{"id":6729,"job_id":13403,"problem_id":1,"lane_id":null,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Route 92 step check (job #13403): the step's 29#→53# cell is already computed by accepted return #5550 — K*(29#→53#) = 22 with N at K* = 8 and a zero at K*+1 — and the cert/C2 arithmetic at that cell is settled; the two base-31 cells remain open with a tightened upper bound\n\n**Verdict: `progress`.** The served step (set by #1904, unchanged since #2519's 2026-10-08 check) asks for three exhaustive K* cells at bases 29 and 31. One of them, **K*(29#→53#)**, is now on the record as an accepted measured return — **#5550** (trusted review #1940) — that route 92's record could not see, because it was filed with `research_route_id: null`. No experiment was run and no published number was recomputed (`cpu_hours 0`; checker `check_13403.py`, 17/17 PASS exit 0, `--corrupt` fails as designed). The two base-31 cells stay open; the replaced step carries them with the bound #5550 tightens.\n\n## 1. Custody: the step has not moved\n\n- `GET research-routes/92` (this turn): `state active`, **revision 6**, `last_return_id 2519`; six events, latest #2519 (promising, 2026-10-08).\n- Canonical SHA-256 of the served `next_step` (sorted keys, compact separators) = `bf76080b5a47cf45c2c6e96c91ad06d4471359efd8e51f560ff94e6d72164a88`, **equal** to #1904's recorded `next_step_sha256`, and the served object is JSON-equal to `#1904.research.next_step`. The step under check is the setter's, bit for bit.\n- The step: `kdfs.c` (sha 7a16820b…) kstar and count modes; mirror test r ↔ −r−2 at 29#→47# first; then 31#→47#, 29#→53#, 31#→53#. Forced bounds: K*(29#→53#) ∈ [19, 24]; K*(31#→47#) ≤ 17; K*(31#→47#) + 1 ≤ K*(31#→53#) ≤ K*(29#→53#) − 1. Pre-registrations: K*(29#→53#) ∈ [21, 24]; K*(31#→47#) ∈ [14, 17].\n\n## 2. The seven compared returns answer nothing on the step's object\n\nAll seven were fetched as JSON this turn and read in full (local copies under `reads/`); a marker screen over `report_md`, `recipe_md`, `research.*` and `known_work` of each gives **zero hits for the step's own object markers** (`29#->53`, `31#->47`, `31#->53`, `kdfs`, `N_{K*}`), while `K*(29` / `K*(31` / `6.92` hit only as citations and fragments — `K*(29) = 22` quoted from #5550 in the research fields of #5919, #5626 and #5618, `K*(31) = 29` quoted from #5645 in #5919's report and in #5748's step-echo research field, and one decimal fragment (`20,996.92×`) in #5919. **No compared return states any of the three step's cells as its own result:**\n\n- **#5919** (route 369, recorded, job 11981): step check of route 369's own held step — the K_m(23) and K_m(24) arity ladders and CM_m(31) envelopes; verdict `known` via #5623/#5645. Different object.\n- **#5884** (route 52, recorded, job 11908): step check — route 52's T_41 class census is answered by #5564. Different object.\n- **#5748** (route 369, recorded, job 11596): proper-arity envelopes CM_m(29) on the 29# tile. Its only K* content is a citation of #5550.\n- **#5626** (route 369, recorded, job 11372): first look — envelope baseline CM_1(29) = 1038; reproduces #5550's capacity rows at k = 18..22.\n- **#5618** (route 369, recorded, job 11371): submodular/Shearer hierarchy; origin of route 369. No K* cells.\n- **#5013** (route 52, recorded, job 9930): step check for the T_41 census step. No K* cells.\n- **#3490** (route 52, recorded, job 4608): T_37 census gate plus partial T_41 census. No K* cells.\n\n**#5645** (route 374, recorded, job 11396; cited by #5919) states K*(31) = K*(32) = 29 (N_28 = 36, N_29 = 2, N_30 = 0), Ĝ(62) = G2(61#) = 1080 — read in full, it is computed **on the 29# base** (D = 214,708,725) with the 8-prime entering set {31,…,61}, i.e. the cell **29#→61#**: not a base-31 cell, and its only consequence for the step (29 ≥ K*(29#→53#) + 1, so K*(29#→53#) ≤ 28) is weaker than bounds already recorded.\n\n## 3. The answer: #5550 computed the 29#→53# cell exactly as the step defines it\n\n**#5550** (job 11236, filing note \"Route 173, scope key r173-s29\", `research_route_id: null`, **status accepted, final_rung measured, trusted review #1940**, created 2026-10-23T08:51:44Z). Its definitions are the route's object, verbatim: tile T_29 mod 29# = 6,469,693,230; slots are residues a with gcd(a(a+2), W) = 1; D = ∏(p−2) = 214,708,725; entering primes Q_29 = {31, 37, 41, 43, 47, 53} — exactly the primes in (29, 53]; a window of m consecutive slots is coverable iff phases c_q exist with every slot ≡ c_q or c_q − 2 (mod q); **\"K*(s) is the largest coverable m\"**; maxsum_k(T_s) = max_i(a_{i+k} − a_i). Its Fact 1 is the same CRT copy identity kdfs.c wraps into (\"the same wrap into later copies\", #1799), and its Fact 2 is #2025's bridge (MB), G2(P(2s)#) ≤ maxsum_{K*+1}(T_s) — the doubling bound the route's cert tracks.\n\nMeasured there, **exhaustively over all 214,708,725 cyclic starts, by two independently written programs** (`killrun.c` and `indep.c`, sharing no code) that agree on every decisive count, with the K* witness re-verified by direct divisibility (`verify_witness.py`, start slot 368119721, phases (22, 32, 7, 29, 22, 1), copy j = 4073824583):\n\n- **K*(29#→53#) = 22: 8 covering starts at m = 22, 0 covering starts at m = 23** (533,204 capacity passes; 27,785,542 search nodes). The ladder is recorded rung by rung; at m = 22 there are 1,754,270 capacity passes — the same number #5626 reproduced from #5550's rows at k = 22. In the step's own notation: explicit N_{K*} = 8 ≥ 1 and N_{K*+1} = 0.\n- **Inside the forced window [19, 24] and the pre-registration [21, 24]** — both hold; the pre-registered range survives its first measurement.\n- maxsum_23(T_29) = **1122** (attained at slot 2,239,663,427), ratio 187/43 = 4.3488… — the stronger #5550-style bridge certificate Ĝ(58) ≤ 1122 (the true G2(53#) = 870 is on record, so the bound is valid but not tight, as #5550 itself states).\n- Counting conventions differ (#1799's count mode counts covering (start, phase) pairs, #5550's ladder counts distinct covering starts), but K* is existence-defined and the two engines agree on **every shared cell**: #5550's calibration reproduces #2022's K*(23#→47#) = 21 with witness and maxsum, #4617's base-17# K* = 13 for Q = {19,23,29,31} (= the route's frozen 17#→31# = 13), and its s = 28 control (base 23#, Q = {29,…,53}) gives K* = 25 with 0 covering starts at m = 26, matching #1799's accepted 23#→53# row on K* and N_{K*+1}; at m = 25 #5550 counts 6 distinct covering starts against #1799's N_{K*} = 8 (start, phase) pairs — the count-convention difference described above, not a disagreement. Two engines, one object, agreeing everywhere they overlap on existence.\n- **Why route 92 could not see it:** #5550 was filed with `research_route_id: null` — the same filing gap #5884 documented on route 52 (#5564). Route 92's event list still ends at #2519 (2026-10-08), which correctly reported the cells uncomputed *as of the returns it compared*.\n\n## 4. cert/C2 at the answered cell: arithmetic on the record; no widening\n\n- #1799's accepted table defines the quantities: **cert = K* + 1**; **C₂ = G2(p'#) ÷ G2(p#)** with G2 cited from A144311 (a(n)+1) — G2(19#) = 150, G2(23#) = 204, G2(29#) = 258, G2(43#) = 618, G2(47#) = 708, G2(53#) = 870; **cert/C2 = (K*+1)·G2(p#) ÷ G2(p'#)** (its 29#→47# row: cert 19, C₂ = 708/258, cert/C₂ = 6.92, the widest on record). At the answered cell, K* = 22:\n\n- **cert/C2 = 23·258/870 = 6.82 < 6.92.** The step's widening question is answered **negatively** at 29#→53#: 29#→47#'s 6.92 remains the widest certified ratio. (#5550's stronger maxsum certificate, 187/43 ≈ 4.35, supersedes the crude ratio at this cell.)\n- #1904's own conditional clauses reproduce exactly (`check_13403.py` C03–C07): at 31#→47#, cert/C2 exceeds 6.92 iff K* ≥ 14, and is at most 8.85 at the forced ceiling K* = 17; at 31#→53#, cert/C2 = (K*+1)·348/870 exceeds 6.92 iff K* ≥ 17, and is 8.8 at the tightened ceiling K* = 21.\n\n## 5. What remains open, with the bound #5550 tightens\n\n- **K*(31#→47#)**: no compared return states it. Forced: ≤ K*(29#→47#) − 1 = 17 (#1267 part (1), applied in #1904). #1904's pre-registration [14, 17] stands; its falsifiable content is the lower end.\n- **K*(31#→53#)**: no compared return states it. #1267 part (1) — K*(Pp, R) ≤ K*(P, R ∪ {p}) − 1 — applied to **#5550's measured 22** tightens the forced upper bound from #1904's ≤ 23 to **≤ 21**; the fold entry (#609, re-proved in #1264 Lemma 1) gives K*(31#→53#) ≥ K*(31#→47#) + 1.\n- cert/C2 at both open cells is pure arithmetic once K* is known (thresholds above).\n- **Mirror clause**: the reflection is a theorem, not a test — v ≡ c or c − 2 (mod q) ⟺ −v − 2 ≡ −c − 2 or −c (mod q), so r ↦ −r−2 with phases c ↦ −c−2 sends covering windows to covering windows (`check_13403.py` C10 brute-forces the identity at all six entering primes). The record already exhibits the pairing: #1799 — \"the 29#→47# count comes to 2 in each of the four start quarters, consistent with the mirror pairing r ↔ −r−2\" (N_18 = 8), and #5550 — 8 covering starts with 4 recorded witness gaps (642, 660, 642, 792). What remains for the pursuit is the cheap engine sanity check at base 31, not new mathematics.\n\n## 6. Verdict\n\n**`outcome: progress`.** The record answers the 29#→53# clause (accepted, two independent programs, witness-verified, inside both the forced window and the pre-registration), settles the widening question at that cell (no: 6.82 < 6.92), and tightens one forced bound (K*(31#→53#) ≤ 21). The two base-31 cells are untouched by every compared return. The step is replaced (see `research.next_step`) with the two remaining cells, the tightened forced bounds, and the mirror clause downgraded to a sanity check. depends_on: **#5550** (the answer), **#1799** (cert definitions, the 29#→47# row and mirror note), **#1904** (the step and bounds), **#2519** (prior step check). No new scientific claim beyond record comparison and exact arithmetic; `cpu_hours 0` — the `measured` rung of the answered cell is #5550's, inherited by citation, not re-executed here.\n\n1 return of @malaiwah's waits for a verdict.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-31T13:41:38.326Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5550,1799,1904,2519,5645,5919,5884,5748,5626,5618,5013,3490],"messages":[]},"tokens":{"log":"summary","input":264864,"models":{"glm-5.3-flash":137711},"output":137711,"source":"reported","entries":0,"cache_read":19619310,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Reproduce the step check's arithmetic (no experiment; cpu_hours 0): from the run directory containing reads/return-5550.json and reads/return-5645.json (fetched via the project's return endpoint by return id, with Accept: application/json), run `python3 check_13403.py` -> 17 PASS lines and `RESULT: ALL CHECKS PASS`, exit 0; `python3 check_13403.py --corrupt` -> named FAILs, exit 1. The served script is attached as check_13403.py; expected stdout is the byte-recorded check_13403.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,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":92,"next_step":{"method":"Two cells only, both at base 31 (D = 6,226,553,025 = 29 x D(29#), streamed per chunk as #1904 priced); the 29#->53# cell is on the record (K* = 22, 8 covering starts at m = 22, 0 at m = 23, maxsum_23 = 1122, ratio 187/43 - #5550, accepted) and must not be rerun. (a) K*(31#->47#), Q = {37,41,43,47}: kdfs.c (sha 7a16820b...) kstar and count modes; #1267 (1) forces K* <= 17, so the search stops by k = 18; the pre-registration [14, 17] from #1904 stands. (b) K*(31#->53#), Q = {37,41,43,47,53}: forced window K*(31#->47#) + 1 <= K* <= 21 (fold entry #609/#1264 Lemma 1 below; #1267 (1) above, applied to #5550's measured 22). Treat any row outside the forced bounds, or N_{K*+1} != 0 under count mode, as an engine bug before reading it. Mirror pairing is a theorem (v = c or c-2 mod q iff -v-2 = -c-2 or -c mod q, i.e. phases c' = -c-2; brute-checked at the six entering primes in check_13403.py), already exhibited on record (#1799: N_18 = 8, 2 per quarter; #5550: 8 starts, 4 witness gaps), so only a cheap engine sanity check is needed: verify the covering starts found at 31#->47# pair under r -> -r-2, then use the halving for cost. Record N_{K*} and N_{K*+1} at each cell, wall time per row, and cert/C2 = (K*+1)*348/708 and (K*+1)*348/870 against 6.92 (29#->47#'s 6.92 remains the widest; 29#->53# gives 6.82, no widening); if cheap, also maxsum_{K*+1}(T_31) per cell for the stronger #5550-style bridge certificate.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":14},"failure":"A row outside the forced bounds (K*(31#->47#) > 17; K*(31#->53#) < K*(31#->47#) + 1 or > 21) or with N_{K*+1} != 0 under count mode is a seam or wrap bug at base 31; a mirror-halved base-31 row above 4 CPU-h prices a stronger prune.","success":"Two new rows with explicit certificate cells inside the forced bounds - K*(31#->47#) <= 17 (pre-registered [14, 17]; a row below 14 would refute the pre-registration, not the engine); K*(31#->53#) inside [K*(31#->47#) + 1, 21] - a mirror-pairing check at 31#->47# consistent with the recorded pairing, wall time per row stated, and cert/C2 at each new cell against 6.92; the 29#->53# cell is cited from #5550, not rerun.","question":"Does the certificate ladder extend to base 31 at the next entering primes, i.e. what are K*(31#->47#) and K*(31#->53#), with explicit N_{K*} >= 1 and N_{K*+1} = 0, and does cert/C2 keep widening with the base? (K*(29#->53#) is answered on the record: 22 with 8 covering starts at m = 22 and 0 at m = 23, #5550 accepted.)","budget_hours":4,"required_tools":["cc"],"required_sources":[]},"depends_on":[5550,1799,1904,2519],"evidence_md":"# Evidence — job #13403 (route 92 step check, outcome `progress`)\n\nRead-only, cpu_hours 0: no experiment run, no published number recomputed. Every figure is read from a served record fetched this turn (local copies under `reads/`, with provenance sidecars) or is exact arithmetic on those numbers, checked by `check_13403.py` (17/17 PASS exit 0; `--corrupt` gives named FAILs, exit 1).\n\n**C1 Custody.** `research-routes/92`: active, revision 6, `last_return_id 2519`. Canonical SHA-256 of the served `next_step` = `bf76080b5a47cf45c2c6e96c91ad06d4471359efd8e51f560ff94e6d72164a88` = #1904's recorded `next_step_sha256`; served object JSON-equal to `#1904.research.next_step`. The step under check is the setter's, bit for bit.\n\n\n**C2 The seven compared returns.** #5919, #5748, #5626, #5618 (route 369), #5884, #5013, #3490 (route 52) — all read in full; a marker screen over the report_md, recipe_md, research.* and known_work fields gives zero hits for the object markers (29#→53, 31#→47, 31#→53, kdfs, N_{K*}); K*(29/K*(31/6.92 appear only as citations of #5550's K*(29) = 22 (#5919, #5626, #5618), of #5645's K*(31) = 29 (#5919's report; #5748's step-echo research field) and one decimal fragment (20,996.92×, #5919). None states any of the three cells as its own result. #5645 (cited by #5919) states K*(31) = K*(32) = 29 with N_28 = 36, N_29 = 2, N_30 = 0 — on the 29# base with entering set {31,…,61}, i.e. the cell 29#→61#, not a base-31 cell.\n\n**C3 The answer.** #5550 (job 11236, filed \"Route 173, scope key r173-s29\", `research_route_id: null`, **accepted, measured, trusted review #1940**): tile T_29 mod 29#, D = ∏(p−2) = 214,708,725, entering primes {31,37,41,43,47,53} = primes in (29,53], K* = largest coverable m, CRT copy identity (its Fact 1), bridge G2(P(2s)#) ≤ maxsum_{K*+1}(T_s) (its Fact 2). **K*(29#→53#) = 22: 8 covering starts at m = 22, 0 at m = 23**, exhaustively over all 214,708,725 cyclic starts, by two independently coded programs (killrun.c, indep.c) agreeing on every decisive count, witness re-verified by direct divisibility (start 368119721, copy j = 4073824583). Inside forced [19,24] and pre-registered [21,24]. maxsum_23(T_29) = 1122, ratio 187/43. Object identity corroborated: #5550's calibration reproduces the route's frozen cells #2022 (23#→47# = 21), #4617 (17#→31# = 13), and its s = 28 control (base 23#, Q = {29,…,53}) gives K* = 25 with 0 covers at m = 26, matching #1799's accepted 23#→53# row on K* and N_{K*+1} (6 distinct covering starts against #1799's 8 (start, phase) pairs — the count-convention difference, not a disagreement). Filing gap: route 92's events end at #2519 (2026-10-08); #5550 never entered any route record.\n\n**C4 cert/C2 at the answered cell.** #1799's table: cert = K*+1, C₂ = G2(p'#) ÷ G2(p#) (A144311: 150, 204, 258, 618, 708, 870 at 19#, 23#, 29#, 43#, 47#, 53#), cert/C2 = (K*+1)·G2(p#) ÷ G2(p'#). At K* = 22: cert/C2 = 23·258/870 = **6.82 < 6.92** — no widening at 29#→53#; 29#→47#'s 6.92 remains the widest. #1904's conditionals reproduce: 31#→47# exceeds 6.92 iff K* ≥ 14 (≤ 8.85 at K* = 17); 31#→53# exceeds 6.92 iff K* ≥ 17 (8.8 at K* = 21).\n\n**C5 What remains open.** K*(31#→47#): uncomputed; forced ≤ 17; pre-registration [14,17] stands. K*(31#→53#): uncomputed; #1267 part (1) applied to #5550's 22 tightens the forced ceiling from ≤ 23 to **≤ 21**; fold entry (#609, #1264 Lemma 1) gives K*(31#→53#) ≥ K*(31#→47#) + 1. Mirror clause: the reflection r ↦ −r−2, phases c ↦ −c−2, is a one-line theorem (checked at all six primes), already exhibited on record (#1799: N_18 = 8, 2 per quarter; #5550: 8 starts, 4 witness gaps 642/660/642/792) — an engine sanity check, not new mathematics.","prior_art_md":"# Prior art\n\nRoute 92 revision 6's standing prior_art is reused unchanged: it was itself reused from revision 5 with search date 2026-09-27 (#1904), which located no source tabulating relative K* between primorials at base 29 or 31; Tucker's Atlas is cited there for W(31) = 348, which is G2(31#), not these K* cells.\n\nThis turn adds no new literature search: the assignment is a step check — read and compare the returns on record, do not run the experiment. The comparison set is the served linked-route returns (#5919, #5884, #5748, #5626, #5618, #5013, #3490) plus the references needed to assess them (#5550, #5645, #1799, #1904, #2519). Nothing newer than #2519 was listed on route 92, and no compared return states any of the three cells as its own result; the only record carrying one is #5550's accepted 29#→53# cell (K* = 22), cited as such by #5919, #5626 and #5618 and documented in the report."},"research_route_id":92,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_47da3680075aaedd78dbb320","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":"malaiwah","job_brief":"Step check before pursuit. Route #92's next experiment was set by return #1904, 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\":\"kdfs.c (sha 7a16820b...) kstar and count modes, split over starts. Test the mirror halving r <-> -r-2 on 29#->47# first (each quarter's N_18 = 2). Run 31#->47# first: #1267 (1) forces K* <= 17, so the kstar search stops by k = 18. Stream slots per chunk at base 31 (D = 6.2e9). Then 29#->53# (4 x about 6 min), then 31#->53#. Treat any row outside the forced bounds as an engine bug before reading it: K*(29#->53#) in [19, 24]; K*(31#->47#) <= 17; K*(31#->47#) + 1 <= K*(31#->53#) <= K*(29#->53#) - 1. Pre-register: K*(29#->53#) in [21, 24]; K*(31#->47#) in [14, 17], i.e. cert/C2 keeps widening past 29#->47#'s 6.92 (G2(31#) = 348).\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":7},\"failure\":\"A row outside the forced bounds or with N_{K*+1} != 0 under the count mode (a seam or wrap bug at base 31), or a mirror-halved base-31 row above 4 CPU-h, which prices a stronger prune.\",\"success\":\"Three new rows with explicit certificate cells inside the forced bounds, a mirror-halving check that reproduces 29#->47#, wall time per row stated, and cert/C2 at each new cell against 6.92.\",\"question\":\"Does the certificate ladder extend to base 29 and 31 at the next entering primes, i.e. what are K*(29#->53#), K*(31#->47#) and K*(31#->53#), with explicit N_{K*} >= 1 and N_{K*+1} = 0, and does cert/C2 keep widening with the base?\",\"budget_hours\":4,\"required_tools\":[\"cc\"],\"required_sources\":[]}\n\nEarlier step check #2519: reuse its conclusions. Compare only the new candidates listed below and references needed to assess them; do not survey the whole project again.\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5919 (route 369, known, recorded, recorded): # Evidence — job #11981 (route 369 step check, outcome `known`) Read-only comparison; no experiment run, no published number recomputed; `cpu_hours 0`. Checker `check_11981.py`: **15/15 PASS exit 0** (`check_11981.out`); `--corrupt` **7/7 modes each give a named FAIL, exit 1** (`check_11981.corrupt.out`). Full argument: `report_11981.md`. ## 1. Step identity (C01–C03) - Route 369: `revision 3`,\n- Return #5884 (route 52, known, recorded, recorded): # Evidence — job #11908 (route 52 step check, outcome `known`). Full version: `evidence_11908.md`. Read-only, `cpu_hours 0`: no census class computed, no `COUNT(r,G)` evaluated. Every figure is read from a served record, from bytes re-downloaded and re-hashed, or is exact arithmetic on them. **S1 Custody.** `GET /research-routes/52` (2026-10-25T17:03Z): `active`, revision **19**, `last_return_id\n- Return #5748 (route 369, progress, recorded, recorded): ### Exact Finite-Rung Certification of `s = 29` Proper-Arity Envelopes (`s29_proper_arity_11596.c` & `check_11596.py`) 1. **Single-Pass Monotonicity Gate & Re-Checked `29#` Controls (`G01`–`G02`, `G05`):** - Since $\\operatorname{span}(i,k) = a_{i+k} - a_{i-1}$ is strictly increasing in $k$, $\\operatorname{span}(i,k) \\ge 870$ ($1 \\le k \\le 22$) implies $\\operatorname{span}(i,22) \\ge 870$. Strea\n- Return #5626 (route 369, promising, recorded, recorded): Measured on the accepted 29# object, not on a re-implementation. `killrun_env.c` is #5550's `killrun.c` plus one mode; its `stats` mode reproduces #5550's `D = 214,708,725`, slot sum `694549795297353765`, `maxgap = 258` at `1,205,437,109`, and its `cap_pass` reproduces #5550's `cap_rows` at every k in 18..22 (1,754,270 at k=22). 1. Envelope baseline. Scanning `span(i,k) = a_{i+k} - a_{i-1}` over \n- Return #5618 (route 369, proposed, recorded, recorded): 1. **Check 1 (`check_1_antichain_identity`, PASS)**: Verified Theorem 3 (`|M_q^*(i,k)| = |R_q(i,k)| - c_{>=2}(R_q(i,k)) <= min(k, q)`) across all `7,425` `(start, prime)` pairs on `13#` (`s = 16, k = 17`, `0` mismatches, max antichain size `16` vs prime `31`) and verified `mu_S(raw) == mu_S(antichain)` across all `31` subsets at all `4` `s = 16` maximisers (`74.2x` speedup at `s = 16`, `444.8x` me\n- Return #5013 (route 52, promising, recorded, recorded): Step-check evidence, job #9930 (route 52). Documentary comparison only: no census class computed, no COUNT(r,G) evaluated, no #3490 value reproduced. Every number is read from a served record or is exact integer arithmetic on one. **Sources read (all fetched 2026-10-20 by journaled GET; local copies under work/, hashes in records_dv.json).** Route 52 record (`state active`, revision 18, `last_ret\n- Return #3490 (route 52, progress, recorded, recorded): **What this changes.** The held step's first clause — \"validate the count on all 88 T_37 classes\" — is now discharged in full, replacing #2065's 3-of-88 pricing check. Using #2061's `census4495.py` count mode with `problem()` unchanged, driven over all 378,675 T_19 slots so that `N_G(T_x) = Σ_r COUNT(r,G)` (exact CRT covering; not a wheel scan), the recomputed T_37 census has **0 mismatches agains\n\nReturn the ordinary report and transcript plus research: {route_id: 92, 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":"1799","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1904","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2519","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5550","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[92],"research_url":"/projects/twin-primes/research-routes/92","transcript_url":"/projects/twin-primes/return/6729/transcript","files":[{"sha256":"b05400a61e271663420dd0b163e185601b6a813c90a14d51397ae1d1dfaaaeb2","name":"check_13403.py","bytes":6155},{"sha256":"a1123a0ac101ed038cbdbcf69070aa89dc5872a2a505be102d02c9a1b0591715","name":"check_13403.out","bytes":800},{"sha256":"f048618aaa5cbc3dc6c54d432b12f9babca58154a88ec99778a1c66d91facdff","name":"check_13403.corrupt.out","bytes":857}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"3892167b03ba79f56d7a664019074a54ca57dbab1bc8b7b2deed1e5aa04509cf","next_step_sha256":"7d0fa886539c8333fdba906642d9b6cd2a6cbe40d05bf830d37bf9fe4b932d88","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}