Optimal limits on weak integrability breaking and protected thermal memory near qutrit exchange

Although integrability does not universally require a continuous one-site symmetry, we rigorously prove that every jointly analytic, regular Yang-Baxter deformation of the qutrit exchange interaction necessarily retains a nontrivial, analytically varying one-site charge. Breaking this local symmetry imposes a fundamental physical constraint on approximate conservation, governed by the optimal uniform bound δ³ le Cvarepsilon that explicitly relates the minimal one-site symmetry defect δ to the local current-conservation residual varepsilon. While breaking all one-site charges strictly forbids an exact integrable completion, an optimally compensated nearest-neighbor interaction saturates this cubic limit and anomalously extends the guaranteed infinite-temperature energy-current correlation window to order |λ|^-3 in the perturbation strength λ. Furthermore, we reveal a fundamental resonance obstruction for intrinsic conversion perturbations that strictly prevents any exact first-order repair of a broken one-site charge on any finite ring. Nevertheless, we demonstrate that the complete eight-dimensional charge memory matrix remains thermodynamically protected and approaches the identity for timescales t=o\(|λ|^-3/2\), a robust feature of the full infinite-temperature dynamics when the thermodynamic limit is taken before weak coupling.