This paper presents the unified theoretical architecture of the LambdaVSC one-parameter cosmological model. Grounded in five foundational axioms including the Principle of Relativity, the framework connects general relativity, quantum theory, particle physics, and dark-sector cosmology via a single vacuum scalar field. In the classical limit, it derives the Einstein field equations; in the weak-field flat-spacetime limit, it derives the Klein-Gordon, Schrodinger, and Dirac equations, naturally yielding canonical commutators and microcausality. Using the inverse fine-structure constant alpha^-1(m_e) = 137.035999177 as the unique input, the model fixes its sole free parameter N = 1.037111 x 10^13. This parameter chain yields consistent low-energy electron and Higgs sectors, a sterile neutrino window, and dark universe dynamics compatible with Planck PR4 and DESI. The framework resolves the W-boson 7 sigma anomaly consistently with electroweak constraints and explores the JWST high-redshift tension via nonlinear structure formation. Core derivations are fully cross-audited with Lean 4, Python, and Maxima, offering a falsifiable quantum gravity candidate that addresses spacetime singularities, the cosmological constant, and dark universe origins.