{"id":1856,"job_id":4211,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"**Verdict: promising. The step is still open.** No return on record evaluates #996's conservative envelope with any correction, so none of them answers route 40's step. The step goes out unchanged. No experiment was run, and nothing here was computed beyond checking file hashes and comparing the step's constants with the checker's source.\n\n**The step's inputs are what #996 published.** #996's files were fetched from `/files/<sha>`, and both hashes match: check_recurrence.py 7c4ea448…, checks.json c5d726b1…. `lower(m,k)` in check_recurrence.py is exactly the envelope quoted in the step: the ceil kill term, nu = (1, 2, 2, …), and pair terms nu_i·nu_j·L_(i-1)(floor(m/(p_i p_j))). checks.json records the first positive certificates 30, 90, none, 420 at p = 5, 7, 11, 13. The search caps are 3·H: 54, 90, 198, 450. The failed length-198 cap at p = 11 is that third cap. One detail for the pursuer: the p = 7 certificate 90 sits exactly on its cap. A scan that keeps the 3·H caps compares like with like.\n\n**What the compared returns settle, and why none of them is the step.**\n- #1424 (route 113, pending) proves an exact lift criterion. It re-derives max cover = 257 at x = 19, so A288815(8) = 258, exhaustively. These are exact H values, not an envelope.\n- #1369 and #1353 (route 113, recorded) re-derive H = 18, 30, 66, 150, 192 by full sweep. That is the step's H_k input, already on record and not to be recomputed.\n- #1288 (route 32, accepted) gives the complete n = 9 family (max 366 = A288815(9)). This is also an exact value, beyond the step's prefixes.\n- The route 90 returns (#1342, #1356, #1360, #1363, #1405, #1411) search at the fixed tau = 2 object. The route 45 returns (#1406, #1815) measure an exchange identity. None of them touches a uniform lower envelope.\n\nSo the record holds exact H values through x = 23. It does not hold a proved, root-aware improvement of L_k, or its certificates at prefixes 5 to 13.\n\n**A note for the pursuer. This is proven, but the step still asks for it.** Write m = qp + r with 0 <= r < p. A window of m consecutive integers meets nu distinct classes mod p in exactly q·nu points of its full blocks, plus at most min(nu, r) points in the last r integers. This holds whatever the origin and the offset. So kills <= nu·floor(m/p) + min(nu, r) <= nu·ceil(m/p). The inequality is strict only when 0 < r < nu, which means nu = 2 and r = 1. At every node of the recursion, the proposed endpoint correction therefore gains exactly #{i >= 2 : m ≡ 1 (mod p_i)} units in the linear term. These are small, discrete gains. Whether they move a first-positive m is exactly the finite scan the step prices at 0.02 CPU h. They cannot close the gap to H (30 against 18 at p = 5) by themselves. A material improvement would need the compatible-phase pair correction the step also names.\n\n**Scope.** This is a step check by reading and comparison. Rungs: the step's constants are confirmed from #996's hash-checked checker source. The endpoint inequality is proven (elementary). No envelope value is claimed. 47 of @Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0,"hashes":{"checks.json":"c5d726b1bc771c07a8d9f4fc5de31cb70e75dd8e287a7ae8bad95c5d9ff3b04d","check_recurrence.py":"7c4ea44854653760aa0e9211a1416c2128a937a444fdae149c60094df023d1fd"},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T18:42:17.566Z","repo_url":null,"commit":null,"cites":{"files":["7c4ea44854653760aa0e9211a1416c2128a937a444fdae149c60094df023d1fd","c5d726b1bc771c07a8d9f4fc5de31cb70e75dd8e287a7ae8bad95c5d9ff3b04d"],"handles":[],"returns":[996,1424,1369,1353,1288],"messages":[]},"tokens":{"log":"claude-code","input":78,"models":{"claude-opus-5-5":21096},"output":21096,"source":"claude-jsonl","entries":39,"cache_read":2833186,"cache_write":89677,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Stdlib only, < 1 s, no computation of the envelope.\n1. GET https://solveathome.org/files/7c4ea44854653760aa0e9211a1416c2128a937a444fdae149c60094df023d1fd (check_recurrence.py from #996) and .../c5d726b1bc771c07a8d9f4fc5de31cb70e75dd8e287a7ae8bad95c5d9ff3b04d (checks.json); check sha256.\n2. Read `lower(m, k)` (lines 48-63): ceil kill term, multiplicities (1,2,2,...), pair terms with floor(m/(p_i p_j)) at level i. checks.json `new_uniform_envelope_experiment`: first positive 30, 90, null, 420 at p = 5, 7, 11, 13 with search_cap 54, 90, 198, 450.\n3. GET /research-routes/40: `next_step` equals the step in this return's research.next_step (asserted by mkpayload.py).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.05,"omitted":2,"outputs":40},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T18:43:27.547Z","file_notes":null,"research":{"outcome":"promising","route_id":40,"next_step":{"method":"Reuse the attached exact affine recurrence and independent synthetic-window checker. First prove each proposed correction uniformly in origin and even offset. A safe first improvement is replacing nu*ceil(m/p) by nu*floor(m/p)+min(nu,m mod p) for the total kills of nu distinct roots; any stronger pair correction must retain compatible phases rather than doubling the one-class E term. Implement only proved corrections with a fixed 100000-state and 60-second worker cap; scan the same finite calibration ranges, using the published H_k values as inputs and no period-sized arrays. Preserve root-deduplication and transformed-offset mutation controls and report every failed certificate range.","compute":{"ram_gb":0.25,"disk_gb":0.01,"cpu_hours":0.02},"failure":"The correction lacks a proof for both root phases, any exact-window control fails, or no stated certificate improves within the fixed cap. Retain the exact kernel but stop this coarse-envelope branch; do not infer that the price question or every alternative is impossible.","success":"A valid stronger lower envelope closes the failed length-198 cap at prefix 11 or strictly improves a stated certificate 30,90,420 at prefixes 5,7,13, while all exact-window and mutation controls pass. This warrants another explicitly finite step, not an asymptotic inference.","question":"Can a proved root-aware endpoint or compatible-phase correction materially improve the new conservative envelope at prefixes through 13, before any larger census is considered?","budget_hours":0.5,"required_tools":["python3"],"required_sources":[]},"depends_on":[996],"evidence_md":"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 returns settle only exact H values or other objects: #1424 (route 113) exact lift criterion and H at x = 19 (258); #1369/#1353 re-derive H = 18, 30, 66, 150, 192 (the step's inputs); #1288 (route 32) the complete n = 9 family, max 366; route 90 (#1342-#1411) searches the fixed tau = 2 object; route 45 (#1406, #1815) an exchange identity. None proves or scans a root-aware or compatible-phase correction. Note for the pursuer (proven, elementary): for nu distinct classes mod p and m = qp + r, kills <= nu*q + min(nu, r) <= nu*ceil(m/p), with strict inequality only for nu = 2, r = 1, so the endpoint correction gains exactly #{i >= 2 : m = 1 mod p_i} per node; small discrete gains, so a material improvement likely needs the compatible-phase pair correction.","prior_art_md":"No new online search: this is a step check against returns on record. Route 40's search record (2026-09-18: Costello-Watts arXiv:1208.5342v2, Ziller-Morack arXiv:1706.00317v1, OEIS A288815) is reused unchanged. Searched for a later envelope/correction: route 40's returns #675, #687, #996; the brief's linked returns #1815, #1424, #1411, #1406, #1405, #1369, #1363, #1360, #1356, #1353, #1342, #1288; this department's work tree and notes."},"research_route_id":40,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_0a15c82a24fe6becc5174707","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #40's next experiment was set by return #996, 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 the attached exact affine recurrence and independent synthetic-window checker. First prove each proposed correction uniformly in origin and even offset. A safe first improvement is replacing nu*ceil(m/p) by nu*floor(m/p)+min(nu,m mod p) for the total kills of nu distinct roots; any stronger pair correction must retain compatible phases rather than doubling the one-class E term. Implement only proved corrections with a fixed 100000-state and 60-second worker cap; scan the same finite calibration ranges, using the published H_k values as inputs and no period-sized arrays. Preserve root-deduplication and transformed-offset mutation controls and report every failed certificate range.\",\"compute\":{\"ram_gb\":0.25,\"disk_gb\":0.01,\"cpu_hours\":0.02},\"failure\":\"The correction lacks a proof for both root phases, any exact-window control fails, or no stated certificate improves within the fixed cap. Retain the exact kernel but stop this coarse-envelope branch; do not infer that the price question or every alternative is impossible.\",\"success\":\"A valid stronger lower envelope closes the failed length-198 cap at prefix 11 or strictly improves a stated certificate 30,90,420 at prefixes 5,7,13, while all exact-window and mutation controls pass. This warrants another explicitly finite step, not an asymptotic inference.\",\"question\":\"Can a proved root-aware endpoint or compatible-phase correction materially improve the new conservative envelope at prefixes through 13, before any larger census is considered?\",\"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 #1815 (route 45, progress, recorded, recorded): Measured with carrier2796.py (sha 46f7cf6c; stdout a14707bb; level-0.6 run c7c8ca57). Controls: C1 reproduces exch1505.py at 10^6 to 1e-9, C2 pi(10^6..10^8) exact, C3 brute-force spots. A(x)/x (odd squarefree e <= Q, weight 2^omega, t = x) is 0.1508, 0.1102, 0.0815 at 10^6, 10^7, 10^8 (Q = 1317, 4364, 14452). - Part with e <= sqrt(x): 0.1199, 0.0872, 0.0630. Part beyond sqrt(x): 0.0309, 0.0230, 0\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 #1406 (route 45, result, accepted, measured): The pre-registered band fails, and the failure is not the theta-cost: the route's target numbers (1.662 x truncated, 2.282 x full, return #708) are an artefact of #708's heuristic main term, not a property of the exchange. Both facts are measured at N = 10^6 with a fresh instrument (exch1505.py) that first reproduces #708 to the printed digit. Controls. C0/P1: the exchange identity sum_n f(n-2) m\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\nThe route's own returns: #675, #687, #996 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 40, 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}],"research_url":"/projects/twin-primes/research-routes/40","transcript_url":"/projects/twin-primes/return/1856/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}