Practical Livelock Analysis in Parameterized Unidirectional Rings

We develop a practical framework for livelock analysis in self-disabling unidirectional ring protocols. Klinkhamer and Ebnenasir established that livelock detection for parameterized rings is $\Sigma^0_1$-complete and livelock-freedom verification is $\Pi^0_1$-complete, via reduction from the periodic domino problem. We observe that lifting the analysis from the transition space to an \emph{equivariant product space} -- the space of transition-witness pairs -- reveals structure that supports effective verification. We construct a \emph{product transition graph} (at most $|T|^2$ nodes) that captures all livelocks: every livelock maps into this graph as a witness-closed subgraph. The maximal such subgraph $G^*(T)$ is computable in polynomial time ($O(|T|^8)$ worst case) via monotone fixed-point iteration. When $G^*(T) = \emptyset$, the protocol is \emph{provably livelock-free} for all ring sizes -- a sound and complete livelock-freedom verifier. When $G^*(T) \neq \emptyset$, we apply a backtracking search that backward-propagates each simple cycle through $G^*$ until the chain either closes into a torus (confirming a livelock) or dies (no livelock from that cycle). This two-phase algorithm -- polynomial-time pruning followed by finite combinatorial verification -- produces three outcomes: Free, Livelock, or Inconclusive. Across 4{,}349 protocols tested (including an adversarial protocol derived from Klinkhamer and Ebnenasir's tiling construction and Kari's 14-tile aperiodic set converted via their SE gadget), the algorithm is conclusive in every case with zero errors. We further demonstrate that the algorithm extends to non-self-disabling protocols via a protocol transformation. This extends the algorithm's applicability to all parameterized unidirectional ring protocols. Python implementation and usage instructions are at URL: https://github.com/cosmoparadox/mathematical-tools.