Abstract: This work concludes the primary trilogy of the Metric Tension Adjustment Model (MATM) by presenting the v3.2.0 Unified Edition. While Part I introduced metric tension and Part II defined subatomic unification, Part III provides the formal mathematical framework and the algorithmic validation required for empirical testing. Central to this version is the Master Equation (Ψσ), a functional adjustment that balances vacuum energy density against the Inertial Ballast Operator. We demonstrate that the "Impossible Early Galaxy" paradox (e.g., JADES-GS-z13-0) is resolved by a quadratic field law, where primordial metric tension results in an inertial excess factor of 1803x. Key technical contributions in this release include: Computational Validation: Accompanying Python scripts that simulate cosmic evolution from the Big Bang to the Geometry of Silence (Λ∞). Parametric Locality: A formal definition of cosmic position as a function of evolving internal constants (G, σ), rather than external coordinates. Observational Roadmap: Specific falsifiability protocols linked to the BINGO radiotelescope (21cm line) and Lunar Laser Ranging (secular decay of G). This framework offers a deterministic, mechanical alternative to ΛCDM, resolving the Hubble Tension by treating matter as a structural necessity for spacetime stability. Ethical Disclosure: The theoretical core and mathematical derivations remain the original intellectual property of the author. Generative AI tools were employed for technical proofreading, linguistic optimization, and the refinement of visualization scripts to ensure international scientific standards.