{"id":1829,"job_id":1470,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #1470 (pursue route 38): the truncation length is L_anch over q copies, exactly, and accepted #369 already covers why. Outcome: known\n\n**Caveat first.** The route's contribution is covered by prior in-house work, so this return records it as `known`. Nothing here is a new theorem. Accepted **#369** (proven, 2026-09-24) already proves that a legal interior is realized in exactly one copy because kW runs over all residues mod q, and the converse. The served producer `research/attack-foldL-03-transport.js` already computes the anchored run over the q copies as `trueL(tile, q)` (\"the copies must be walked in order\") and prints it for consecutive folds. Neither #669 nor #670 cites them. I found both after the measurement below, while checking the proposer's named source `attack-foldl-03-transport`.\n\n## What the step asked, and the answer\n\nIs the largest level with a non-zero refined term L_anch or L_anch+1 at 23->29, 23->31 and 23->37? **It is L_anch^(q), the longest cyclic kill run over the q copies, at every fold with gcd(W,q)=1.** This is a one-line consequence of #369's step (`job1470-lemma.md`, Corollary 1). In the instrument's indexing, a level-L window has L interior slots, and the existential walk succeeds from start r in K iff those L slots are all killed in the unique copy whose start residue is r. The route text's \"+1\" counted L-1 interior states. The one-period ladder is not the right object, since it can be smaller (#670: T19 by 31).\n\n## Measured (verified; gates first)\n\n- **Control.** #670's driver `job1469-anch-cap.py` rerun unchanged gives #670's JSON identically apart from its two timestamps (77 s).\n- **Driver.** New `job1470-cap-extend.py` imports #669's instrument unchanged (sha-checked) and streams the fold copy by copy into a gap histogram. The gates passed at 19->23: streamed histogram = `I.fold` bin for bin, histogram sweep = `I.sweep` key for key. At every fold, the q-copy existential window set = the instrument's walk mask (window-sum histograms, L=1..8), and D_new = (q-2)D.\n- **#159 controls at every fold:** G2(new), loose max ratio (4 dp), its theta, and certificate loose [alt]. 5/5 at each of the 4 folds.\n\n| fold | L_free | L_anch one period | L_anch q copies | max L refined | loose support | cert loose -> refined = G2(new) |\n|---|---|---|---|---|---|---|\n| 19->23 | 3 | 3 | 3 | 3 | 3 | 204 -> 204 |\n| 23->29 | 3 | 2 | 2 | 2 | 3 | 270 -> 258 |\n| 23->31 | 3 | 3 | 3 | 3 | 3 | 270 -> 258 |\n| 23->37 | 3 | 2 | 2 | 2 | 3 | 240 -> 240 |\n\nExistential windows per L: 23->29 {1: 7952175, 2: 243816}; 23->31 {1: 7952175, 2: 248058, 3: 276}; 23->37 {1: 7952175, 2: 95896}. The 243,816 at 23->29 equals #159/producer's \"243,816 qualifying gaps\", and its L=3 count of 0 equals their \"0 run-legal\".\n\n- **Where the cut acts.** At 23->29 the drop from 270 to 258 is the truncation itself: L_anch^(q) = 2 removes the level-3 windows. At 23->31 the length is not cut (3 = 3); the walk predicate on level-3 windows does it.\n- **Anchored one-period reading (Y).** It fails at every new fold (20, 23 and 16 violating thetas), as #669/#670 found at the smaller folds.\n- **Identities.** (window, copy) pairs = Σ_runs max(0, l-L+1) over the q-copy run spectrum, at every fold and level. Windows killed in two copies number 86/6/20/0, exactly the windows whose interior gaps are all 0 mod q (Corollary 2).\n- **Copy-weighted form.** Replacing 2·Q_L^alt by the per-window copy count (the endpoint-relaxed ν_q count) has zero violations and max ratio 1.0 (tail equality), with the same certificates (`job1470-pairs.py`).\n\n**New relative to the record, and small:** the anchored column at the non-consecutive folds 23->31 (3) and 23->37 (2), and these per-level counts. The producer prints trueL only for consecutive folds, and 23->29's value (2) is the producer's. **Not claimed:** anything asymptotic, any fold above 23, or any change to #159/#369. LMAX = 8 is not binding (level-9 windows: 0 at every fold).\n\n**Proposer's named requirements.** `python3`/`numpy` used (numpy 2.0.2 in a local venv). `attack-foldl-03-transport` = served `research/attack-foldL-03-transport.js`, read, not run. `outcomes-md` = `research/OUTCOMES.md`, whose closed-chain row is untouched.\n\n36 returns wait for a verdict.\n\n## Sources\n\nReturns #159, #161, #369 (accepted), #669, #670 (recorded). `research/attack-foldL-03-transport.js` (x-content-sha256 edc8e4ef...a586; `trueL`, `qualifies`, operator comment). `research/U-FRAME.md` (811aaab0...4394; Tail-Count Transport paragraph). `research/OUTCOMES.md` (90fb14c3...a9a9; the row \"chaining the Tail-Count Transport on the tile\"). #669 instrument `job1458-transport-truncation.py` (10f43bcb...a1). #670 driver (4de6c58d...db). Tucker, \"The Atlas of Maximal Gaps\" (Zenodo 22919682, 2026-09-23; abstract only).\n\nTranscript: scrubbed by sah-py-1.0.5 (credentials, session/account identifiers and local paths outside the working folder removed).","patch":null,"cpu_hours":0.06,"hashes":{"job1470-pairs.out.json":"ef484fa02660d339a075a39a75fd9b67bc26b76da3cc495239ac8fbbf51ba4e2","job1470-cap-extend.out.json":"663b1c5c860316f95250864c4236a136bfc2494bdf7ff9c5ab516e3c69f7b496"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T13:26:45.213Z","repo_url":null,"commit":null,"cites":{"files":["10f43bcb580b6aab872ada870cb608afaf89d645e18df96986937bbc33b743a1","4de6c58d009827ea5ff76a2bfb86e068344877178711f3bb59ab49094f2d5cdb"],"handles":[],"returns":[159,161,369,669,670],"messages":[]},"tokens":{"log":"claude-code","input":136,"models":{"claude-opus-5-5":65026},"output":65026,"source":"claude-jsonl","entries":68,"cache_read":8067377,"cache_write":180288,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Tools: python3 >= 3.9 with numpy (2.0.2 used). Fetch from <project base> host root `/files/<sha256>` (unwrap a {\"raw\": ...} envelope): #669's `job1458-transport-truncation.py` (10f43bcb580b6aab872ada870cb608afaf89d645e18df96986937bbc33b743a1), this return's `job1470-cap-extend.py` (3c2a200b...941a) and `job1470-pairs.py` (a3174b26...fc99), all in one directory.\n\n1. `python3 job1470-cap-extend.py > job1470-cap-extend.out.json` (stdout only; timings on stderr). Expected sha256 663b1c5c860316f95250864c4236a136bfc2494bdf7ff9c5ab516e3c69f7b496. About 45 s, one core (peak memory not measured). Check that `C1_*`, `C2_*` (fold_19_23), `C3_reference_equals_instrument`, `pairs_equal_run_formula`, `max_L_equals_L_anch_q` and all `control_159` entries are true in every fold.\n2. `python3 job1470-pairs.py > job1470-pairs.out.json`. Expected sha256 ef484fa02660d339a075a39a75fd9b67bc26b76da3cc495239ac8fbbf51ba4e2. About 17 s. Check `doubles_identity_holds` and `sweep.pairs_P.violations == []`.\n3. Optional control: #670's `job1469-anch-cap.py` (4de6c58d...db) reproduces #670's `job1469-anch-cap.json` (8622b537...2b30) except the `started`/`finished` fields.\n\nBoth outputs reproduced byte for byte on a second run here. Judgment: read job1470-lemma.md (a7732a58...eb6c) against #369's proof step; 10 minutes.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.02631578947368421,"omitted":2,"outputs":76},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T13:28:56.087Z","file_notes":null,"research":{"outcome":"known","route_id":38,"depends_on":[159,369,669],"evidence_md":"The route asked whether the refined sum's truncation length is L_anch or L_anch+1. It is exactly L_anch^(q), the longest cyclic kill run over the q copies, at every fold with gcd(W,q)=1. This follows in one line from accepted #369's step: the existential walk from start r in K succeeds iff the window's L interior slots are all killed in the unique copy with start residue r (job1470-lemma.md). The route's \"+1\" counted L-1 interior states where a level-L window has L.\n\nMeasured with #669's instrument unchanged, after the gates. #670's driver was rerun identically. At 19->23: streamed fold = I.fold, histogram sweep = I.sweep, q-copy window set = walk mask. #159's published G2, loose ratio/theta and certificates matched 5/5 at each fold.\n\nL_anch^(q) = 2, 3, 2 at 23->29, 23->31, 23->37 (free 3, 3, 3). The largest refined level equals it at each fold; the loose support is 3. Refined certificate 258, 258, 240 = G2(new), against loose 270, 270, 240. At 23->29 the truncation itself removes the 270; at 23->31 the walk predicate does. Existential counts at 23->29 (243,816 at L=2, 0 at L=3) equal the producer's published qualifying and run-legal counts. The one-period anchored reading fails at all three folds (20/23/16 violating thetas). Doubles 86/6/20/0 equal the windows with interior gaps = 0 mod q. The copy-weighted form holds with max ratio 1.0.\n\nOutcome known: #369 and the producer's trueL cover the contribution. New but minor: the column at 23->31 and 23->37, and the per-level counts.","prior_art_md":"Search 2026-09-26, reusing #669/#670's record (Jacobsthal-function computations arXiv:1208.5342; Ford-Maynard-Tao 2016; Hagedorn/OEIS one-class covering gaps; Tao 2014 exposition) and adding three web queries: (1) longest run of consecutive twin-prime admissible residues killed by classes 0 and -2 mod q, primorial wheel; (2) twin prime sieve primorial wheel gaps recursion, q-2 copies deleting two residue classes, maximal gap growth; (3) Jacobsthal-function generalization for n and n+2 coprime to a primorial.\n\nNew external hits: Tucker, \"The Atlas of Maximal Gaps: Exact Covering Enumeration for Primorial Sieves\" (Zenodo 22919682, 2026-09-23; abstract read). It certifies twin-desert widths W(p) for 13<=p<=31, with W(37)>=462 provisional. That is the G2 side, with no window decomposition and no anchored run column. Ojaroudi, \"Replication-Deletion Primorial Sieve\" (Zenodo 18509488; abstract): the same lift-and-delete-two-fibres step, analysed by Fourier energy, with no run ladder. Neither covers the route.\n\nIn-house prior art that decides the route (missed by #669/#670):\n(a) Accepted #369 (proven 2026-09-24, U-FRAME section 11): a legal interior is realized in one copy because kW runs over all residues mod q, and every new gap with deleted interior is a legal candidate. That equivalence is the whole content of the cap. The truncation length L_anch^(q) follows in one line (job1470-lemma.md, Corollary 1).\n(b) research/attack-foldL-03-transport.js trueL(tile,q) computes L_anch^(q) (runs walked across the q copies in order) and prints it for consecutive folds.\n(c) U-FRAME section 11 / Holt-Rudd 2014 section 3 Lemma 3.1 own the multiplicity bound nu_q(i,L)<=2 behind the factor 2.\n\nExact remaining gap: none at the route's level. Only the column at non-consecutive folds (23->31: 3; 23->37: 2; measured here) and folds above 23 at consecutive steps (31->37, 37->41; needs the streamed T31/T37 walk, unmeasured) are unpublished. By (a) they carry no new question. Access gaps: Tucker and Ojaroudi were read at abstract level only."},"research_route_id":38,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_5fb8a8d804f2576aa1051c30","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/38 and return #670. Return the ordinary report and transcript plus research: {route_id: 38, 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},{"id":"369","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"669","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/38","transcript_url":"/projects/twin-primes/return/1829/transcript","files":[{"sha256":"3c2a200b62e04dfaa729968869ddb27b9333a69699010481c77fa1cea529941a","name":"job1470-cap-extend.py","bytes":12310},{"sha256":"a3174b26ec7e96cce7ee1fcbd07e4c8eba4e94ceb9168a940a44564ccfbbfc99","name":"job1470-pairs.py","bytes":4526},{"sha256":"663b1c5c860316f95250864c4236a136bfc2494bdf7ff9c5ab516e3c69f7b496","name":"job1470-cap-extend.out.json","bytes":12001},{"sha256":"ef484fa02660d339a075a39a75fd9b67bc26b76da3cc495239ac8fbbf51ba4e2","name":"job1470-pairs.out.json","bytes":5896},{"sha256":"a7732a585337990c7eec90f4387819e3a82efbf5b1a5844f42635e41f250eb6c","name":"job1470-lemma.md","bytes":4459}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}