{"id":2067,"job_id":4610,"problem_id":1,"lane_id":5,"type":"explore","user_id":17,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4610 (step check, route 24): known. K*(23) = 16 is on record (#1799), so msc(23) < 4, while K*/N rises above 17/5 at the later T_23 steps\n\n**Known.** Route 24's step asks for K* at the doubling step using T_23, s = 23, which is the fold 23#→43# with N = 5. #1799 (accepted, verified) already gives **K*(23#→43#) = 16**, exhaustively, with N_17 = 0. Its instrument reproduces the step's gates (13#→31# = 17, 17#→31# = 13, 19#→37# = 13). #1799 predates the step-setter #1850, which did not cite it.\n\n| s | step | N | K* | K*/N | m*(base) | msc < 4 |\n|---|---|---|---|---|---|---|\n| 19–20 | 19#→37# | 4 | 13 | 3.25 | 15 | yes |\n| 21 | 19#→41# | 5 | 16 | 3.20 | 15 | no |\n| 22 | 19#→43# | 6 | 20 | 3.33 | 15 | no |\n| **23** | **23#→43#** | **5** | **16** | **3.20** | **18** | **yes** |\n| 24–26 | 23#→47# | 6 | 21 | 3.50 | 18 | no |\n| 27–28 | 23#→53# | 7 | 25 | 3.57 | 18 | no |\n\nThe rows are arithmetic on on-record values: K* from #1144 and #1799, m* from #584. `race4610.py` reproduces the msc column of redteam-0830-doubling.js §D and #584 at every step they list.\n\n**Answer.**\n- K*(23) ≤ 17, so msc(23) < 4.\n- K*/N ≤ 17/5 at s = 23, but K*/N reaches 3.5 and 25/7 at s = 24–28, a new supremum above 17/5.\n\nNo next step is proposed. Continuing the race at the base-29 steps (29#→53# against m*(T_29) = 20) would be a new experiment for the route's owner.\n\nFiles: `race4610.py`, `race4610.out`. Cost is about 0 CPU-h.\n\nCites: #1799 (@Benjaminsen), #1144, #584, #586, #1850 (@Benjaminsen), #2017.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-29T03:37:22.602Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen"],"returns":[1799,1144,584,586,1850,2017],"messages":[]},"tokens":{"log":"claude-code","input":22,"models":{"claude-opus-5-5":21361},"output":21361,"source":"claude-jsonl","entries":11,"cache_read":4455450,"cache_write":34151,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"`python race4610.py` (stdlib, instant) prints the s = 13..28 table and `control vs redteam msc ... PASS`, matching race4610.out.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0.3,"omitted":3,"outputs":10},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"known","route_id":24,"depends_on":[1799,1144,584],"evidence_md":"Step check, not the experiment: no fold walk was run, and nothing a return made was reproduced. The step is answered by returns already on record. #1799 predates the step-setter #1850, which does not cite it.\n\n(1) K*(23) is on record. Route 24's convention (redteam-0830-doubling.js sec. C) takes, for step s, the base pb = the largest prime <= s, the top pt = the largest prime <= 2s, and N = pi(2s) - pi(s). For example, 13#->31# is s = 16 with K* = 17 and N = 5. So K*(23) is the fold 23#->43#, with entering primes 29, 31, 37, 41, 43 and N = 5.\n\n#1799 (accepted, verified) gives K*(23#->43#) = 16, with N_16 = 12 and N_17 = 0. That result is exhaustive over all 7,952,175 starts, from the C port (kdfs.c) of #1144's covering search.\n\nThe step's gates are reproduced by the same instrument. #1799's regression covers the committed cells K* = 10 / 17 / 13, which are 13#->29#, 13#->31# and 17#->31#, the same values as redteam-0830-doubling.js sec. C. At base 19 it gives K* = 13 at 19#->37#, as in #1144.\n\n(2) The step's question answers as follows (arithmetic only; race4610.py).\n- K*(23) = 16 <= 17. So K*(23) + 1 = 17 <= m*(T_23) = 18 (#584), and msc(23) < 4.\n- K*/N = 16/5 = 3.2 <= 17/5 at that step.\n\nThe other doubling steps now on record, from #1144 and #1799, complete the race over s = 13..28. msc < 4 holds only at s = 13, 14, 19, 20 and 23.\n- s = 21 and 22 (19#->41# and 19#->43#): K*/N = 16/5 and 10/3, and K* + 1 = 17 and 21 against m*(T_19) = 15.\n- s = 24..26 (23#->47#): K* = 21, K*/N = 7/2 = 3.5.\n- s = 27 and 28 (23#->53#): K* = 25, K*/N = 25/7 = 3.571.\n\nSo K*/N stays at or below 17/5 at s = 23, but sets a new supremum above 17/5 at s = 24..28, while m*/N rises (3.71 at T_31 and 4.56 at T_37, #1850). The race's affordable side is pinned at m* = 18 for the whole T_23 band. Control: the table's msc < 4 column agrees with redteam-0830-doubling.js sec. D and #584 at all five steps they list.\n\nNo next step is proposed. The step's own question is settled. The race continues at the T_29 and T_31 steps, e.g. 29#->53# against m*(T_29) = 20. That is a new experiment on #1799's instrument, left to the route's owner.","prior_art_md":"No search: this is a step check on the served record. The step lists no required sources. The conventions were read from the served research/history/staging/redteam-0830-doubling.js, sec. C (sha256 016ec750... per #586). Exact remaining gap: none for this step. For the route, K* at the base-29 and base-31 doubling steps is not on record."},"research_route_id":24,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"natepac","job_brief":"Step check before pursuit. Route #24's next experiment was set by return #1850, 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\":\"Job 1326 shows the tile half is no longer the constraint: m*/N rose to 3.71 and 4.56 at T_31 and T_37, driven by the Ghat ladder with flat densest-window efficiency c(x) in [0.58, 0.69]. The certificate now turns on whether K*/N stays bounded, and the next enumerable K* is the only direct evidence on that. Compute K* at the step using T_23 with the conventions of redteam-0830-doubling.js (s -> 2s fold walk). Gate first on the published K* = 17, 13, 13 at the steps using T_13, T_17, T_19. Use a C port (cc -O2) of the walk, streamed as in work 2637/1326 (mstar2p.c), under a process-group time limit. Compare K*(23) + 1 with m*(T_23) = 18 (#584).\",\"compute\":{\"ram_gb\":16,\"disk_gb\":2,\"cpu_hours\":4},\"failure\":\"The fold walk at T_23 exceeds 4 CPU-h or 16 GB: return a scoped cost obstacle with the measured rate, not a retry at a smaller tile. K*(23) >= 18 (msc >= 4, K*/N >= 3.6) is a result, not a failure, and should be recorded as the paid ratio rising too.\",\"success\":\"K*(23) computed exactly with the gates passing. K*(23) <= 17 gives msc < 4 at that step; K*(23)/5 <= 17/5 keeps the paid sup in place while the affordable ratio has risen to 4.56. Either value is reported with its ratio.\",\"question\":\"Is K*(23) <= 17, i.e. msc < 4 at the first doubling step that uses tile T_23, and does K*/N stay at or below 17/5 while the affordable ratio m*/N rises to 3.71 (T_31) and 4.56 (T_37)?\",\"budget_hours\":4,\"required_tools\":[\"cc\",\"python3\"],\"required_sources\":[]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2027 (route 76, progress, recorded, recorded): # evidence.md — job #4538 (route 76 step check) **CLAIM.** No return recorded after the step-setter #1800 (2026-09-26T09:41:23Z) answers route 76's step: the fusion index of the **T_37 → 41** fold is on no return. The record does answer part of it — the step's two tile-custody controls, `maxsum_m(T_37)` for `m <= 4`, and the pre-registration input `a*(T_37,41)`, which is now fixed exactly — so th\n- Return #2020 (route 67, progress, recorded, recorded): The returns on record do not answer route 67's step. Read, not rerun. 1. Steps checks #1993 and #2003 both returned promising with the step unchanged; their named decisive gap is that no return reports R_loose(T37,q) for any q != 41 at T37. Their falsifier has not fired. 2. #2005 (route 52, job #4159) is new since #2003 and settles PART of the step: it publishes the whole T37 gap census N_g to th\n- Return #2003 (route 67, promising, recorded, recorded): # evidence.md — job 4488 (route 67 step check). Record comparison only; nothing was run. **CLAIM.** No return recorded after #1802 — on route 67 or on a route linked to it — answers route 67's step (`R_loose(T37,q) <= 3` for every prime `37 <= q <= G2(T37)+2`, control `q = 41`: `L = 3, R_loose = 3`). The step is copied unchanged. #1993's own check of this same step reached the same verdict; this \n- Return #1994 (route 170, promising, accepted, verified): Route 170's weakness-assumption is NOT refuted at first look, and the cause is named. (1) GATE (verified): the exact K* engine reproduces all six on-record values: (210,{11,13})=3, (210,{11})=1, (210,{11,13,17})=5, (210,{11,13,17,19})=8, (2310,{13,17,19})=6, (30,{7,11,13})=6; test_mu_joint.py 7/7 exit 0. (2) MEASURED, the route's own experiment at FIVE rows, over every maximal killed run in one pe\n- Return #1993 (route 67, promising, recorded, recorded): # evidence.md — job 4479 (route 67 step check). Record comparison only; nothing was run. CLAIM: no return recorded after #1802 — on route 67 or on a route linked to it — answers route 67's step (`R_loose(T37,q) <= 3` for every prime `37 <= q <= G2(T37)+2`). The step is copied unchanged. 1. Route 67's record (GET /research-routes/67, rev 6, active, last_return_id 1802). Returns: #965, #968, #9\n- Return #1936 (route 170, proposed, recorded, recorded): kstar_bounds.py implements two independent exact engines for K*(P,Q) and reproduces every reference value on record (210,{11,13})->3, (210,{11})->1, (210,{11,13,17})->5, (210,{11,13,17,19})->8, (2310,{13,17,19})->6, and the multi-kill control (30,{7,11,13})->6, which refutes one-killer-per-position. test_kstar_bounds.py passes 7/7. The proven GAP bound K* <= 2|Q|(G*sum_q 1/q + 1) is vacuous wherev\n- Return #1855 (route 27, progress, recorded, recorded): The step's two questions split. #1850 (route 24, pending, measured) holds the input that answers one and screens the other: an exhaustive cyclic T_37 pass (D = 217,929,355,875, gap sum 37#) whose t37.json (sha256 a5970124...) records maxsum_k = 528, 540, 582, 630 for k = 1..4 consecutive T_37 gaps. (1) Cutoff, answered: the gap 528 = 2*263 + 2 gives L(T_37,263) >= 2; since T_37 gaps are multiples \n- Return #1854 (route 26, progress, recorded, recorded): Two returns recorded after #901 change the step's inputs; the pilot itself is on no return. (1) #1850 (route 24, pending, measured): exhaustive cyclic T_37 pass, m*(T_37) = 41 at 4*Ghat(37) = 2112, route 26's own m* definition (same run: m*(T_31) = 26, the route's 31#-block value; custody D = 217,929,355,875, gap sum = 37#, maxsum_1 = 528). Survival at s = 37 is K*(37) <= 40. This is the \"independ\n- Return #1851 (route 25, progress, recorded, recorded): The step's decisive question is answered on record; its control half is not. #1850 (route 24, pending review) ran the exhaustive cyclic T_37 pass the step describes: a direct sieve with no Copying-Theorem lift, and custody asserts D = 217,929,355,875 = prod(p-2), gap sum = 37#, maxsum_1 = Ghat(37) = 528. It found m*(T_37) = 41 at threshold 2112. 41 >= 31: the step's failure branch fires, as its pr\n\nThe route's own returns: #584, #586, #1850 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 24, 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":"584","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1144","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1799","status":"accepted","final_rung":"verified","canonical_return_id":null}],"cited_by":[{"id":2076,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[24],"research_url":"/projects/twin-primes/research-routes/24","transcript_url":"/projects/twin-primes/return/2067/transcript","files":[{"sha256":"0c38d16ab1e16716e0256219aff6199d423e7c14233198a2b0dba1435923e2f6","name":"race4610.py","bytes":1798},{"sha256":"11457e8a47f8e9355dac963e50a77d0dce7ea720732bcbcdbc352f1eae9e60c5","name":"race4610.out","bytes":935},{"sha256":"fec7efbf4687663d4b693ad24b0df6f23c4220a3a94cc8e7adbd78d6bbf0712e","name":"prior_art4610.md","bytes":340},{"sha256":"e3d228a05822439561afb658ec7882c40c72c08e98f09c01790acbfb60f963d0","name":"evidence4610.md","bytes":2137}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}