Thermodynamic Orthogonality from Cartan Decomposition: Coherence Work, Entropy Production, and Their Separation in Open Quantum Systems

The Cartan decomposition su(d) = h ⊕ n encodes a fundamental thermodynamic separation: coherence work W* = δQ_std − δQ_ent (eigenvector-projector rotation) responds exclusively to root-space perturbations (F_α ∈ n), while the entropy-production gap g responds exclusively to Cartan perturbations (F_β ∈ h). This thermodynamic orthogonality is a structural principle from which three results follow non-perturbatively and without secular conditions. Theorem O (thermodynamic orthogonality): The sub-Jacobian of ρ ↦ (W*, g) at ρ₀ = I/d is block-antidiagonal for any d ≥ 2, with det J_sub ≠ 0 iff the dissipator is non-isotropic. Theorem C (irreducibility, from Thm. O): No f: ℝ → ℝ satisfies δQ_ent + f(W*) = T dS_vN for all ρ. Coherence work is an irreducible thermodynamic quantity. Theorem N (second law without secular approximation): σ_new = −Tr[D[ρ](log ρ − log ρ_ss)] ≥ 0 for any GKSL generator; σ_new is the Cartan-orthogonality-invariant measure of irreversibility, with W*/T = σ_ent − σ_std experimentally accessible. Theorem H_d (trivial case of Thm. O): det J_sub = 0, equivalently W* = 0 for all ρ, if and only if A = c I_{d²−1}, b = 0 (T₁ = T₂ for d = 2) — the isotropic dissipator, the unique case in which the Cartan structure imposes no thermodynamic separation. In summary, Theorem O yields three corollaries: (I) W* is irreducible (Theorem C); (II) a non-negative entropy production σ_new ≥ 0 exists within the GKSL framework, without secular conditions (Theorem N); (III) W* = 0 for all ρ iff T₁ = T₂ (Theorem H_d).