Inverse-Casimir Geometry of Nonunital Multipole Transport in Linear-Spin Lindbladians

## Overview This preprint studies the canonical Gorini--Kossakowski--Sudarshan--Lindblad (GKLS) class generated by jump operators linear in the generators of a fixed irreducible spin-j representation. Writing the positive Kossakowski matrix as $C=R+iA$, with R real symmetric and Aab=εabcbc, separates two sharply different dynamical roles: the real-symmetric sector preserves irreducible tensor rank, while the vector b is exactly the nonunitality vector and carries all inter-rank transport. ## Main results For the Hilbert--Schmidt projectors PL onto irreducible tensor rank L, define the neighboring Schrödinger-picture transport blocks FL=PL+1 DC PL,L=PL DC PL+1. They satisfy the exact scaled-adjoint relation RL=-L/L+2FL^, so all nonzero singular values, and more generally all unitarily invariant norms, carry the universal directional factor $(L+2)/L$. This directed imbalance selects the inverse adjoint Casimir W=( K²)⁻¹=∑L=1²ʲ1/L(L+1)PL on the traceless operator space. The internal nonunital transport is W-skew, and multiplicity freeness makes W, up to an overall scale, the unique positive $SU(2)$-invariant metric with this property. For the full homogeneous traceless generator, the Hamiltonian and nonunital sectors are W-skew, whereas the real-symmetric Kossakowski sector is W-self-adjoint and dissipative. This yields an exact Lyapunov identity and contraction in the inverse-Casimir norm. The scalar sector remains exceptional: nonunitality generates a one-way scalar-to-dipole source, which obstructs any positive $SU(2)$-invariant extension of the skew geometry through the scalar mode. Under the standard finite-spin Stratonovich--Weyl correspondence, W(-ΔS²)⁻¹ on the mean-zero band-limited sphere. Thus the selected operator metric is exactly the finite-spin H⁻¹(S²) Green geometry, not merely a semiclassical limit. Complete positivity supplies the sharp resource bound 2| b|≤ TrC, with equality precisely for rotated rank-one circular J_± channels. ## Scope and verification The inverse-Casimir reciprocity, uniqueness, dissipative decomposition, contraction, scalar-boundary obstruction, spherical realization, and complete-positivity extremality are established analytically in the manuscript. Computational checks are used as regression and consistency tests rather than substitutes for proof. Supporting verification code, canonical computational outputs, and figure-generation material are archived separately on Zenodo at DOI **10.5281/zenodo.22096401**. ## Version status This record is **Version v1.0** of the public preprint. It corresponds to the manuscript's final preprint-compatible scientific and notation freeze dated September 3, 2026.