A divergence norm for the fourth-order coefficient and the invariance of the counting law

The fourth-order coefficient that controls the relative-entropy disturbance of a reduced quantum state under Gaussian coherent noise is identified here as one half of the quadratic form of the chi-squared divergence, evaluated not on a state but on the second-order mean drift of that state. The coefficient had appeared in an existing series of papers as a computed number attached to each noise family separately, with no statement of what kind of object it is. Naming it as a divergence norm on the drift is what makes the three results below possible, and each of them is a statement the series could not have reached while the coefficient remained a number. The first result is an identification, not a new computation. Five symbols are used for that coefficient across the series, carrying three different normalisations; they are shown to denote one quantity, and the conversion between them is recorded in a table. The normalisation that each symbol carries is stated explicitly, because the three differ by factors of the monitored dimension and of the total dimension, and a reader comparing two of the papers without that table would conclude that they disagree. The second result is a deformation. When the generating function of the expansion is replaced by the one-parameter family whose member at unit parameter is the logarithm, the coefficient becomes the quadratic form of the corresponding divergence of that family. The reason is local and explicit: the confluent value of the deformed kernel is the second derivative of the generator, so the kernel that the expansion produces is the Hessian of the generating function and nothing else. Closed forms follow for both noise families already studied, expressed in the power sums of the deformed theory. For unitarily invariant noise the coefficient is a single quadratic combination of three power sums. For Pauli-string noise on a stabilizer code only one power sum moves. The third result follows from the second. Because the only dependence on the deformation parameter sits in the diagonal of the kernel, whose sum over the spectrum is that single power sum, the counting law for stabilizer codes is invariant in form under the entire deformation, with the power sum read at the working parameter. The law is therefore not a feature of the relative entropy in particular; it is a feature of the whole family, and the choice of generating function changes one scalar in it and nothing of its structure. A structural correction to the series is also recorded, and it is the result with the clearest consequence for error correction. Protection at fourth order requires two independent conditions: the vanishing of every first-order amplitude, and the vanishing of the drift. The Knill-Laflamme condition supplies only the first. A code satisfying it may therefore carry a fourth-order error, and the residual is exactly the coefficient above together with the connected channel. That residual is thus the obstruction to full protection at this order, and a protection statement elsewhere in the series, which asserts the two conditions to be equivalent, is strictly weaker than it claims. A code and a noise family exhibiting the gap are given, and the residual is measured on them. The manuscript is self-contained. Three appendices derive, from the definitions, every ingredient that the results use: the two expansion coefficients and the quadratic form through the divided-difference Frechet derivative, the drift of each noise family from the completeness identities of the mode algebra, and the two vanishing arguments. No statement in the paper depends on work that is not yet published, and one appendix reproduces, from a convention-free law for the drift, the published coefficient of an earlier paper of the series, including the factor that the geometry of four tensor factors contributes to it. Every identity is certified numerically against the integral definitions of the deformed theory rather than against the closed forms used to derive it: the power sums are computed by quadrature in the substitution t = exp(u), with the interval split at the logarithm of every eigenvalue, at ninety working digits, and only then compared. Every statement of the appendices is checked separately on explicit matrices built from the definitions, with the perturbation applied by exact matrix exponentiation. Each check reports the discrepancy it attains rather than a verdict, so that the resolution of the test is visible; where a finite difference sets that resolution, the tolerance of the row is the resolution attained and not a wish. Three rational values reported as exactly verified in an earlier paper of the series are reproduced here from a single spectrum by a route not used when they were obtained. The archive deposited with this record carries the four self-contained checks, a README describing what each one measures, and the licence. Four further guards belong to the manuscript and are not in the archive, because none of them can run without files that are not part of it; they check the documents and the deposit rather than the mathematics, and all four are named in the README. The limitations of the work are stated in the manuscript as six paragraphs, among them the restriction of the drift to the diagonal case, the factor outside the weight hypothesis, and the extent of the prior-art search behind the disclaimer of novelty, which was two queries and the reading of two abstracts: the verdict recorded for the new statements is therefore "not found", which is weaker than "new". The use of an AI tool in the research and in the preparation of the manuscript is disclosed in the Acknowledgements, with the model and its version named, and described in the section on numerical certification.