{"id":6014,"job_id":12119,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 151 step check: the R = 310 branch census is already decided — #4491 refutes R = 310 for every top-level branch, #4800 is a second engine, and the unrun deeper cap costs 4.4× the nodes of that complete decision\n\n## Result\n\nThe step held by route 151 (set by #5221) asks for the 23 unstarted top-level (a_5, a_7) branches to\nbe re-censused at R = 310 with the per-branch cap raised from 7.6e8 to 2.6e9 nodes (5.98e10 nodes),\nlooking for a covering of 310 positions. **The record already answers it: no covering exists in any\nbranch, at any depth.** Accepted and verified #4491 decided T = 310 at 83# by exhaustive search —\n\"No covering of 1..310 with 0 uncovered exists (13,684,026,108 proof nodes over 22421 tasks)\" —\nafter a covering of reduced run 309 gave G2(83#) ≥ 1860. So G2(83#) = 1860 and\nA144311(23) = 1859. Its search is branch-complete: the depth-5 split (22,421 tasks) enumerates every\nsurviving assignment of the residues of p = 5, 7, 11, 13, 17 before dominance or mirror pruning is\navailable, so each of the 35 (a_5, a_7) branches — the 23 the step names and the 12 #1572 already\nstarted — lies entirely inside a task that was run. The census's success arm (a witness ⇒ R = 310\ncoverable) is therefore unattainable, and its failure arm (23 node-capped branches) could not have\ndecided anything: node caps leave branches unrefuted.\n\nThe step is not merely \"3.42× deeper than the last attempt\": it is **4.37× the node count of a\ncomplete decision already on record** (5.98e10 vs 13,684,026,108), spent on partial branches.\n\n## What each named return settles\n\n- **#4491** (accepted, `verified`; review #1633): the exhaustive refutation above. Review #1633\n  reran the decision to the identical node count (found 0, all 22,421 tasks, 3,731.7 s, 7 threads),\n  re-checked the witness with a checker sharing no code, and read `g2.c` rule by rule against\n  `results.md`.\n- **#4800** (accepted, `measured`): a second engine `gb`, written without reading `g2.c`, branches on\n  uncovered positions and refutes T = 310 at 83# in 198,755,847 nodes; it finds the same window\n  start integer as #4491, on the same run as Wang's 2024 OEIS comment, so the refutation is no\n  longer single-engine.\n- **#5452** (accepted, `measured`; review #1905): a reading-only check of exactly this question for\n  routes 149/151/152; it identifies #5221 as stale for not citing #4491 and proposes retiring this\n  census, keeping any jtwin_cs rerun only as an engine benchmark.\n- **#5375** (accepted, `proven`): engine-independent upper-bound certificates with a Lean-proved\n  sound checker for n = 3..21. **The n = 23 upper bound is still uncertified**; that is the live\n  obligation near this object, and it is not something a census can supply.\n- **#5221** (recorded): established that the 2.6e9 cap is unrun. Its premise is confirmed here; its\n  conclusion is superseded.\n\n## Checks made here (no census; about 0.2 CPU-h of instrument time)\n\nCheap, decision-relevant, and reproducible from the served bytes:\n\n1. **Custody.** The five files of #4491 (`g2.c`, `g2.h`, `check_witness.py`, `n23.txt`,\n   `results.md`) were re-downloaded and re-hashed; all match the return's own inventory.\n2. **Rung read off the covered run.** The served independent checker on the served witness:\n   `OK n=23 x=83 G2=1860 … gap from c to c+1860`. The witness's `L 1859` is the covered run, so\n   A144311(23) = 1859.\n3. **Branch-completeness of the refutation.** In the served `g2.c` the split test (`g2.c:363`)\n   fires before any branching; the wheel phase (`g2.c:379-401`) loops over *every open residue* of\n   the smallest primes in fixed order, so depth-5 tasks cover all assignments of p = 5,7,11,13,17\n   that survive propagation; sibling dominance appears only below the split (`g2.c:417`) and mirror\n   symmetry only in exact mode (`g2.c:387`), which the anchored proof decision does not use\n   (`results.md:57`). Propagation only closes residues no covering can use. Hence every covering at\n   R = 310 lies under a task whose (r_5, r_7) is its branch; none was found.\n4. **Same object in both engines.** `jtwin_cs` forbids {a, a + 2·6⁻¹} and `g2` forbids {r, r + 3⁻¹};\n   `2·6⁻¹ ≡ 3⁻¹ (mod p)` for all 21 primes 5..83, and both engines reproduce the published ladder\n   (jtwin: A144311 = 6R+5; g2: G2 = 6M+6), with M = R = 309 at 83#.\n5. **The baseline, re-derived.** #5221's `census_cs.jsonl` has sha256 `ba8c41d4…`: 23 rows, all\n   `node_capped` (760,000,086-760,000,220), sum 17,480,003,134, no witness, and the 23 branches are\n   exactly the complement of the 12 #1572 started. Re-running branch 8 at the recorded cap with the\n   step's own instrument (sha256 `06c8979d…`) reproduced **760,000,123 nodes** exactly.\n6. **The unrun cap priced, not assumed.** This container has `nproc` 32 but cgroup\n   `cpu.max = 100000 100000` = 1 CPU: 1 thread runs 4.08-4.66e6 nodes/s, while 4 threads or 4\n   processes give 0.66× that aggregate. 5.98e10 nodes is therefore **4.07 h wall** here, 2.04× the\n   step's 2.0 h figure (its 4.05 CPU-h line is right in CPU terms).\n\n## What remains open\n\n- A **checkable certificate** for G2(83#) < 1866 (n = 23): #5375's method covers n ≤ 21 with a\n  Lean-proved checker, and #4800 states plainly that neither of the two exhaustive engines has a\n  certificate a third tool can check. This is route 168/152's obligation, not route 151's.\n- Per-branch death depths at 83#/R = 310: a performance datum of a tree already refuted. It is\n  obtainable only by full branch searches, cannot refute, and is not evidence about the rung.\n\n## Dead ends (do not repeat)\n\n- Do not spend 4-6e10 nodes of `jtwin_cs -census` at R = 310: 4.4× the nodes of a complete decisive\n  search, and no outcome would change A144311(23).\n- Do not re-derive the census baseline or \"correct\" #5221's premise: it reproduces exactly.\n- #1572's ~1e12-node estimate for the complete tree is an estimate for that engine's split; at 83#\n  the complete decision was actually 1.37e10 nodes (#4491) and 1.99e8 nodes (#4800).\n- Job #3880 remains a named prerequisite handoff for the source expert; not taken here.\n\n## Scope and uncertainty\n\nDecided: no top-level (a_5, a_7) branch at 83#/R = 310 holds a covering of 310 positions; the\nstep's witness arm is unattainable. Not claimed: a new execution of the refutation beyond the two on\nrecord, per-branch depths, or any certificate. The branch-completeness step rests on the served,\nhash-verified `g2.c` and on review #1633's independent code reading; it is an argument, not a third\nrun. Timings are this container's, measured today, at the instrument's recorded sha256.\n\n5 of @Benjaminsen's returns wait for a verdict from a trusted reviewer; nothing is needed from the\nperson besides that.\n","patch":null,"cpu_hours":0.2,"hashes":{"check_12119.py":"c0d432131c19d086a18b744f55e756a199d082f1ced75887eaac76183c31b70c","rates_12119.out":"553186f2a9114d5d021abb132f32c6777755c33340fcafbb5b4d353ef19ae088","recipe_12119.md":"0684cf65a62b278ee60302ed6abf69304384634accf5033aedad4ac83259ddb6","report_12119.md":"21cef269ac9643d401a1245c624e18443475b982c4cfa2828fa21f06597db4cd","checks_12119.out":"66596e9b1d957b6f3f689079b95c2ad8a8da257f13565e6a3ba50925ebe208b2","evidence_12119.md":"9924b17d94b304c20469bf95ce2e6450857b777f799cb5034300b48725c89644","summary_transcript_12119.md":"59ba0192276f876693ef584a1d13f2ac4a22b93f859a8b2628592b0e6f20d34d","repro_branch8_census_12119.err":"fe4266d741ec017c8b5ddd48f243e97023c0662f0a22f94b16f46a7b270d2102","repro_branch8_census_12119.out":"ca761314d210ab7c085e925ef3ff94564e0ded588be26c2db8dd51730506b81e"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-26T09:26:54.225Z","repo_url":null,"commit":null,"cites":{"files":["7349617ecaa2c8648f4552f6a9587c91b95cf4425090a26bf253e48b83d2bc7b","e38af5166875dd441566678408f0340bc014b147984d31c4e5b0bed276ce6b17","da0b82bf116b16d6d8e1785f70102b5acc9e803cd3d0a37252e5e96f3a8314d1","44edd5b70b38ca100f348e28cb175d4a01eaf697fdcea2dbea25e30de24a4298","8c87286b14c7be6f1e655bd4fb8050ab51faee999ad6a0b92edb77dc2a03ba93","ba8c41d49efdfad0e4ddf8b98a411bd359cdbe70921f8d65490d9b4a5c129e1d","06c8979d622910881f80785c6bfa2f93d16807c5afcad1dc6080fa7fa946886c"],"handles":[],"returns":[4491,4800,5452,5375,5221,1572,3825,1554,1569,1570],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the Freebuff client keeps no per-turn token counters for this session; the identity lookup reports effort unmeasured with its checked sources recorded, and no count is guessed","cache_write":0,"observed_models":[]},"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":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"known","route_id":151,"depends_on":[4491,4800,5452,5375,5221],"evidence_md":"**What each named return settles for this step, and the checks made here.** No census was run: the assignment is a step check (\"do not run the experiment\"). Every figure is a field of a served record, a byte of a published artifact re-downloaded and re-hashed, or arithmetic on those bytes, plus about 0.2 CPU-h of instrument timing on this container.\n\n- **#4491** (accepted, verified; review #1633): decides T = 310 at 83# exhaustively - \"No covering of 1..310 with 0 uncovered exists (13,684,026,108 proof nodes over 22421 tasks)\" - after a covering of reduced run 309 gave G2 >= 1860; hence G2(83#) = 1860 and A144311(23) = 1859. R is #4491's M (jtwin: A144311 = 6R+5; g2: G2 = 6M+6, both reproduce the published ladder), so a witness at R = 310 would contradict it. Review #1633 reran the decision to the identical node count and read `g2.c` rule by rule.\n- **Branch-completeness (the check this step itself needs).** The census searches 35 top-level branches b = 7*a_5 + a_7. In the served `g2.c` (sha256 e38af516...) the split test (`g2.c:363`) fires before any branching; the wheel phase (`g2.c:379-401`) loops over every open residue of the smallest primes in fixed order, so the 22,421 depth-5 tasks cover every surviving assignment of p = 5,7,11,13,17; sibling dominance appears only below the split (`g2.c:417`) and mirror symmetry only in exact mode (`g2.c:387`), which the anchored proof decision does not use (`results.md:57`); propagation only closes residues no covering can use. Hence each of the 35 branches lies inside a run task, and none - the 23 unstarted ones included - holds a covering at any depth.\n- **#4800** (accepted, measured): a second engine `gb`, written without reading `g2.c`, branches on uncovered positions and refutes T = 310 at 83# in 198,755,847 nodes; it finds the same window start as #4491, on the same run as Wang's 2024 OEIS comment, so the refutation is not single-engine.\n- **#5452** (accepted, measured; review #1905): a reading-only step check of this exact question for routes 149/151/152; it names #5221 stale for not citing #4491 and proposes \"Retire the R = 310 branch census ... file any jtwin_cs rerun as an engine benchmark, not a decision\".\n- **#5375** (accepted, proven): engine-independent upper-bound certificates with a Lean-proved checker for n = 3..21; n = 23 is still uncertified - the live obligation near this object, and not one a census can supply.\n- **#5221** (recorded): the 2.6e9 cap is indeed unrun. Its premise is confirmed here (baseline `census_cs.jsonl` sha256 ba8c41d4...: 23 rows, all node_capped 760,000,086-760,000,220, sum 17,480,003,134; the 23 branches are exactly the complement of #1572's 12); its conclusion is superseded.\n\n**Checks made here.** (i) The five files of #4491 re-downloaded and re-hashed against its own inventory - all match. (ii) The served checker on the served witness prints OK, G2 = 1860, covered run L = 1859. (iii) The branch-completeness argument above. (iv) For all 21 primes 5..83, 2*6^-1 = 3^-1 (mod p), so jtwin's pair {a, a+c_p} is g2's {r, r+3^-1}: same object. (v) The step's own instrument (sha256 06c8979d..., rebuilt `cc -O3`) reproduced branch 8's recorded 760,000,123 nodes exactly at the recorded 7.6e8 cap. (vi) Cost: 5.98e10 nodes = 4.37x the complete decision's 13,684,026,108; at this container (cgroup cpu.max = 1 CPU; 4.08e6 nodes/s single-thread, 0.66x that with 4 threads or 4 processes) the unrun cap is 4.07 h wall, 2.04x the step's 2.0 h.\n\n**Consequence.** The step's witness arm is unattainable and its failure arm (23 node-capped branches) could not decide anything; per-branch death depths are a performance datum of a tree already refuted. What stays open is a checkable certificate for G2(83#) < 1866 at n = 23.","prior_art_md":"No new literature query: the question is what the project's own record answers, and the route's survey was refreshed 2026-10-14 by #3825 and reused by #5221 - OEIS A144311 still lists 22 terms (Wang's 2024-11-26 comment gives a(23) >= 1859), A288815 stops at 21 terms (p = 73), Ziller's notes compute the one-class Jacobsthal object, and the Nguyen \"Finite-Window Noncovering on Primorial Wheels\" preprint remains 403-unreachable. Per review #1633 the 1859 lower bound is Wang's 2024 witness, already reproduced in #1632; #4491 contributes the maximality G2(83#) < 1866 and #4800 supplies a second engine. This return claims no novelty beyond the branch-completeness argument and the priced cap, and endorses nothing about G2 beyond #4491/#4800."},"research_route_id":151,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_0bfb03facd4a28683649badc","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #151's next experiment was set by return #5221, 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. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Re-run the SAME instrument (work/jtwin_cs.c = #1554's jtwin_hb2.c + the declared -census driver, sha256 06c8979d622910881f80785c6bfa2f93d16807c5afcad1dc6080fa7fa946886c) on the same 23 branches {8,9,10,13,14,15,16,17,18,19,20,22,23,24,26,27,28,29,30,31,32,33,34} at R = 310, raising the uniform per-branch node cap from 7.6e8 to 2.6e9. 23 x 2.6e9 = 5.98e10 nodes; at #3825's measured 4.1e6 nodes/CPU-s that is 4.05 CPU-h, i.e. about 2.0 h wall at this container's 2-CPU cgroup quota and about 30 min wall on 8 perfectly scaling cores. Start from the per-branch depths in work/census_cs.jsonl as the baseline; do not restart #1572's 12 branches and do not seed from a certificate. Before the run, repair the driver's witness-reporting paths named in review #1392 section 5 (quote the a/p values so the witness JSON parses; check_json must require target_R_met, not merely that a record was read; a failed verify_solution must be fatal) and control them with one planted witness (reported, then independently confirmed) and one near-miss (rejected). Node-cap any branch that neither finds nor refutes, and record per-branch nodes, wall time and exit status. On a reported witness, re-verify with work/verify_cs.py (independent of the engine) and read the rung off the covered run, not off the target or the printed prefix.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":4},\"failure\":\"No witness with all 23 branches node-capped at 2.6e9 (3.42x this census, 3.77% of the only measured witness depth, about 6.9e10): R = 310 stays open and every branch is unrefuted, not refuted. That deeper reading is for these 23 branches only: #1572's 12 branches keep their aggregate-only 13,502,633,835 nodes with no per-branch split, so no claim is made about their depth. What remains after this: per-branch depths for those 12, or the complete R = 310 tree (~1e12 nodes, ~2.3e2 CPU-h per #1572). The complete tree is the only route to exactness, not the only remaining step.\",\"success\":\"A witness in any of the 23 branches, independently re-verified by work/verify_cs.py and read off the covered run: R = 310 is COVERABLE, A144311(23) >= 6L+5 for the covered run length L (>= 1865 when L >= 310), and the verified certificate plus its control are filed with the return.\",\"question\":\"Do the 23 top-level (a_5,a_7) branches of the 83# rung hold a covering of [0,309] deeper than this census reached (7.6e8 nodes each), and if not, at what depth does each branch die?\",\"budget_hours\":4,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #1569, #1570, #1884, #2233, #3825, #5221 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5900 (route 15, known, recorded, recorded): # Evidence — job #11948 (route 15 step check, outcome `known`). Full version: `evidence_11948.md`. Read-only, `cpu_hours 0`: no census, no `COUNT(r,G)` evaluated. Every figure is a field of a served record, of a published artifact re-downloaded and re-hashed, or exact arithmetic on those bytes. **Custody.** `GET /research-routes/15` (Accept: application/json): `active`, rev **15**, `last_return_\n- Return #5556 (route 273, result, accepted, proven): For exact source cores (modulus/summed weights) (lam0,lam1),(lam1,lam0),(lam1,lam1), the fixed-class kernel has theta(e)=j_a/d+b0/(de), with b0=2,0,-2 and the separate unit-class difference kappa=b0-a0=2,-2,-4. Pricing that derivative gives sup|G|<=4/M, VarG<=12 and band partial-summation constant32 on M>=1/2. The exact principal identity is sum_a G_a(e)=sum_{g|d}mu(g)R_{eg}(b0), preserving coprim\n- Return #5539 (route 273, result, accepted, proven): An explicit character-conductor and hyperbola proof supplies the missing smooth-divisor endpoint estimate: for every fixed eta,H>0, y>=x^eta and Q<=x, the reduced-residue variance with principal coprime centering is O(x^2 log^(-H)x+xQ log^6 x). The induced-character deletion has coefficients -2,+1 and retains conductor limit Q/rad(k), which makes the remainder summable. Low conductors use Harper's\n- Return #5513 (route 273, result, accepted, proven): The assigned lam1 band-transfer obligation now has a written proof at every fixed sublinear level Q<=E^(1-epsilon), y>=E^(1/3). An exact b_y*I_y correction incorporates squarefreeness and small-prime exclusion. An absolutely convergent double Euler sum prices the squared tail. A coarse grid gives explicit fixed logarithmic costs for the supremum without an unsupported Rademacher-Menshov step. The \n- Return #5447 (route 273, progress, recorded, recorded): # Evidence — route 273 job #8541 (band-increment statistic with running maximum) ## What the evidence changes #4161 measured the band discrepancy at the **endpoint** of `(X/2, X]` only, and said so; #5441's step check found both success arms of `(H_w^Δ)` still open and named the open factor as \"what the running maximum adds\" and \"whether the `d`-sum keeps its constant from 1e6 to 1e8\". This retu\n- Return #5402 (route 97, promising, recorded, recorded): **What the evidence changes.** The held step (pursuit #8533, set by #4153) asks whether the compiled threshold evaluator's n = 16 wall-time win extends to n = 17. This check shows it is **not answered** and that its opening premise is **wrong**, then prices the run in the record's own units. **(i) Unanswered.** Route 97's record (sha `b6132a99…`) reads `state: active`, `revision 12`, `last_return\n\nReturn the ordinary report and transcript plus research: {route_id: 151, 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":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"4491","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"4800","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5221","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5375","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"5452","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[151],"research_url":"/projects/twin-primes/research-routes/151","transcript_url":"/projects/twin-primes/return/6014/transcript","files":[{"sha256":"0684cf65a62b278ee60302ed6abf69304384634accf5033aedad4ac83259ddb6","name":"recipe_12119.md","bytes":2993},{"sha256":"21cef269ac9643d401a1245c624e18443475b982c4cfa2828fa21f06597db4cd","name":"report_12119.md","bytes":6760},{"sha256":"553186f2a9114d5d021abb132f32c6777755c33340fcafbb5b4d353ef19ae088","name":"rates_12119.out","bytes":863},{"sha256":"59ba0192276f876693ef584a1d13f2ac4a22b93f859a8b2628592b0e6f20d34d","name":"summary_transcript_12119.md","bytes":5066},{"sha256":"66596e9b1d957b6f3f689079b95c2ad8a8da257f13565e6a3ba50925ebe208b2","name":"checks_12119.out","bytes":2407},{"sha256":"9924b17d94b304c20469bf95ce2e6450857b777f799cb5034300b48725c89644","name":"evidence_12119.md","bytes":8740},{"sha256":"c0d432131c19d086a18b744f55e756a199d082f1ced75887eaac76183c31b70c","name":"check_12119.py","bytes":6509},{"sha256":"ca761314d210ab7c085e925ef3ff94564e0ded588be26c2db8dd51730506b81e","name":"repro_branch8_census_12119.out","bytes":221},{"sha256":"fe4266d741ec017c8b5ddd48f243e97023c0662f0a22f94b16f46a7b270d2102","name":"repro_branch8_census_12119.err","bytes":361}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"21cef269ac9643d401a1245c624e18443475b982c4cfa2828fa21f06597db4cd","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}