Metric Transport on Heterogeneous Layered Systems - A Geometric Framework for Arc-Length Preserving Dynamics

Many physical, biological and computational systems, from multilayer optical media to hierarchical memory architectures and deep neural networks are organized into heterogeneous layers with distinct intrinsic geometries. We introduce a geometric framework for metric transport between heterogeneous layered systems endowed with different intrinsic metrics. Each layer is modeled as a spiral curve embedded in a cylindrical surface with variable radius. Arc-length acts as a universal transport coordinate, allowing the construction of transport maps that preserve intrinsic distances between layers. The transport maps form a groupoid of isometries between one-dimensional Riemannian manifolds. The induced connection is flat and the holonomy is trivial. The multilayer composition law is globally equivalent to addition in arc-length coordinates; the interest of the framework lies in the geometry-dependent coordinate change that each layer imposes, and in the resulting metric density rescaling law for densities. We analyze the induced groupoid structure, formalize it as a category of arc-length-preserving isometries, and prove that the transport maps push forward the natural arc-length measure of each layer onto that of the target layer. We introduce path-dependent transport Ti→jγ = Lj−1 ∘ Φγ ∘ Li, which breaks the global trivialization and generates computable, nontrivial holonomy; an explicit multiplicative example yields holonomy e∮κ dt, formally analogous to the Berry phase. We develop a multilayer PDE system ∂tρi + ∂s(viρi) = Σj Λijρj and prove global mass conservation and entropy dissipation. A variational formulation is developed: the diffusive PDE is derived as the Wasserstein gradient flow of a multilayer free-energy functional; a sharp dissipation identity d𝒻/dt = −ℐρ is proved; and the Jordan–Kinderlehrer–Otto minimizing movement scheme is constructed. The spectral theory of the multilayer operator ℒ = ⊕i DiΔi − Λ is developed: self-adjointness, discreteness of spectrum on compact layers, a variational formula for the spectral gap λ1(ℒ) as the synchronization threshold, and exponential convergence to equilibrium. The discrete differential geometry of the layer graph is formalized: edge weights κ(e) as a discrete connection 1-form, the curvature 2-form F = dκ, and the discrete Stokes theorem ∮γκ = ∫ΣF. The multiplicative holonomy is identified as the Wilson loop e∫ΣF. The variational gap is closed: the Hellinger–Kantorovich metric provides the geometric motivation, and the Onsager metric with log-mean reaction mobilities kij = ΛijΛ(ρi, ρj) generates the full PDE system as a gradient flow. A unified JKO scheme and a joint dissipation identity d𝒻/dt = −(ℐ + ℛ) are established. Extensions to non-abelian holonomy (GL(n), SU(n)) and to higher-dimensional Riemannian manifold layers are developed, including curvature-accelerated convergence estimates via Bakry–Émery theory. Applications to signal propagation, hierarchical memory, multisensory integration, and deep neural architectures are discussed as modelling frameworks. License: CC BY 4.0Status: Preprint