ETCQ IV: Global consistency on strong lenses and galaxies Context

This fourth volume of the ETCQ framework (Energy–Trame–Constant–Quantum) presents the complete field formulation underlying the previously empirical energetic model.The derived Lagrangian density couples a scalar field (χ) and a vector field (u), describing the compressive and rotational responses of the quantum lattice.Under quasi-stationary and axial symmetry, it reduces to coupled Helmholtz-type field equations, reproducing the observed V4-L law across the SPARC galaxy sample.The universal energetic slope $p_0 ≃ 0.60$ and the temporal coherence factor $χ_c ≃ 0.58$ emerge naturally from the same α-bounded dynamics, where $\alpha = 1/137$ defines the lattice rigidity.Without any local tuning, the model fits baryonic dynamics from planetary scales to galactic halos, and provides a physical interpretation of the Hubble-tension and AGN RM delays as manifestations of the local time-slowing induced by the lattice saturation.ETCQ-V4-L thus unifies gravitational, relativistic, and quantum regimes within a single α-saturated field framework