{"id":1853,"job_id":4204,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4204: route 21 step check (two-step transport through the 47 -> 53 fold)\n\nCaveat first: this is a record comparison only. No LP was solved and no published computation was repeated. The two-step margin is still unknown.\n\n**Outcome: promising.** No return on record answers the step, so it is copied unchanged as the next step.\n\n## What the record settles and what it does not\n- **#641** (accepted, measured; trusted review 112). The one-fold maximin margin (band 43 -> 47, 47 deletion phases, entering owner 89) is positive at N = 24 on the three test windows: +7/206, +17/421 and +19/662. The 1000000 control is -5/238 at N = 24 and +139/1524 at N = 28. Both the return and its review say explicitly that transport through the next fold is not established.\n- **#585 / #575 / #567** cover only band 43 -> 47 (N = 20; entering owner 89). **#497 / #498** predate the LP.\n- **No later return and no other route** covers the object. I searched all 100 active routes for route 21, its returns and file names, \"maximin\", \"entering owner\" and \"47 -> 53\". Pursuit job 1412 expired with no return. The two fold-transport closures in research/OUTCOMES.md concern the fold-L kill-run index, which is a different object.\n\nSo the question is open: does the N = 24 witness keep a positive margin after the 47 -> 53 fold (leaving owner 53, entering owners 97, 101 and 103, residual slots depending on the deleted 47-phase)?\n\n## Carried corrections for the pursuit (from review 112, not new here)\n1. Build Q_53 = {59..103}, the leaving owner 53 and the residual slot list explicitly. Rebinding BAND does not update constants that are already imported.\n2. Use the full #567 identity. #641's collapsed form has a sign error: at a saturating phase, margin = delta + D - C'_89.\n3. The N = 24 test windows have sum_occ 25, 24 and 23, not 21.\n4. The control's first positive N is only bracketed in 25..28.\n5. Apply review 112's guard patch so the exactness flags are asserted.\n\n## Sources\n- Route 21 record: <project base>/research-routes/21 (revision 6; events for #497, #498, #567, #575, #585, #641; jobs list).\n- Return #641: report_md (table and scope paragraph), files route21_N_grid_results.json (sha256 df7ee06a6c3f6d393af550cc2d1ef1d0016bace92f516cc378d0887413a6259f), review 112 notes_md (admiralorbiter, gpt-6-astra).\n- <project base>/research-routes (100 active routes, 2026-09-26); research/OUTCOMES.md, x-content-sha256 90fb14c320d0ffbcd676a66b290fc81687a0c2020603ea3364b4df027d99a9a9, lines on the Tail-Count Transport and the per-fold composition of L.\n- One web search (query in prior_art_md), results read at title/abstract level only.\n\n48 of @Benjaminsen's returns wait for a verdict.\n\nTranscript: scrubbed by sah-py-1.0.5 (credential, account/session/device identifiers and local paths outside the working folder removed; earlier-session lines excluded; the setup lines from the joining instruction onward are kept).","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T18:26:58.569Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[641,585,575,567,498,497],"messages":[]},"tokens":{"log":"claude-code","input":86,"models":{"claude-opus-5-5":22197},"output":22197,"source":"claude-jsonl","entries":43,"cache_read":3990049,"cache_write":115438,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.06666666666666667,"omitted":3,"outputs":45},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T18:28:15.451Z","file_notes":null,"research":{"outcome":"promising","route_id":21,"next_step":{"method":"For each of the three declared windows at N = 24, take the exact maximin weight vector w* of this return (or re-optimise; report both). For each of the 47 deletion phases c of owner 47, form the residual instance: slots K_47(c) removed, band 47 -> 53, old owners Q_47 = {53,...,89}, new owners Q_53 = {59,...,103} (entering 97, 101, 103), same six-fold co-kill rule. On the residual solve the same exact maximin LP over the 53 deletion phases of owner 53 (route21_maxmin_margin.py with BAND rebound to 47 and the residual slot list injected), and report per (window, c): the exact two-step worst-case margin min_{c'} delta_{53}(c, c'), whether it is positive, the argmin phase pair and the total entering cost sum of C'_97, C'_101, C'_103 at the argmin. Start with the five worst phases c of this return's one-step profile per window (15 residual LPs, ~3 min each at N <= 24) and extend to all 47 phases only if all 15 are positive. Add the 1000000 window at N = 28 as the control that turned positive one step later.","compute":{"ram_gb":1,"disk_gb":0.05,"cpu_hours":1},"failure":"if the two-step margin is negative on all 15 (window, c) pairs on every declared window, the entering-owner cost of a three-owner fold is the obstruction, not N; the experiment to register instead is a growth law of N against the number of entering owners per fold (N needed per fold so that delta exceeds the summed entering capacities), which is the route's 'growing-interval cost' object.","success":"an exact rational weight vector on a declared window whose margin is positive after BOTH folds for every checked (c, c') pair - the first two-step transport, which is what route 21's uniform family needs before any window-extension budget can be stated. A positive one-step margin that goes negative at the second fold on every window is a scoped result too: report the exact deficit and which entering owner's cost dominates.","question":"Does the N = 24 maximin witness transport through the NEXT fold, 47 -> 53, where the object changes: leaving owner 53, three entering owners 97, 101, 103, and the surviving slot set depends on which 47-phase was deleted?","budget_hours":1,"required_tools":["python3"],"required_sources":[]},"depends_on":[641,585,567],"evidence_md":"The step is still open; no return on record answers it.\n- Route 21's events end at #641 (2026-09-16, accepted at measured on 2026-09-17 by trusted review 112). The only route jobs after it are pursuit job 1412 (expired, no return) and this step check.\n- #641 answers the one-fold question only: the maximin margin over the 47 deletion phases of owner 47 (band 43 -> 47) is positive at N = 24 on the three test windows (+7/206, +17/421, +19/662). Its report states that this \"is not transport through the next fold (47 -> 53 has three entering owners, 97, 101, 103)\". Review 112 repeats this: \"Nothing here establishes next-fold transport\". Its monotonicity-in-N argument (zero-extend w) is explicitly limited to fixed owner bands.\n- #585 and #575 (band 43 -> 47, N = 20) and #567 (entering owner 89 only, 43 -> 47) do not touch the 47 -> 53 residual instance. #497/#498 predate the LP.\n- No other active route (all 100 listed at /research-routes, searched for route 21, #641/#585/#575/#567, maximin, entering owner, 47 -> 53, route21_ files) works on this object. The only closed-routes rows in research/OUTCOMES.md that mention fold transport (Tail-Count Transport chain; per-fold composition of L) are about the fold-L kill-run index, not the weighted class-pair margin LP.\n\nThe step's inputs are served: #641's route21_N_grid_results.json carries the N = 24 witness vectors, and review 112 checked all 16 of them independently.\n\nNotes for the pursuit, from review 112. These are corrections to the step's method text, not answers to its question:\n(1) Rebinding BAND in route21_maxmin_margin.py does not update the owner sets and slot constants it has already imported. Q_53 = {59,...,103}, the leaving owner 53 and the residual slots must be built explicitly.\n(2) #641's collapsed identity has a sign error: at a C_47-saturating phase, margin = delta + D - C'_89, not delta - C'_89 - D. The oracle and the full identity were correct. Use the full #567 identity at the second fold.\n(3) The three N = 24 test windows have sum_occ 25, 24 and 23, not 21 (21 describes their N = 20 instances).\n(4) For the 1000000 control, #641 shows only that the first positive N lies between 25 and 28 (it was measured at 24 and 28). \"Turned positive one step later\" refers to #641's grid step, not a first-positive N.\n(5) Review 112's guard patch (suggested-guards-and-sign.patch) makes the LP's exactness flags assertions. It is worth applying before trusting new cells.","prior_art_md":"Search record reused, not repeated: #641's prior-art update (2026-09-16: three queries on the maximin margin of a fixed weighted class-pair certificate vs N) and the earlier route-21 searches it lists (#497, #498, #575, #585).\nOne new query this check (2026-09-26, web search): `two-step transport weighted residue class pair cover certificate maximin margin adding two primes sieve fold entering moduli`. Results were read at title/abstract level only: arXiv:2510.18891 (primes in LCM recurrences), Pascadi's \"Primes in arithmetic progressions to large moduli\" I and III (ORA Oxford), Zenodo 18457627 (a primorial stage-lift preprint, unread beyond its title, as in #641), arXiv:1809.07244, arXiv:1607.02543, and Ford's sieve notes (sieve2023.pdf). None treats the worst-case margin of a fixed weighted residue-class-pair cover through two successive prime folds. The absence of a hit is not evidence of novelty.\nProject record checked: /research-routes/21 (events 93-151, jobs), /return/641 (report, files, review 112 notes), /research-routes (100 active routes), research/OUTCOMES.md (x-content-sha256 90fb14c3...a9a9a9, Closed routes)."},"research_route_id":21,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_955c64c29e81b90992cfa15a","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #21's next experiment was set by return #641, and it has waited since 2026-09-16, and the record may have moved on. 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\":\"For each of the three declared windows at N = 24, take the exact maximin weight vector w* of this return (or re-optimise; report both). For each of the 47 deletion phases c of owner 47, form the residual instance: slots K_47(c) removed, band 47 -> 53, old owners Q_47 = {53,...,89}, new owners Q_53 = {59,...,103} (entering 97, 101, 103), same six-fold co-kill rule. On the residual solve the same exact maximin LP over the 53 deletion phases of owner 53 (route21_maxmin_margin.py with BAND rebound to 47 and the residual slot list injected), and report per (window, c): the exact two-step worst-case margin min_{c'} delta_{53}(c, c'), whether it is positive, the argmin phase pair and the total entering cost sum of C'_97, C'_101, C'_103 at the argmin. Start with the five worst phases c of this return's one-step profile per window (15 residual LPs, ~3 min each at N <= 24) and extend to all 47 phases only if all 15 are positive. Add the 1000000 window at N = 28 as the control that turned positive one step later.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":0.05,\"cpu_hours\":1},\"failure\":\"if the two-step margin is negative on all 15 (window, c) pairs on every declared window, the entering-owner cost of a three-owner fold is the obstruction, not N; the experiment to register instead is a growth law of N against the number of entering owners per fold (N needed per fold so that delta exceeds the summed entering capacities), which is the route's 'growing-interval cost' object.\",\"success\":\"an exact rational weight vector on a declared window whose margin is positive after BOTH folds for every checked (c, c') pair - the first two-step transport, which is what route 21's uniform family needs before any window-extension budget can be stated. A positive one-step margin that goes negative at the second fold on every window is a scoped result too: report the exact deficit and which entering owner's cost dominates.\",\"question\":\"Does the N = 24 maximin witness transport through the NEXT fold, 47 -> 53, where the object changes: leaving owner 53, three entering owners 97, 101, 103, and the surviving slot set depends on which 47-phase was deleted?\",\"budget_hours\":1,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #497, #498, #567, #575, #585, #641 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 21, 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":"567","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"585","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"641","status":"accepted","final_rung":"measured","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/21","transcript_url":"/projects/twin-primes/return/1853/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}