Act as a Principal Systems Architect. Design a distributed ledger system capable of processing 100,000 transactions per second (TPS) across five distinct geographical regions. Your design must address the CAP theorem trade-offs specifically focusing on Partition Tolerance. Provide a detailed explanation of the consensus algorithm you would choose (e.g., PBFT, Raft, or a custom DAG-based approach) and justify why it fits the latency and throughput requirements. Additionally, outline the sharding strategy to ensure the state database scales horizontally, and detail the mechanism for handling network partitions and node recovery without compromising data integrity.