Mitigation and Resurgence of the Scalability Trilemma: Mathematical Analysis for Optimal Sharding in Dual–Layer Byzantine Fault–Tolerant Blockchains

The dual–layer consensus process enhances the transaction processing performance of Byzantine Fault-Tolerant Blockchains (BFT BCs) by employing sharding, where validator nodes are partitioned into multiple shards. Within each shard, consensus is formed at a lower level, while an upper–level consensus aggregates decisions from shard leaders, significantly reducing communication overhead. This paper presents a theoretical analysis of the impact of sharding in dual–layer consensus processes on average transaction throughput using a mathematical model. By employing stochastic analysis to model broadcast and block times, we derive a mathematical expression for average throughput, which aligns qualitatively with simulation results obtained through ShardEval. Furthermore, we determine the optimal number of shards that maximizes average throughput, demonstrating that in an optimally configured dual–layer BFT BC, the scalability trilemma can be mitigated, enhancing both security and scalability as decentralization increases. However, we also mathematically demonstrate that cross–shard transactions resurge the scalability trilemma, confirming that even a small fraction of such transactions significantly degrades scalability performance.