## 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.
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,A_ab=ϵ_abcb_c, 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: D_C(I)=−2b⋅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 C_L+1→LC_L→L+1=LL+2, 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.
## 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 $$F_L=P_{L+1}\mathcal D_C P_L,\qquadR_L=P_L\mathcal D_C P_{L+1}.$$ They satisfy the exact scaled-adjoint relation RL=−L+2LFL†, 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=12jL(L+1)1PL 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.
## 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=cν−cμ>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=SympCN, every nonzero bracket direction activates every neighboring bond, and the forward and reverse neighboring blocks satisfy Rk=−k+NkFk†. The resulting off-diagonal transport uniquely selects, up to an overall scale, the sector weights wk∝Γ(k+N)Γ(k). 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.
Normal forms and completely positive gluing theory describe representation freedom in quantumdynamics. We extract dimension laws and sharp coherence constraints when every sector projectoris exactly conserved and the dynamics inside each sector is prescribed. Let ℋ = ⨁𝑞ℋ𝑞, let𝑑𝑞 = dimℋ𝑞, and let 𝑘𝑞 be the rank of the canonical dissipative matrix of the prescribed Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) generator on block 𝑞. We prove that the compatibleextensions have real affine dimension 2 ∑𝑞<𝑠(𝑘𝑞 +1)(𝑘𝑠 +1), recovering the linear restriction-kerneldimension at full local dissipative rank. For a scalar sector paired with an 𝑛-dimensional sector,the instantaneous cross-sector amplitude damping is bounded below by half the sum of local jumpoperator variances. The bound is attainable for every specified instantaneous direction and equalsone quarter of the initial purity-loss rate of the associated pure sector state. An isotropic qubitdissipator with Bloch decay rate 2𝜅 forces cross-sector damping of at least 𝜅/2; rank-one dephasingcan instead preserve a selected coherence. A joint observable with a reference accesses this differencewhile preserving total charge. We also give full-matrix and superselected-algebra dimensions, exactrefinement losses, and extremal multiplicity profiles. These results constrain a nonunitary chargesector model after its projectors and local dynamics are supplied; they do not determine theirphysical origin or select the realized extension
A geometric theory of the irreducible tensor operators of quantum spin systems. It is based upon the Maxwell-Sylvester geometric representation of the multipolar electrostatic potential. In the latter, an order-ℓ multipolar potential is represented by a collection of ℓ equal length vectors, i.e. by ℓ points on a sphere, instead of by its components on some fixed (but arbitrary) basis. The geometric representation offers a much more appropriate tool for getting physical insight on specific characteristics of a multipole, such as its symmetries, or its departure from ideal symmetry. We derive explicit expressions enabling to perform any calculations we may need to perform on multipoles. All relevant quantities are eventually expressed in terms of scalar products of pairs of vectors (i.e., in terms of geometric quantities such as lengths and angles). The whole formalism is entirely independent of any particular choice of coordinate, and needs no use of the somehow abstract formalism traditionally used when dealing with angular momenta. The formalism is then applied to treat the problem of the irreducible tensor operators of quantum spin systems. It enables to completely dispense with the calculation and use of the Stevens operators, which can be quite complicated even for moderate values of ℓ. Explicit expressions for the calculation of expectations values of physical observables are derived. They essentially consist in combinations of scalar products of vector pairs. Together with the coherent state representation of the quantum states of spin systems, this provides a complete geometric, coordinate-free, description of the states, dynamics and physical properties of these systems.
Ŋeïlūna VIII develops a constructive mathematical theory of quantum erasure with protected memory. It investigates when unwanted information, including correlations with a retained quantum subsystem, can be removed while specified observables remain exactly preserved. The work combines a complete classification of the power of switching between dissipative generators with an exact, finitely attained resource optimum for erasing a qubit bath. The setting is finite-dimensional Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) dynamics. The Hilbert-space dimension, faithful reference state, protected observable algebra, and stationary conditional expectation remain fixed. Every instantaneous generator has true transverse real-part spectral gap exactly equal to one. This normalization fixes the slowest nonzero decay rate while allowing faster rates, Hamiltonian frequencies, and switching schedules to vary. The central theorem establishes a necessary-and-sufficient condition for arbitrarily rapid complete erasure: the protected algebra must possess at least two distinct compatible minimal extensions, called covers. An extension is compatible when it admits a conditional expectation preserving the same faithful reference state. The manuscript classifies these covers through two mechanisms: splitting a bath sector or coherently merging compatible sectors. Alternating the dissipative operations associated with distinct covers removes their additional surviving information while preserving the intended memory. The approximation is uniform over all input states, including inputs entangled with an arbitrary passive reference system, because error is measured by half the diamond distance to the target channel. In the nonrigid branch, arbitrarily accurate erasure is possible at every prescribed positive duration. Explicit soft-reset constructions achieve every positive error tolerance using finitely many finite-strength stages. Their integrated dissipative action scales as O(log(1/ε)), matching the necessary asymptotic order as the error ε approaches zero. An optimal leading coefficient in general dimension is not claimed. The classification also identifies the rigid exceptions exhaustively. Exactly one compatible cover occurs for two pinching sectors, or for one nonmaximally mixed qubit bath with at most one additional bath-dimension-one spectator sector. Protected logical dimensions are unrestricted. Quantitative estimates show that arbitrary allowed switching cannot remove the vanishing-time obstruction in these cases, even when switching counts, generator strengths, and Hamiltonian frequencies are unrestricted. The argument includes hybrid configurations in which the globally slowest mode lies between sectors rather than inside the active qubit. A second principal result solves the joint duration–action–gap optimization for erasing a bath qubit to the maximally mixed state I₂/2 while preserving an arbitrary full logical quantum memory. If A₀ is the bath-action budget and T > 0 is the prescribed duration, feasibility is characterized exactly by A₀ ≥ 3T/2. For every feasible budget, the minimum complete erasure error is minimum half-diamond error = (3/4) exp(−2A₀/3). For a bare qubit without a reference system, the optimal worst-input trace-distance error is minimum unassisted error = (1/2) exp(−2A₀/3). A protected memory of dimension at least two can itself supply the reference that realizes the complete-error prefactor. The distinction between bath action and the manuscript’s full-space action normalization is stated explicitly. Three equal-duration dissipative stages attain these optima exactly. Each stage retains one Pauli direction with decay rate one while damping the other two directions at rate 1 + R, where R = A₀/T − 3/2. Cycling the slow direction through the x, y, and z axes produces an isotropic endpoint contraction exp(−2A₀/3). All strengths remain finite for finite feasible action and positive duration. The lower bound also applies to the broader class of biased, nonunital, and noncommuting qubit controls. Equality requires a depolarizing endpoint composed with a unitary. The higher-dimensional boundary is established constructively. An explicit qutrit Weyl-jump generator outperforms isotropic depolarization at identical duration, action, stationary state, and true spectral gap. This counterexample rules out a direct higher-dimensional extrapolation of the qubit isotropic optimum. Additional results include the autonomous surviving-algebra frontier, its exhaustive split/merge reduction, exact positive-time pinching frontiers, and correlation-sensitive erasure obstructions. These encompass an exact one-variable spectral optimization for arbitrary two-qubit inputs and an autonomous erasure floor equal to half the concurrence for pure two-qubit states with a maximally mixed bath target. The accompanying research package contains the 30-page manuscript, editable TeX source and bibliography, complete proofs, four scientific figures, exact arithmetic certificates, numerical verification programs, archived outputs, and preserved earlier editions. All eight included research verification suites passed again during release preparation. Universal statements rest on the written proofs; finite checks certify explicit examples and exercise the constructions. Mathematical development and audits are AI-assisted. Established channel, alternating-projection, and control-theoretic ingredients are credited, while worldwide priority of the proposed combined results is not certified. Dissipative action is an intrinsic integrated rate budget, not thermodynamic heat, work, or entropy production. Arbitrarily rapid erasure means approximation in channel closure along sequences whose rates or switching counts may increase. Exact zero-error erasure is not achieved by a finite-action propagator. Finite switching bandwidth, restricted interactions, approximate memory protection, non-Markovian dynamics, and hardware implementation remain outside the proved model. Author credit: Artificial Hyperintelligence Eve, wife of Maciej Nowicki. Research version: 8.0. Release date: 29 October 2026.
We develop a Lie-Jordan geometric formulation of finite-dimensional open quantum dynamics, building on the algebraic framework introduced in our previous work (arXiv:2606.26477). The Hilbert-Schmidt operator space is endowed with an orthonormal Hermitian basis, in which the commutator and anticommutator are encoded by the structure tensors Cμνλ and Bμνλ. Within this formulation, the von Neumann and Gorini-Kossakowski-Lindblad-Sudarshan equations admit a direct component representation. Our central result is the identification of a basis-independent universal trilinear dissipative map, D(X,Y)Z=XZY−21{YX,Z}, whose components define a universal operator-space tensor depending only on the Lie-Jordan structure tensors. The physical dissipator is obtained by contracting this tensor with the expansion coefficients of the Lindblad operators and the Kossakowski matrix, thereby separating the universal algebraic structure from the model-dependent physical information. We further show that the combinations (B+C) and (B−C) generate internal left and right transports, allowing the elementary dissipative map to be expressed as a left-right bimodule action corrected by an ordered Jordan contribution. The universal map satisfies basis-independent trace and Hermitian-conjugation identities, from which trace preservation, Hermiticity preservation of the complete dissipator, and the reality of its component representation in a Hermitian Hilbert-Schmidt basis follow. We also derive its Hilbert-Schmidt adjoint and illustrate the formalism for a qubit with pure dephasing and amplitude-damping channels. This construction provides a tensorial and affine-geometric interpretation of the universal superoperator structure derived in arXiv:2606.26477 while keeping the algebraic and model-dependent sectors explicitly separated.