Closed Forms for the Quartic Commutator Channel and the Diagonal Mixed Term, and the Kubo-Mori Geometry of the Fourth-Order Coefficient

The fourth-order coefficient of the mean relative-entropy disturbance of a reduced encoded state under Gaussian coherent noise was assembled in closed form in previous work, but with two objects left as sums over spectral indices: the quartic commutator channel C, given there only as a four-index contraction of the third divided difference of the logarithm, and the diagonal part b_1 of the mixed channel. Both are closed here by one method, and the consequences are read off. The method. Substitute the integral representation of the divided differences before performing the index sums, which turns every Kronecker delta into a coincidence of resolvent labels, and then integrate by parts in the ring Q[d_1][t, m_1, ..., m_4] carrying the single derivation dm_k/dt = -k m_(k+1), where m_k(t) = sum_u (p_u + t)^(-k). Two structural facts make it usable, and ignoring either costs the solution. First, the integrand of chi = sum_u 1/p_u is exact, m_2 = -d(m_1)/dt, so chi is not an independent integrand but a boundary value; including it in the ansatz leaves its own coefficient undetermined. Second, no per-monomial convergence bound may be imposed on the antiderivative: only the total need converge, and the correct antiderivative contains four individually divergent monomials whose divergences cancel. Imposing the bound cuts the ansatz space from 111 monomials to 55 and the system then has no solution at all. The results are C = 1 - 2 Xi + (d_1^2/2 + 1/3) chi - (2/3) T - (d_1/6) S_w, b_1 = d_1^2 (Xi - chi), each with an explicit antiderivative, so no coefficient is fitted: the three coefficients of C emerge from the linear system as unknowns, and the additive constant and the chi term emerge from the two boundary values. No index contraction beyond the four functionals Xi, chi, T and S_w survives anywhere in the fourth-order coefficient. The functional S_w = sum_uv w_uv is not new to the theory: the connected channel already needs it at order d_2^2, in the form Conn-A_22 = 1 - Xi + D(Xi - 3)/4 + d_1 d_2^2 S_w/2. What is new is that the same functional returns in the d_2-independent part through C, that the two weights collect into (d_1/6)(3 d_2^2 - 1) S_w, that the coefficient needs exactly four spectral functionals and no fifth, and that S_w is not a linear combination of the other three. The last is proved for every d_1 >= 3, analytically: a degeneration lemma, sending one eigenvalue to zero and separating the coefficient space by the functional form of the log-coefficient, confines the kernel to two dimensions; an expansion about the degenerate spectrum then closes it, with discriminant d_1 (d_1 - 2)^2 / 4, which vanishes exactly at d_1 = 2. The same statement proves the irreducibility of the third functional T, which the preceding paper had established by least squares; and at d_1 = 2 the two relations S_w = chi + 2 Xi - 4 and S_w = -2T - 2chi are exhibited, which is why four functionals collapse to two there. The geometry. All four functionals are contractions of the Kubo-Mori (Bogoliubov-Kubo-Mori) metric g and of the Bogoliubov-Kubo-Mori skewness tensor tau, the Amari-Chentsov tensor of quantum information geometry, on two canonical directions: the identity 1 and the maximal-coherence operator J of the Schmidt eigenbasis, which is d_1 times the projector on the flat Schmidt superposition. Explicitly chi = g(1,1) = g(1,J), S_w = g(J,J), Xi = g(rho.J, J) and T = l_2[rho.J, J, J], with rho.J the Jordan product, and the numerical coefficients are contractions too: g(rho,rho) = 1 and g(rho,1) = d_1, the rank. The normalisation is fixed, not assumed, by the relative-entropy expansion S(rho + sX || rho) = s^2 g(X,X)/2 - s^3 tau(X,X,X)/6 + O(s^4). The directions are not a choice. In the two index patterns in which it enters, the mode two-point function of the invariant ensemble equals 2 (rho.J)_uv J_vu - 2 rho_uv rho_vu and 2 (rho.J)_uv 1_us - 2 rho_uv rho_vs identically in the spectrum, for every d_1; and a free index sum is a pairing with J since sum_uv X_uv = Tr[X J], while a delta-identified pair is a pairing with 1. Those two facts do not depend on the order of the expansion, so any contraction of the mode two-point function against any kernel l_r, at any order, lands in the span of {1, rho, J, rho.J}. In that language the identity and the pure coherence operator K = J - 1 are Kubo-Mori orthogonal, g(1,K) = 0, and S_w - chi = norm(K)^2: the fourth functional is the squared Kubo-Mori length of the coherence direction and chi is its classical Fisher part. Coherence enters through two functionals rather than one, since Xi - d_1 = g(rho.K, K) is purely off-diagonal as well. Every channel is then a rational-weighted sum of metric and skewness contractions on the four directions, with d_2 the only number entering from outside the geometry: G = g(Lambda,Lambda)/2, A_13 = -(5/6) D sum_a g(J_a,J_a), and b_1 = d_1^2 g(rho.J - 1, J). Sharper still, the Amari-Chentsov tensor enters the whole coefficient exactly once, at order d_2^0 and with weight -1/6, which is precisely the weight the relative-entropy expansion assigns to its own cubic term; everything else, at every order in d_2, is metric. Corollaries. The chain d_1^2 <= Xi <= d_1 sqrt(chi) <= chi holds for every spectrum, with equality at any one place exactly at a degenerate spectrum; the left link is from the preceding paper, the other two are new, and the consequence is that b_1 <= 0 always and vanishes only at degeneracy. The coherence length is bounded sharply, norm(K)^2 >= d_1^2 (d_1 - 1) and hence S_w >= d_1^3, saturated at the degenerate spectrum and nowhere else. The d_2-independent bracket collapses to (d_1^2/2 - 1/6) chi + T/3 - (d_1/6) S_w, in which Xi and the additive constant cancel exactly, so the constant term is independent of the functional that carries the whole second-order theory. At a degenerate spectrum C = 1 - (5/3) d_1^2 + (2/3) d_1^4, which gives 40 at d_1 = 3, and at d_1 = 2 the bracket equals 2 + (chi - Xi) and is therefore at least 2. Certification. Three independent routes to the full coefficient agree at every configuration tested: the raw five-channel sum evaluated from the mode objects on an explicit orthonormal Hermitian basis, the assembled equation of the preceding paper as printed, and the fully explicit form derived here. Three numbers published before these derivations existed are reproduced by them: C = 7.0492 at d_1 = 2 with p = (0.7, 0.3), C = 40.000 at d_1 = 3 with a degenerate spectrum, and B_1 = -10.2986 at d_1 = 2, d_2 = 2. The reduction of the four-index sum is an exact symbolic identity in the spectrum and in the integration variable; the antiderivative identity holds as a polynomial, not numerically; and the geometric form of C agrees with the four-index sum to 10^(-79). Two limitations are stated. Everything here is for the unitarily invariant ensemble, and nothing survives for a structured family such as a Pauli family satisfying the Knill-Laflamme conditions on a stabilizer code, where the first-order mode objects vanish and C = 0 identically. And one identity, an exact relation among the five functions holding on geometric spectra at d_1 = 3 and explaining why a least-squares test of reducibility is weak there, is established numerically with exact rational coefficients and held-out parameter values rather than symbolically; nothing else depends on it. One evaluative sentence of the preceding paper is corrected: the coefficient needs four spectral functionals, not three, and the fourth is printed in that paper's own assembled equation. The undercount is in the count. No equation, no theorem, no table entry and no value of that paper is affected, because C was computed there from its four-index sum directly and b_1 from its defining double sum directly. An expectation for order eps^6 is recorded and labelled as one: the directions are already forced for the part of the expansion built from first-order mode objects alone, while the two-point structures of the higher mode objects and the count of contraction patterns available to l_4 and l_5 remain open. Nothing at sixth order is computed here. This note is posted here for the first time. It closes the two residual index sums of Closed Forms for the Remaining Fourth-Order Channels of Gaussian Coherent Noise, and a Definitional Correction (Zenodo DOI 10.5281/zenodo.23684111), and corrects one evaluative sentence in it. The functional S_w it identifies geometrically was introduced in A Complete Fourth-Order Expansion of Reconstruction and Logical Errors for Arbitrary Gaussian Coherent Noise (Zenodo DOI 10.5281/zenodo.23644208), and the functionals Xi and chi in A chi-squared divergence law for the fourth-order breakdown of entanglement wedge reconstruction in Gaussian holographic codes (Zenodo DOI 10.5281/zenodo.23621356). The scripts reproducing every number in the note are deposited as metric_skewness_scripts.zip. They require Python 3.10+ with NumPy, SciPy, mpmath and SymPy; no other dependencies, CPU only. Five of them are reused unmodified from the deposit of the preceding paper, which is deliberate: the new closed forms are checked against code written before they existed.