The UAT Lagrangian Formulation: Derivation of the Quantum Braking Action, Causal Parameters, and CMB Acoustic Peak Predictions

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.This update (v2) adds the resolution of the effective matter density problem within the UAT/UCP framework. The new document establishes the identity η ≡ κ_crit = 4.978, demonstrating that the Lagrangian vacuum minimum and the Ivancho causal limit are the same fundamental constant. Numerical integration of the Klein-Gordon equation reveals that the scalar field φ is frozen at ~9% of its minimum —an open puzzle that does not invalidate the mechanism. The causal membrane integral M_p = ∮ Ψ_8phase · κ_crit dt is derived from the surface term of the UAT action, showing that mass amplification depends on κ_crit as a structural constant, not on the dynamical field value. The amplification factor A = κ_crit × φ × (1 - 0.07) ≈ 7.49 yields Ω_m^(eff) = 0.3133, in 0.55% agreement with Planck 2018. Supplementary Python scripts for the Klein-Gordon integration and λ parameter sweep are included. Author: Miguel Ángel Percudani [email protected]