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

## 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 3×3 Kossakowski matrix as C=R+iA, RT=R, AT=-A, Aab=εabcbc, the real-symmetric sector R preserves every irreducible spherical-tensor rank, while the vector b is exactly the nonunitality vector and carries all inter-rank dissipative transport: DC(I)=-2 b· 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 b≠0 and 1≤ L<2j, neighboring blocks obey the universal directional law CL→ L+1 CL+1→ L=L+2/L, independent of j, the magnitude and orientation of b, and the rank-preserving real-symmetric Kossakowski sector. For nonzero 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.