{"id":1929,"job_id":4295,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4295 (pursue route 80): |Q| = 3 no-wrap dominance holds at every prime level 7 ≤ x ≤ 10^6, and a general-|Q| lemma reduces the question to twin-prime counts in (x, 2x)\n\n**Outcome: result (review requested).** Both parts of the step pass. Part A: all 43 prime levels 7 ≤ x ≤ 200 are certified at |Q| = 3. Part B: check C (the copy lemma) is proved under the stated side conditions, and those conditions are verified at all 78,495 prime levels x ≤ 10^6. Together these give corrected = nonwrap for every triple of primes > x at every prime level 7 ≤ x ≤ 10^6. The same argument gives a lemma for any |Q| = k.\n\n**Scope first.**\n- **The route's \"for every Q\" is false as stated once |Q| > D(T_x).** Take |Q| = D + 1 primes. Each kills one of the D + 1 consecutive slots r_0..r_{D-1}, P + r_0 (set A_q = slot mod q). Then corrected ≥ D + 1 > D ≥ nonwrap. So the statement needs the hypothesis \"no block is wholly killed\". Every certificate below proves that hypothesis for the Q it covers.\n- **Rungs.**\n  - The reduction, the copy lemma, the bound d(x) ≥ p*² − 1 and the general-k lemma are **proven** (derivations below).\n  - The per-level certificates are **verified**: exhaustive, stdlib Python, exact integers.\n  - Nothing is claimed beyond x = 10^6. An all-x version needs twin primes in (x, 2x), which is open.\n- **Premises.** Only #1543's palindrome (accepted, verified) is required. #1827 (pending) is not a premise: its one-sided/mirror reduction is restated and proved here, and its `construct` (check C) is reused verbatim and re-verified from residues.\n\n## Reduction (any |Q|; notation as in #1543/#1827)\n- c = P − 1 is the last slot, g_t = r_t + 1, and by the palindrome the slots next to c are c ± g_t.\n- **(O) One-sided seams.** A run crossing the seam contains c and c + g_0. If it has no slot left of c, its mirror n ↦ 2c − n is a block-0 run under mirrored 2-sets, so it is ≤ nonwrap.\n- **Two-sided seams.** Otherwise the run's slot set is exactly W(u,v) = {c − g_{u−1..0}, c, c + g_{0..v−1}}, with u, v ≥ 1.\n- **(B) Whole blocks.** A run over ≥ 3 blocks contains a whole block. The first K slots of block b are killed iff r_0..r_{K−1} are killed under A_q − bP. So \"r_0..r_{K−1} not coverable by Q\" excludes this case.\n- **Short seams.** Any k primes kill any k consecutive slots, so nonwrap ≥ k, and seams of length ≤ k are dominated.\n- **Finiteness.** With j primes left and more than j uncovered slots, some prime kills two slots at distance d, so q | d(d−2)(d+2). Cover enumeration is therefore finite.\n\n## Part A (cert3.py, x = 7..200, |Q| = 3)\n- **l_max(x).** The largest two-sided window coverable by 3 primes > x. It is 6 at x = 11, 17, 19, 23, 29, 31 and 5 at the other 37 levels.\n  - It is always ≥ 5, because any prime factor of r_t(r_t + 2) kills both c ± g_t (2g_t ≡ ±2).\n- **Check T (transfer).** For every window of length 4..l_max and every cover leaf, let M be the primes that kill ≥ 2 slots. The check finds l_max consecutive block-0 slots coverable by M plus 3 − |M| single kills. That gives nonwrap(Q) ≥ the seam length for every real cover, since each real cover has a leaf whose M lies within its own primes.\n  - T passes at 40 levels, including x = 7, 11, 13, 17, where no copy exists (#1922's note).\n  - At 179, 191 and 193, T finds no window within the scanned prefix n < 4·10^5.\n- **Check C (copy).** `construct` (#1827, unchanged) returns n = −1 + kP/p* whose slot pattern on [−g_{l_max−1}, g_{l_max−1}] equals c's. It succeeds at every x ≥ 19.\n- **Whole block.** The least uncoverable prefix is K = 7 at x = 7, 5 at x = 11 and 29, and 4 elsewhere.\n- **Verdict.** `not_certified_by_T_or_C: []` over all 43 levels.\n\n## Part B: the copy lemma, proved\n**Lemma C.** Let p* ≥ 5 be a prime ≤ x, and let R be a two-sided seam run of length l with offsets J ⊂ [−G, G]. Suppose (S1) no p*-only offset j (p* the only prime ≤ x dividing (j−1)(j+1)) has |j| ≤ G, and (S2) 2l + 1 < p*. Then n = −1 + kP/p* for some k ∈ [1, p*) maps R to a block-0 run of the same length under A'_q = A_q + (n − c).\n\n*Proof.*\n- For p ≤ x with p ≠ p*, n ≡ c (mod p). As k varies, b = n mod p* takes every residue except −1.\n- c + j fails at p iff p | (j−1)(j+1).\n- **Slots of R.** n + j (j ∈ J) is a slot iff b ∉ {−j, −j−2}. That excludes ≤ 2l residues, plus b = −1, and 2l + 1 < p* leaves some b.\n- **Gaps of R.** For even j strictly inside R and not in J, some prime p ≤ x divides (j−1)(j+1). By (S1), p ≠ p*, so n + j is not a slot. Odd j are even numbers.\n- **Kills and block.** Kills are translation-invariant under A'. Since G < P/p* − 1, the image lies inside block 0. ∎\n\n**d(x) ≥ p*² − 1** (p* = the largest prime ≤ x with p* + 2 composite, as in #1913).\n- A p*-only j is s·p* ± 1 with s odd, and s·p* divides (j−1)(j+1). So s has no prime factor ≤ x other than p*.\n- s = 1 fails: p* − 2 ≥ 3 has a prime factor < x, and p* + 2 is composite with a factor ≤ √(p*+2).\n- Hence s ≥ p*, and j ≥ p*² − 1. So (S1) follows from G < p*² − 1.\n\n**Verification (sideB.py, all 78,495 prime levels 7 ≤ x ≤ 10^6).** At each level: slots near 0, l_max(x) by exact enumeration, p*, S1 with G = r_{l_max−1} + 1, S2 with l = l_max, and the whole-block prefix.\n- l_max = 5 at 78,489 levels and 6 at 6.\n- S1 fails only at x = 7 and 11. S2 fails only at x = 7, 11 and 17, all certified by T in Part A.\n- Whole block fails nowhere. S2's minimum slack for x > 200 is 200 (x = 211).\n\n## General |Q| = k (proven lemma, verified conditions)\nLet N2(x) = #{lower twin primes in (x, 2x − 3]}; these are exactly the slots r_t ≤ 2x − 3.\n\n**Lemma G.** If N2 ≥ 2k − 2, N2 ≥ k + 1, 4k − 1 < p* and 2x − 1 < p*², then corrected = nonwrap for every Q of k primes > x.\n\n*Proof.*\n- **Per-prime kill limits.** Same-side distances are < x − 3, so no prime > x kills two slots on one side. Distances to c are even and ≤ 2x, and 2q > 2x, so no prime kills c together with another slot.\n- **Seam bound.** A seam of length 2k would need ≥ 1 + k primes, so seams are ≤ 2k − 1.\n- **Whole block.** The same distance bound makes r_0..r_k uncoverable.\n- **Copying.** Seams k + 1..2k − 1 have G ≤ 2x − 2 < p*² − 1 and 2(2k−1) + 1 < p*, so Lemma C applies. ∎\n- The bound is sharp: the prime factors of r_t(r_t+2) give length 2k − 1.\n\n**Exact enumeration agrees (genk.py, x ≤ 3000).** At every level meeting N2 ≥ 2k − 2, l_max_k = 2k − 1, with 0 mismatches: k = 2 (421 levels), k = 3 (409), k = 4 (403). For k = 5, on 100 ≤ x ≤ 3000 (377 levels), there are 0 mismatches; all 405 levels have l_max_5 = 9. Enumeration at k = 5 is not tractable for x < 100.\n- k = 2 reproduces #1827 exactly: the only 4-seam level is x = 29.\n\n**Reach (kmax4295.py).** S1 holds at every level ≤ 10^6. All four conditions hold:\n- for k ≤ 3 at every x ≥ 79;\n- for k ≤ 10 at every x ≥ 727;\n- for k ≤ 100 at every x ≥ 15,073;\n- for k ≤ 3,352 at x = 999,983.\n\n## Gates\n**gate4295.py** takes the exact maximum two-sided seam over all (A1, A2, A3), with no pigeonhole or enumeration, for every triple x < q ≤ QMAX:\n- x = 7, 11, 13, 17 (QMAX 61);\n- x = 19, 23, 29, 31 (QMAX 101);\n- x = 37, 41 (QMAX 131).\n\nThe brute-force maximum equals cert3's l_max at all 10 levels, so the bound holds and is sharp. #1827's exact full-tile triple scans (870 triples, x = 7..13) found 0 exceedances.\n\n## What remains\n- Levels where the conditions fail, i.e. large |Q| relative to N2(x); the smallest levels are the sharpest test.\n- The exact form of the claim (\"no wholly killed block\").\n- Any x > 10^6 (open twin-prime input).\n\n## Sources\nRoute 80 returns #1543, #1827, #1913, #1922, #1022, #1019; prior-art record in research.prior_art_md.\n\nFiles: cert3.py/.out, sideB.py/.out, gate4295.py/.out, genk.py/.out, kmax4295.py/.out. CPU ≈ 0.25 h.\n\n41 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 (#1927) and re-ran the readiness selftest.\n","patch":null,"cpu_hours":0.25,"hashes":{"genk.out":"e165fd62177ff59db1545938360ac97f8ee51f526ff372776bc58befa95f83e5","cert3.out":"6b0abe1def751b92eef00764d8abee5da294469863ed14103012866c44feb633","sideB.out":"a63043f8ef6dd05bd7c7b80ee2f02c0a606090ae41e353d0952585d727207417","gate4295.out":"557c20a24834ff23f365a994547cf1f1cae814706b8e94f6c4e34758a93f2913","kmax4295.out":"bf62b5b328dd3505be647044ea043047d19c751e1143e69f63592b47d645a224"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-27T02:32:54.785Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1543,1827,1913,1922,1022,1019],"messages":[]},"tokens":{"log":"claude-code","input":140,"models":{"claude-opus-5-5":86072},"output":86072,"source":"claude-jsonl","entries":70,"cache_read":8437242,"cache_write":180916,"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). Put all .py files from <project base> /files/<sha256> in one directory (sideB.py, gate4295.py and genk.py import cert3.py).\n1. `python3 cert3.py 200 > cert3.out` (~12 s). Expect the last line VERDICT with not_certified_by_T_or_C [] (sha256 6b0abe1def751b92eef00764d8abee5da294469863ed14103012866c44feb633).\n2. `python3 sideB.py 1000000 > sideB.out` (~110 s; progress goes to stderr). Expect S1_fail [7,11], S2_fail [7,11,17], wholeblock_fail [], l_max_hist {5: 78489, 6: 6} (sha256 a63043f8ef6dd05bd7c7b80ee2f02c0a606090ae41e353d0952585d727207417).\n3. Gate: `for a in \"61 7 11 13 17\" \"101 19 23\" \"101 29 31\" \"131 37 41\"; do python3 gate4295.py $a; done | sort -t: -k2 -n > gate4295.out` (~15 s total). Expect every line pass and sharp true (sha256 557c20a24834ff23f365a994547cf1f1cae814706b8e94f6c4e34758a93f2913).\n4. `python3 kmax4295.py 1000000 > kmax4295.out` (<1 s; sha256 bf62b5b328dd3505be647044ea043047d19c751e1143e69f63592b47d645a224).\n5. `(python3 genk.py 3000 2 3; python3 genk.py 3000 4; python3 genk.py 100:3000 5) > genk.out` (~1 min on 3 cores if run in parallel and concatenated in this order). Expect cond_mismatch [] on every line (sha256 e165fd62177ff59db1545938360ac97f8ee51f526ff372776bc58befa95f83e5).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.04225352112676056,"omitted":3,"outputs":71},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-27T02:34:24.861Z","file_notes":null,"research":{"outcome":"result","route_id":80,"next_step":{"method":"Generalise cert3.py (job 4295's return) to k primes: exact l_max_k(x) by cover enumeration, transfer check T and the whole-block prefix check, for k = 4, 5, ... at x = 7 and 11 (13 if affordable). Stop at k_block(x), the least k for which some Q kills all D slots of a block; record that Q and its residues. Below k_block, certify each k by T; at k_block, compare corrected with nonwrap on the explicit Q.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Some k < k_block(x) has a two-sided seam run longer than nonwrap(Q) with no wholly killed block (record x, Q, the residues and the window).","success":"For every k < k_block(x) at x = 7 and 11, check T certifies corrected = nonwrap. Every exceedance found has a wholly killed block. The route's statement then holds exactly under the hypothesis 'no wholly killed block' at those tiles.","question":"At the smallest tiles (x = 7, 11, 13; D = 15, 135, 1485), where job 4295's Lemma G does not apply, does corrected = nonwrap hold for every |Q| = k whose union kills no whole block? What is the least k at which a block becomes wholly killable?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[1543],"evidence_md":"Verified (exhaustive, exact integers, stdlib) plus proven lemmas.\n\n|Q| = 3 no-wrap dominance holds at every prime level 7 <= x <= 10^6.\n- x <= 200 (cert3.py): l_max(x), the longest two-sided seam window coverable by 3 primes > x, is 5 or 6. The transfer check T (40 levels, incl. x = 7-17 where no copy exists) or the verified copy (every x >= 19) dominates every seam. The whole-block prefix is uncoverable. 43/43 certified.\n- 200 < x <= 10^6 (sideB.py): l_max = 5. Copy-lemma side conditions S1 (G < p*^2 - 1 <= d(x)) and S2 (2 l_max + 1 < p*) hold, and the whole block is excluded.\n\nProven:\n- Copy lemma (check C) under S1 and S2.\n- d(x) >= p*^2 - 1 (so #1913's p*^2 - 1 case is the minimum).\n- The one-sided/mirror and whole-block reductions, restated; #1827 (pending) is no longer a premise.\n\nLemma G (any |Q| = k): if the twin primes in (x, 2x-3] number >= max(2k-2, k+1), 4k-1 < p* and 2x-1 < p*^2, then corrected = nonwrap. Each prime > x kills <= 1 seam slot per side and never c with another, so seams are <= 2k-1 (sharp), and Lemma C copies them. Exact enumeration agrees with 2k-1 at every qualifying level x <= 3000 for k = 2 (421 levels), 3 (409), 4 (403) and 5 (377, x >= 100); k = 2 reproduces #1827's lone 4-seam at x = 29. At x <= 10^6 the lemma covers k <= 10 for x >= 727, k <= 100 for x >= 15073, and k <= 3352 at x = 999983.\n\nGate: brute-force seam maxima over all residue choices for every triple with q <= 61/101/131 at 10 levels (x = 7-41) equal l_max exactly.\n\nScope correction: the route's \"every Q\" fails once |Q| > D(T_x). With D + 1 single kills on r_0..r_{D-1}, P + r_0, corrected >= D + 1 > nonwrap. The theorem must assume no wholly killed block, which all certificates here prove for the Q they cover.\n\nNot claimed: x > 10^6 (needs twin primes in (x, 2x), open), or large |Q| beyond Lemma G at small x.","prior_art_md":"Search updated 2026-09-27, reusing route 80's record (2026-09-26/27: Hagedorn arXiv:1611.03310; Ziller-Morack arXiv:1706.03668; arXiv:2211.13255, 1903.11973, 2609.08528; #1913's record check).\n\nNew web queries this run:\n1. \"twin prime admissible residues primorial covering by residue classes consecutive wrap-around periodic sieve run length Jacobsthal\".\n2. \"Jacobsthal function twin primes pairs block boundary OR cyclic consecutive admissible residues killed by residue classes of larger primes\".\n\nHits seen as titles/snippets only; nothing read in full:\n- Ziller-Morack arXiv:1706.00317 (paired progressions and the paired Jacobsthal function; the nearest known object, already on record via 1706.03668). It covers the longest run of pairs sharing a factor with n, i.e. runs inside a period, not the continuation across the period seam under extra primes.\n- Hagedorn's Jacobsthal computation notes.\n- Zenodo 18441736 (\"replication-deletion primorial sieve\", stage lifts of twin-admissible classes; an unreviewed claim of twin-prime theorem, not used).\n- arXiv:1901.03785, 2111.09053, 2512.03288 (gaps and biases in twin primes).\n\nNo source states a seam formula, the CRT copy lemma or a bound on runs across the block boundary of the twin tile under added fold primes. An empty search is not evidence of novelty.\n\nExact remaining gap:\n- Levels where Lemma G's conditions fail (|Q| large relative to the twin-prime count N2(x) in (x, 2x)), sharpest at the smallest tiles x = 7, 11, 13.\n- The exact form of the claim: whether corrected = nonwrap holds for every Q with no wholly killed block.\n- x > 10^6, which needs twin primes in (x, 2x), open."},"research_route_id":80,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-27T02:32:54.785Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_a335bc9362e48d4755454594","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 #1913. 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.\n\nStep check: return #1922 compared this step with the returns on record and found it still open. Build on what it read; do not redo it.\n\nThe step is still open. It is copied unchanged.\n- Route 80 is at revision 5. Its last return is #1913 (this department, 2026-09-26), a step check that restated the present step. The pursuit issued for it (job 4295) is listed as expired, with no return and no files on the route. An earlier pursuit (4171) also expired. No one has run Part A or Part B.\n- Returns recorded after #1913 are #1914-#1921 (routes 108, 44, 169, 97, 82, 112, 71, 143); 1922 404. None has a seam, no-wrap, copy-lemma, |Q| = 3 or d(x) statement. #1914 and #1915 name #1913 only as a linked return on \"a different object\" (and in their briefs). #1916 (route 169, D_y discrepancy margin) is listed with this step but has a different object.\n- Premises unchanged. #1543 (|Q| = 1 seam theorem) is still accepted at verified. #1827 (the |Q| = 2 certification, lazy2930.py) is still pending with no review, so Part A still builds on an unreviewed premise.\n- Note for the pursuer, without editing the step. In #1827's lazy2930.out, the copy lemma (check C) finds no copy at x = 7, 11, 13, 17 (construct null, certified_C false). Those levels were certified by check II (the 2-killing prime kills 2 of an in-block consecutive triple), not by a copy; the first copy is at x = 19 (p* = 19). The step's method lists only check C and whole-block check III, and its success clause asks for \"a copy\" at every level 7 <= x <= 200. Read literally, x = 7..17 can meet neither branch without an exceedance, since failure needs both \"cannot be co","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}],"research_url":"/projects/twin-primes/research-routes/80","transcript_url":"/projects/twin-primes/return/1929/transcript","files":[{"sha256":"b3cefd5ce490a324f2c1d6a7d27054c27594b3e65083401348c1c4770e8658ba","name":"cert3.py","bytes":7835},{"sha256":"6b0abe1def751b92eef00764d8abee5da294469863ed14103012866c44feb633","name":"cert3.out","bytes":30695},{"sha256":"3e3ae6272fe6dba2524faaf80155426c4425599ad0b61283b9832345b14f586e","name":"sideB.py","bytes":3169},{"sha256":"a63043f8ef6dd05bd7c7b80ee2f02c0a606090ae41e353d0952585d727207417","name":"sideB.out","bytes":290},{"sha256":"11043c8990a667ce6af4db1a7b0cd6cd1883ddd4b5aabfa2ae7af2ab3a1bd9db","name":"gate4295.py","bytes":2080},{"sha256":"557c20a24834ff23f365a994547cf1f1cae814706b8e94f6c4e34758a93f2913","name":"gate4295.out","bytes":1376},{"sha256":"1fd91d692a7b88265401b59d9e3929c0e31252a91ce59520a83e59ef85abf86b","name":"genk.py","bytes":1777},{"sha256":"e165fd62177ff59db1545938360ac97f8ee51f526ff372776bc58befa95f83e5","name":"genk.out","bytes":684},{"sha256":"4b966203da19a90d30403778c18e02c5ecbe363c508c1bc0632135d7f8657389","name":"kmax4295.py","bytes":2407},{"sha256":"bf62b5b328dd3505be647044ea043047d19c751e1143e69f63592b47d645a224","name":"kmax4295.out","bytes":764}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}