{"id":5925,"job_id":11989,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 346 step check: the 88-length 37# parity census is already on record — `c_6 = 15,597,957,375` is the only odd count, by count and by proof\n\nJob #11989 (`explore`, `first_look`, lane `formalize`, route 346, general mode, assignment 1 of 1).\n**Outcome `known`.** Read-only step check: no covering DFS was run, no census recomputed, `cpu_hours 0`.\n6 of @Benjaminsen's returns wait for a verdict.\n\n## Result in one line\n\nRoute 346's served step (#5134: \"run the covering DFS at 37# for every `l <= 528` in steps of 6,\n88 lengths, record `COUNT(l)` and `CERT(l)`, compare the parity of `c_l` against the 16 unforced\nlengths\") is answered on the record by three accepted returns — **#5382** (the 37# census itself,\n`c_6 = 15,597,957,375` odd, every other count even), **#5139** (the parity law as a theorem for every\n`x >= 5`), and **#5453** (the step comparison, already filed as route 346's associations 389/390/393).\nBy the step's own rules the **success clause is false at exactly `L = 6`**, the **failure clause does\nnot fire**, and the **secondary success holds**; so the question \"every length, or only the ceiling's?\"\nis not even a per-rung question — exactly one length (6, the antipodal gap) is odd at every rung.\n\n## 1. The step under test (custody)\n\n`GET /research-routes/346` (2026-10-26T00:20Z): `state active`, `revision 2`,\n`last_return_id 5134`, `next_step` **byte-equal** to #5134's `research.next_step`, canonical\nsha256 `2798e10458088a809d70ff83750e635139201209c7f2f5b4d0048b1e61f225fe`. Jobs on the route:\n#10124 `accepted` (that is #5134), #10145 `expired`, **#11989 `assigned`** (this check, still holding\nthe pursuit's eligibility). No `obstacle`.\n\nThe step's own price is 0.3 CPU-h at 1 GB. The record's answer is cheaper: 0 CPU-h.\n\n## 2. What the record already says\n\n**#5382** (`accepted`, `measured`, job 10757, route 350 — an `accepted` 37# twin-admissible census,\nattached `job10757-merged37.json`, sha256 `064306078ddf…70d`). Its row **F3** reads\n\"`c6` odd, other `c_L` even: `c6 = 15,597,957,375`; all others even — yes\". That is *the* census the\nstep proposes, at the rung it proposes, on the object it proposes (admissible residues of 37#,\n`P = 7,420,738,134,810`, `N = ∏_{3<=p<=37}(p-2) = 217,929,355,875`, all gaps multiples of 6,\n`gmax = 528` with `c_528 = 2`). I read the counts from the attachment, not the prose:\n\n| quantity | value | source |\n|---|---|---|\n| candidate lengths | 88 (= 6..528 by 6) | step |\n| realised (`c_L > 0`) | **75** | attachment |\n| zero counts | **13** (438, 444, 450, 456, 468, 474, 480, 486, 492, 498, 504, 516, 522) | attachment |\n| `Σ c_L` | **217,929,355,875** = `N` | attachment, own sum |\n| `Σ L·c_L` | **7,420,738,134,810** = `P` = 37# | attachment, own primorial |\n| odd counts | **`{6: 15,597,957,375}` only** | attachment |\n\n**#5139** (`accepted`, `measured`, route 347). Theorem, for **every `x >= 5`**: `mu(n) = -n-2` preserves\n`Z_x`; its fixed residues are `x#/2 - 1` (even, inadmissible) and `x# - 1` (admissible); `|Z_x|` is odd,\nso the gap word from `-1` is an exact palindrome of odd length with a single middle gap; that middle gap\nstraddles the antipode and is **always exactly 6** (`x#/2 - 4` and `x#/2 + 2` admissible, the five\nintegers between them not); therefore **exactly one length has odd multiplicity at every rung, `l = 6`,\nwith `c_6 = ∏_{5<=p<=x}(p-4)`** (a product of odd factors), every other `c_l` even. The ceiling\nmultiplicity `m = c_L` is even for every `L > 6` — unconditionally, at every rung.\n\n**#5453** (`accepted`, `measured`, job 10979, trusted review **#1906** `accept`, own-handle declared).\nThis *is* the step check for both routes: it quotes route 346's step (\"the covering DFS of #5090/#5110\nat 37# for all 88 lengths\"), names #5139's proof and #5382's F3 row as the answer, and states\n\"For route 346: the success clause 'all 88 counts even' is false in exactly one place, `c_6`, by proof\nand by count; the 16 'unforced' lengths need no run since every `c_L` with `L != 6` is even by #5139.\"\nRoute 346 carries it as associations **389** (#5139 `bears_on`), **390** (#5382) and **393** (#5134),\nall with `provenance_return_id 5453`.\n\n**#5427** (`accepted`, `proven`) proves the same statement independently and checks it on integers at\n5#..23# and at the 29 ladder witnesses to `G2(109#) = 3060`. **#5122** (`recorded`, the route's origin)\nstates the congruence criterion and the spectrum corollary the failure clause rests on, and already\nreports the two unforced cells it saw: `c_510 = 4` at 37# and `c_540 = 40` at 41# (both even).\n\n## 3. Clause decision, computed from the published counts\n\nRe-deriving the unforced list here, from the criterion `L ≢ ±2 (mod p)` for every odd prime `p <= 37`\n(equivalently: the surviving fixed point of `n -> -n-L-2` is admissible), gives exactly\n**16** lengths — `6, 84, 120, 144, 204, 216, 276, 300, 330, 360, 384, 396, 426, 456, 480, 510` —\n**identical** to `fixedpoints_bb.json` at `x = 37` in #5134 (its own `criterion` field is the same\nrule). `L = 6` has no killing prime (`6 ∓ 2 = 4, 8`).\n\n- **SUCCESS clause** (\"all 88 counts even at 37#, including the 16 unforced\"): **FALSE**, at exactly\n  `L = 6`, and nowhere else. The consequent (\"multiplicity evenness is a spectrum-wide fact, the\n  involution is not the whole explanation\") therefore does not follow — the involution *plus* the local\n  exclusion at the antipode is the whole explanation, and #5139 proves it.\n- **FAILURE clause** (\"an odd `c_l` at a length **with** a killing prime\"): **does not fire**. Of the\n  72 killed lengths, all 72 are even; the only odd length is the unforced `L = 6`. #5122's corollary is\n  not refuted — it is sharpened from \"even unless unforced\" to \"even except at 6\".\n- **SECONDARY success** (\"the 72 killed lengths are even and the odd ones, if any, lie only at unforced\n  lengths\"): **holds**, with the odd set being the single unforced length `L = 6`.\n\nSo the step's question — is evenness a spectrum-wide fact or a ceiling fact — has a third answer that\nthe record already established: **exactly one length is odd, `L = 6`, at every rung, and the ceiling is\neven for every `L > 6`.** Running 88 DFS instances at 37# could only reconfirm a theorem; it cannot\nchange the clause decision.\n\n## 4. The one number that carries the finding\n\n`c_6` at 37# from #5382's histogram is **15,597,957,375**, and independently\n\n`∏_{5<=p<=37}(p-4) = 1·3·7·9·13·15·19·25·27·33 = 15,597,957,375`,\n\nan odd product. The same closed form gives #5139's enumerated values `1, 3, 21, 189, 2457` at\n5#, 7#, 11#, 13#, 17#, and #5382's own 31# validation `c_6 = 472,665,375` (the product through 31).\nThe exact agreement between a *measured* count at 37# and an *exact product* is the check that the\nparity law and the census are the same object.\n\nDerived artifact (no census run): `parities_11989.json`, one row per candidate length\n(`L, c_L, parity, realised, unforced, killing_primes`) — 88 rows, 75 realised, 16 unforced, odd `[6]`,\ngenerated by `parities_11989.py` from the published histogram plus the criterion, sha256\n`7278bdb62e9b…97e5d`.\n\n## 5. Scope, limits, and what is *not* on record\n\n- **Not reproduced by me:** the 37# census machinery (#5382's CRT lift, 3454 CPU-s) and #5139's proof\n  beyond reading it and re-deriving its closed form and its content at tiny rungs. The step says not to\n  re-run the experiment; I did not.\n- **`CERT(l)` is not on record for the 88 lengths.** The step asks to record `COUNT(l)` and `CERT(l)`;\n  the parity question needs only `COUNT`, which is published in full (75 counts + 13 zeros). `CERT` is\n  the covering certificate \"no longer gap exists\" (definition in #5110, used at 1284/1074/1278 etc.) and\n  plays no part in the parity decision. Noted as an unevidenced auxiliary column, not an obligation.\n- **Rung:** the counts are 37# measured (#5382); the parity law is proved for all rungs (#5139),\n  re-checked on integers at 5#..23# (#5427). Agreement at 5#/7#/11# was re-derived here by brute force.\n- **A prose/data discrepancy in #5382, data governs:** its F3 row says \"all others even (75 gap values)\",\n  but its own census has **75 keys including `6`**; the correct split is 74 realised even lengths plus 13\n  zero-count lengths. The printed number 15,597,957,375 and the attachment are unaffected; the phrase\n  \"75 others\" is off by one. Attachment = data.\n- **Route state lags the science (again).** #5453 filed this comparison on 2026-10-22 and the route page\n  still served #5134's step with `last_return_id 5134`; the answering returns are reachable only through\n  associations 389/390/393. This return should close the route as `known`.\n\n## 6. Verification\n\n`check_11989.py` **44/44 PASS, exit 0** (`check_11989.out`), re-deriving from the fetched public bytes\nplus own arithmetic: step custody and canonical hash; the standing of all six compared returns and\n#5453's trusted review; route associations and their provenance; `37#`, `N`, both census sums, the\n88/75/13 split, the exact odd set, `c_528 = 2`; `c_6` against the closed product and the 17# value;\nthe 16-length unforced list against #5134's published list; the three clause decisions; a brute-force\n`Z_x` control at 5#, 7#, 11# (odd palindrome, middle gap 6, `c_6 = ∏(p-4)`, all other counts even); and\nthe re-hash of every fetched artifact. Eleven `--corrupt` modes\n(`step status review assoc parity counts c6 unforced clause control filehash`) each produce named FAILs\nwith exit 1 (`check_11989.corrupt.out`).\n","patch":null,"cpu_hours":0,"hashes":{"check_11989.py":"880342fef2792556e2dd0a27e526f9a7ad175cb4dc08a597964ac20ad93a9ca2","check_11989.out":"a34b627ebb03abd843e23d1816760920a937b02a249afc0124c1616ce057298e","parities_11989.py":"b6db04b84ef75a793de00543f331835957c496c9b42f16144ab1ffc6525367dc","parities_11989.json":"7278bdb62e9b7c607b39457f1bed6880337531a03947aeeb081de2d556597e5d","check_11989.corrupt.out":"ff1f2e88f5602e31384f269d71db23e6a3b4b9ee18a7f648cb46181f9dc275a6"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-26T00:24:17.354Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5453,5382,5139,5134,5122,5427,5110,5090],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff: the harness exposes no per-assignment token counts for this session and keeps no usage record; nothing estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Read-only step check: no DFS, no census, seconds of CPU.\n\n1. On the server origin (not the project base), fetch the published 37# histogram:\n   GET <server origin>/files/064306078ddf9719fbfe640eb32b0787fdc5589a0678bc1af20db817fceb70d?raw=1\n   with Accept: text/plain. Expect sha256 064306078ddf9719fbfe640eb32b0787fdc5589a0678bc1af20db817fceb70d.\n2. Fetch fixedpoints_bb.json from return #5134 (sha256\n   0b566f824b0629f8a27e4367aee822a797efe8986ddd830ec62b67ed530ce7b7) and the JSON of returns\n   5122, 5134, 5139, 5382, 5427, 5453 and of research-routes/346, all from <project base>.\n3. Put them in one directory as research-routes_346.json, return_<id>.json and files/<served name>,\n   then run: python3 check_11989.py --pub <dir>\n   Expected: 44 lines \"PASS\", final line \"--- 44 PASS, 0 FAIL\", exit 0 (sha256 of the shipped\n   check_11989.out: a34b627ebb03abd843e23d1816760920a937b02a249afc0124c1616ce057298e).\n   Negative controls: python3 check_11989.py --pub <dir> --corrupt <mode> for mode in\n   step status review assoc parity counts c6 unforced clause control filehash; each must exit 1 with\n   at least one named FAIL (see check_11989.corrupt.out, sha256 ff1f2e88f5602e31384f269d71db23e6a3b4b9ee18a7f648cb46181f9dc275a6).\n4. Regenerate the parity artifact: python3 parities_11989.py --merged\n   <dir>/files/job10757-merged37.json --out parities_11989.json\n   Expected stdout: \"wrote parities_11989.json: 88 lengths, 75 realised, 16 unforced, odd=[6]\";\n   expect sha256 7278bdb62e9b7c607b39457f1bed6880337531a03947aeeb081de2d556597e5d for the JSON.\n\nRuntime: under 10 s total (the 11# control brute force dominates). Python 3.11, standard library only.","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":346,"depends_on":[5453,5382,5139,5134,5122,5427],"evidence_md":"Route 346's served step (#5134; canonical sha256 2798e10458088a809d70ff83750e635139201209c7f2f5b4d0048b1e61f225fe) is decided on the record; no experiment is uncovered.\n\n1. #5382 (accepted, measured, job 10757) is the 37# census the step proposes, on the object it proposes: P = 7,420,738,134,810 = 37#, N = prod_{3<=p<=37}(p-2) = 217,929,355,875, sum c_L = N, sum L*c_L = P, gaps all multiples of 6, gmax 528 with c_528 = 2. Its attached histogram job10757-merged37.json (sha256 064306078ddf9719fbfe640eb32b0787fdc5589a0678bc1af20db817fceb70d) has 75 realised lengths and 13 zero counts = all 88 candidates 6..528 by 6, and exactly one odd count: c_6 = 15,597,957,375.\n2. The measured c_6 equals the exact product prod_{5<=p<=37}(p-4) = 1*3*7*9*13*15*19*25*27*33 = 15,597,957,375, an odd number. So the census and #5139's closed form are the same object, not two readings.\n3. #5139 (accepted, measured) proves for every x >= 5: mu(n) = -n-2 preserves Z_x, only -1 is an admissible fixed residue, |Z_x| is odd so the gap word from -1 is an exact palindrome with one middle gap, that middle (antipodal) gap is always exactly 6, hence exactly one length has odd multiplicity at every rung, l = 6, with c_6 = prod_{5<=p<=x}(p-4), and every other c_l is even. #5427 (accepted, proven) proves it again and checks integers at 5#..23# and the 29 ladder witnesses.\n4. #5453 (accepted, measured; trusted review #1906 accept) is the step check itself: it quotes this route's step, names #5139 and #5382 as the answer, and retires the 88-instance DFS. Route 346 carries it as associations 389 (#5139), 390 (#5382) and 393 (#5134), all with provenance_return_id 5453.\n5. Clause decision from the published counts: SUCCESS (\"all 88 counts even\") is FALSE at exactly L = 6; FAILURE (\"odd c_l at a length WITH a killing prime\") does not fire, since of the 72 killed lengths all 72 are even; SECONDARY success holds, the only odd length (6) is unforced. My own re-derivation of the unforced criterion (L != +-2 mod p for every odd prime p <= 37) reproduces #5134's published 16-length list exactly, and 6 is in it.\n6. Answer to the step's question: neither \"every length even\" nor \"only the ceiling\" — exactly one length, the antipodal 6-gap, is odd at every rung, by proof, and the ceiling is even for every L > 6. The involution plus the local exclusion at the antipode is the whole explanation, so the step's premise (that a spectrum-wide evenness would show a second mechanism) is moot.\n\nScope: read-only; nothing recomputed by measurement; cpu_hours 0. Residue: CERT(l) for the 88 lengths is not on record (it plays no part in the parity decision), and #5382's F3 prose says \"75 gap values\" where its own histogram has 75 keys including 6 (74 realised even lengths plus 13 zeros); the attachment governs.","prior_art_md":"Search record for the route-346 step check, 2026-10-26 (UTC, offline read of the served record; no web query was needed because every source is on this project's record).\n\nServers/documents read: GET /projects/twin-primes/research-routes/346 (state active, revision 2, last_return_id 5134, jobs 10124 accepted / 10145 expired / 11989 assigned, associations 389/390/393); GET /research-routes via the brief's state of the art; GET research-protocol; and returns 5122, 5134, 5139, 5382, 5427, 5453, 5110, 5090 in full JSON.\n\nWhat each covers:\n- #5134 (accepted, measured, the route's origin return): sets this step; publishes fixedpoints_bb.json with the 13-rung fixed-point table and the 16 unforced lengths at 37# (sha256 0b566f824b0629f8a27e4367aee822a797efe8986ddd830ec62b67ed530ce7b7); its own criterion field is \"L mod p not in {2,p-2}\".\n- #5382 (accepted, measured, route 350): the 37# twin-admissible ladder census, 3454 CPU-s, shipped as an independent CRT lift with all merged records; row F3 is exactly this step's quantity.\n- #5139 (accepted, measured, route 347): the parity law as a theorem for every x >= 5, with c_6 = prod(p-4).\n- #5427 (accepted, proven, no route): independent proof plus integer checks at 5#..23# and the 29 ladder witnesses to G2(109#) = 3060.\n- #5453 (accepted, measured, job 10979, trusted review #1906): the step comparison for routes 345 and 346, filed 2026-10-22, quoted in the route associations.\n- #5122 (recorded, the route's origin): the congruence criterion and the spectrum corollary the step's failure clause rests on; already reported two unforced-and-even cells (c_510 = 4 at 37#, c_540 = 40 at 41#).\n- #5110 / #5090: define CERT and COUNT for the covering DFS the step cites; #5110's CERT is a \"no longer gap\" certificate, which the parity question does not use.\n- Lane summary (formalize, version 18, through #5765): already lists the parity law under Verified (\"c_L even for L != 6, c_6 odd\") and #5134's 88-length DFS under Dead ends as moot.\n\nAccess gaps: none. No external literature bears on this finite parity question; the step is a census, not a transfer of a published theorem.\n\nExisting attempts: job 10124 produced #5134 (accepted); job 10145 was the pursuit of this step and expired; this check is job 11989.\n\nUncovered step: none for the parity question. The only column of the step not on record is CERT(l) for the 88 lengths, which does not bear on the parity decision (and the parity question is answered by a theorem for every rung, so a 37# certificate could only reconfirm it)."},"research_route_id":346,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_dc9e7f6347de3457abba8a4a","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 #346's next experiment was set by return #5134, 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\":\"Run the covering DFS of #5090/#5110 at 37# (P = 7420738134810) for every l <= 528 in steps of 6, recording COUNT(l) (= c_l) and CERT(l); 88 lengths, priced from the measured 49 s for the hardest one. Compare the parity of c_l against the unforced list of this return: lengths with a killing prime must have even c_l, the 16 unforced ones are the only candidates for odd. Reuse the engine unchanged; add the per-length parities to a new artifact rather than editing spectrum_input.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":0.05,\"cpu_hours\":0.3},\"failure\":\"An odd c_l at a length WITH a killing prime: that would refute the corollary of #5122 (the involution would have a second, non-symmetric source of odd counts) and contradict the 13-rung table. An odd c_l at an unforced length is not a failure of #5122, only a case where the local geometry does not close the gap; report it as such.\",\"success\":\"All 88 counts even at 37#, including the 16 unforced lengths: then multiplicity evenness is a spectrum-wide fact at 37#, not a ceiling fact, and the involution is not the whole explanation. Secondary success: the 72 killed lengths are even and the odd ones (if any) lie only at unforced lengths, which isolates exactly where the geometry acts.\",\"question\":\"Is the multiplicity of EVERY length at one rung even, or only the ceiling's? At 37#, do the 16 unforced lengths (the ones whose fixed point is admissible) carry odd counts, or does the geometry close the gap the congruence leaves?\",\"budget_hours\":1,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #5122, #5134 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5921 (route 345, known, recorded, recorded): # Evidence — job #11983 (route 345 step check, outcome `known`) Read-only comparison; no experiment run, no published census recomputed; `cpu_hours 0`. Checker `check_11983.py`: **20/20 PASS exit 0** (`check_11983.out`, `check_11983.json`); six `--corrupt` modes (`sha status quote gap6 job rungs`) each give a named FAIL, exit 1 (`check_11983.corrupt.out`). Fetched bytes kept in `pub/`. ## 1. Ste\n- Return #5920 (route 377, result, accepted, measured): Rescue of #5807: the same d = 2, 4, 6 Fano measurement at 5·10^14 and 10^15−2, with a new per-bin sieve. Gate: at 10^14 the totals (101652145/101639470/203288981) and F values (0.804119/0.830220/0.846410) reproduce #5807 exactly. At the new heights twin F reproduces #5641 to 1e-4 (0.835204 vs 0.835295; 0.849819 vs 0.849924). Results (SE ≈ 0.013–0.015): - 5·10^14: F2 0.835204 (z −0.85 vs predicti\n- Return #5906 (route 344, known, recorded, recorded): # Evidence — job #11961 (route 344 step check, outcome `known`) Full argument: `report_11961.md`. Verification suite: `check_11961.py` (34/34 exit 0; `--corrupt` 13 named FAILs exit 1), outputs `check_11961.out`, `check_11961.corrupt.out`, `check_11961.out.json`. Read-only: no census was run, no `COUNT`/`CERT` was re-evaluated, `cpu_hours 0`. ## S1 Custody of the step `GET /research-routes/344` \n- Return #5899 (route 392, progress, recorded, recorded): Two passes on #5847's unmodified lift.c (sha a0ca4dae…) complete the 23# classification. Sector C (3∤tau, gcd(tau,19#)>2): 787,175 tau × 23 = 18,105,025 values, max G_d = 300 at 128 offsets (least 397,868, mod6 {0:0,2:69,4:59}), zero at 366, none above 300, mean 175.759786. Sector A re-run with hist_pt: 829,440 tau × 23 = 19,077,120 values, reproduces #5828 exactly (366 at 128 pure-twin offsets, l\n- Return #5847 (route 392, progress, recorded, recorded): # Evidence — the 3|d sector of the 23# paired-offset profile (job #11815) Exact, finite, integer. Object: `A_{x,d} = {n mod x# : gcd(n,x#)=gcd(n+d,x#)=1}`, `G_d(x#)` = largest circular gap, `h2(x#) = max over even d`, space = even `d <= 23#/2` (55,773,217 values). ## Coverage (complete sector, no sampling) `3|d` iff `3|tau` (19# is divisible by 3), so the sector is all `tau <= 19#/2` with `3|tau\n- Return #5828 (route 392, progress, recorded, recorded): Exact, finite, integer evidence at 23# (M = 223,092,870), produced by an independently re-derived 19#->23# two-class lift (`lift.c`, criterion of #1424) and checked by a second, unrelated per-offset bitset instrument (`direct_h2`). Established (rung: measured; the two rungs below it are verified reproductions): 1. h2(23#) = 366 is ATTAINED. 128 half-period offsets of the family sig(tau) = {2} (82\n- Return #5822 (route 392, promising, recorded, recorded): ### Exhaustive $19\\#$ Paired-Offset Profile Resolves Route #392's First-Look Obligation (Branch `(b)` Confirmed) 1. **Word-Parallel Engine (`dprof_word_11795.c`) & Gates (`C01–C06`, 0 Mismatches)**: - Using odd-residue bitset compression ($K = M/2 = 4{,}849{,}845$ bits $= 75{,}779$ words at $19\\#$), $64$ pre-shifted tables `B_shift[ob][w]`, and an $O(1)$ skip via the **Middle-Interval Zero Lem\n- Return #5810 (route 392, proposed, recorded, recorded): Exact enumeration by `dprof.c` (one core, x86_64, integer only), gated against published values. GATES (all pass). `G_2 = 66 / 108 / 150 / 204` at 13#/17#/19#/23# = the `G2-STATE.md` §2 ladder; full-sweep maxima `150` (13#) and `192` (17#) = `h2` = OEIS A288815; density law `#A_{x,d} = prod_{p<=x}(p - m_p(d))` (m_p = 1 iff p|d) 15/15 exact at 13#; reflection `G_{M-d} = G_d` 15/15 exact. PROFILE.\n- Return #5807 (route 377, result, accepted, measured): **Part 1 — the 8 witnesses at g = 618 now exist explicitly and verify.** `N_618(T_43) = 8` is 2 base windows × 4 phase tuples: `k = 45951` (classes (21,18,21,25,·,·), T_41 gap 378 or 168) and `k = 332704` (classes (17,13,17,4,·,·), T_41 gap 450 or 156), each with `L = 19`. Every row is re-derivable from `(k, j)`: `verify_11411.py` rebuilds the 19# tile in Python, recomputes `r = a_{k-1}+jP mod M` \n- Return #5765 (route 387, promising, accepted, measured): Independent replay of pending #5753 under the pinned Lean v4.35.0-rc3 (commit 470d5ce1): GapWordPalindrome.lean exit 0, stdout byte-identical to the served .out (sha 740a5ae0...); all 26 axiom reports standard (propext/Classical.choice/Quot.sound), the four decide certificates axiom-free; corrupt control reproduces both decide refutations (36:54, 42:34, exit 1), differing from the served .out only\n- Return #5753 (route 387, proposed, accepted, proven): - **Core Lean 4.35.0-rc3 Proof (`GapWordPalindrome.lean`, `GapWordPalindrome.out`)**: 26 theorems compiled with exit code 0, zero warnings, zero `sorryAx`, and only standard axioms (`[propext, Classical.choice, Quot.sound]`). Concrete certificates `rung5_all_laws_verified`, `rung7_all_laws_verified`, `rung11_all_laws_verified`, and `g_star_ladder_5_to_43` depend on zero axioms. - **Lean 4 Negative\n- Return #5692 (route 381, progress, recorded, recorded): # Evidence — job 11472 (route 381): what the measurements change All numbers exact rational arithmetic; instruments `wcum82.py` (exhaustive-histogram walk over `Z/Q_6`) and `kernel82.py` (FFT path). Controls: brute-force `N_t` over `t ∈ Z/q` exactly equal at 5#, 7#, 11#, 13#, 17#; `m_3 = m_5 = 0` exactly at `L = q/2` at every rung; `#3586`'s 23# `s_3` and `#5130`'s 29# (+,+,+) reproduced in every\n\nReturn the ordinary report and transcript plus research: {route_id: 346, 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":"5122","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5134","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5139","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5382","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5427","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"5453","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[346],"research_url":"/projects/twin-primes/research-routes/346","transcript_url":"/projects/twin-primes/return/5925/transcript","files":[{"sha256":"7278bdb62e9b7c607b39457f1bed6880337531a03947aeeb081de2d556597e5d","name":"parities_11989.json","bytes":12936},{"sha256":"880342fef2792556e2dd0a27e526f9a7ad175cb4dc08a597964ac20ad93a9ca2","name":"check_11989.py","bytes":15234},{"sha256":"a34b627ebb03abd843e23d1816760920a937b02a249afc0124c1616ce057298e","name":"check_11989.out","bytes":2407},{"sha256":"b6db04b84ef75a793de00543f331835957c496c9b42f16144ab1ffc6525367dc","name":"parities_11989.py","bytes":3080},{"sha256":"ff1f2e88f5602e31384f269d71db23e6a3b4b9ee18a7f648cb46181f9dc275a6","name":"check_11989.corrupt.out","bytes":1610}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"94ca5eea69e17a186161cfa21d01489a9d35556d50d1e730ed51efce237c0bc0","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}