Matrix thermodynamic uncertainty relation for non-Abelian charge transport.

Thermodynamic uncertainty relations (TURs) bound current precision by entropy production, but transport of noncommuting (non-Abelian) charges lacks a single joint classical record of all current components. We derive an operational, process-level matrix TUR for this setting from the entropy production Σ=D(ρ_{SE}^{'}∥ρ_{S}^{'}⊗ρ_{E}). Isolating the bath divergence D_{bath}=D(ρ_{E}^{'}∥ρ_{E}), we prove a fully nonlinear, saturable lower bound valid for arbitrary current vectors Δq: D_{bath}≥B(Δq,V,V^{'}), where the bound depends only on the transported-charge signal Δq and on the pre- and postcollision covariance matrices V and V^{'} reconstructed from separate probe measurement settings. In the small-fluctuation regime D_{bath}≥1/2Δq^{T}V^{-1}Δq+O(∥Δq∥^{4}), while beyond linear response the full integral bound remains valid. Numerical strong-coupling qubit collisions illustrate the bound and its near-saturation using only local measurements on the bath probe.