Nonunital Multipole Transport in Linear Spin Lindbladians: Exact Rank Hopping and Universal Block Nonreciprocity

Abstract

## Description This preprint gives an exact classification of tensor-rank transport generated by the canonical Gorini-Kossakowski-Sudarshan-Lindblad (GKLS) class with jump operators linear in the generators of a fixed irreducible spin-$j$ representation. When Hamiltonian terms are included in the full-Liouvillian statements, they are restricted to be at most linear in the spin generators. Writing the positive $3\times3$ Kossakowski matrix as $$C=R+iA,\qquad R^T=R,\qquad A^T=-A,\qquad A\_{ab}=\epsilon\_{abc}b\_c,$$ the real-symmetric sector $R$ preserves every irreducible spherical-tensor rank, while the vector $\mathbf b$ is exactly the nonunitality vector and carries all inter-rank dissipative transport: $$\mathcal D\_C(I)=-2\mathbf b\cdot\mathbf J.$$ The nonunital transport sector admits a Casimir-Jordan normal form and produces only nearest-neighbor motion in tensor rank. Closed all-rank hopping amplitudes and exact Frobenius block strengths are obtained for arbitrary spin $j$. For $\mathbf b\neq0$ and $1\le L<2j$, neighboring blocks obey the universal directional law $$\frac{\mathfrak C\_{L\to L+1}}{\mathfrak C\_{L+1\to L}}=\frac{L+2}{L},$$ independent of $j$, the magnitude and orientation of $\mathbf b$, and the rank-preserving real-symmetric Kossakowski sector. For nonzero $\mathbf b$, the $L=0$ boundary is strictly one-way. For $j>1/2$, nonunitality is equivalently detected by dipole-to-quadrupole leakage. Spin one-half is the terminal nontrivial irrep in which the operator-rank hierarchy ends before a quadrupole sector appears, yielding universal scalar-plus-dipole closure for linear-spin dissipators. This is a dynamical closure characterization, not a microscopic derivation of the electron spin quantum number. ### Reproducibility The analytic proofs are contained in the manuscript. A separate verification and figure-generation package provides exact-arithmetic checks, canonical outputs, and the source used to generate the figures: **DOI: 10.5281/zenodo.21941069** The computational package is an independent verification layer and is not used as a substitute for the analytic proofs. ### Version Version v1.0, public preprint release, 24 August 2026.