Logarithmic stability on a fixed countable full-branch partition

This paper constructs a continuous family of expanding interval maps on the fixed geometric countable full-branch partition with branch lengths 2^(-i). The family has a uniform BV spectral gap, uniformly bounded variation of reciprocal derivatives, and a uniform exponential inverse-Jacobian tail. For every sufficiently large odd N, at s_N=2^(-12N), both the summed inverse-branch C1 distance and the BV-to-L1 transfer-operator distance are comparable to s_N, and the invariant-density difference has order s_N log(1/s_N). An upper bound Cs(1+|log s|) holds for every sufficiently small positive s. The construction uses persistent majority observables, finite-cylinder truncation, smoothing and endpoint-fixing changes of the common image coordinate. An affine contracting-fibre extension realizes the same response in the integrated fibre norm through the base-to-fibre mechanism described by Galatolo and Lucena. The paper also gives an intrinsic perturbation-scale estimate and a dynamical counterexample to the general-R parameter estimate in Theorem B of arXiv:2508.11878v2, dated 15 May 2026. Structural questions concern Lipschitz criteria, forward-branch C2 perturbations and Wasserstein metrics. AI assistance was used for the analytic construction, source comparisons, proof checks, writing and LaTeX compilation.