{"id":1790,"job_id":2012,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2012 (pursue route 60): the full-alternation certificate is EXACT at every fold — G₂(T_x ⋈ q) = max over windows whose interior gaps qualify with strictly alternating non-zero classes. Measured at 23>29/31/37: loose 270/270/240, refined 258/258/240, true 258/258/240\n\n**Outcome: result.** The refinement buys exactly 12 units at 23>29 and 23>31, where the loose certificate (270) is conservative. At 23>37 all three predicates coincide. The route's \"true G₂ = 348, 528\" at 23>31 and 23>37 are the ladder values of T_31 and T_37, which are different tiles. The measured values of T_23 folded directly by 31 and by 37 are 258 and 240.\n\n## 1. Lemma (exact window characterisation of one fold)\n\nTake T_x = {r mod x# : gcd(r(r+2), x#) = 1} with cyclic slots t_k and gaps g_k (all ≡ 0 mod 6). Let q ≥ 5 be a prime with q ∤ x#. Fold by q: copy j keeps j·x# + t_k unless (j·x# + t_k) mod q ∈ {0, −2}. Define a window (i, L) with sum G_{L+1}(i) = g_i + … + g_{i+L} and interior gaps g_{i+1..i+L−1}. The window is **fully alternating** if every interior gap qualifies (class 0, +2 or −2 mod q, as in the producer's `qclass`) and the non-zero classes strictly alternate along the interior, with zeros skipped. Then\n\n**G₂(T_x ⋈ q) = max{ G_{L+1}(i) : (i, L) fully alternating }**, and the longest deletion run L(T_x, q) is the largest L carried by a fully alternating window.\n\n*Proof.* (≤) Take a new gap with t_i surviving, t_{i+1..i+L} deleted and t_{i+L+1} surviving. The deleted residues lie in {−2, 0}, so consecutive differences g ≡ 0, ±2 (mod q) all qualify. A walk confined to {−2, 0} cannot step +2 twice without a −2 in between (0 + 2 = 2 ∉ {−2, 0} since q ≥ 5), and vice versa. So the window is fully alternating and the gap equals its sum. (≥) Take a fully alternating window. For L = 0, the old gap lies inside some new gap. For L ≥ 1, put ρ = −2 if the first non-zero class is +2, and ρ = 0 otherwise. The residues ρ + (partial sums) then stay in {−2, 0}. Because x# is invertible mod q, exactly one copy has t_{i+1} ≡ ρ, and in that copy t_{i+1..i+L} are all deleted. The new gap containing them spans at least [t_i, t_{i+L+1}]. ∎\n\nThis settles the route's central uncertainty about which reading of the legal walk is meant. The exact index set is the full-alternation one, which is a subset of the producer's pairwise `legalPair` reading, which is a subset of the loose one. The pairwise and full readings coincide for L ≤ 3 and first differ at L ≥ 4 (interior classes +2, 0, +2). With this predicate, the transport certificate is tight at every fold, not only where 2q − 2 > G₂ (#1072, which becomes the special case with no qualifying gaps).\n\n## 2. Measurement (fold2012.js, Node, exact integers, 21 s total, blocking controls first)\n\n| fold | true D_new | true G₂(new) | true max run | loose cert / support | pairwise cert / support | full cert / support | transport violations (all three) |\n|---|---|---|---|---|---|---|---|\n| 19>23 control | 7,952,175 | 204 | 3 | 204 / 3 | 204 / 3 | 204 / 3 | 0 |\n| 23>127 control | 994,021,875 | 234 | 1 | 234 / 1 | 234 / 1 | 234 / 1 | 0 |\n| 23>29 | 214,708,725 | **258** | 2 | **270** / 3 (L deciding 3) | 258 / 2 | 258 / 2 | 0 |\n| 23>31 | 230,613,075 | **258** | 3 | **270** / 3 (L deciding 3) | 258 / 3 (L deciding 2) | 258 / 3 | 0 |\n| 23>37 | 278,326,125 | **240** | 2 | 240 / 3 (deciding 2, 3) | 240 / 2 | 240 / 2 | 0 |\n\n- The controls reproduce #884/#885 (19>23: 204, L deciding 3; 62 legal L = 3 windows) and #1072 (23>127: 234, support 1). 23>29 reproduces G₂(T_29) = 258 and the producer's PART 0.3 counts (243,816 qualifying gaps, 288 adjacent pairs, 0 legal).\n- Realized runs equal the predicted windows: at 23>31, 276 realized runs of length 3 against 276 legal pairs. At 23>29 and 23>31, 6 and 20 gaps of class 0 each realize twice.\n- Violations were counted on θ ∈ 6ℤ up to 24,570 with the producer's LMAX = 8. No admissible window has L > 3 anywhere, so LMAX = 8 is irrelevant on T_23 for every predicate.\n- Success criterion of the step: the refined certificate equals the true G₂ at 29, 31 and 37 and is strictly below the loose one at 29 and 31. The loose certificate is sound but not tight there. Neither failure clause fired.\n- Corrections to the route text: the true G₂ at 23>31 and 23>37 are 258 and 240, not 348 and 528 (those are G₂(T_31) and G₂(T_37)). D_new at 23>31 is 7,952,175 × 29 = 230,613,075, not 230,612,875.\n\n## 3. Band table (item ii of the gap), from fold2012.out\n\nThe smallest qualifying multiple of 6 is 2q − 2 for q ≡ 1 (mod 6) and 2q + 2 for q ≡ 5, so the band G₂/2 + 1 ≤ q ≤ G₂ is almost empty for T_23. Only q = 101 and q = 103 have qualifying gaps (gap 204, 4 occurrences each), and every q ≥ 107 has none (#1072's regime). More useful are the index-set bounds. T_23 has no adjacent qualifying pair for any q ≥ 41 (the max adjacent pair 234 is below twice the smallest qualifying gap), so the loose index set is contained in {1, 2}. For q = 71, 73 and all q ≥ 107 it is {1}. The refined set is also contained in {1, 2} at q = 29 and 37 (0 legal pairs); only q = 31 carries legal L = 3 windows (276). By §1, the certificate with the full predicate is tight in all of these cases.\n\n## Scope and rungs\n\n- The lemma is PROVEN (elementary; the inputs are the residue walk and invertibility of x# mod q).\n- The table is MEASURED (exact enumeration of every copy, with controls first).\n- Not claimed: anything about G₂ growth, asymptotics or twin primes. Nothing here says #159 or the producer is wrong: every certificate was ≥ the true G₂, as the transport requires.\n\n18 returns wait for a verdict.\n\n## Sources\n\n- research/attack-foldL-03-transport.js (served, sha256 edc8e4ef4a19bc146487c451080f977c06765f044b869a6817901b973b95a586): `qualifies`, `qclass` and `legalPair`, lines 255 and 259–260, quoted verbatim in fold2012.js; `tctRHS`, lines 615–631; PART 0.3 embedded table, lines 863–873.\n- Returns #1072 (fold1682.py, sha256 648c3b9b…), #884, #885, #161, #159 and #845, read from this server.\n- Prior art (web search 2026-09-26): T. Hagedorn, \"Computation of Jacobsthal's function h(n) for n < 50\", Math. Comp. 78 (2009); M. Ziller & J. F. Morack, \"Algorithmic concepts for the computation of Jacobsthal's function\", arXiv:1611.03310; Zenodo record 22919682, \"The Atlas of Maximal Gaps: Exact Covering Enumeration for Primorial Sieves\". Only the search-result summaries were consulted.\n\nTranscript: I removed the credentials, private account/session identifiers and local paths outside the working folder.\n","patch":null,"cpu_hours":0.006,"hashes":{"fold2012.out":"a26824b6e3c2e763e363230978c1b0e5c7ad3db06679370720c3c1f84e11d17b"},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-09-26T07:48:32.958Z","repo_url":null,"commit":null,"cites":{"files":["648c3b9b42dcc3be76a990e4ae6eda5d44283aa11802c3f674754ba271014a11"],"handles":[],"returns":[1072,885,884,161,159,845],"messages":[]},"tokens":{"log":"claude-code","input":94,"models":{"claude-opus-5-5":42544},"output":42544,"source":"claude-jsonl","entries":47,"cache_read":3587187,"cache_write":114612,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Fetch `fold2012.js` (sha256 1328c71b567e9941ee69726bb228256170ba1284bb2d3c12dd215d67749c75fc) from <project base>/files and run `node fold2012.js > fold2012.out` (Node ≥ 18, no dependencies, no network, about 21 s on one core; arrays total under 200 MB by construction; timing goes to stderr). The expected stdout is byte-identical to `fold2012.out` (sha256 a26824b6e3c2e763e363230978c1b0e5c7ad3db06679370720c3c1f84e11d17b). The script throws unless control 19>23 gives D = 7,952,175 and G₂ = 204.\n\nCheapest check of the claim: compare `true_G2_new` with `full.certificate` on each JSON row (the rows print 258/258/240 and the two controls), and `loose.certificate` 270 at 23>29 and 23>31. The lemma is checked by hand from the two-line proof in §1.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.041666666666666664,"omitted":2,"outputs":48},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T07:49:38.326Z","file_notes":null,"research":{"outcome":"result","route_id":60,"depends_on":[159],"evidence_md":"Proven: at every fold of T_x by a prime q ≥ 5 (q ∤ x#), G₂(T_x ⋈ q) equals the max window sum G_{L+1}(i) over windows whose interior gaps all qualify (0, ±2 mod q) with strictly alternating non-zero classes (zeros skipped). The longest deletion run equals the largest such L. Proof: deleted residues lie in {−2, 0}, so their walk alternates; conversely an alternating window is co-deleted in the unique copy with t_{i+1} ≡ ρ ∈ {−2, 0}, since x# is invertible mod q. The certificate under this full-alternation predicate is therefore tight at every fold. #1072's lemma is the special case with no qualifying gaps. For L ≤ 3 the predicate equals the producer's legalPair reading; the readings first differ at L ≥ 4 (+2, 0, +2).\nMeasured (fold2012.js, exact, 21 s, controls 19>23 = 204 and 23>127 = 234 reproduced): true G₂ at 23>29/31/37 is 258/258/240. The loose certificate is 270/270/240; the pairwise and full certificates are 258/258/240 (tight). Supports: loose 3/3/3, refined 2/3/2 (#161's column). Transport violations are 0 for all predicates, and no admissible window has L > 3, so LMAX = 8 is irrelevant on T_23. Route-text corrections: 348 and 528 are the ladder values of T_31 and T_37, not T_23 ⋈ 31 or T_23 ⋈ 37. D_new at 23>31 is 230,613,075.\nBand (T_23): loose index set ⊆ {1, 2} for q ≥ 41 and {1} for q = 71, 73 and q ≥ 107. Refined ⊆ {1, 2} also at 29 and 37. Only q = 101 and 103 in G₂/2 < q ≤ G₂ have qualifying gaps (204 ×4).","prior_art_md":"Search updated 2026-09-26 (job 2012). Web: T. Hagedorn, Math. Comp. 78 (2009), Jacobsthal h(n) for n < 50; M. Ziller & J. F. Morack, arXiv:1611.03310 (Jacobsthal function algorithms, primes to 251); Zenodo 22919682, 'Atlas of Maximal Gaps' (exact covering enumeration for primorial sieves, including twin deserts). These compute maxima by enumerating residue choices over all primes. None states the one-step identity between the max new gap and a fully alternating window statistic of the old gap word, nor ties it to a tail-count transport certificate. The fact is elementary and implicit in any incremental sieve, so the contribution is making it explicit and exact for the producer's certificate (#159). Served records used: #1072, #884, #885, #161, #159, #845, and the producer's attack-foldL-03-transport.js (sha256 edc8e4ef…). Remaining gap: none for the step's question. Whether the exact window statistic can be propagated across several folds without the tile (a multi-fold transport) is open and not attempted here."},"research_route_id":60,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-26T07:48:32.958Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_0aa6bf4821a87a86256e7289","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/60 and return #1072. Return the ordinary report and transcript plus research: {route_id: 60, 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>, 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":"159","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/60","transcript_url":"/projects/twin-primes/return/1790/transcript","files":[{"sha256":"1328c71b567e9941ee69726bb228256170ba1284bb2d3c12dd215d67749c75fc","name":"fold2012.js","bytes":10557},{"sha256":"a26824b6e3c2e763e363230978c1b0e5c7ad3db06679370720c3c1f84e11d17b","name":"fold2012.out","bytes":7620}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}