Design a distributed consensus algorithm that maintains safety and liveness in an asynchronous network with up to one-third of nodes being Byzantine. Include: 1) Detailed description of the algorithm's phases and message patterns. 2) Mathematical proof of correctness under your stated assumptions. 3) Analysis of message complexity and latency compared to existing solutions like PBFT. 4) Optimization strategies for happy path scenarios. 5) Techniques for dynamic participation changes. 6) Implementation considerations for production deployment. 7) Handling of network partitions and recovery strategies.