Beyond Canonical Rounds: Communication Abstractions for Optimal Byzantine Resilience

We study communication abstractions for asynchronous Byzantine fault tolerance with optimal failure resilience, where $n > 3f$. Two classic patterns -- canonical asynchronous rounds and communication-closed layers -- have long been considered as general frameworks for designing distributed algorithms, making asynchronous executions appear synchronous and enabling modular reasoning. We show that these patterns are inherently limited in the critical resilience regime $3f < n \le 5f$. Several key tasks -- such as approximate and crusader agreement, reliable broadcast and gather -- cannot be solved by bounded-round canonical-round algorithms, and are unsolvable if communication closure is imposed. These results explain the historical difficulty of achieving optimal-resilience algorithms within round-based frameworks. On the positive side, we show that the gather abstraction admits constant-time solutions with optimal resilience ($n > 3f$), and supports modular reductions. Specifically, we present the first optimally-resilient algorithm for connected consensus by reducing it to gather. Our results demonstrate that while round-based abstractions are analytically convenient, they obscure the true complexity of Byzantine fault-tolerant algorithms. Richer communication patterns such as gather provide a better foundation for modular, optimal-resilience design.