A Lock-Free, Fully GPU-Resident Architecture for the Verification of Goldbach's Conjecture

Abstract

We present a fully device-resident, multi-GPU architecture for the large-scale computational verification of Goldbach's conjecture. In prior work, a segmented double-sieve eliminated monolithic VRAM bottlenecks but remained constrained by host-side sieve construction and PCIe transfer latency. In this work, we migrate the entire segment generation pipeline to the GPU using highly optimised L1 shared-memory tiling, achieving near-zero host-device communication during the critical verification path. To fully leverage heterogeneous multi-GPU clusters, we introduce an asynchronous, lock-free work-stealing pool that replaces static workload partitioning with atomic segment claiming, enabling $99.7$% parallel efficiency at 2 GPUs and $98.6$% at $4$ GPUs. We further implement strict mathematical overflow guards guaranteeing the soundness of the 64-bit verification pipeline up to its theoretical ceiling of $1.84\times 10^{19}$. On the same hardware, the new architecture achieves a $45.6\times$ algorithmic speedup over its host-coupled predecessor at N = $10^{10}$. End-to-end, the framework verifies Goldbach's conjecture up to $10^{12}$ in $36.5$ seconds on a single NVIDIA RTX 5090, and up to $10^{13}$ in $133.5$ seconds on a four-GPU system. All code is open-source and reproducible on commodity hardware.

14 pages, 4 figures, 3 tables. The presented work details a major architectural overhaul: migration of the segmented sieve to GPU L1 shared memory and the implementation of a lock-free multi-GPU work pool. Source code available at: https://github.com/isaac-6/goldbach-gpu