Yang-Baxter Integrability and Exceptional-Point Structure in Pseudo-Hermitian Quantum Impurity Systems

We develop a mathematically controlled framework for Yang--Baxter integrability in pseudo-Hermitian quantum impurity systems arising from periodic driving of a Dirac-like bath. The effective impurity Hamiltonian possesses a dynamically generated $\PT$ symmetry and exhibits exceptional points (EPs) where it becomes non-diagonalizable. We construct the Yang--Baxter generator as a rank-one operator on the two-particle contact space, built from biorthogonal impurity eigenvectors, and prove that it satisfies the Temperley-Lieb relations. Its standard Baxterization gives an $R$-matrix, an RLL relation, an RTT structure,and a commuting family of transfer matrices. At the exceptional point(EP), the semisimple biorthogonal eigenvector construction is replaced by a Jordan-chain contact vector, while the Hamiltonian itself develops a nilpotent Jordan block. Within this framework we derive biorthogonal Bethe equations and show that the Gaudin matrix becomes defective at the EP, establishing that the smallest singular value $\sigma_N(G)\to0$ at the EP while remaining $\OO(1)$ at the Kondo critical point,providing a sharp algebraic diagnostic. We further prove that Bethe rapidities exhibit square-root coalescence and $\mathbb{Z}_2$ monodromy at the EP, reflecting the underlying Jordan structure, and that the effective pseudo-Hermitian Hamiltonian emerges from the periodically driven microscopic system by adiabatic coarse graining of off-shell angular-momentum modes, with corrections controlled by the auxiliary-mode gap.