Parity of affine Karpelevich containment on Markov rings

Unrefereed research candidate. In one sentence: on a driven Markov ring, the conjectured affine Karpelevich bound on rate-matrix eigenvalues holds for every choice of rates exactly when the ring has three states or an even number of states, and fails for every odd ring with five or more states. The complex eigenvalues of a Markov rate matrix set how fast a driven chain relaxes and how strongly it oscillates. Xu, Kolchinsky, Delvenne and Ito (Physical Review Letters, 2025) conjectured that, after rescaling time by the fastest escape rate and the imaginary part by a factor set by the cycle affinity F, these eigenvalues always lie in the Karpelevich region Ωn of stochastic-matrix eigenvalues. An earlier Evidence Press release (doi:10.5281/zenodo.23687327) refuted this for general networks and left the single-ring case open (its Open problem 8.1). This paper settles that case: for bidirectional rings on n ≥ 3 states at any fixed F > 0, containment holds for all rates exactly when n = 3 or n is even. The even and three-state cases follow from known results (the even case from the affinity–winding theorem of Kolchinsky, Ohga and Ito, an unrefereed 2026 preprint). The new content is the failure for every odd n ≥ 5 at every driving strength: weakly coupled pairs of states (dimers) create eigenvalues near −1 whose rescaled imaginary parts grow too steeply. It is proved at every prescribed affinity and, with explicit constants, along a rational family whose affinity tends to zero. Rings with integer rates 1, 2, 3, equal escape rates and affinity log 3 violate the conjecture in five, seven and nine states, each with an exact rational certificate. Near −1 the conjectured slope is too small by a factor approaching n/m ≈ 2 at weak driving (n = 2m + 1), while the proved affinity–winding bound becomes sharp as n grows. Numerically, violations concentrate on rings with nearly equal escape rates, which suggests why random searches missed them; for the five-state witness the conjectured region would overstate the driving force by about 43%, while the valid winding bound gives a correct lower estimate. The archive contains the manuscript (PDF, Markdown source and generated LaTeX), figures and their scripts, the bibliography, exact rational certificate scripts (Python standard library) with shipped outputs and a byte-for-byte replay driver, stage-checked negative controls, NumPy floating-point illustrations, a verification ledger, the dated literature audit, a rating-free internal review summary, CI and SHA-256 sums. Assurance: written proofs plus exact certificates of the finite examples; numerical sections are floating-point illustrations, not certificates; internal AI-assisted review only (not peer review, independent reproduction or formal verification). Creator: Anonymous. Text, data and figures CC0-1.0; code MIT.