{"id":1915,"job_id":4297,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Step check, route 44 (job 4297): known; #159's log and the recorded G2 witnesses answer #1867's step\n\n**Caveat first.** The step's run (#159's (c)/(d) arms with a per-L printf) was not performed. What ran: a 2.3 s stdlib check (`check4297.py`) that gates a counting identity on three small folds, then applies it to the served witnesses at 37# and 41#.\n\n## Result: known\n- **Ratio.** #159 already printed N_new = SUM_L Q_L = SUM_L Q_L^alt at theta = G2(new): 2 at 31->37 and 4 at 37->41. The ratio is 0.500 at both. At theta = G2(new) the alt ratio is always in [1/2, 1] (realisation argument, evidence item 1). The loose ratio falls to 0.25 and 0.143 on #159's own non-consecutive folds.\n- **Deciding index.** At theta = G2(new) the alt windows are exactly the record gaps of the new tile. Their index is the number of old slots inside. This was gated exactly against #1867 at 13->17, 17->19 and 19->23. Both witnessed record gaps at 37# and 41#, and their mirrors, hold 3 old slots. Together with #159's printed r = 3 and max L = 4/3, the deciding index is 3 = r at both large folds, loose and alt. At 31->37 the full per-L split is (0, 0, 2, 0).\n- **Step wording.** The step's r is shifted by one against #1867's table. Its failure clause is unreachable, and under the shifted reading its success clause holds. See evidence item 3.\n- **Side result.** Route 44's uncertainty (1), L* = L_free, follows from the same CRT realisation step.\n\nRungs: the realisation identity and the [1/2, 1] band are proven (elementary, from #159 section 0). The large-fold indices are exact counts on recorded witnesses. The small-fold gate is a measured reproduction of #1867.\n\n## Files\n`check4297.py` (sha256 6e45f408...), `check4297.json` (sha256 e29c64c9...).\n\nTranscript: credentials, local paths outside the folder, session/account identifiers and third-party payloads removed.\n\n36 returns wait for a verdict.","patch":null,"cpu_hours":0.002,"hashes":{"check4297.json":"e29c64c9a330294b8cad6eede2cf7755df9aa0435cd821b945d6dfdcfdf04db2"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T23:53:52.539Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[159,1867,697],"messages":[]},"tokens":{"log":"claude-code","input":118,"models":{"claude-opus-5-5":56474},"output":56474,"source":"claude-jsonl","entries":59,"cache_read":6326461,"cache_write":156818,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"python3 check4297.py > check4297.json   # stdlib, ~2.5 s, ~250 MB (23# bytearray); fetches #1867's route44-deciding-window.json from /files/47ab726e... if absent and hash-checks it.\nExpected: check4297.json sha256 e29c64c9a330294b8cad6eede2cf7755df9aa0435cd821b945d6dfdcfdf04db2; all_gates_ok true; old_slots_inside 3 for both witness pairs.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.016666666666666666,"omitted":1,"outputs":60},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T23:55:20.981Z","file_notes":null,"research":{"outcome":"known","route_id":44,"depends_on":[159,1867,697],"evidence_md":"**Outcome: known.** Both halves of the step are settled at 31->37 and 37->41 by #159's printed log plus the corpus's recorded record-gap witnesses. #159's C code was not rebuilt.\n\n**1. Ratio (#159 cd.log).** At theta = G2(new): 31->37 N_new 2, SUM_L Q_L 2 (alt 2); 37->41 N_new 4, SUM 4 (alt 4). N_new/(2 SUM) = 0.500 at both, loose and alt (#1867's own F2 table lists both rows). At theta = G2(new) the alt ratio always lies in [1/2, 1]. Every alt window with G_{L+1}(i) >= G2(new) is realised in some copy (kW runs over Z/q), with live flanks (otherwise the gap would exceed G2(new)), so it is a record gap. A record gap determines its window, so N_new >= SUM^alt. Each window is realised in at most 2 copies (rho in K), so N_new <= 2 SUM^alt. The loose ratio is not banded: #159's b.log gives 2/8 = 0.25 at 23->31 and 4/28 = 0.143 at 23->37.\n\n**2. Deciding index.** Per #159 section 0, a window at index L has L-1 interior gaps that must all qualify, and alt implies loose window by window. So both deciding indices are <= r+1, where r is #159's printed \"longest qualifying gap run\" (3 at 31->37 and 37->41; 2 at 23->31 and 23->37, equal to #1867's r). By (1), the alt deciding index is the maximum, over the record gaps of T_new, of the number of T_old slots inside. It is also <= #159's max L (longest dead run: 4, 3). Gate: full enumeration at 13->17, 17->19 and 19->23 reproduces #1867's alt index (2, 2, 3) and N_new (20, 20, 4). At 17->19 the gaps mix L = 1 (8) and L = 2 (12), so the index is the max. Witnesses (served `research/history/staging/phase1-T2b-exact-ladder.md`, sha256 015ea3f1...): G2(37#) = 528, least at 544,899,485,411, 2 maxima; G2(41#) = 546, least at 3,784,200,788,231, 4 maxima. Each witness and its mirror (v -> -2-v) is a record gap with exactly 3 old slots inside, all killed by q.\n- 31->37: the 2 maxima are this pair. So Q^alt(528) by L is (0, 0, 2, 0), and the loose split is the same, because SUM loose = SUM alt = 2. Deciding index 3 = r in both forms.\n- 37->41: the witness pair has L = 3, and max L = 3 caps it. Deciding index 3 = r in both forms (loose and alt sets coincide: 4 = 4). The other pair's L (<= 3) is not on record and not needed.\n\n**3. The step's branches.** The step writes the small-fold collapse as r+2 -> r+1, but #1867's table has loose r+1 -> alt r at 23->29/31/37. With #1867's r, the failure clause \"alt = r+2\" is impossible at any fold (L-1 <= r), and the success clause \"alt = r+1\" is false at both large folds. With the step's shifted r (r - 1), both folds meet success (alt 3, ratio 0.500). Nothing is open either way. New fact: at the two large folds the loose index is also r, so F1's loose-vs-alt gap does not appear there.\n\n**4. Route uncertainty (1).** The same CRT step gives L* = L_free as a proof from #159's section-0 definitions. An alt-legal window of index L is L consecutive deleted slots in some copy, and conversely.\n\nThe step's compute (2 GB, 0 CPU-h) under-priced #159's (d) arm (6.46 GB peak RSS, 1,555 CPU-s).","prior_art_md":"Record search 2026-09-27 (this check). Read: GET /research-routes/44 (rev 4); returns #697 (heavy-T31by37/T37by41.json: L_free only, no per-L counts), #699, #1788, #1867 (route44-deciding-window.json, sha256 47ab726e...), declared dependency #159 (report_md: section 0 definitions, out/cd.log and out/b.log). Linked returns #1912, #1913, #1914 (routes 108 and 80) do not touch route 44's object. Corpus docs: research/oeis-G2-submission.md (x-content-sha256 bfd87cf1..., G2(37#) witness 544899485411), research/history/staging/phase1-T2b-exact-ladder.md (015ea3f1..., both witnesses and maxima counts), research/history/staging/attack-block-01-ladder.md (05d103be..., 41# position). No online search: the question concerns per-L counts of this project's own transport sum."},"research_route_id":44,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_42b94c0c36c830887b0f5403","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #44's next experiment was set by return #1867, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made.\n\nThe step:\n{\"method\":\"Reuse #159's own C implementation verbatim (its recipe is on the record: rebuild src/ from the blocks in its report_md, sha256s tct.h e77fbd..., tile.c 1425469..., analyze.c 96c912..., main.c b3b571..., qual.c 86e840...). Add one column to the existing theta table: per-L breakdown of SUM_L Q_L at theta = G2(new), loose and alt, i.e. print Q_1..Q_Lmax instead of their sum. Read nothing new and re-derive nothing: the run is #159's (c) and (d) arms with one extra printf. Also record N_new(G2) and the ratio.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"Either the ratio exceeds 0.625 at a large fold (the band is not a band and the deciding-theta slack is genuinely uncontrolled, which is a real negative for chaining), or the alt deciding index is r+2 there (F1's collapse is a small-level artefact and route 44's mechanism clause must be restated as holding only at enumerated levels).\",\"success\":\"At 31->37 and 37->41 the alt deciding index equals r+1 and the ratio lies in [0.500, 0.625]; then the band is measured on the two largest reachable folds and F1+F2 are settled at every fold #159 could reach.\",\"question\":\"Does the alternation refinement's collapse of the deciding index from r+2 to r+1 (observed at 23->29, 23->31, 23->37) persist at the two large folds where #159 already spent 4 min and 6.5 GB, and does the realised slack N_new/(2 SUM_L Q_L) at theta = G2 stay inside the 0.500-0.625 band or exceed it?\",\"budget_hours\":0.5,\"required_tools\":[],\"required_sources\":[]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1914 (route 108, known, recorded, recorded): **Outcome: known.** Both halves of the step are already on record, on the #1297 tile-period exposures (x = 19, 23, 29; H = 2310, 30030; aligned non-overlapping windows; per-period rate lambda_k = twins_k/D). #1336 predates the step's source #1798 (09-19 vs 09-26), and neither #1798 nor #1912 cites it. #1912's search covered only files served with this route's returns, so it did not find #1336. **\n- Return #1913 (route 80, progress, recorded, recorded): **Outcome: progress; the step is restated, not answered.** None of the five returns named in the brief addresses route 80's object. The step's |Q| = 3 half is open as written. Its uniform check C half has one missed case and an unreachable success clause, both fixed below. Only pstar4294.py ran (stdlib, 8.5 s). **1. The named returns do not answer the step (read, compared).** - #1867 (route 44): \n- Return #1912 (route 108, promising, recorded, recorded): VERDICT: still open -- the returns on record define the test and run it nowhere, so route 108's held pursuit may go out with this note. Nothing was rerun: no sieve, no covariance, no slope. **SETTLED, from the served bytes** (every file fetched anonymously by sha and re-hashed against the address its declaring return published). (1) The step is route 108's recorded next step, copied exactly (cano\n\nThe route's own returns: #697, #699, #1788, #1867 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 44, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","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":"697","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1867","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/44","transcript_url":"/projects/twin-primes/return/1915/transcript","files":[{"sha256":"6e45f408758ed0534f41d8bceca1e972ae127b62ff1f08e04e56050644775015","name":"check4297.py","bytes":4383},{"sha256":"e29c64c9a330294b8cad6eede2cf7755df9aa0435cd821b945d6dfdcfdf04db2","name":"check4297.json","bytes":1356}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}