Word-Level Bulk Marking for GPU Verification of Goldbach's Conjecture beyond 4x10^18

Abstract

The exhaustive verification of Goldbach's conjecture has stood at 4x10^18 since 2014, established by Oliveira e Silva, Herzog and Pardi using roughly 770 one-core CPU years on distributed clusters. Recent GPU frameworks have improved throughput substantially but have not extended the frontier, reaching 10^13. We report an in-progress verification of the interval [4x10^18, 4.5x10^18) on a single six-year-old four-GPU workstation. The verification kernel replaces per-integer primality lookups with word-level bulk marking: maintaining a bitmap of unresolved even integers and clearing it by shifted conjunction against the segment's prime bitmap, unsolved &= ~(P >> j_p), where the shift j_p is independent of the segment start. No primality test is performed at any stage, which removes the Miller-Rabin witness-set ceiling; the sole limit is 2^64 index arithmetic. Sieving uses shared-memory tiling with a bucket structure whose per-tile occupancy proves predictable to within 0.9% (3 sigma), permitting the counting pass to be eliminated. Because the hardware lacks ECC memory and the run spans months, we describe a correctness protocol combining full dual-path recomputation on the GPU with an architecturally independent CPU sample, exploiting the fact that max p_min(n) is invariant under all implementation parameters and therefore serves simultaneously as checksum, cross-machine fingerprint and scientific output. Two findings from the thermal work may be of wider use. At a fixed power limit, each degree Celsius of room temperature costs 0.80% of throughput (correlation -0.855), so ventilating the room overnight is worth about eight days over the run. Replacing hand-tuned per-card power limits with a closed loop targeting a fixed die temperature raised throughput by 18% while reducing its hour-to-hour variation by a factor of three, and removes the need for seasonal adjustment by hand. Aggregate throughput is 3.09x10^10 integers/s including dual-path verification, equivalent to approximately 96 core years of the 2014 computation at roughly $600 of electricity. The paper claims no novelty for the segmented sieve, the shared-memory tiling, the correctness protocol, or the choice of fingerprint. An appendix records thirteen judgements overturned during the work, seven by measurement and six by literature search; a further section withdraws a benchmark reported in version 1, which timed library start-up rather than steady-state throughput. Progress as of this deposit: 1.35x10^16 integers verified (2.7% of the interval), max p_min(n) = 8627, zero dual-path disagreements. No counterexample encountered. Status: work in progress. The verification is ongoing, with estimated completion February 2027. This record will be updated with the final results, the full distribution of p_min(n) over the interval, and the complete source code on completion.