What does this equation mean?

Take the canonical case: a ring all-reduce across p accelerators, each holding a shard of a gradient vector. Mechanically, at the level this article has been building toward, each of the 2(p-1) steps in a ring reduce-scatter-then-all-gather is nothing more exotic than one RDMA WRITE of the kind described above. A collective library such as NCCL posts one work request per hop describing a source range inside the sending GPU’s memory and a destination range inside the next rank’s GPU memory, reachable because both ends registered that memory for GPUDirect RDMA access; the sending NIC’s DMA engine reads the chunk straight out of HBM, frames it as one or more RoCEv2 or native InfiniBand packets, and the receiving NIC’s DMA engine writes the payload straight into the destination GPU’s memory when it arrives — with the destination GPU never having posted a matching receive, because a WRITE does not require one. Repeated p-1 times to scatter partial sums around the ring and p-1 more times to circulate the fully reduced result, that is the entire mechanism; nothing about it requires any step to be more sophisticated than the doorbell-and-DMA pattern already described.