Rank-Transition Holonomy and Geometric Hysteresis in Directional Relational Manifolds

We develop a global transport theory for rank-changing Directional Relational Manifolds(DRM). In a DRM, smooth metric transport acts inside constant-rank strata, while possiblynon-invertible transition maps act when a trajectory crosses a rank-changing boundary. For aclosed hybrid trajectory, the ordered composition of intrastatum transports and interstratumtransition maps defines a rank-transition holonomy operator.Unlike ordinary holonomy, this operator need not be invertible. We prove that a closed base-space loop may return to its initial point and initial effective dimension while failing to return toits initial relational state. In particular, if one transition in the loop has rank smaller than theinitial fiber dimension, then the total holonomy has deficient rank and therefore cannot equalthe identity. This yields a form of geometric hysteresis: the final effective state depends notonly on the endpoint, but also on the ordered history of dimensional collapses and reactivations.We show that this phenomenon may occur even when every regular stratum is flat and everyintrastatum parallel transport is trivial. The resulting memory is therefore generated by ranktransitions rather than by ordinary curvature. We introduce quantitative hysteresis invariantsbased on rank defect, norm defect, energy loss, and commutators of transition sequences. Wefurther prove that dissipative transition loops generate a contraction semigroup rather than aholonomy group. Explicit examples demonstrate irreversible rank-2 → 1 → 2 cycles, path-orderdependence, and dimensional memory.The theory isolates a specifically geometric phenomenon enabled by DRM: non-invertibleholonomy induced by dimensional change.