid tracks baryons adiabatically. Finally, environmental variation in a0a0 predicts ∼5%∼5% lensing scatter in the dynamical-mass regime. This work sharpens the quantitative picture of PMG: while the covariant formalism is self-consistent, the weak-lensing amplitude deficit and early-universe condensate density remain central challenges for future resolution.