Description:This repository contains the complete theoretical and mathematical foundation of the Unified Applicable Time (UAT) and Unified Causal Principle (UCP) framework. It transitions the UAT from a phenomenological model resolving the Hubble tension to a rigorous scalar-tensor field theory. The core of this release is the derivation of the UAT Lagrangian, demonstrating that the dynamics of cosmic expansion are governed by a causal scalar field ϕ non-minimally coupled to gravity. Repository Contents: The UAT Lagrangian (Main Paper): Derives the exact values of the non-minimal coupling (ξ=−0.2810) and the vacuum self-coupling (λ=3.08×10 −112 in Planck units). These parameters are not arbitrary but rigidly fixed by the causal limit (κ crit =4.978) and the 7% thermal calibration margin at recombination. Observational Predictions (CMB): Contains the theoretical predictions for the displacement of the CMB acoustic peaks. The UAT framework predicts a structural shift of Δℓ/ℓ=+5.85% and an alteration in the shift parameter R to 1.7750, driven by the modified sound horizon (r d ≈141.00 Mpc). Parameter Derivation Code: The Python script executing the numerical closure of the UAT parameters. Methodological Note for Future Implementation:The analytical closure presented here forms the definitive theoretical groundwork of the UAT. Full dynamic integration and confrontation with high-precision cosmological datasets (like the full Planck likelihoods) exceed the capabilities of standard numerical libraries. A complete validation pipeline will require the explicit integration of this scalar-tensor Lagrangian into advanced Boltzmann Solvers (such as CLASS or CAMB). This repository establishes the exact theoretical metrics and target parameters required for those future modifications. Author: Miguel Ángel Percudanimiguel_percudani@yahoo.com.ar