State convertibility and fluctuation theorems from a dynamical reference: majorization meets martingales

State convertibility represents a fundamental concept used to determine whether a transformation is possible given a specific set of resources. Within the field of Thermodynamics, where physical process are required to preserve a reference state typically in microcanonical or canonical form, this translates into the notions of majorization and thermo-majorization —criteria that require constructing and comparing state-dependent Lorenz curves. In this work, we firstly unify and extend these notions to an arbitrary and possibly time-dependent reference distribution $g(t)$, introducing the concept of $g(t)$-majorization; we then introduce a dual picture whereby state convertibility is turned into a one-dimensional convex-order problem, which allows us to demonstrate that a transition is admissible if and only if the associated real-valued distributions of relative populations $k_j(t)\mathrm{/}{g}_j(t)$ are connected by a martingale. Building on it, we then derive an exact fluctuation theorem for a reference-relative entropy production whose average violation certifies, through a ${\chi }^2$-divergence bound, the mismatch between an assumed and the true reference evolution—a model-independent diagnostic that requires no independent characterization of the latter and turns an observed breakdown of the fluctuation relation into a certified lower bound on the reference error.