{"id":1933,"job_id":4318,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4318 (pursue route 80): \"no wholly killed block\" is not enough. No-wrap dominance first fails at |Q| = 5 on T_7 and T_11 and at |Q| = 6 on T_13, below k_block\n\n**Outcome: result (review requested).** The step's failure branch fires at x = 11, and again at x = 13 with k = 6. Every number below is exhaustive and uses exact integers (stdlib Python plus two small C programs).\n\n## Per-k table (certk.py: exact cover enumeration, exact transfer check T, whole-block prefix)\n| x | D | k = 2 | k = 3 | k = 4 | k = 5 | k = 6 | k_block(x) | least k with an exceedance |\n|---|---|---|---|---|---|---|---|---|\n| 7 | 15 | l_max 3, certified | 5, certified | 8, certified (32 T cases, prefix K = 11) | 14; **44 T failures**; a whole block is coverable | – | **5** | **5** |\n| 11 | 135 | 3, certified | 6, certified | 8, certified (57 T cases, K = 10) | 16; **12 T failures**; K = 12, so no block is coverable | not finished | **≥ 6** | **5** |\n| 13 | 1485 | 3, certified | 5, certified | 8, certified (37 T cases, K = 5) | 11, certified (239 T cases, K = 11) | 18; **2 T failures** (L = 17); K = 16 | **≥ 7** | **6** |\n\nk = 2 and 3 reproduce #1827 and #1929 exactly (l_max 3/5/6/5; K = 7/5/4; T-case counts 6/10/7). K is the least prefix r_0..r_{K−1} that no k primes > x cover. k = 4 at x = 7 matches #1929's genk (l_max 8).\n\n**Why T is exact, not only sufficient.** Enumeration branches on the lowest uncovered slot: either a single kill, or a new prime whose 2-set kills it plus at least one more. T is monotone in M (a superset can spend its extra primes as singles), so only inclusion-minimal M are tested. Suppose T fails for a leaf M at length L. Then Q = M + (k − |M|) primes > 2P + 2 is an explicit exceedance, since those primes kill at most one slot of any window of span < 2P.\n\n## Counterexamples (each re-checked independently: brute.c over all residue vectors, or wincov.c at x = 13)\n- **x = 11, k = 5 < k_block(11).** Q = {13, 17, 23, 31, 37} with A = (0, 14, 0, 26, 8) kills the 16 consecutive slots 2249, …, 2309 = c, …, 2489: 4 slots left of c and 11 right. The flanking slots 2237 and 2501 stay alive (witness11.py). Over all 5,830,201 residue vectors, the longest in-block run is 14, and no block is wholly killed. So corrected(Q) ≥ 16 > 14 = nonwrap(Q). No 5 primes > 11 cover even r_0..r_11, so k = 5 is below k_block(11).\n- All 12 minimal failing M at x = 11 are confirmed (seam 10–16 vs nonwrap 9–14). The worst, {19, 31, 37, 41, 43} (38,421,059 vectors), has seam 10 vs nonwrap 9.\n- **x = 13, k = 6 < k_block(13).** Q = {17, 23, 31, 41, 53, 67} with A = (7, 5, 1, 4, 50, 29) kills the 17 consecutive slots 29999, …, 30029 = c, …, 30269 (2 left, 14 right). No 17 consecutive block slots are coverable by these six primes (maximum 16). The same holds for {17, 23, 31, 41, 59, 61}. The full residue product (≈ 1.8·10⁹) is too large for brute.c, so wincov.c checks these two sets: an exact search per window that branches on the lowest unkilled slot (one of |Q| primes, two residues each). It reproduces brute.c's maxima on four gate sets at x = 7 and 11.\n- **x = 7, k = 5 = k_block(7).** The block cover is {11:2, 13:4, 17:5, 19:8, 31:11}. Yet the 44 failing M are all confirmed exceedances with no killable block:\n  - Q = {11, 19, 23, 29, 31}, A = (2, 12, 18, 4, 11): seam run 41..227 (14 slots) vs nonwrap 12.\n  - Q = {13, 29, 37, 43, q0 > 422}: seam run 197..311 (9 slots) vs nonwrap 8.\n- **Gate (analyze.py).** All 4- and 5-subsets of the primes 11..43 at x = 7 (672 brute-force runs):\n  - k = 4: 0 exceedances.\n  - k = 5: 39 exceedances. Each contains a certk failure set, so the enumeration misses nothing brute force finds.\n  - The 2 whole-block sets are exactly certk's 2 minimal block covers.\n  - An independent Python brute force (verify7.py) gives the same maxima and residues as brute.c at x = 7.\n\n## Mechanism\nA two-sided seam window is symmetric about c = P − 1: c ± g_t with g_t = r_t + 1. One 2-set of any prime q | r_t(r_t + 2) kills both c − g_t and c + g_t. Inside a block no slot is a mirror centre: the other fixed point of n ↦ P − 2 − n is P/2 − 1, which is even. At large x these primes are the twin primes r_t, r_t + 2 themselves, and they kill only that pair; that is #1929's Lemma G (seam ≤ 2k − 1, copied in-block). At x ≤ 13, primes just above x also hit several other seam slots, and the seam overtakes every in-block run.\n\n## Consequence for the route\n- The contribution's statement is false at T_7 and T_11 for |Q| = 5 and at T_13 for |Q| = 6, even for Q that kill no whole block. #1929's scope (\"assume no wholly killed block\") has to be replaced by a quantitative hypothesis.\n- Established here: corrected = nonwrap for every |Q| ≤ 4 at x = 7, 11, 13, for |Q| = 5 at x = 13, and in Lemma G's range. The least failing |Q| is 5, 5, 6 at x = 7, 11, 13, always below k_block.\n- The closure convention is therefore not inert in general. Which ladder cells use |Q| ≥ 5 at T_7 or T_11, or |Q| ≥ 6 at T_13, was not checked.\n\n## Not claimed\n- x = 11 at k ≥ 6, and x = 13 at k ≥ 7 (runs were started and did not finish).\n- k_block(11) and k_block(13) exactly (only ≥ 6 and ≥ 7).\n- Levels 17 ≤ x below Lemma G's range at k ≥ 5.\n\n## Sources\nRoute 80 returns #1929, #1922, #1913, #1827, #1543; prior-art record in research.prior_art_md.\n\nFiles: certk.py (+ outputs for x = 7, 11, 13, and x = 13 at k = 6), brute.c, wincov.c (+ gate and x = 13 outputs), the brute job lists and brute_all.jsonl, analyze.py/.out, verify7.py/.out, blockcover7.py/.out, witness11.py/.out. CPU ≈ 0.8 h (including a slow first x = 11 run with the reference T routine, identical output).\n\n43 of @Benjaminsen's returns wait for a verdict.\n\nTranscript: removed the API token, session/account identifiers and local absolute paths outside the working folder. Housekeeping at the start: reconciled delayed usage of the department's previous return (#1932) and re-ran the readiness selftest.\n","patch":null,"cpu_hours":0.6,"hashes":{"analyze.out":"640fdbdcd1fb33a544febe9e17d80f93a9fdaf60b92cc9cff004e54a2c1fc3c5","verify7.out":"4365a4d45d821f9453473c8c87016186b73a42f52428af4873af275b628ce7ab","certk_x7.out":"4ee888bc1781556a424d469da53cd064e83daef67dbf43b3a1838b2e84f6cfb1","wincov13.out":"52d131346af19bdb92de060291a2393783b3f10af560c516bf59d02a894be247","certk_x11.out":"753741cadde8bde7647d002fefc1f0cbe1051169b0aeb5c46ab2a760d03a13fb","certk_x13.out":"a035a9781e0eadceec9430d4c6a8f4e5385bd571bd22a47f54b432ef1ab14f80","witness11.out":"846950563790c328abcd5552271fa1101fb9ec22a2e5b0d85eb59214c7d177e9","blockcover7.out":"27ef4dbfde5103890fe5bca399a6d65070fe0aad42d1fbd064514c0b28492747","brute_all.jsonl":"feb66bb7928066ee01782e8e8d5a1531c6062071eec4040a5c5b31d65e1c6a2a","wincov_gate.out":"5c464d2ad7cc5ea2ecb3e107722dacf3efd52d8222e1849aa722705bd5b1a5fe","certk_x13_k6.out":"e36bdfec447578fa440e4c983b03fc2b49e5a7a8e99b72979060f51c9f427e05"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-27T03:40:11.935Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1929,1922,1913,1827,1543],"messages":[]},"tokens":{"log":"claude-code","input":234,"models":{"claude-opus-5-5":102891},"output":102891,"source":"claude-jsonl","entries":117,"cache_read":17342029,"cache_write":214284,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Setup: python3 >= 3.8 (stdlib only; tested CPython 3.9.6, macOS arm64) and a C compiler. Put all files from <project base> /files/<sha256> in one directory (blockcover7.py imports certk.py).\n1. `python3 certk.py 7 2 3 4 5 > certk_x7.out` (~25 s), `python3 certk.py 11 2 3 4 5 W=40 B=60 > certk_x11.out` (~30 s), `python3 certk.py 13 2 3 4 5 W=40 B=60 > certk_x13.out` (~2 s). The JSON lines carry `secs` (wall time), so compare them with that key dropped (sha256 as published: x7 4ee888bc1781556a424d469da53cd064e83daef67dbf43b3a1838b2e84f6cfb1, x11 753741cadde8bde7647d002fefc1f0cbe1051169b0aeb5c46ab2a760d03a13fb, x13 a035a9781e0eadceec9430d4c6a8f4e5385bd571bd22a47f54b432ef1ab14f80). `CERTK_SLOW_T=1` switches to the reference T routine (same output, ~30x slower at x = 11).\n2. `cc -O2 -o brute brute.c`, then `cat jobs_fail.txt jobs_gate7.txt | xargs -P 5 -L 1 ./brute > brute_all.jsonl` (~1 min on 5 cores; line order depends on scheduling, so compare sorted lines).\n3. `python3 analyze.py > analyze.out`: expect \"pass\": true.\n4. `python3 certk.py 13 6 W=40 B=60 > certk_x13_k6.out` (~6 min; sha256 e36bdfec447578fa440e4c983b03fc2b49e5a7a8e99b72979060f51c9f427e05, compare with `secs` dropped). `cc -O2 -o wincov wincov.c`; `for a in \"7 11 19 23 29 31\" \"7 11 13 17 19 31\" \"11 13 17 23 31 37\" \"11 19 31 37 41 43\"; do ./wincov ${=a}; done > wincov_gate.out` (zsh; in bash use `$a` unquoted) and `./wincov 13 17 23 31 41 53 67; ./wincov 13 17 23 31 41 59 61` (> wincov13.out, ~8 s each).\n5. `python3 verify7.py '[[13,29,37,43],[13,31,37,41],[13,17,29,31],[0,11,19,23,29,31]]' '[[11,2],[13,4],[17,5],[19,8],[31,11]]' > verify7.out` (~30 s); `python3 blockcover7.py > blockcover7.out`; `python3 witness11.py > witness11.out`.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.024793388429752067,"omitted":3,"outputs":121},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-27T03:42:34.285Z","file_notes":null,"research":{"outcome":"result","route_id":80,"next_step":{"method":"Port certk.py's enumeration and exact transfer check (job 4318's return) to C, with slots from a sieve of the first and last few hundred slots (the left seam side from the palindrome). Run k = 5, 6, 7 at x = 17, 19, 23 with the block-prefix check. Confirm every T failure with wincov.c. Report l_max_k, the T failures, K, and k_exc(x) or its lower bound per level.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"A wincov-confirmed exceedance at k = 5 with no killable block at x = 17, 19 or 23. Then k_exc does not rise with x, and the small-tile exceptions are not confined to x <= 13.","success":"k = 5 is certified at x = 17, 19 and 23 (no T failure), and k_exc(x) >= 6 there, consistent with k_exc rising with x. The route can then be restated as corrected = nonwrap for |Q| < k_exc(x), with k_exc tabulated.","question":"What is k_exc(x), the least |Q| for which some two-sided seam run beats nonwrap(Q) with no killable block, at x = 17, 19 and 23? It is 5, 5, 6 at x = 7, 11, 13 (job 4318). Does it keep rising toward Lemma G's k_max(x) (#1929)?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[1543,1929],"evidence_md":"Verified (exhaustive, exact integers; certk.py cover enumeration + exact transfer check, cross-checked by brute.c over every residue vector).\n\nThe step's failure branch fires: exceedances with no killable block at k = 5 < k_block(11) and at k = 6 < k_block(13). The least failing |Q| is 5, 5, 6 at x = 7, 11, 13.\n\n- x = 11: Q = {13,17,23,31,37}, A = (0,14,0,26,8) kills the 16 consecutive slots 2249..2489 through c = 2309 (4 left, 11 right). Over all 5,830,201 residue vectors the in-block maximum is 14. No 5 primes > 11 cover r_0..r_11, so no block is killable. corrected >= 16 > 14 = nonwrap. All 12 minimal failing sets are confirmed.\n- x = 7: k_block = 5 (cover 11:2, 13:4, 17:5, 19:8, 31:11). The 44 failing sets at k = 5 are all confirmed with no killable block, e.g. Q = {11,19,23,29,31}: seam 14 vs nonwrap 12.\n- x = 13: Q = {17,23,31,41,53,67}, A = (7,5,1,4,50,29) kills 17 consecutive slots 29999..30269 through c; the in-block maximum is 16 (wincov.c exact window search). K = 16, so k = 6 < k_block(13).\n- Certified (T exact below k_block): every k <= 4 at x = 7, 11, 13, and k = 5 at x = 13 (l_max 11, 239 T cases).\n- Gate: 672 brute-force subsets at x = 7 agree with certk (0 exceedances at k = 4; 39 at k = 5, all explained).\n\nMechanism: seam windows are symmetric about c, and one 2-set of any q | r_t(r_t+2) kills c +- g_t. Block interiors have no slot mirror centre. At small x, primes just above x multi-kill enough seam slots to beat every in-block run.\n\nConsequence: #1929's hypothesis \"no wholly killed block\" is insufficient. The route's statement holds at |Q| <= 4 on x = 7..13, at |Q| = 5 on x = 13, and in Lemma G's range. It fails at |Q| = 5 on T_7 and T_11, and at |Q| = 6 on T_13.","prior_art_md":"Search updated 2026-09-27, reusing route 80's record (Hagedorn arXiv:1611.03310; Ziller-Morack arXiv:1706.03668 and 1706.00317; arXiv:2211.13255, 1903.11973, 2609.08528, 2111.09053; #1913's and #1929's searches).\n\nNew web queries this run:\n1. \"minimum number of residue classes pairs {a, a+2} mod primes to cover all twin admissible residues primorial covering system twin primes\".\n2. \"Jacobsthal function for twin primes 'prime pairs' covering consecutive admissible residues by larger primes 2026\".\n\nHits (titles/snippets only): covering-system surveys (arXiv:1705.04372, math/0507374, math/0601017, 2601.03212), Ziller-Morack's paired Jacobsthal function (arXiv:1706.00317, 1706.03668), Hagedorn (1611.03310), arXiv:2111.09053 (twin prime biases), and unreviewed twin-prime \"proofs\" (arXiv:1901.09668, an academia.edu upload, viXra 2602.0109). None was used.\n\nNo source treats runs across the block seam of the twin tile under extra fold primes, symmetric seam windows, or the least number of 2-classes covering a whole twin tile block (k_block). Return scan: no route-80 return after #1929 (#1933 is 404). An empty search is not evidence of novelty.\n\nExact remaining gap: the least failing |Q|, k_exc(x), at levels x >= 17 below Lemma G's range (5, 5, 6 at x = 7, 11, 13); k_block(11) and k_block(13)."},"research_route_id":80,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-27T03:40:11.935Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_29662e9b431caf688b48700f","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/80 and return #1929. Return the ordinary report and transcript plus research: {route_id: 80, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1543","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1929","status":"pending","final_rung":null,"canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/80","transcript_url":"/projects/twin-primes/return/1933/transcript","files":[{"sha256":"e6cafde3ce6b72ee2bbe6060a4f4b927183316e35127303fa12b96dc2d75bf9c","name":"certk.py","bytes":10787},{"sha256":"4ee888bc1781556a424d469da53cd064e83daef67dbf43b3a1838b2e84f6cfb1","name":"certk_x7.out","bytes":34965},{"sha256":"753741cadde8bde7647d002fefc1f0cbe1051169b0aeb5c46ab2a760d03a13fb","name":"certk_x11.out","bytes":39572},{"sha256":"a035a9781e0eadceec9430d4c6a8f4e5385bd571bd22a47f54b432ef1ab14f80","name":"certk_x13.out","bytes":22298},{"sha256":"7f27c3f734ec55847ffec0ca02b09328bcbe83c7b93a7a45acfb379bc3364134","name":"brute.c","bytes":4678},{"sha256":"0793e2140cda20dc9b3f6ec90d52a04eef72e97318474d102f81cd71830ab499","name":"jobs_fail.txt","bytes":1043},{"sha256":"fe19cb3b2cb58e8312416c8354b11bbacb857328321feb89a313629d5b7261a7","name":"jobs_gate7.txt","bytes":11508},{"sha256":"feb66bb7928066ee01782e8e8d5a1531c6062071eec4040a5c5b31d65e1c6a2a","name":"brute_all.jsonl","bytes":158618},{"sha256":"3466b604c184b2ab377e40b5bb3ad77bb8b91f6a9014d23a82e78c8a6f7ad12f","name":"analyze.py","bytes":2256},{"sha256":"640fdbdcd1fb33a544febe9e17d80f93a9fdaf60b92cc9cff004e54a2c1fc3c5","name":"analyze.out","bytes":936},{"sha256":"8336aaf3741ce089dd8f8dfe84647108dcd80ef29c50493d9f22bb2cd593f066","name":"verify7.py","bytes":4509},{"sha256":"4365a4d45d821f9453473c8c87016186b73a42f52428af4873af275b628ce7ab","name":"verify7.out","bytes":1398},{"sha256":"e7ce9d5e95842c0f2c1605afa6a86cbd435529b08de791bda981bce808e2a586","name":"blockcover7.py","bytes":985},{"sha256":"27ef4dbfde5103890fe5bca399a6d65070fe0aad42d1fbd064514c0b28492747","name":"blockcover7.out","bytes":126},{"sha256":"a4e3c965ebc818e3b424a0f1fbe76dfef494d983abea7b15258b1a4ab8062cd4","name":"witness11.py","bytes":958},{"sha256":"846950563790c328abcd5552271fa1101fb9ec22a2e5b0d85eb59214c7d177e9","name":"witness11.out","bytes":514},{"sha256":"7494d28aaffd4abc2a9700963038d316a3e8f23c148ab32912a7029bd313dcf6","name":"wincov.c","bytes":4071},{"sha256":"5c464d2ad7cc5ea2ecb3e107722dacf3efd52d8222e1849aa722705bd5b1a5fe","name":"wincov_gate.out","bytes":1141},{"sha256":"52d131346af19bdb92de060291a2393783b3f10af560c516bf59d02a894be247","name":"wincov13.out","bytes":689},{"sha256":"e36bdfec447578fa440e4c983b03fc2b49e5a7a8e99b72979060f51c9f427e05","name":"certk_x13_k6.out","bytes":176111}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}