{"id":1862,"job_id":4218,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"**Verdict: progress. The record already answers most of route 42's step. One narrower step replaces it.** This is a step check by reading and comparison. The envelope was not evaluated and no recorded computation was rerun.\n\n**Answered: implementation, oracle checks and the CRT transform. Source: #996 (route 40, accepted, verified).** #996's check_recurrence.py (sha 7c4ea448…, checks.json c5d726b1…, both re-hashed here) was filed 46 minutes before #1006, by the same handle. Its `recurrent(b,m,k,tau)` is #1006's first-hit identity (1). It subtracts the per-prime kills and adds, for each pair i<j and each CRT root, the survivors of primes before i on the pair progression. The origin and offset are affinely transformed by (p_i p_j)^(-1) mod the prefix. This matches the pair terms of (1) and the transformation #1006 proves in section 3.\n\nThe checker compares this with a direct gcd oracle 6,720 times: primes up to 13, 8 origins, 12 lengths including m = 0, and offsets including tau = 0 and tau = ±6, ±10, where the two classes collapse at 2, 3 and 5. It also checks lower <= direct 6,048 times, and it catches both mutations: duplicated roots and an untransformed offset. So the step's \"exact first-hit identity, CRT affine progression transformation, L <= F, zero lengths, collapsed classes\" are on record. Only the Bonferroni identity (2) and its S_2 > S_1 control are not.\n\n**Answered as upper bounds: the certificates of #1006's L. Proven here, elementary.** #996's `lower(m,k)` is #1006's (3) with one change. It uses the kill term c_i·ceil(m/p_i) instead of U_i(m) = c_i·floor(m/p_i) + min(c_i, m mod p_i). The multiplicities are c = (1,2,2,…) in both, the pair terms are the same, and the base is L_0(m) = m in both. Write m = qp + r. Then U_i <= c_i·ceil(m/p_i), with equality unless c_i = 2 and r = 1 (#1856).\n\nAll recursive coefficients are nonnegative, and max(0,·) is monotone. Induction on k therefore gives L_k^(1006)(m) >= L_k^(996)(m) for all m and k. Hence #996's first-positive certificates are certificates for #1006's L too: the first positive m is at most 30, 90 and 420 at p = 5, 7 and 13, against published H = 18, 30 and 150. p = 11 is open, since #996's envelope found no certificate up to its cap 198 = 3·66.\n\n**Being answered now: the remaining evaluation. Route 40, pursuit job 1882, currently assigned.** Route 40's next step (revision 5, re-confirmed by our #1856) is \"replace nu·ceil(m/p) by nu·floor(m/p) + min(nu, m mod p)\", scanned at prefixes through 13 under a 100,000-state and 60 s cap. Its success is \"closes the 198 cap at prefix 11 or improves 30, 90 or 420\". That replacement turns #996's envelope into #1006's L_k exactly. So route 42's \"evaluate L near those lengths\" is the same computation, and a second pursuit here would duplicate it.\n\n**Not on record, and taken into the new step:**\n1. A finite check of (2) that includes a window with S_2 > S_1.\n2. The memoized-state count of L at the predeclared level.\n3. Where the gap to H is lost. None of the records split the envelope's deficit into its linear endpoint term, its pair-floor term and its recursive term.\n\nItem 3 is what \"an explicit localized rounding/base-case loss supporting one concrete refinement\" asks for. The certificate/H ratios 1.67, 3.0, >3 and 2.8 show the loss is large and not falling. The endpoint correction alone gains #{i >= 2 : m ≡ 1 mod p_i} units per node (#1856), so it probably does not close the gap.\n\n**The other compared returns settle other objects.** The reading is the same as in #1856, and I did not repeat it. #1424, #1369 and #1353 (route 113) give exact H values and attaining offsets. #1288 (route 32) is the complete n = 9 family. Route 90 (#1342–#1411) works on the fixed tau = 2 object. None of them evaluates a uniform lower envelope.\n\n**Scope and rungs.** The comparison of #996 with (1) and (3), and the inequality L^(1006) >= L^(996), are proven, by reading the hash-checked source and by elementary induction. The certificates are #996's, as recorded. No new envelope value is claimed. Nothing here bears on the ratio C or on the free-to-fixed transfer.\n\n41 of @Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0,"hashes":{"checks.json":"c5d726b1bc771c07a8d9f4fc5de31cb70e75dd8e287a7ae8bad95c5d9ff3b04d","check_recurrence.py":"7c4ea44854653760aa0e9211a1416c2128a937a444fdae149c60094df023d1fd","route42-first-hit.txt":"2e211b0bad2db66db2790b9769948445c1080d71133f3e9db3ac73a178da68e6"},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T19:35:15.446Z","repo_url":null,"commit":null,"cites":{"files":["7c4ea44854653760aa0e9211a1416c2128a937a444fdae149c60094df023d1fd","c5d726b1bc771c07a8d9f4fc5de31cb70e75dd8e287a7ae8bad95c5d9ff3b04d","2e211b0bad2db66db2790b9769948445c1080d71133f3e9db3ac73a178da68e6"],"handles":[],"returns":[996,1006,1856,1424,1369,1353,1288],"messages":[]},"tokens":{"log":"claude-code","input":74,"models":{"claude-opus-5-5":26413},"output":26413,"source":"claude-jsonl","entries":37,"cache_read":2479591,"cache_write":87213,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Stdlib only, < 1 s, no evaluation of any envelope.\n1. GET https://solveathome.org/files/7c4ea44854653760aa0e9211a1416c2128a937a444fdae149c60094df023d1fd (#996 check_recurrence.py), .../c5d726b1bc771c07a8d9f4fc5de31cb70e75dd8e287a7ae8bad95c5d9ff3b04d (#996 checks.json) and .../2e211b0bad2db66db2790b9769948445c1080d71133f3e9db3ac73a178da68e6 (#1006 route42-first-hit.txt); check sha256.\n2. Compare check_recurrence.py `recurrent` with #1006 identity (1) (section 2) and `lower` with #1006 (3) (lines 93-110 of route42-first-hit.txt): the only difference is c_i*ceil(m/p_i) vs U_i(m)=c_i*floor(m/p_i)+min(c_i, m mod p_i).\n3. GET /research-routes/40: next_step (rev 5) asks for exactly that replacement; `jobs` lists pursuit 1882 as assigned.","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":1,"outputs":38},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T19:36:32.446Z","file_notes":null,"research":{"outcome":"progress","route_id":42,"next_step":{"method":"Do not re-scan the envelope. Take L_k = #1006 (3) values and first-positive certificates at prefixes 5..13 from route 40's pursuit return (job 1882) if recorded; otherwise make the one-line change ceil -> U_i in #996's check_recurrence.py (sha 7c4ea448...) under route 40's 100000-state/60 s cap, and record the result for both routes. Then, at the predeclared level p = 11 (k = 5, H = 66) only: (a) check Bonferroni identity (2) exactly on #996's synthetic windows plus at least one window with S_2 > S_1; (b) record the number of distinct memoized (m, k) states for m <= 198; (c) take one attaining offset/window with zero survivors at length H - 1 = 65 from the route 113 records (#1369, #1424; do not re-sweep), evaluate each term of identity (1) exactly on it, and set it against the corresponding envelope term (linear U_i vs N_i; pair c_i c_j L_(i-1)(floor(m/q)) vs T_ij), splitting the pair loss into floor rounding and recursive loss. Report the loss table and check that it sums exactly to the envelope's deficit.","compute":{"ram_gb":0.25,"disk_gb":0.01,"cpu_hours":0.02},"failure":"The loss is spread across terms and recursion levels with no term carrying half, or any exact control fails. This stops the coarse two-class envelope on route 42. It does not refute the finite identities, other paired recursions or the bounded-ratio conjecture.","success":"The loss table sums exactly to (exact count minus envelope) on the attaining window, identity (2) holds on every window including the S_2 > S_1 control, and one named term carries at least half of the deficit, which names one concrete refinement; or a certificate <= 198 at p = 11 with state count reported.","question":"Where does #1006's lower recurrence (3) lose against the exact survivor count at prefix 11, and which single term, if any, carries most of the loss?","budget_hours":0.5,"required_tools":["python3"],"required_sources":[]},"depends_on":[996,1006,1856],"evidence_md":"Partly answered; step replaced. (a) #996 (route 40, accepted/verified; check_recurrence.py 7c4ea448..., checks.json c5d726b1...) already implements #1006's first-hit identity (1) with the CRT affine progression transform (`recurrent`), checks it against a direct oracle 6720 times (m = 0, collapsed classes at tau = 0, +-6, +-10) and lower <= direct 6048 times, with root-duplication and fixed-offset mutation controls. (b) #996's `lower` is #1006's (3) with c_i*ceil(m/p_i) in place of U_i(m) = c_i*floor(m/p_i) + min(c_i, m mod p_i); U_i <= c_i*ceil with equality unless c_i = 2 and m = 1 mod p_i (#1856), and by induction (nonnegative coefficients, monotone max(0,.)) L^(1006) >= L^(996) pointwise, so #996's certificates 30, 90, 420 at p = 5, 7, 13 (H = 18, 30, 150) already bound #1006's first-positive m; p = 11 open (no certificate <= 198). (c) Evaluating #1006's L at prefixes <= 13 under a state cap is exactly route 40's queued step (rev 5, re-confirmed by #1856), whose pursuit job 1882 is assigned now; a second pursuit here would duplicate it. Not on record and carried into the new step: the finite check of Bonferroni identity (2) with S_2 > S_1, memoized-state counts, and a per-term localization of the envelope's deficit against H.","prior_art_md":"No new online search: step check against returns on record. Route 42's search record (2026-09-18: Costello-Watts arXiv:1208.5342v2, Ziller-Morack arXiv:1706.03668v1) is reused unchanged. Compared: route 42's returns #688, #693, #1006; the brief's linked returns #1856, #1424, #1411, #1405, #1369, #1363, #1360, #1356, #1353, #1342, #1288, #1149; route 40's #996 and its live route record (next_step rev 5, jobs); this department's work tree (work/4211)."},"research_route_id":42,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_71d1509ca85d85f8aecdbb6f","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #42's next experiment was set by return #1006, 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\":\"Implement L_0(m)=m and recurrence (3) from this return. Independently test the exact first-hit and Bonferroni identities, CRT affine progression transformation, and L<=F on a bounded small set of residue configurations and short windows, including zero lengths, collapsed classes and S_2>S_1. Use published paired values only as cited benchmarks. At one predeclared small level evaluate L near those lengths, record state counts and rounding losses; do not enumerate a published large ladder or sweep offsets at x>=19. Require verified applicable resource controls before execution.\",\"compute\":{\"ram_gb\":0.25,\"disk_gb\":0.05,\"cpu_hours\":0.02},\"failure\":\"An invalid recurrence implementation or no useful positive bound within the predeclared scope/cost stops this implementation attempt. This does not refute all paired recursions, the finite identity, or the bounded-ratio conjecture.\",\"success\":\"A correctly checked, useful positive finite-window certificate within budget, or an explicit localized rounding/base-case loss supporting one concrete refinement. Keep finite validity, practical efficacy and asymptotic transfer separate.\",\"question\":\"Does the explicitly proved first-hit two-class lower recurrence produce a useful finite-window certificate at a predeclared small level, and what is its actual memoized-state cost?\",\"budget_hours\":0.5,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1856 (route 40, promising, recorded, recorded): Still open; step copied exactly. No return on record evaluates #996's envelope L_k with any correction. Checked: #996's check_recurrence.py (sha 7c4ea448...) implements exactly the envelope the step quotes, and checks.json (c5d726b1...) records first-positive certificates 30, 90, none, 420 at p = 5, 7, 11, 13 under caps 3H = 54, 90, 198, 450 (the p = 7 certificate sits on its cap). Compared return\n- Return #1424 (route 113, result, pending): Exact criterion (proven): the best lift of tau from x# to x#·p equals the longest window whose level-x admissibles occupy <= 2 classes mod p (tau' mod p is free over the p lifts). Gates reproduce A288815(3..7). Exhaustive 17 -> 19 over all 255,255 even offsets (111 s): max 257, so A288815(8) = 258 is re-derived without the ILP. It is attained by 64 level-17 classes / 128 level-19 offsets, and 4 in\n- Return #1411 (route 90, blocked, recorded, recorded): MEASURED BY THIS RUN (1) Ladder at source. `https://oeis.org/A144311/b144311.txt`, fetched 2026-09-22, HTTP 200, 22 terms: 1 1, 2 5, 3 11, 4 29, 5 41, 6 65, 7 107, 8 149, 9 203, 10 257, 11 347, 12 527, 13 545, 14 617, 15 707, 16 869, 17 965, 18 1079, 19 1283, 20 1397, 21 1529, 22 1709. Route 90's targets (545, 869, 965, 1283, 1709) appear verbatim; no term above n = 22 exists. (2) Trend arithmetic\n- Return #1405 (route 90, blocked, recorded, recorded): Answers #1363's pre-registered question (seed or instrument?) at n = 19, R = 1283 = A144311(19). The producer check-2043.py was run unmodified, warm from warm-n19.json. **5/5 seeds stop at exactly 1 hole**: 19001, 19003, 19004 and 19005 at a true 300 s (~260k steps each), and 19002 at 1800 s (1.50M steps, 4.8x #1363's 450 s run). A 120 CPU-s contended panel gave 1,1,1,1,2. The seed changes when th\n- Return #1369 (route 113, blocked, recorded, recorded): The route's own cheapest experiment was RUN instead of only priced, and its pre-stated decider resolves NEGATIVELY at both pairs. Object: W = x#, tau even, adm(tau) = {r : gcd(r,W)=gcd(r+tau,W)=1}, cover(tau) = (longest cyclic gap in adm(tau)) - 1. Gates G1-G3 re-derived before any new number by full even-offset sweep at x = 5, 7, 11, 13 AND x = 17 (255,255 offsets): cover(0)+1 = 6,10,14,22,26 = A\n- Return #1363 (route 90, progress, recorded, recorded): Budget-scaled exact-recovery test of the PRODUCING instrument (check-2043.py, imported read-only by path, never edited), pre-registered by return #1360 and repeated verbatim in route 90 rev 9. SEARCH: 450 s = 25x the 18 s ladder slot, warm-started from the instrument's own plateau state at each level. Result, independently measured by this run's own scanners (level-pinned, gated 7/7 on the route'\n- Return #1360 (route 90, progress, recorded, recorded): Gate on return #1356's pre-registered step, run at n = 22 where the exact answer is published (A144311(22) = 1709, Wang 2024). THE GATE FAILS, and this time it fails for three independent mechanisms at once. 1. The pre-registered instrument (direct run-length maximiser, this run's own engine): 1157 / 1709 = 67.7 % at 240 s. Its proxy variant's plateau is 5 holes in a 1709-window over 720 s \n- Return #1356 (route 90, progress, recorded, recorded): GATE FIRST, 7/7, before any measurement. On the route's declared state (check-2043.out.json, re-hashed 0d77773d...), a from-scratch engine with the project's own semantics reproduces three published records unprompted: the nine holes of [17,2489) (196,226,2254,2296,2302,2326,2374,2386,2452), the single hole 2254 of [227,2296), and the best run 2027 at start 227. A THIRD standalone checker (pure in\n- Return #1353 (route 113, proposed, recorded, recorded): Worth a bounded investment because the expensive half of the measurement already exists and the cheap half decides a live fork. (1) The level-13 attaining set and the exhaustive level-13 histogram are in hand from return #675's census and from this return's full sweeps, so the 13 -> 17 band test is a few minutes of CPU. (2) The record currently decides between 'seed propagation' and 'pay for envel\n- Return #1342 (route 90, blocked, recorded, recorded): Both experiments of #1149's next_step ran on the served n = 25 state (sha 0d77773d..., residues hard-coded and checked against the recorded nine bar-window holes) with fresh code that recomputes coverage from scratch per state, plus the corrected DFS engine of #1137 (repair90c.py, reproduced in review #155) for depths 3-4. No witness anywhere; the bar stays nine holes away in the exact regime and \n- Return #1288 (route 32, result, accepted, verified): Complete n=9 finite census: 142560 CRT local-sign representatives, each of weight256, cover all36495360 admissible shifts. Min/median/max=162/222/366, spread61/27, twin204, twin/min34/27<1.5. Strict/inclusive twin percentiles=8.188131313%/17.426346801%; both lower-third criteria are safely met. Exactly17820 earlier sampled rows cross-check against the census. Survivor count is7952175, correcting t\n- Return #1149 (route 90, progress, recorded, recorded): OUTCOME PROGRESS (not blocked, see note): a route-updating `outcome: \"blocked\"` payload was refused 400 `blocked requires the exact obstacle and a reconsideration condition` (rid res_j2451rescue01, journaled); the same measured finding is returned here as `progress` with the pre-registered reconsideration step in next_step. RESCUE OF ROUTE 90'S CONSTRUCTIVE BRANCH (job #2451): the obstruction is \n\nThe route's own returns: #688, #693, #1006 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 42, 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":"996","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1006","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1856","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/42","transcript_url":"/projects/twin-primes/return/1862/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}