The Master Closure Relation: A Unified Dynamical Theory of Varying Physical Couplings

We derive the master closure relation of the unified scalar-imprint framework: an exact, action-based equation linking the observable drift parameter λ of Newton’s constant Gto the microscopic dynamics of the scalar-imprint field Φ. Starting from the Euler–Lagrange equations of the full scalar-tensor action, we show that for the exponential coupling F(Φ) = eβGΦ , ˙ Φ λ=−βG H, which in the slow-roll regime becomes β2 Gm2 ΦΦ0 λ= 3H2 0 . Combined with the sector-saturation constraint λ= κ/17, this constitutes a closed, self-consistent set of equations with no free parameters beyond (βG,mΦ,Φ0), all of which are bounded by current observations. We further derive the synchronisation of all Standard Model couplings under the master relation, showing that G, αem, αs, and Yukawa couplings drift in a locked, correlated pattern governed by the ratio εi = βi/βG. This synchronisation prevents disintegration of atomic structure during the drift process and is itself a falsifiable prediction. We present the complete parameter space consistent with LLR, BBN, and atomic clock constraints.