The Minimal Lagrangian Formulation and Experimental Falsifiability of the TGIU Framework

The foundational TGIU field equation is:\begin{equation}\nabla_{\mu}J_{\mathrm{CST}}=\lambda_{\mathrm{TEF}}(\Phi-S),\label{eq:TGIUfundamental}\end{equation}where:\begin{itemize}\item $J_{\mathrm{CST}}$ — the real informational rotation operator (Curvature--State Transform);\item $\Phi$ — informational coherence density;\item $S$ — informational entropy density;\item $\lambda_{\mathrm{TEF}}$ — Trans--Equilibrium Force coupling constant.\end{itemize} %------------------------------------------------------------\subsubsection{1. Constructive Principle} \noindentEquation (\ref{eq:TGIUfundamental}) expresses a first-order informational continuity. To generate it variationally, one seeks a Lagrangian whose Euler–Lagrange equations reproduce this balance. %------------------------------------------------------------\subsubsection{2. The Minimal TGIU Lagrangian} \noindentThe simplest sufficient form is:\begin{equation}\boxed{\mathcal{L}_{\mathrm{TGIU}}=\frac{1}{2}\,(\nabla_\mu J_{\mathrm{CST}})(\nabla^\mu J_{\mathrm{CST}})-\lambda_{\mathrm{TEF}}\,J_{\mathrm{CST}}(\Phi - S).}\end{equation}