Irreversibility from Self-Reference: Gradient Flow and an H-Theorem for a Self-Referential Statistical Operator Framework

This paper extends the self-referential statistical operator framework introduced in the companion work “Emergence of Tsallis Statistics from a Self-Referential Nonlinear Operator: A Variational Framework” (Zenodo DOI: 10.5281/zenodo.20151216). We investigate perturbative stability beyond the local kernel approximation, convergence of the iterative dynamics, and establish an H-theorem within the local kernel approximation through a gradient-flow formalism. Numerical evidence for monotonic free-energy dissipation and convergence is presented, together with an analysis of the non-perturbative self-coupling regime and re-entrant phase behavior. The work provides a dynamical and irreversibility foundation for the proposed self-referential nonextensive statistical mechanics framework.