History-Dependent Gravity XXVI:Confinement as Memory-Induced Reversal of the Polyakov-Loop Renormalization Group Flow

This paper presents the 26th installment of the History-Dependent Gravity (HDG) series, focusing on the nonperturbative mechanism of color confinement in pure Yang-Mills theory. Utilizing the functional renormalization group (FRG), we analyze an effective average action augmented by a scale-dependent temporal memory kernel Kk(□)Kk(□). While the FRG flow naturally admits an infrared fixed point yielding a confining gluon propagator D(p)∼Aˉ/p4D(p)∼Aˉ/p4 and a linear static potential V(r)=σrV(r)=σr, we rigorously prove that this p−4p−4 enhancement is insufficient to trigger confinement on its own. Specifically, it merely rescales the one-loop Weiss potential, preserving its deconfining minimum. The breakthrough of this work lies in demonstrating that the temporal memory kernel K(ω)∼Aˉ/ω2K(ω)∼Aˉ/ω2 fundamentally deforms the FRG beta function of the Polyakov-loop effective potential. This memory-induced contribution, Δβa2HDG>0Δβa2HDG>0, actively reverses the renormalization-group trajectory. It drives the quadratic center-symmetric coupling a2(k)a2(k) from negative values in the ultraviolet (deconfined phase) to positive values in the deep infrared (confined phase, ⟨L⟩=0⟨L⟩=0). We provide a rigorous derivation of this flow reversal, supported by scaling analysis and Litim regulator evaluations. Phenomenologically, the framework predicts a string tension of σ1/2≈430 MeVσ1/2≈430 MeV and a deconfinement transition temperature Tc≈270 MeVTc≈270 MeV, both in excellent agreement with state-of-the-art lattice QCD simulations. This establishes temporal nonlocality not merely as a correction, but as the primary driver of the confinement phase transition.