Dynamical Confinement from Temporal Memory: Infrared Gluon Dynamics and Chiral Transition in HDG

We present an analytic infrared solution of the gluon Dyson–Schwinger equation (DSE) within the History-Dependent Gravity (HDG) framework. In this approach, temporal memory effects are encoded in a spectral density ρ(μ;T)ρ(μ;T), which is inherited directly from the quark sector and modifies the quark-gluon vertex. We demonstrate that this single memory kernel dynamically generates a Gribov-type infrared gluon propagator, D(p2;T)∝p2/(p4+γ4(T))D(p2;T)∝p2/(p4+γ4(T)), where the confinement scale γ(T)γ(T) is determined self-consistently by the spectral density and the infrared quark mass. By explicitly evaluating the Euclidean Schwinger function via contour integration, we prove analytically that reflection positivity is violated for all temperatures below the chiral transition, establishing that the gluon cannot exist as a physical asymptotic state. Furthermore, we derive a linearly rising static quark-antiquark potential V(r)∼σrV(r)∼σr, with a string tension σ(T)∝γ2(T)σ(T)∝γ2(T). Crucially, because the same spectral kernel governs both the quark gap equation and the gluon DSE, the confinement scale vanishes exactly at the chiral restoration point, yielding a parameter-free dynamical locking of the critical temperatures: Tc=TχTc=Tχ. This work provides a unified, first-principles description of confinement and chiral symmetry breaking driven by non-Markovian temporal memory, offering concrete, testable predictions for finite-temperature lattice QCD simulations.