(GQR41: Paper C): Accessibility–Ordering Invariants and Geometry-Controlled Cumulative Transport in Dirac Fluids

We extend the Accessibility–Ordering Invariant (AOI) and Quantum Geometry Engine(QGE) framework to geometry-controlled Dirac transport, motivated by hBN–grapheneheterostructures with programmable width, edges, and spatial structure. Rather than in-voking Dirac field theory or hydrodynamic closure, transport is treated as a geometry- andordering-gated dynamical process in which accessibility updates do not commute in time.We show that directional transport asymmetry emerges from accessibility ordering alone(AB versus BA protocols), even when all instantaneous parameters are held fixed. Thisordering produces a finite AOI-gap that accumulates dynamically, persists under disorderand probe variation, and exhibits non-additive responses under combined structural controls.Introducing an edge-mediated formulation reveals a second, distinct memory channel.Cumulative boundary energy and charge fluxes generate a Wiedemann–Franz–type proxythat exhibits robust, monotonic ratchet-like accumulation under cyclic driving, in contrast tothe more irregular and partially reversible ordering memory captured by the AOI-gap. Ge-ometry, particularly ribbon width and spatial gradients or domains, governs the availabilityand amplification of both memory channels.The coexistence of an ordering-sensitive AOI memory and a geometry-amplified cumula-tive transport memory constitutes a falsifiable prediction for programmable Dirac devices.These results identify accessibility ordering and geometry as independent control axes fordirectional transport and long-term memory in Dirac fluids.