Directional Relational Manifolds: Stratified Variable-Rank Geometry, Rank-Transition Holonomy, Geometric Hysteresis, and Energy-Controlled Transitions

Abstract

We introduce Directional Relational Manifolds (DRM), a stratified variable-rank geometricframework for systems whose effective degrees of freedom may emerge, collapse, or reorganizewith state. The construction starts from a smooth finite-rank vector bundle E → M, a smoothsymmetric positive-semidefinite bundle metric g, and an anchor ρ : E → TM. The effectiverelational fiber and effective dimension areEp = Ep/ Ker gp, dDRM(p) = rank gp.The base is decomposed into constant-rank strata. On each stratum, the effective quotientfibers form a smooth metric vector bundle on which connections, curvature, parallel transport,energy, and admissible dynamics are defined. Rank-changing trajectories are modeled as hybridgeometric paths: invertible metric transport acts inside strata, while possibly non-invertibletransition maps act at rank-changing events.The global contribution is a theory of rank-transition holonomy. For a closed hybrid path,the ordered composition of intrastatum parallel transports and interstratum transition mapsdefines an endomorphism of the initial effective fiber. Unlike ordinary holonomy, this operatorneed not be invertible. We prove that if a closed loop based at an n-dimensional effective fibercontains a transition of rank r < n, then the total hybrid holonomy has rank at most r andtherefore cannot be the identity. Thus a trajectory may return to the same base point and thesame nominal dimension without returning to the same relational state. This produces geometrichysteresis: path-dependent memory generated by dimensional collapse and reactivation. Theeffect may occur even when every regular stratum is flat and every intrastatum transport istrivial.We further establish an energy criterion for rank transitions. A transition is dissipative exactlywhen its metric operator norm is at most one, or, equivalently in matrix form, J⊤G+J ⪯ G−.Compositions of dissipative transitions and metric parallel transports are globally non-expansiveand generate a contraction semigroup rather than a holonomy group. Rank defect, norm defect,energy loss, spectral persistence, and transition commutators are introduced as quantitativehysteresis invariants. Explicit examples exhibit irreversible rank-2 → 1 → 2 cycles, orderdependent collapse maps, and memory in flat strata. Under constant-rank, nondegeneracy,anchor-isomorphism, and compatibility hypotheses, DRM reduces to Riemannian geometry; withan injective anchor it reduces to sub-Riemannian geometry; and with the Fisher informationmetric it reduces to Fisher–Rao information geometry.