QFI–DF: Quotient Flow Invariants under Drift and Flow

We recast QFI–DF as a general dynamical theory of Quotient Flow Invariants (QFI) rather than of quantum Fisher information. A QFI is an idiom-portable curvature/information assignment living on quotient flows: finite (or finitely presented) Markov kernels obtained by pushing an underlying process through an observable partition. Building on the invariant schema from QFI (Primitives)—isomorphism invariance, refinement monotonicity, and gap- damped lumpability stability—we analyze how QFI evolves under discrete drift (time-varying kernels) and continuous flow (semigroups/generators). We prove: (i) monotonicity under admissible data-processing drifts; (ii) conservation regimes when drifts commute with the invariant structure; (iii) gap-damped relaxation rates; (iv) QFI as a Lyapunov functional for broad flow classes; and (v) regeneration-based infinitely-often validity guarantees that close the loop for practical scheduling. This DF layer thus supplies the temporal hinge between the schema (Primitives) and the spectral instrumentation (SM: CGF/rate functions and Murphy horizons). Quantum mechanics appears as a corollary idiom: the quotient-flow invariant reduces to the standard SLD–QFI when the idiom is Hilbert-space with CPTP coarse-grainings.