Closed Forms for the Remaining Fourth-Order Channels of Gaussian Coherent Noise, and a Definitional Correction

The complete fourth-order coefficient of the mean relative-entropy disturbance of a reduced encoded state under Gaussian coherent noise decomposes into five channels. Closed forms were previously available for two of them. This note evaluates the remaining three for the unitarily invariant ensemble, so that the whole coefficient is explicit for every bulk dimension d_1, every geometric dimension d_2 and every spectrum. The delta1-delta3 channel, reported in earlier work as measured but not evaluated, is A_13 = -(5/3) D (Xi - 1), with D = (d_1 d_2)^2 the total Hilbert-space dimension and Xi the Kubo-Mori spectral functional of the second-order theory. The proof uses only the two operator identities sum_a A_a^2 = D and sum_a A_a M A_a = Tr(M), and is independent of the code. The channel is negative for every spectrum, and not merely because Xi > 1: since the logarithmic mean is dominated by the arithmetic one, Xi >= d_1^2 with equality exactly at a degenerate spectrum, so A_13 <= -(5/3) D (d_1^2 - 1). The quartic commutator channel C is obtained in closed form as a contraction of the divided-difference kernel l_3 with two copies of the mode-sum tensor N, and is proved independent of d_2. The mixed channel B splits as B_1 + B_2. The remainder B_2 is the one that resisted: it is obtained here in closed form for every d_1, every d_2 and every spectrum, B_2 = 3 Xi - chi/2 - 2 + T, with chi = sum_u 1/p_u and T = sum_uvs l_2(u,v,s) p_u, where l_2 is the second confluent divided difference of the logarithm. T is a third spectral functional, and it is shown not to be a function of the other two: the best affine fit to {Xi, chi, 1} over forty interior spectra leaves worst residuals 2.86, 8.34 and 13.80 at d_1 = 3, 4, 5, against values of abs T reaching 82, 297 and 542. T is nevertheless not an alien object. In the variables m_k(t) = sum_u (p_u + t)^(-k) and h(t) = sum_u p_u/(p_u + t), the three functionals are Xi = integral h m_1 dt, chi = integral m_2 dt and T = -integral h m_1^2 dt, so T is the next member of one family. T and chi moreover scale alike, both as p^(-1), so no counting of degrees separates them; only the mode sum does. The derivation has two halves. The double mode sum Q_uvs = sum_ab (J_a)_uv (J_b)_vs (K_ab + K_ba)_su closes on the bulk spectrum alone, by four completeness identities: two that the preceding paper already prints, and two that carry an uncontracted mode operator and are what close the terms of K carrying A_a A_b Phi. Twenty-four cases are involved, and they are tabulated in the derivation note deposited in this record. The resulting divided-difference contractions then collapse by two identities proved from the integral representation of the Kubo-Mori kernels, F_4 = -d_1^2/2 and F_3 = d_1^2/2 - Xi. Three statements that could previously only be checked numerically follow as corollaries: B_2 is independent of d_2 and of the code; at a degenerate spectrum B_2 = -(1/2)(d_1^2 - 1)(d_1^2 - 4); and at d_1 = 2 it reduces to 2 (Xi - chi), that collapse being an accident of the dimension, since T there equals -(3/2) chi - w_12. Consequently the assembled coefficient is explicit at every configuration, its d_2-independent constant collapsing to 2 Xi - chi/2 - 1 + T + C. At d_1 = 2, d_2 = 2 and p = (0.7, 0.3) it reproduces a direct sampling measurement to 0.13 standard errors, at d_1 = 3 with a degenerate spectrum to 0.45, and the formula for A_13 reproduces retrodictively, to 0.55 standard errors, a measurement published before this derivation existed. Every closed form is certified against exact mode sums on an explicit orthonormal Hermitian basis of the full space, link by link rather than only end to end, and the formula for B_2 is tested on held-out spectra it never saw. An erratum to the preceding paper is also recorded. As printed, its third-order mode object L_abc = R([A_a,[A_b,[A_c,rho]]]) is anti-Hermitian, which would force A_13 to vanish identically for every ensemble, every code and every spectrum, contradicting the nonzero delta1-delta3 word measured in earlier work. The third-order deformation carries i^3, and restoring that factor removes the contradiction. That paper's own numerics evidently carried the factor, because it reports nonzero values of A_13 on two families that are not stabilizer codes; no value reported in any of the preceding papers is affected, and what requires correction is one definition as set. Two limitations are stated. Everything here is for the unitarily invariant ensemble, and none of the operator arguments survives for a structured mode family such as a Pauli basis on a stabilizer code. And a discrepancy with a published measurement at d_1 = 3, d_2 = 12 is located at a definite configuration but not adjudicated: the coefficient gives -9492 at the degenerate endpoint there, while the estimator used in this note cannot be carried to d_2 = 12 at the coupling used throughout, the Taylor control eps times the norm of W reaching 1.44. This note is posted here for the first time. It supplies the three missing channels of the five-channel decomposition of A Complete Fourth-Order Expansion of Reconstruction and Logical Errors for Arbitrary Gaussian Coherent Noise (Zenodo DOI 10.5281/zenodo.23644208), corrects one printed definition in it, and evaluates in closed form the delta1-delta3 word measured 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 degeneracy measurement it compares against is reported in Breakdown and Persistence of the Reconstruction-Error Hierarchy in Approximate Holographic Codes (DOI 10.13140/RG.2.2.19676.14723). The derivation note carrying the twenty-four case tables is deposited in this record as B2_derivation_note_2026-10-27.pdf. The scripts reproducing every number in the note are deposited as closed_forms_scripts.zip. They require Python 3.10+ and NumPy; no other dependencies, CPU only.