The Fourth-Order Coefficient of the alpha-Entropy Disturbance under Gaussian Coherent Noise: A Graded Module and Closed Forms

A pure encoded state of an approximate quantum code is perturbed by exp(i eps W) with W drawn from a centered Gaussian ensemble, and D_alpha(eps) denotes the disturbance of a reduced state measured by the Bregman divergence of the alpha-entropy Tr f_alpha, with f_alpha(x) = (x^alpha - x)/(alpha - 1). This paper derives the complete fourth-order coefficient of the mean disturbance in closed form, for the whole family 0 < alpha < 2, and identifies what the coefficient is made of. A companion paper established the case alpha = 1, where the Bregman divergence of x log x is the relative entropy; the question answered here is whether the closure obtained there is a property of the logarithm or a property of the construction. The mechanism is a grading. In the ring of resolvent traces, assign weight -1 to the integration variable of the divided-difference integral representation and weight k to the k-th trace m_k. Under p -> lam p, t -> lam t a weight-w monomial scales as lam^-w, so weight is a genuine grading; and the integration-by-parts relation of the weighted ring is weight-homogeneous, so the computation splits into sectors that do not communicate. In the trace basis the relevant operator is D_alpha(mu) = (alpha - n) mu + L(mu) on a monomial of decay order n, where L raises the order by one and preserves the weight, and every D_alpha-exact integrand is annihilated by the integral. That last statement replaces the integration by parts of the companion paper and is better behaved: in the trace basis the decay of a product is manifest, so a single order hypothesis kills the boundary term outright. The floor of each weight sector turns out to be a single-factor integral, and those have an elementary closed form: they are power sums pi_v = alpha sum_u p_u^(alpha - v), one per weight, with the exponent fixed by the difference k - j of the trace sk,j and not by k and j separately. Two of the three that occur, pi_0 and pi_1, collapse to the constants 1 and d_1 exactly at alpha = 1. The numbers 1 and d_1 appearing in the companion paper's channel formulas are therefore not constants of the problem but values of two spectral functionals at one point of the family, and that is why four functionals suffice at the logarithm and six are needed elsewhere. No search inside the ring of resolvent traces can produce sum_u p_u^alpha, which is why the logarithmic method does not transfer as it stands. All five channels are then evaluated. Three of them - the drift, the delta_1 delta_3 channel and the connected delta_2 delta_2 channel - require no integration by parts at all: every contraction that occurs in them is one of four elementary sums, and their entire generalisation is the replacement of the two constants by pi_0 and pi_1. The mixed channel needs reductions of order three and the quartic commutator channel of order four. The assembled coefficient is a polynomial in alpha, d_1 and d_2 over the six functionals pi_0, pi_1, pi_2, Xi_alpha, S_alpha and T_alpha, and it reduces to the companion result coefficient by coefficient as alpha -> 1. Two members of the family are exceptional, and in each case the exception explains something rather than merely recording it. At alpha = 1 the coefficient of Xi_alpha in the dimension-independent part has a double zero; the companion paper observed that Xi drops out there and could not say why, and a tangency is the answer. At alpha = 1/2 the total weights of the coherence functional S_alpha and of the skewness-carrying functional T_alpha both vanish, for every pair of code dimensions, because both are proportional to 2 alpha - 1; the positivity of the coherence weight, which at the logarithm looks like a law, is in fact the statement alpha > 1/2. The coefficient is then read geometrically. The six functionals are contractions of the Hessian of the alpha-entropy and of its third derivative on four operator directions - the identity, the state, the maximal-coherence operator of the Schmidt eigenbasis and its state-dressing - and those directions are not chosen but read off the mode two-point function, which the Gaussian ensemble fixes identically in the spectrum. Among the identities, pi_0 is the squared length of the state itself and pi_1 its inner product with the identity, which is the geometric content of the two missing generators. A Pythagoras decomposition splits S_alpha into a classical and a coherent part, orthogonal for every member of the family, and the coherent part is bounded below sharply by alpha (d_1 - 1) d_1^(3 - alpha), with equality exactly at a degenerate spectrum; the proof is one application of Jensen to a segment representation of the weight and replaces a mean inequality that holds only at alpha = 1. The skewness tensor enters the whole coefficient exactly once, with weight -(2 alpha - 1)/6, while the divergence assigns weight -1/6 to its own cubic term for every alpha: the two agree at alpha = 1 and nowhere else, so the coincidence recorded in the companion paper is a property of the logarithm and not a structural identity. The paper is explicit about what it is not. The functional expanded here is the Bregman divergence of the alpha-entropy, not the Petz quasi-entropy of f_alpha and not the Tsallis or Renyi divergence of order alpha; those differ already at second order, their Hessian being alpha sum x^2 / p against alpha sum x^2 p^(alpha - 2) here, and the three coincide only at alpha = 1. Nor is the Hessian a monotone metric away from the logarithm: a monotone weight is homogeneous of degree -1 in the two eigenvalues and this one is homogeneous of degree alpha - 2, so the family sits inside the Hessian geometry of a convex entropy and meets the monotone-metric framework at a single point. The alpha = 1/2 statement is an identity in the assembled coefficient and carries no claim about any affine connection. The range 0 < alpha < 2 comes from convergence alone, and the channel evaluations after the third section use the completeness identities of the unitarily invariant ensemble. One negative statement, that three particular products do not individually reduce to their sector floor, rests on a finite search and not on a proof, and the class that obstructs them is left as an open question. Every closed form is certified by routes that share nothing with the derivations. Each channel is recomputed from its original index-sum definition with the metric weight summed directly over the spectrum; the appendix, which proves the four generator-independent inputs taken over from the companion paper rather than citing them, is rechecked in explicit models with every mode sum performed by brute force; the three finite enumerations behind it are reproduced term by term, one of them in exact rational arithmetic; the geometric dictionary is rechecked with the metric evaluated as a weighted trace integral; and the identification of the expanded functional is itself tested, against both candidates, at vanishing perturbation. In the course of that work two defects of the companion paper were found and are recorded here: the grouped form of its appendix table does not transport values for three word pairs, which leaves its collected sum and every result unchanged, and the proof of its kernel lemma names the wrong mechanism for the vanishing of the first-order term. Both are corrected in the current version of that deposit. This paper builds on 10.5281/zenodo.23758559, which it does not supersede: that paper proves the case alpha = 1 and this one the family, and the appendix here makes the present work independent of it for every step of every proof. The verification scripts are included in the attached archive under the MIT licence.