Bounded swaps in pass-and-swap queues: counterexamples to a conjecture of Dorsman and Gardner, and a proof for bipartite swapping graphs with odd caps under positive service rates

Unrefereed research candidate. Dorsman and Gardner (Queueing Systems 107, 2024) studied closed tandems of two pass-and-swap queues in which a service completion may trigger at most w swaps, and conjectured that, when the swapping graph is (w+1)-colourable, every state whose placement orientation has a directed path longer than w edges is transient.

This package shows that the conjecture fails for every cap w ≥ 2. Under positive service rates, the cycle C_n has a recurrent class of excessive height if and only if n ≡ 1 (mod 2w), and that class is unique; for n = 2w+1 it has (2w+2)!/(w!)^2 states and is closed for every admissible rate function. Two odd cycles joined by a path give a further family, and an exact census of all connected graphs with at most seven vertices finds 1355 violating (graph, cap) pairs, none bipartite and none at w = 1. Conversely, the conjecture holds for bipartite swapping graphs with odd caps under positive rates; exact computation finds no counterexample for any cap on bipartite graphs with at most nine vertices (Conjecture 7.2). On the counterexample class of C_{2w+1}, homogeneous rates give an explicit product-form law, while for additive per-job rates scaling the original product weights by one constant per orientation is stationary only when rates are constant within each queue.

The archive contains the manuscript (PDF and LaTeX), standard-library Python verifiers with exact arithmetic, JSON certificates, a one-command byte-for-byte replay, semantic negative controls and SHA-256 sums. Assurance: written proofs; producer-side replay of all 15 certificate runs, byte-identical, including a clean replay from a separate copy; internal AI-assisted review and AI-assisted adversarial review. This is not peer review, independent reproduction or formal verification. Creator: Anonymous. Prose, manuscript and data CC0-1.0; code MIT.