Casimir-Gap Transport and Projective Spectral Geometry in Lie-Algebraic Lindbladians

## Overview This preprint studies antisymmetric Kossakowski transport generated by compact Lie-algebraic jump operators in finite-dimensional Gorini--Kossakowski--Sudarshan--Lindblad (GKLS) dynamics. The central question is whether directed transport between representation sectors can itself select an operator-space geometry in which the transport becomes reciprocal or skew-adjoint. For a compact simple Lie algebra, the antisymmetric coefficient tensor admits an exact orthogonal bracket--kernel decomposition. The bracket projection carries all nonunitality and satisfies a Casimir--Jordan identity, whereas the bracket-kernel component is unital and Hilbert--Schmidt skew-adjoint. ## Main results Projection between Casimir eigensectors gives the universal scaled-adjoint law Fν→μ=-Δμν-CadΔμν+CadFμ→ν^, where Δμν=c_ν-c_μ>0 is the Casimir gap and Cad is the adjoint Casimir. The critical gap Δμν=Cad produces a one-way transport edge, including the universal scalar-to-adjoint boundary. For the multiplicity-free symmetric-power hierarchy Vₚ=Symᵖ CN, every nonzero bracket direction activates every neighboring bond, and the forward and reverse neighboring blocks satisfy Rₖ=-k/k+NFₖ^. The resulting off-diagonal transport uniquely selects, up to an overall scale, the sector weights wₖ∝Γ(k)/Γ(k+N). Under the exact equivariant harmonic correspondence with CPN-1, this selected metric becomes a projective spectral multiplier. On nonconstant projective harmonics it is identified with the sign-corrected derivative, at the constant-kernel point, of a centered and rescaled projective cosine-transform intertwining family. Its inverse has the same scalar Gamma-ratio multiplier, when indexed by harmonic degree, as the corresponding order-N conformal spherical intertwiner. For N≥3, a dabc-mediated same-sector Jordan block obstructs any positive $SU(N)$-invariant skew metric for the full traceless bracket transport. Consequently, within the symmetric-power family, $SU(2)$ is exceptional: the bracket kernel disappears, the same-sector obstruction is absent, and the transport-selected metric reduces to inverse Casimir and extends over the entire traceless internal bracket dynamics. ## Interpretation and scope The Gamma-function sequence, projective harmonic correspondence, cosine-transform spectrum, and conformal spherical multipliers are established mathematical structures. The new mechanism developed here is their dynamical selection by Lie-algebraic Lindblad transport, together with the representation-theoretic obstruction that prevents the $SU(2)$ inverse-Casimir geometry from extending generically to higher rank. The separate $SU(2)$ prototype, including its dissipative decomposition, contraction theory, exact spherical Green geometry, and complete-positivity extremality, is developed in: **Lee Byoungwoo, “Inverse-Casimir Geometry of Nonunital Multipole Transport in Linear-Spin Lindbladians,” Zenodo Version v1.0 (2026), DOI 10.5281/zenodo.22266973.** ## Version status This record is **Version v1.0** of the public preprint. The manuscript is presented without internal research-version or freeze labels.