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 $Σ^0_1$-complete and livelock-freedom verification is $Π^0_1$-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.

翻译：我们为自禁用单向环协议开发了一个实际的活锁分析框架。Klinkhamer和Ebnenasir通过从周期性格子问题归约证明，参数化环的活锁检测为$Σ^0_1$-完全问题，而活锁自由验证为$Π^0_1$-完全问题。我们观察到，将分析从转移空间提升至一个等变乘积空间——即转移-证据对空间——可以揭示支持有效验证的结构。我们构建了一个乘积转移图（最多$|T|^2$个节点），该图捕获了所有活锁：每个活锁都映射为此图中的一个证据封闭子图。最大此类子图$G^*(T)$可通过单调不动点迭代在多项式时间内计算（最坏情况$O(|T|^8)$）。当$G^*(T) = \emptyset$时，协议对所有环规模均可证明是活锁自由的——这是一个既完备又正确的活锁自由验证器。当$G^*(T) \neq \emptyset$时，我们采用回溯搜索，通过$G^*$反向传播每个简单环，直到链闭合形成环面（确认存在活锁）或终止（该环不产生活锁）。这一两阶段算法——多项式时间剪枝后接有限组合验证——可产生三种结果：Free、Livelock或Inconclusive。在测试的4,349个协议中（包括从Klinkhamer和Ebnenasir拼贴构造导出的对抗性协议，以及通过其SE构件转换的Kari 14个非周期拼贴集），该算法在每个案例中均得到结论且零错误。我们进一步证明，该算法可通过协议转换扩展到非自禁用协议。这将其适用范围扩展至所有参数化单向环协议。Python实现及使用说明见URL：https://github.com/cosmoparadox/mathematical-tools。