How fast can a driven Markov network oscillate? A four-state counterexample to the affine Karpelevich conjecture, and the exact dimension at which it fails

Unrefereed research candidate. The eigenvalues of a continuous-time Markov rate matrix set how fast its distribution relaxes and how strongly it can oscillate. Xu, Kolchinsky, Delvenne and Ito (Phys. Rev. Lett. 135, 257102, 2025; Supplement Eq. (S3), read on arXiv:2507.08938v2) conjectured that these eigenvalues fill the classical Karpelevich region of stochastic matrices, stretched by the largest escape rate and flattened vertically by a factor set by the largest cycle affinity per step. This package refutes that conjecture. An integer four-state rate matrix, with about 8.4 kBT of driving around its strongest cycle, has the eigenvalue −4745 + 450√3 i, whose rescaled image lies 0.0087 outside the region; one integer inequality proves it. For at most three states the containment holds, so four is the smallest failing size. The mechanism is weak resetting: it moves eigenvalues by O(ε) but relaxes the affinity factor by order ε2/3. This gives counterexamples near a whole boundary arc in four states, an explicit interval 0 < ε ≤ 1/100, exactly certified five- and six-state instances (data printed in an appendix and re-checked by a stand-alone checker), and an asymptotic construction in every dimension n ≥ 4. Scope: every counterexample uses the complete graph. Whether the conjecture holds for single-cycle networks, the setting in which it was tested, remains open; a small numerical search found no violation, which is evidence, not proof. The violations are thin. The proved affinity ellipse and Farey-polygon bounds are unaffected. The archive contains the 20-page manuscript (PDF and LaTeX), standard-library Python verifiers, exact JSON receipts and certificate data, a stand-alone certificate checker, semantic negative controls, CI and SHA-256 sums. Assurance boundary: written proofs; exact and interval-arithmetic certificates; a producer-side replay from a clean copy; internal AI-assisted review, including correctness audits by a different model before and after revision, a targeted proof audit and a verification re-review. This is not peer review, independent reproduction or formal verification. Creator: Anonymous. Prose, manuscript and data CC0-1.0; code MIT.