A proof of the fourth-order counting law for coherent Pauli noise on stabilizer codes, and the hypothesis it needs

For a stabilizer code carrying one tracked logical qubit, perturbed by a centered Gaussian ensemble of Pauli Hamiltonian modes, the fourth-order coefficient of the mean relative-entropy disturbance of the reduced logical state was conjectured by the present author to be the pure counting expression (1/2) N_ac (N_ac + 2) _0^2 sum_u 1/p_u, with N_ac the number of modes anticommuting with the bare logical Zbar; it was certified rational-exactly at three configurations and reported as an exactly verified conjecture. This note proves it. Both second-order kernels are evaluated in closed form by Pauli algebra alone. The diagonal kernel is K_aa = (1 - eps_a) _0 sigma_z, with eps_a the commutation sign of the mode with Zbar, which yields the induced drift, its quartic coefficient, and the diagonal part of the connected channel unconditionally - using neither the Knill-Laflamme conditions nor the code distance. The off-diagonal kernel carries a sign factor f_m = 1 + eps_am eps_bm - eps_bm eta_ab - eps_am multiplying an expectation that vanishes unless the product of the two modes lies in the normalizer of the stabilizer group. This gives a necessary and sufficient vanishing criterion and, from it, the explicit hypothesis the counting law needs: 2w < d_N(S), where w is the mode weight and d_N(S) the weight of the lightest non-identity normalizer element. The hypothesis covers the two distance-three certifications, explains why the law fails on the distance-two [[4,2,2]] code, and is sufficient but not necessary - the degenerate Shor code violates it yet satisfies the law, because the sign factor annihilates the offending pairs. Every kernel identity is certified mode by mode and pair by pair: 732 modes and 1,602 pairs across four codes, with no exception. This note is posted here for the first time. It proves a conjecture stated in A Complete Fourth-Order Expansion of Reconstruction and Logical Errors for Arbitrary Gaussian Coherent Noise (Zenodo DOI 10.5281/zenodo.23644208) and resting on the stabilizer-code framework of Knill-Laflamme Violations, Induced Logical Dephasing, and Deterministic Danger Maps for Coherent Noise on Stabilizer Codes (Zenodo DOI 10.5281/zenodo.23622023). The scripts reproducing every number in the note are deposited in this record as counting_law_scripts.zip. They require Python 3.10+ and NumPy; no other dependencies, CPU only. Version 2 adds counting_law_scripts.zip, which an upload error left out of version 1. The note itself is unchanged: the PDF is byte-for-byte the file published in version 1.