History-Dependent Gravity Spectral Kernel in the DSE/BSE Framework: Pion Sector and a Numerical No-Go Behavior for Rainbow-Ladder Vector Mesons

We investigate the application of the History-Dependent Gravity (HDG) spectral kernel to non-perturbative Quantum Chromodynamics (QCD) and hadron spectroscopy. Using the Dyson-Schwinger and Bethe-Salpeter equations (DSE/BSE) formalism within the rainbow-ladder (RL) truncation, we map the HDG temporal memory kernel onto the quark-gluon interaction. We rigorously demonstrate that the HDG kernel preserves the Axial Ward-Takahashi Identity (AXWTI). By fixing the kernel parameters (αeff=30.0αeff=30.0, ω=0.5ω=0.5 GeV) via the Goldstone theorem condition, the model successfully predicts a realistic pion decay constant (fπ≈93fπ≈93 MeV) and satisfies the Gell-Mann--Oakes--Renner (GMOR) relation, yielding a physical dynamical quark mass B(0)≈0.38B(0)≈0.38 GeV. However, we identify a fundamental limitation in the vector channel. We establish a numerical no-go behavior: within the RL truncation, no Yukawa-type HDG kernel can simultaneously produce realistic dynamical chiral symmetry breaking and bind the ρρ-meson. Attempts to enhance the infrared interaction to force ρρ-meson binding drive the dynamical quark mass to unphysical values (B(0)>3B(0)>3 GeV), which catastrophically suppresses the two-quark propagator structure in the BSE kernel. This tension indicates that the temporal nonlocality encoded in the HDG framework cannot be consistently realized solely through propagator dressing. A viable HDG realization must avoid IR over-enhancement and requires beyond-rainbow-ladder extensions, specifically the incorporation of memory effects into dressed quark-gluon vertices. This upload includes the LaTeX manuscript and the Python numerical implementation used to generate the results.