Non-Semisimple Integrability and Dynamical Limits

Abstract: We investigate Jordan structure and dynamical limits in six families of integrable spin-$\frac{1}{2}$ chains obtained from the boost automorphism method. Local operator algebras connect nilpotent evolution on finite periodic chains to central defects in spectral additivity, polynomial unipotent $R$-matrices and constant braid limits. For conservative endpoint erosion, we determine the characteristic and minimal polynomials and show that the Jordan partition can change while both polynomials remain unchanged. We also identify the deletion limits that distinguish the parity obstruction to removing signs by diagonal similarity in Class 5 from the parity nilpotent dependence in Class 6. Matching expansions and fragmentation recursions yield exact amplitudes and survival probabilities for processes with killing. In addition, we establish criteria excluding conservative Markov realizations and determine which conservative completions of specified reaction densities admit regular $R$-matrices of difference form.