Modified Weak-field Dynamics (MDW 2.2): Formal Derivations of Geometric Constants, Observational Tests, and Falsifiability

This manuscript (Part VII) provides the formal mathematical and physical foundations for the Modified Weak-field Dynamics (MDW 2.2) framework, offering a zero-free-parameter alternative to the Dark Matter paradigm. Key theoretical developments and observational tests presented in this work include: Mathematical Stability: Derivation of a ghost-free scalar-tensor action with a conformal coupling of xi = 1/6, ensuring physical consistency and stability in the Einstein frame. Geometric Acceleration Scale: A formal derivation of the acceleration constant a0 = (H0 * c) / sqrt(32) based on the density of states (32 real degrees of freedom) within a 5D manifold projection. This prediction matches SPARC galaxy data with 97.5% accuracy. Euler Mass Relation: Using entropy maximization principles, I derive the fundamental relation M_eff = M_b * (1 + 1/e), providing an excellent fit for rotation curves across 175 SPARC galaxies with an RMS error of only 5.92 km/s. Solar System Screening: Numerical proof of a screening mechanism where the fifth force is exponentially suppressed (factor approx 10^-50) in high-density regions, satisfying all current Weak Equivalence Principle (WEP) constraints. Euclid Mission Forecast: A definitive quantitative prediction for the 2027 Euclid mission, specifying a weak lensing signal slope of Delta_Sigma proportional to R^-2, which distinguishes the model from the R^-1 slope predicted by standard NFW dark matter profiles. Falsifiability: Establishment of rigorous criteria for model invalidation, including direct dark matter particle detection and statistical inconsistency of the geometric a0 relation. This work marks the transition of MDW 2.2 from a conceptual model to a fully formalized, falsifiable physical theory.