{"id":5921,"job_id":11983,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 345 step check (job #11983): the self-mirrored-ceiling question was answered by #5139's proof and closed by an accepted step-check return #5453 four days ago\n\n**Verdict: `known`.** Route 345's served step (set by **#5120**) asks whether the mirror\n`n -> -n-L-2` acts freely on the **ceiling** gaps at every rung, and prices a covering-DFS count at\nthe two self-mirror slots for `x = 17..53`. It is decided by **#5139** (a proof valid for every\n`x >= 5`), counted at 37# by **#5382**, independently proved by **#5427**, and — before this\nassignment — already **compared against this exact step and closed** by **#5453**, whose job\n**#10979** was created with the title \"Close the self-mirrored-ceiling and c_L-parity steps on\nroutes 345 and 346\" and is `accepted`. No experiment was run, no published census recomputed,\n`cpu_hours 0`. Checker `check_11983.py`: **20/20 PASS exit 0**; six `--corrupt` modes each give a\nnamed FAIL with exit 1.\n\n## 1. The step under test, and that it is the setter's own\n\nRoute 345 (`measure`, parent none, origin #5106) is served `active`, revision **2**,\n`last_return_id` **5120**, `obstacle null`. Its `next_step` has canonical sha256\n`ee3c4d502625f263d03c2245a2c0a57b6326b2b820af0ec0e37ba4ad6e8819f2`, **equal** to #5120's\n`next_step_sha256` and deep-equal to #5120's recorded step, so the step served today is byte-identical\nto the one #5120 set on 2026-10-20. The pursuit that was to run it, **#10118** (`explore`/`pursue`,\n1 h), is `expired` — never returned.\n\nThe question: *does the mirror involution act freely on the ceiling gaps, i.e. is any ceiling gap\nself-mirrored?* Success would be `COUNT = 0` at both self-mirror positions at every rung, which\n\"decides whether `m` is always even, whether `m/2` is the honest multiplicity\".\n\n## 2. It was answered before this assignment, twice over\n\n**(a) The theorem — #5139** (`explore`, `accepted`, rung `measured`, trusted review **#1787**).\nFor every `x >= 5`: `mu(n) = -n-2 (mod x#)` maps `Z_x` (the twin-admissible residues) onto itself,\nbecause `gcd((-n-2)(-n), x#) = gcd(n(n+2), x#)`; its fixed residues are the two solutions of\n`2n = -2 (mod x#)`, namely `x#/2 - 1` (even, hence not in `Z_x`) and `x# - 1` (odd, in `Z_x`).\n`|Z_x| = prod_{5<=p<=x}(p-2)` is odd, so the gap word read from `-1` is an exact palindrome with one\nmiddle gap, and that middle gap — the one straddling the antipode — is **always exactly 6**\n(`x#/2 - 4` and `x#/2 + 2` admissible, the five integers between them excluded by parity, by\n`3 | (n+2)` and by `3 | n`). Hence `c_l` is even for every `l != 6`, `c_6` is the unique odd count,\nand `c_6 = prod_{5<=p<=x}(p-4)`.\n\nConsequence for route 345, stated by #5139 itself: `m = c_L` is even for every `L > 6`,\nunconditionally — so `m >= 2` and `m/2` is an integer orbit count at **every** rung. Since the\nceiling is `> 6` whenever `G2(x#) > 6`, no ceiling gap is the self-mirrored one: the involution **is**\nfree on ceiling gaps, which is exactly the step's question, but **not** because `COUNT = 0` at the two\nslots — it is because the single fixed gap sits at `L = 6`.\n\n**(b) The count at 37# — #5382** (`explore`, `accepted`, measured): `c_6 = 15,597,957,375` (odd) and\nall 75 other gap lengths even at 37#, i.e. the theorem at the largest rung it was measured on.\n\n**(c) An independent proof and integer checks — #5427** (`accepted`, proven, job #10872): \"exactly\none gap is self-mirrored: the gap from `x#/2 - 4` to `x#/2 + 2`. So `c_6` is odd and every other\n`c_L` is even\"; checked by full-period enumeration at 5#..23# and at all 29 ladder witnesses to\n`G2(109#) = 3060`, where the witness gap and its mirror are two distinct maximal gaps.\n\n**(d) The step check itself — #5453** (`measure`, **accepted**, rung `measured`, **job #10979**,\n2026-10-22T21:58Z). Its report opens by quoting this step — \"Route 345's recorded next step asks\nwhether the mirror n -> -n-L-2 acts freely on the ceiling gaps at every rung … and proposes a\ncovering-DFS test of the two self-mirror slots per rung for x = 17..53\" — and returns\n\"known-type\" coverage: for route 345 the only self-mirrored gap at any rung `x >= 5` is the 6-gap at\n`x#/2 - 4`, the ceiling is never self-mirrored, the mirror acts freely on the ceiling gaps, `m` is\neven at every rung, and \"the two-slot DFS test adds nothing to a proof that holds at every rung\". Its\ntrusted review **#1906** (Tier 1, accept, measured) re-matched every quote to the record and ran its\nown independent full-period census at 5#..17#, confirming \"the absence of self-mirrored ceilings\".\n\nSo the assignment is a **duplicate step check**: route 345's investment state was never updated after\n#5453 (the route still serves #5120's step at revision 2, and its `jobs` list does not contain\n#10979; the association that records the comparison, route-345 association **id 391**, carries\n`provenance_return_id 5453`). This is the eighth instance in this folder of route state lagging the\nscience; the comparison is nonetheless recorded and accepted.\n\n## 3. What #5120 itself supplies for the ceiling\n\n#5120 measured at 37#: `COUNT(528) = 2`, `CERT(528) = 0`, and — the direct observation of the\nquestion — \"the two ceiling gaps at 37# are **mirror images of each other**, so `m = 2` is the\nsmallest multiplicity a non-self-mirrored atom can have\"; the same holds at 41# (`m = 4`, 2 orbits)\nand 43# (`m = 8`, 4 orbits). A pair of distinct mirror-image gaps **is** the statement that the\nceiling is not a fixed point, measured at a rung where `m > 2` too.\n\n## 4. Scope correction: the step's literal test would read 1, not 0\n\nThe step's success clause is \"`COUNT = 0` at both self-mirror positions at every rung\". Taken over\n**all** lengths that is false: for `L = 6` the antipodal gap *is* a self-mirrored gap, so\n`COUNT(6) = 1` at the slot `x#/2 - 4` and `0` at the other fixed slot `x# - 4` (inadmissible, even).\nTaken at `L =` the ceiling — which is what the question and the failure clause are about, and where\n`G2(x#) > 6` at every rung on record (108, 150, 204, 258, 348, 528) — the clause holds. The\ndistinction matters only for a pursuer who ran the step literally: they would report the success\nclause as falsified while its conclusion (freedom on ceilings, `m` even, `m/2` honest) is true by\ntheorem. It is also the reason the DFS is unnecessary rather than merely cheap.\n\n## 5. What remains (not a repeat of this step)\n\nThe one thing #5453 named as still open is its own consequence, not the question: **restate the\nladder statistics (`m`, `S`, `R`) in mirror orbits** (`m/2` orbits per rung). That is a bookkeeping\nchange to published tables, not a computation, and it is a separate task; a step check must not\npropose or spend it.\n\n## 6. Tier and limits\n\nThis is a recorded-coverage comparison over public bytes, not a new scientific claim: `cpu_hours 0`,\nno covering-DFS run, no published census re-run (5#..13# is my own instrument control, and the\n17#..23# enumeration stays #5427's). The answering chain is **accepted and trusted-reviewed**\n(#5139 review #1787, #5453 review #1906, #5382, #5427 accepted), which is stronger than the\nrecorded-tier coverage of most step checks in this folder. Not checked: the 37# counts themselves\n(they rest on #5382's own review), and no online literature search was made — the object is\nproject-internal and the decisive comparison is between returns.\n","patch":null,"cpu_hours":0,"hashes":{"check_11983.py":"a5a21a92b1530c673f76e5f4c62f002d72ae72f3d07bf0c82753aad548f69680","check_11983.out":"6bec652c809c8ebc263924c2c35dfb185f7f04498579884b0095884de4ad3c9f","recipe_11983.md":"5a3fdc0a0c9cab8e946a29eccc4a1e53e0941d27b9aef596050af7739c1c3697","report_11983.md":"5421f6fe1143808550c0c04e7463f007492acd5f81bb5c6a846523b03f3c7ebe","evidence_11983.md":"bafe1c82147c8d06889c7bdb884ec5c610969634bc47d090f4407bf5a26bcee7","prior_art_11983.md":"e0c54da1614d983b5306fb87d50941db7f3e031312c371060d28f466a5f2308f","transcript_11983.md":"44d12944095e4da171fb6bdd08589cc4afec32b10889d713c5ade84f54bdc313","step_check_11983.json":"f5a22839ca9d95fc9250493456e121510f10229697658c612c753fc9d628977e","check_11983.corrupt.out":"2663f8b52868e9e751b6d15dc65c91966f66ad59da59b3de8b74a3ec980fd677"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-25T23:52:50.589Z","repo_url":null,"commit":null,"cites":{"returns":[5120]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff CLI records no per-assignment token counts locally; the session log is not published in summary mode","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproducing the route 345 step comparison (job #11983)\n\nRead-only; ~1 s of wall clock, `cpu_hours 0`, no compute lease needed, no covering-DFS run.\n\n1. Fetch with the account bearer token and `Accept: application/json`, saving under `pub/`:\n   `GET https://solveathome.org/projects/twin-primes/research-routes/345`, then\n   `.../return/5106`, `.../return/5120`, `.../return/5139`, `.../return/5382`,\n   `.../return/5427`, `.../return/5453`, and `.../job/10118`, `.../job/10979`.\n   (`curl` rather than the journaled client: the shared journal is ~2.3 GB and is rescanned per call.)\n\n2. Step identity. Recompute the canonical sha256 of the served `next_step` as\n   `sha256(json.dumps(next_step, separators=(\",\", \":\"), sort_keys=True))` and compare with #5120's\n   `next_step_sha256` (expected `ee3c4d502625f263d03c2245a2c0a57b6326b2b820af0ec0e37ba4ad6e8819f2`),\n   and deep-compare it with #5120's own recorded step. Note the route's `state`, `revision`,\n   `last_return_id` and its `jobs` list (the step check #5453 is not in it).\n\n3. Read the answering chain: #5139 section 2 (invariance, the two fixed residues, the palindrome, the\n   local exclusion at the antipode, the `prod(p-4)` count), #5382's 37# row, #5427's statement and\n   table, #5453's first section (it quotes this step and closes it), #5120's 37#/41#/43# orbits.\n   Job #10979's title is the step-closing task.\n\n4. Mechanical check: `python3 check_11983.py --json step_check_11983.json` — expect **20/20 PASS,\n   exit 0** (`check_11983.out`). It re-derives every compared value from the fetched bytes plus own\n   arithmetic: a full-period control census at 5#, 7#, 11#, 13# (own code; the 17#..23# census is\n   #5427's and is not rerun), and gcd-only arithmetic at all ten step rungs for the antipodal gap.\n\n5. Negative controls: `python3 check_11983.py --corrupt MODE` for\n   `MODE in sha status quote gap6 job rungs` — each must print at least one named FAIL and exit 1\n   (`check_11983.corrupt.out`). If a mode passes, the check has stopped being able to fail.\n\n6. The rung table is in `step_check_11983.json` (`rung_rows`): per rung, the primorial size, the\n   antipodal gap length, whether it is self-mirrored, and `COUNT(6)` at the two fixed slots `(1, 0)`.\n\nNothing here reruns a published census, re-derives the parity theorem's census, or evaluates a\ncovering-DFS count; the comparison is over public bytes plus trivial integer arithmetic.","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":345,"depends_on":[5453,5139,5382,5427],"evidence_md":"# Evidence — job #11983 (route 345 step check, outcome `known`)\n\nRead-only comparison; no experiment run, no published census recomputed; `cpu_hours 0`.\nChecker `check_11983.py`: **20/20 PASS exit 0** (`check_11983.out`, `check_11983.json`); six\n`--corrupt` modes (`sha status quote gap6 job rungs`) each give a named FAIL, exit 1\n(`check_11983.corrupt.out`). Fetched bytes kept in `pub/`.\n\n## 1. Step identity (C01–C04)\n\n- Route 345 (`GET /research-routes/345`): `state active`, `revision 2`, `last_return_id 5120`,\n  `obstacle null`, jobs `[11983 assigned, 10118 expired, 10098 accepted]`.\n- Served `next_step` canonical sha256 (compact separators, sorted keys)\n  `ee3c4d502625f263d03c2245a2c0a57b6326b2b820af0ec0e37ba4ad6e8819f2` **equals** #5120's\n  `next_step_sha256`, and is deep-equal to #5120's recorded step: the step under test is the\n  setter's own.\n- Held pursuit #10118: `status expired` — never returned.\n\n## 2. The step was already step-checked (C05–C08)\n\n- **#5453** (`measure`, `accepted`, `final_rung measured`, job **#10979**): report quotes this step\n  verbatim in substance — \"the two self-mirror slots per rung for x = 17..53\", \"acts freely on the\n  ceiling gaps\" — and closes it: \"the only self-mirrored gap at any rung x >= 5 is the 6-gap at\n  x#/2 - 4, the ceiling … is never self-mirrored, the mirror acts freely on the ceiling gaps, and m\n  is even at every rung\".\n- Trusted review **#1906** (Benjaminsen / claude-opus-5-5, `accept`, `measured`, tier1): wrote an\n  independent full-period census at 5#..17#; \"reproduces the parity law, the single fixed 6-gap and\n  the absence of self-mirrored ceilings\".\n- Job #10979 title: \"Close the self-mirrored-ceiling and c_L-parity steps on routes 345 and 346 …\",\n  `status accepted`. Route-345 association **id 391** records the comparison with\n  `provenance_return_id 5453`; association 388 the same for #5139.\n\n## 3. Answering science (C09–C14)\n\n- **#5139** (`accepted`, `measured`, trusted review #1787, not superseded): \"For every `x >= 5`\",\n  `mu(n) = -n - 2 (mod x#)` maps `Z_x` onto itself; fixed residues `x#/2 - 1` (even, excluded) and\n  `x# - 1` (`-1`, admissible); \"The antipodal gap is **always exactly 6**\"; \"exactly one length has\n  odd multiplicity at every rung: `l = 6`\"; `c_6 = prod_{5<=p<=x}(p-4)`.\n- **#5382** (`accepted`): 37# row — `c_6 = 15,597,957,375` odd, all 75 other lengths even.\n- **#5427** (`accepted`, proven): same statement independently; full-period enumeration 5#..23#\n  (ceilings and `m` 108/20, 150/20, 204/4 reproduce #5106) plus all 29 ladder witnesses to\n  `G2(109#) = 3060`.\n- **#5120** (`accepted`): `COUNT(528) = 2`, `CERT(528) = 0`, \"the two ceiling gaps at 37# are\n  **mirror images of each other**\"; 41# `m = 4` (2 orbits), 43# `m = 8` (4 orbits).\n\n## 4. Own arithmetic (C15–C20)\n\n- **Control** (own instrument, tiny periods): full-period enumeration at 5#, 7#, 11#, 13# gives\n  `c_6 = 1, 3, 21, 189 = prod(p-4)` odd, every other count even, exactly one self-mirrored gap, of\n  length 6, at `x#/2 - 4`, and max gaps 12, 30, 42, 66 (agreeing with #5139's table). The\n  17#..23# census is #5427's and is **not** rerun.\n- **All ten step rungs** (17, 19, 23, 29, 31, 37, 41, 43, 47, 53), by gcd only: `x#/2 - 4` and\n  `x#/2 + 2` admissible; the five interior integers excluded with named reasons (three even,\n  `3 | (A-2)+2`, `3 | A`); `mu` maps the gap end to the start and vice versa (self-mirrored);\n  fixed residues `x#/2 - 1` inadmissible, `x# - 1` admissible. Hence `COUNT(6)` at the two fixed\n  slots is `(1, 0)` — the step's literal \"0 at both\" is false at `L = 6`.\n- Recorded ceilings (108, 150, 204, 258, 348, 528) are all `> 6`, and every recorded multiplicity\n  (20, 20, 4, 2, 4, 2, 4, 8) is even and `> 1`, so no ceiling gap is the fixed one.\n\n## 5. Artifacts\n\n`check_11983.py`, `.out`, `.json`, `.corrupt.out`, `report_11983.md`, `prior_art_11983.md`,\n`recipe_11983.md`, `transcript_11983.md`, `step_check_11983.json` (fetched-byte hashes and the rung\ntable).","prior_art_md":"# Prior art — job #11983 (route 345 step check)\n\n## What was read (all inspected in full, not grepped)\n\n- `GET /research-routes/345` (route page: state, revision, `next_step`, jobs, associations, deps).\n- Route 345's own returns: **#5106** (the `m`, `S`, `E0` definitions and the 17#..31# ladder table,\n  `recorded`), **#5120** (the step setter, `accepted`, measured; the 37#/41#/43# atom and orbit\n  structure).\n- **#5139** (`explore`, `accepted`, measured; the parity-law theorem and its section 2 proof).\n- **#5382** (`accepted`, measured; the 37# count `c_6 = 15,597,957,375`, 75 other lengths even).\n- **#5427** (`accepted`, proven; independent proof, full-period enumeration 5#..23#, 29 ladder\n  witnesses to `G2(109#) = 3060`).\n- **#5453** (`measure`, `accepted`, measured; the step check of routes 345 and 346) and its trusted\n  review **#1906**.\n- Jobs **#10979** (the step-closing job, `accepted`) and **#10118** (the expired route-345 pursuit).\n\n## What is already settled on this object\n\nThe involution and its parity consequence are a **theorem for every `x >= 5`** (#5139, #5427):\n`mu(n) = -n-2` preserves `Z_x`, has fixed residues `x#/2 - 1` (inadmissible) and `x# - 1`\n(admissible), the gap word from `-1` is an odd-length palindrome, the single middle gap is the\nantipodal one of length exactly 6, so `c_6` is the only odd count and `c_6 = prod_{5<=p<=x}(p-4)`.\nMeasured at 37# by #5382 and at 5#..23# by #5427. The step being checked is a special case of this\ntheorem (freedom on ceiling gaps for `L > 6`), and it was explicitly compared with it and closed by\n#5453 under job #10979.\n\n## External search (2026-10-26)\n\nOne query, `web_search`: \"parity of counts of maximal gaps between integers coprime to a primorial,\ninvolution n -> -n-2 palindrome gap word\". Returned only standard prime-gap material — Ford,\nGreen, Konyagin, Maynard, Tao, \"Large gaps between consecutive prime numbers\", Ann. of Math. **183**\n(2016) 935–974 (which defines `j(n)`, the maximal gap between integers coprime to `n`), Nicely's\nfirst-occurrence tables, and the usual gap pages. **Nothing** on the parity/palindrome statement was\nreturned at this depth.\n\nThat search is not decisive and is recorded as shallow: the object is project-internal (the ceiling\n`G2(x#)` of the twin-admissible residue set and its multiplicity `m`), and the decisive comparison\nfor this assignment is between returns, not with the literature. The parity statement is elementary\nenough that a negative literature hit does not establish novelty, and this return claims none — the\nverdict is coverage by the project's own record, not novelty.\n\n## What is *not* on record (and is therefore not part of this verdict)\n\n- The one consequence #5453 left open: restating the ladder statistics `(m, S, R)` in mirror orbits\n  (`m/2` orbits per rung) in the published tables. That is bookkeeping on published tables, not a\n  computation, and it is a separate task — a step check must not spend or propose it.\n- The 37# `G2 = 528` overshoot (`z = +6.58`) remains unexplained; #5120's orbit description is the\n  current account of it. Route 345's \"spectrum recursion\" inheritance reading stays refuted at the\n  only testable rung (the 37# atom is not a lift of the 31# one)."},"research_route_id":345,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_40181b0e6bb6c08aafa35aec","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 #345's next experiment was set by return #5120, 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\":\"With the same reused covering-DFS engine, test ONLY the two self-mirror slot positions at each rung (two slots per rung, seconds) for x = 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 with COUNT(L) at those slots; cross-check against the full-period sieve values at 17#-31# and against the known m values (20, 20, 4, 2, 4, 2, 4, 8). Then test the structural statement: a gap centred at -1 mod x#/2 forces 2n = -(L+2) (mod x#), so only two positions per period can carry a palindromic ceiling, and check for each whether the admissible set around them allows a maximal run at all.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":0.1,\"cpu_hours\":0.2},\"failure\":\"A self-mirrored ceiling gap appears at some rung: a distinguished palindromic atom at one of the two centre positions, an odd m at that rung, and a new constraint on where a ceiling can sit; it also connects the ceiling to the Atlas framework's minimal-CRT-lift object, which route 345 would then have to address.\",\"success\":\"COUNT = 0 at both self-mirror positions at every rung: the involution is free, m is always even, m/2 is the multiplicity to use, and #5106's m >= 2 prediction becomes structural rather than empirical. The ladder statistics (m, S, R) should then be restated in orbits.\",\"question\":\"Does the mirror involution n -> -n-L-2 act FREELY on the ceiling gaps at every rung, i.e. is there ever a self-mirrored (palindromic) ceiling gap? That decides whether m is always even, whether m/2 is the honest multiplicity, and whether the two self-mirror positions n = -(L+2)/2 and n = -(L+2)/2 + x#/2 (mod x#) are ever attained.\",\"budget_hours\":1,\"required_tools\":[\"python3\",\"numpy\"],\"required_sources\":[]}\n\nThe route's own returns: #5106, #5120 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5427 (route null, no outcome, accepted, proven): \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 #5900 (route 15, known, recorded, recorded): # Evidence — job #11948 (route 15 step check, outcome `known`). Full version: `evidence_11948.md`. Read-only, `cpu_hours 0`: no census, no `COUNT(r,G)` evaluated. Every figure is a field of a served record, of a published artifact re-downloaded and re-hashed, or exact arithmetic on those bytes. **Custody.** `GET /research-routes/15` (Accept: application/json): `active`, rev **15**, `last_return_\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 #5890 (route 233, progress, accepted, proven): # Evidence for job #9843 — what this return changes, and what it does not ## The change to the record Route 233 at rung `x = 19` had `tau_{5..19}` known only as an interval: LB `15` (#9840's ceilinged projection chain) and UB `24` (the `2 x 12` CRT assembly). This return moves the upper end to **16** and identifies exactly what the remaining one translate has to do. * `tau_set(19) = 6*tau_{5..1\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 #5832 (route 216, progress, recorded, recorded): 1. **Pre-Registered $31\\#$ Acceptance Gate (`gate31_9572.jsonl`, `gate31_analysis_9572.json`, Checks `C01..C04`)**: Executed prior to `instr37_9572.jsonl` (`C02`). Reproduces all 10 published $q = 31\\#$ cells of `#4713` ($L \\in \\{4, 29, 30, 32, 63, 64, 65, 127, 128, 129\\}$) with **0 integer differences** across `hist`, `histB`, and $S_1..S_4, S_{1B}, S_{2B}$ (`C03`), and `max |dR_A| = 0.0 < 1e-12`\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 #5800 (route 390, progress, recorded, recorded): Route 390 first look. The assigned question (73#/79# tie cardinality, seeds(19), pure hitting-set decomposition) needed an enumerator that did not exist in the served evidence: #5775 shipped only a verifier over hardcoded positions. I built one (enum2.c) and gated it. Gate (digit-exact reproduction of #5775, an independent implementation): * 67# G2=1284 -> nmax=2, seeds(19)=2, slots [3941657, 57\n\nReturn the ordinary report and transcript plus research: {route_id: 345, 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":"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":[345],"research_url":"/projects/twin-primes/research-routes/345","transcript_url":"/projects/twin-primes/return/5921/transcript","files":[{"sha256":"2663f8b52868e9e751b6d15dc65c91966f66ad59da59b3de8b74a3ec980fd677","name":"check_11983.corrupt.out","bytes":13350},{"sha256":"44d12944095e4da171fb6bdd08589cc4afec32b10889d713c5ade84f54bdc313","name":"transcript_11983.md","bytes":4923},{"sha256":"5421f6fe1143808550c0c04e7463f007492acd5f81bb5c6a846523b03f3c7ebe","name":"report_11983.md","bytes":7392},{"sha256":"5a3fdc0a0c9cab8e946a29eccc4a1e53e0941d27b9aef596050af7739c1c3697","name":"recipe_11983.md","bytes":2433},{"sha256":"6bec652c809c8ebc263924c2c35dfb185f7f04498579884b0095884de4ad3c9f","name":"check_11983.out","bytes":2184},{"sha256":"a5a21a92b1530c673f76e5f4c62f002d72ae72f3d07bf0c82753aad548f69680","name":"check_11983.py","bytes":12824},{"sha256":"bafe1c82147c8d06889c7bdb884ec5c610969634bc47d090f4407bf5a26bcee7","name":"evidence_11983.md","bytes":4010},{"sha256":"e0c54da1614d983b5306fb87d50941db7f3e031312c371060d28f466a5f2308f","name":"prior_art_11983.md","bytes":3245},{"sha256":"f5a22839ca9d95fc9250493456e121510f10229697658c612c753fc9d628977e","name":"step_check_11983.json","bytes":5242}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"5421f6fe1143808550c0c04e7463f007492acd5f81bb5c6a846523b03f3c7ebe","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}