CN-shared certificate — 2026-08-15

Status: exact finite replay of cn_shared_analyzer.py / verify_cn_shared.py

Claim boundary: certificate for the statements in CN-SHARED-THEOREM.md. Not Erdős-Straus, not López-all-primes, not universal DSC-P.

Replayable commands

python3 research/erdos-straus/cn_shared_analyzer.py \
  --out /tmp/cn-shared-output --j-limit 160 --k-limit 1500 --program-k-limit 200
python3 research/erdos-straus/verify_cn_shared.py \
  --out /tmp/cn-shared-output --j-limit 120 --k-limit 180

Official machine replay

Artifacts in certificates/:

• certificates/cn-shared-report.md
• certificates/cn-shared.json
• certificates/cn-shared-independent-verifier.json

Independent verifier: VERIFIED.

SHA-256 (certificates/SHA256SUMS):

aed869f84500e109b1477d08cc24c7486bbe60397040bbe209ae285aada2a52e  cn-shared.json
741958030cfed41f56a7fb9e7faf728ca97110954bd21e016c27d9410fd8b0b0  cn-shared-report.md
2239aeb27765bc7515a7cfaa0dae83ebb0350a746bcae7eb718c97b82c124c78  cn-shared-independent-verifier.json

Scan	Result
Totient-ratio, odd q ≤ 5000	10,071 checks; min ratio 2; 0 fails
Standard-L pairs j ≤ 160, mixed r	980,238 pairs; 0 fails; 0 C1 fails
Program L, hard r, k ≤ 200	7,184,124 pairs; 0 fails
Exact complementary q=3 on standard L	124 unrestricted covers; 0 on L=840 and L=2520
Admissible tight q ≤ 9 pairs, k ≤ 1500	73,814 candidates; 1,365,201 escapes; 21 complementary fails; 0 novel fails
Tight q ≤ 9 triples, k ≤ 1500	3,994,891 checks; 0 fails
Independent pair-mask recheck	306,564 pairs; 0 disagreements
Independent program sample k ≤ 180	5,186,154 pairs; 0 fails

The 21 admissible complementary covers

All have q1=q2=3 and contain layer 205.

k	h	t	pair
465	289	1828	25, 205
608	1	2429	205, 322
972	1	3885	52, 205
972	169	3815	52, 205
972	361	3879	52, 205
1050	169	2099	205, 556
1050	529	4191	205, 556
1180	1	2359	205, 322
1180	1	4717	205, 322
1180	361	4711	205, 322
1180	529	4711	205, 322
1271	361	4919	52, 205
1271	529	4919	52, 205
1310	1	2619	70, 205
1310	121	4584	70, 205
1310	169	5237	70, 205
1310	529	5231	70, 205
1466	1	5861	205, 322
1466	1	2931	205, 322
1466	169	5861	205, 322
1466	529	5855	205, 322

Pair counts: (205,322)×9, (52,205)×5, (70,205)×4, (205,556)×2, (25,205)×1.

Unrestricted counterexamples (not program-admissible)

Complementary q=3 covers exist for arbitrary (L,r). First recorded examples include L=55440, j=7,520, r=653 with R={2} and {1}. These show unrestricted C2-shared is false. They are not Type A/B candidates.

Absorption check

Every recorded case with q_205=3 and R_205 ≠ ∅ has 39 | L and is covered by T_10, as proved in CN-SHARED-THEOREM.md §6.