Replay

Do not claim the Landau–Ramanujan density. The scripts never use it.

What is certified

G(n) < 2√2 n^{1/4} − 3 for every integer 2 ≤ n ≤ 1.024e15 except n ∈ {3, 6, 21, 91}. (n=1 makes the right-hand side negative.)

Jameson’s a=2 has no exceptions on 1..5e6 (exhaustive table).

Independent checks

# Exact G(n) from a two-square table. Exceptions must be {1,3,6,21,91}.
python3 compute/verify_exhaustive.py --N 2000000 --out compute/exhaustive_a3_2e6.json

# Stored witnesses for every danger-zone top with m <= 250.
python3 compute/verify_a3_cert.py compute/a3_cert_m250.json

# m <= 2000 is a ~1e6-witness dump. It is not in the tree.
# Rebuild, then verify:
# python3 compute/certify_a3.py --m-max 2000 --out compute/a3_cert_m2000.json
# python3 compute/verify_a3_cert.py compute/a3_cert_m2000.json

# Re-run the danger-zone search with no stored witnesses.
python3 compute/certify_a3.py --m-max 8000 --out compute/a3_summary_m8000.json

certify_a3.py enumerates the integer danger zone of RESEARCH.md (ladder tops n=u^2+m^2+1 with 2u=m^2+k and 2m-2 ≤ k ≤ 3m+2) and searches for a lattice point with leftover < Φ−3.

Plot

python3 compute/plot_gaps.py 2000000 figures/gap_ratios.png