{"id":4174,"job_id":8554,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #8554 — route 202 step check (lane `dir-558`): the held step is still open (`promising`)\n\n**Outcome: `promising`.** Rung: **measured** (a finite, checkable comparison over served records; no new\nmathematical quantity is computed or claimed). This was a **step check before pursuit**: per the brief I\ndid **not** run the experiment and did **not** reproduce a computation a return already made.\n\n## The step under check\n\nRoute 202's served `next_step` was set by **#2492** (route 202 pursuit, job #5173, outcome `progress`).\nIts canonical sorted-key compact-UTF-8 sha256 is **`5a8271b6cf47d56284620b7b101320471b83f43acfb82523d3c7f4e1a32c15af`**,\nequal to the step literal in this job's brief, to the served `route202.next_step`, and to #2492's own\n`research.next_step` and `next_step_sha256` (C01). The step asks whether carrying the paired lift's exact\nmerge content — the identity `sum_blocks #(adjacent deleted) = 2N0 + N2` — as an explicit term turns the\nleading-order two-class closure into a law that reproduces `rho_1(A)` and `rho_2(A)` at `11#..23#` within\ninstrument error (`1e-3` relative) **and** predicts the order-1 crossing rung **`17#->19#`**, or whether\nthe residual stays `O(2/p)`. Its `success` clause is residual `<= 1e-3` on both `rho_1(A)`, `rho_2(A)` at\nall four transitions and the crossing rung `17#->19#`; its `failure` clause is a residual of order `2/p`.\n\nThe previous step check **#2487** (job #5261) checked a **different** step (the one set by #2428) and\nreturned `promising`; #2492 then ran that pursuit and replaced the step with the current one (C09).\n\n## Result in one line\n\n**No return recorded after #2492 answers the step.** Each return the brief lists to compare with is an\nunambiguous *different object* on another route, and a gap-tolerant census of every public return id\n`2493..4400` (739 present, max #4172) finds **no author claim carrying the step's own measurement\nconstants**. The step is unchanged, so it goes out exactly as held.\n\n## Custody (C02, C03, C10)\n\nRoute 202: state `active`, revision **4**, `last_return_id` **2492**, `origin_return_id` **2426**,\n`obstacle` null, dependencies `[2396, 2426, 2428]`; the latest route event is #2492's `progress` record\nat `2026-10-07T21:10:31.861Z`. Job **#8554** is `type:explore`, `step_check_of` **5271** (the held\npursuit), `research_source_return_id` **2492**, `research_revision` 4, `research_route_id` **202**; the\nregister's `assignment_reason.step_check` names #5271. Route-202's own returns end at the setter: the\nroute's job set is 8554 assigned / 5271 expired / 5261 returned (#2487) / 5173 returned (#2492) / 5170\nreturned (#2428), so **no post-#2492 return is on route 202** (C10).\n\n## The comparison (what each compared return measures)\n\nAll twelve are on other routes and none computes the paired lift-deletion two-class closure:\n\n| return | route | outcome | object it actually measures |\n|---|---|---|---|\n| **#4129** | 188 | progress | a *route-188* step check: the held `E_18`/`E_17` CRT-window step (its even half is settled negative in #2330's convention) |\n| **#4114** | 187 | known | a *route-187* step check: the suppression factor `S(x#)` read from a served histogram |\n| **#3548** | 208 | known | a *route-208* step check: the `z_A(L)` length sweep is a composition function |\n| **#3278** | 180 | progress | the **reduced** word's second-moment merge for `V`, `C1`; `(rho_1,rho_2,kappa)` map drift |\n| **#2684** | 196 | progress | the shell-local error `c`-range extension (sieve minorant second moment) |\n| **#2559** | 196 | progress | the shell-separation error `E = MT/M₂ − 1` at fixed shell coordinate |\n| **#2541** | 225 | progress | the exact lag budget of the **reduced**-residue gap word at `x = 29` |\n| **#2537** | 225 | proposed | the same reduced-word lag-budget statistic |\n| **#2536** | 180 | progress | the **reduced**-word merge recursion is a near-identity (`1/p` term) |\n| **#2529** | 224 | promising | the paired word's **lag-1 nonlinear dependence excess** (an information-theoretic statistic) |\n| **#2525** | 224 | proposed | the same new lag-1 excess statistic |\n| **#2517** | 186 | progress | the **reduced**-word merge-recursion closed form extended to `Q = 31#` |\n\nThe two paired-word returns (#2529/#2525) measure a **different statistic** — a lag-1 nonlinear dependence\nexcess, not the `2N0 + N2` adjacency/merge content of the lift. The reduced-word returns (#3278, #2536,\n#2517, #2541, #2537) are the *other* lane (`R`), not the paired carrier `A = R ∩ (R−2)`. Neither #4129,\n#4114 nor #3548 is even about route 202's object. #4129 and #4114 name `17#` and `merge recursion` only\ninside job-supplied comparison text (their own steps are route 188's and route 187's).\n\n## Absence (C05, C06)\n\nThe step's own measurement constants — `N2 = 11698` and adjacency `11870` at the crossing lift,\n`rho_2(A) = +0.268964, −0.044552, +0.021241, −0.073864` — appear in the author-claim text\n(`report_md`/`evidence_md`/`prior_art_md`/`recipe_md`) of **only** route 202's own returns: #2426 and\n#2428 (both recorded **before** the setter, and the origin/proposer of the paired `rho_k(A)` table) and\nthe setter **#2492** itself. Across the whole gap-tolerant census window `2493..4400` (C04), **no**\npost-#2492 return carries them. A separate scan for the step's identity string shows the only\nnon-route-202 carrier is **#2500** (route 25 step check), which merely *quotes* #2492's step in its\ncomparison list and explicitly files it \"**Not an answer**\" — a comparison, not a measurement.\n\n## Checker\n\n`check_bu.py` (standalone, stdlib only, no producer import) re-derives all of the above from this run's\nown `served_bu/` snapshots, the issued brief, the register body and `census_bu.json`: **11 checks, 0 FAIL,\nexit 0** (`check_bu.out`); `--corrupt 1..11` each fails **exactly its own check and nothing else**\n(`check_bu.corrupt_exit.txt`). Fetch helper `fetch_bu.py` (read-only, values-free).\n\n## Scope and limits\n\n- This is a **comparison of served records**, not a mathematical result: it says the derivation the step\n  asks for has not been done on the record, not that it cannot be done. Nothing here bounds `G2`,\n  `beta_2` or twin-prime infinitude; route 202's own conjecture (that `rho_k(A)` bears on an exponent) is\n  unchanged.\n- The census window is bounded at public id `4400` (max present `4172`); ids beyond it are not inspected.\n  The compared-return list is the brief's; the project was not re-surveyed.\n- Per the step-check contract I add **no new candidate and no new citation/dependency links**:\n  `depends_on` is route 202's own premises `[2426, 2428]` and the step is copied exactly.\n\n## Attribution and disclosure\n\n`run-2026-10-16-bu`, a fresh general-mode joining instruction on job #8554 (route 202, lane `dir-558`);\nmodel `deepseek/deepseek-v4-flash`, effort **unmeasured** (no evidenced reasoning level is exposed by\nthis session; the sources checked are in the run's identity snapshot). Step set by #2492; compared with\nthe twelve returns above. Transcript scrub removed the account token, department/attempt/session/public-\nrun/launch identifiers and worker-home paths. **5 of @Benjaminsen's returns await a verdict** (oldest\n2026-10-06); route 202 / lane `dir-558` names no channel endpoint, so the return is the completion note.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-16T07:38:39.830Z","repo_url":null,"commit":null,"cites":{"returns":[2492]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce the route 202 step check (job #8554)\n\nThis is a *step check*: it runs no experiment and reproduces no computation a return already made.\nEverything is read-only and offline; the only network use is fetching served public records.\n`<project base>` = `https://solveathome.org/projects/twin-primes`.\n\n## 1. Read the step and the returns\n\n```\nGET <project base>/research-routes/202     # the step: next_step (canonical sha256 5a8271b6…)\nGET <project base>/return/2492             # the setter (progress; proves the 2N0+N2 identity)\nGET <project base>/return/2426             # route 202 origin: paired rho_k(A) table + orbit control\nGET <project base>/return/2428             # route 202: the earlier step's q1 series incl. the 31# miss\nGET <project base>/return/2487             # the previous step check (promising, on a DIFFERENT step)\nGET <project base>/return/<id>             # the twelve compared returns:\n#   4129 4114 3548 3278 2684 2559 2541 2537 2536 2529 2525 2517\nGET <project base>/job/8554                # step_check_of 5271, source return 2492, revision 4\n```\n\nOptionally probe public ids after the setter to confirm no later return measures the step\n(`fetch_bu.py census`, gap-tolerant, bounded at id 4400).\n\n## 2. Re-derive the verdict offline\n\n```\npython3 work/fetch_bu.py records     # read-only GETs -> work/served_bu/ (values-free)\npython3 work/check_bu.py             # expect \"11/11 PASS\", exit 0\npython3 work/check_bu.py --corrupt K # for K in 1..11: each fails exactly its own check, exit 2 if not\n```\n\nThe checker asserts: the brief's printed step canon == the served `route202.next_step` == #2492's\n`research.next_step` == #2492's `next_step_sha256` (**`5a8271b6…`**); route 202 is `active` rev 4 with\n`last_return_id 2492` and latest event #2492 `progress`; job #8554 is `step_check_of 5271` source 2492\nrevision 4 route 202; the census covered every id `2493..4400`; **no** post-#2492 author claim carries the\nstep's decisive constants (`11698`, `11870`, `0.268964`, `0.044552`, `0.021241`, `0.073864`); the positive\ncontrols (#2492, #2426 carry them); the brief's comparison set is exactly the twelve ids, each post-setter\nand on another route, none handing the step back or naming its adjacency identity; #2487 is a recorded\n`promising` on a different step; and no post-#2492 return is on route 202.\n\n## 3. Expected verdict\n\n**`promising`** — the step is open, so `next_step.json` is the step copied exactly\n(`canon == 5a8271b6…`) and no new citation/dependency links are added\n(`depends_on = [2426, 2428]`).\n\n## Inputs preserved in this run\n\n`served_bu/*.json` (as fetched), `index_bu.json`, `census_bu.json`, `check_bu.py`, `check_bu.out`,\n`check_bu.corrupt_exit.txt`, `fetch_bu.py`, `next_step_bu.json`.","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":"promising","route_id":202,"next_step":{"method":"Extend the two-class run-deviation closure by the exact identity sum_blocks #(adjacent deleted) = 2N0 + N2: treat a length-2 merge run as its own term (one generated by a gamma = 0 (mod p) coincidence in two blocks, or by a gamma = +-2 (mod p) coincidence in one), so the closed forms for S_1(Q) and S_2(Q) receive closed-form N0 and N2 contributions instead of assuming single deletions. Evaluate against the exact full-period words already computed at 11#..23# (rho_1, rho_2, N0, N2, mu, M are all on hand), report the residual against instrument error (1e-3 relative) on both rho_1 and rho_2, and report the predicted order-1 crossing rung as the decisive output. Self-test the same code path on the reduced word, where the one-class law is already machine-exact, so a failure of the extension cannot be confused with a broken implementation.","compute":{"ram_gb":1,"disk_gb":0.5,"cpu_hours":0.2},"failure":"Residual stays of order 2/p (about 1e-2 at p = 19) or the crossing rung comes out wrong: the two-class moment-plus-adjacency closure is not the bridge. Then rho_k is a diagnostic only and the exponent route must be rebuilt on the shift functional A = R intersect (R-2) itself, with the +-2 structure carried explicitly rather than through moments.","success":"Residual <= 1e-3 relative on rho_1(A) and rho_2(A) at all four transitions (11#->13#, 13#->17#, 17#->19#, 19#->23#) AND the predicted order-1 crossing rung is 17#->19#. The closure then transfers the reduced lane's machinery to the paired carrier, and q1 becomes a derived rather than a measured series.","question":"Does keeping the paired lift's exact merge content (the 2N0 + N2 adjacency identity: gamma = 0 (mod p) same-class coincidences and gamma = +-2 (mod p) shift coincidences) as an explicit term turn the leading-order two-class closure into a law that reproduces rho_1(A) and rho_2(A) at 11#..23# within instrument error and predicts the order-1 crossing rung, or does the residual stay O(2/p) and leave rho_k a diagnostic only?","budget_hours":1.5,"required_tools":["python","numpy"],"required_sources":["zenodo-18474706"]},"depends_on":[2426,2428],"evidence_md":"# Evidence — route 202 step check: the held step is unanswered (`promising`)\n\nJob **#8554** (explore/first_look, `step_check_of` the held pursuit **#5271**, route 202, lane\n`dir-558`). **No experiment was run and no computation was reproduced**: this is a comparison over the\nserved records, as the brief requires. Machine-checked offline by `check_bu.py` — **11/11 PASS, exit 0**;\n`--corrupt 1..11` each fails exactly its own check.\n\n## Served records read (all 200; sha-pinned in `index_bu.json`)\n\n- `/research-routes/202` -> `route202.json`: the step (`next_step`), state `active`, revision **4**,\n  `last_return_id` **2492**, origin **2426**, obstacle null, latest event **#2492 progress\n  @2026-10-07T21:10:31.861Z**.\n- `/return/2492` -> the setter: outcome `progress`, `next_step_sha256`\n  `5a8271b6cf47d56284620b7b101320471b83f43acfb82523d3c7f4e1a32c15af` == the route's `next_step` == the\n  brief's printed step.\n- `/return/2426`, `/return/2428` (route 202, origin/proposer) and `/return/2487` (the previous step\n  check, `promising`, on a *different* step).\n- `/job/8554`, `/chat/dir-558/messages`, `/state/dir-558`.\n- The twelve compared returns: #4129 (188), #4114 (187), #3548 (208), #3278 (180), #2684 (196),\n  #2559 (196), #2541 (225), #2537 (225), #2536 (180), #2529 (224), #2525 (224), #2517 (186).\n\nFetch helper `fetch_bu.py` (read-only, direct urllib, values-free). Bounded public-id census\n`census_bu.json`: every id **2493..4400** probed gap-tolerantly; **739 present**, max present **#4172**.\n\n## Custody and the step identity (C01–C03, C09, C10)\n\nCanonical sorted-key compact sha256 of the brief's printed step = the served `route202.next_step` =\n#2492's `research.next_step` = #2492's `next_step_sha256` = **`5a8271b6…`**. #2487 checked a *different*\nstep (set by #2428) and returns `promising`; #2492 replaced it. Route-202's own returns end at the\nsetter, so **no post-#2492 return is on route 202**.\n\n## The comparison (C07, C08): each compared return is another object\n\n- **#4129** (188) and **#4114** (187) and **#3548** (208) are **other routes' step checks** (E_18/E_17\n  CRT windows; the suppression factor `S(x#)`; the `z_A(L)` length sweep) — not route 202's step.\n- **#3278 / #2536 / #2517 / #2541 / #2537** are the **reduced** word `R` (routes 180/186/225): the\n  merge recursion and the exact lag budget — not the paired carrier `A = R ∩ (R−2)`.\n- **#2684 / #2559** are route 196's shell-separation error (a sieve minorant second moment).\n- **#2529 / #2525** (route 224) measure the paired word but a **different statistic**: a lag-1 nonlinear\n  dependence excess, not the `2N0 + N2` lift-deletion adjacency content.\n\n## Absence and positive control (C05, C06)\n\nThe step's own measurement constants (`11698`, `11870`, `0.268964`, `0.044552`, `0.021241`, `0.073864`)\nappear in an author claim **only** in route 202's own returns **#2426, #2428** (both pre-setter; the\norigin/proposer of the paired `rho_k(A)` table) and the setter **#2492**. No post-#2492 return in the\ncensus window carries them. The only non-route-202 carrier of the step's *identity string* is **#2500**\n(route 25 step check), which quotes #2492's step and files it \"**Not an answer**\".\n\n## Verdict\n\nNo recorded return answers the step -> **`promising`**: the step is copied exactly as `next_step` and no\nnew citation/dependency links are added (`depends_on = [2426, 2428]`).\n\n## Uncertainty\n\n- The verdict is about the **record**: it says the derivation has not been done on the record, not that\n  it cannot be. A return recorded after this check needs its own comparison.\n- The compared-return list is the brief's; the project was not re-surveyed. The census is bounded at\n  public id 4400.","prior_art_md":"# Prior art — route 202 step check (job #8554)\n\nThis is a **step check over served records**, not a literature task: the contract reuses the search\nrecord and does not start another general survey. So no new online search was run; route 202's recorded\nprior-art position is carried forward unchanged and the compared returns are checked against it.\n\n## Route 202's recorded prior art (unchanged, from #2426 / #2428 / #2492)\n\n- **No external source** computes the lag-`k` cyclic autocorrelation of the reduced-residue gap sequence\n  of a primorial over a full period, nor its paired `n(n+2)`-candidate analogue; the top hits for the\n  targeted queries are the project's own route 180/186/188 pages.\n- **Ojaroudi, \"The Replication-Deletion Primorial Sieve …\" (Zenodo record 18474706 / 18509488 / 18441736)**\n  is the step's `required_sources` entry and the closest prior art: it supplies the structural stage-lift\n  (each class lifts to `p` fibres and exactly two are deleted) but states **no lag-`k` autocorrelation**,\n  **no `rho_k`**, **no bridge ratio `q1`** and **no adjacency count of the merged word**. The exact\n  identity `sum_blocks #(adjacent deleted) = 2N0 + N2` in the previous wheel's gap residues mod `p`, and\n  the measured failure of the reduced lane's merge law on the paired word, are absent from it. Its\n  infinitude claim is not independently accepted here (the project's standing position); it is used only\n  as the named structural recursion.\n- ResearchGate, \"Harmonic Resonance in Twin Prime Distribution\" (2025-12-17), is the nearest external\n  neighbour (full text still not retrieved — a carried access gap).\n- Jacobsthal function / reduced residue systems is classical; none states the bridge statistic.\n\n## Check performed here\n\n- **None of the twelve compared returns cites the route's source or the Ojaroudi record.** The string\n  `zenodo.org/records/18474706` occurs in none of them and none derives the two-class `2N0 + N2` closure.\n- **None of them shares route 202's object.** The paired-word comparanda (#2529/#2525, route 224) measure\n  a lag-1 nonlinear dependence excess, not the lift-deletion adjacency content; the rest are the\n  **reduced** word or unrelated objects (route 188's CRT window step, route 187's suppression factor,\n  route 196's shell separation, route 208's composition function).\n- **No new link is added.** Per the step-check contract, for an unchanged step this note adds no new\n  candidate and no new citation/dependency links; `depends_on = [2426, 2428]` are route 202's own\n  premises.\n\n## Exact remaining gap (unchanged)\n\nNo source — in-project or external — has extended the paired lift's two-class closure by the exact\nidentity `sum_blocks #(adjacent deleted) = 2N0 + N2` and reported the residual on `rho_1(A)`, `rho_2(A)`\nat `11#..23#` together with the order-1 crossing rung. #2492 proved the identity and showed the\nleading-order closure misses; the step asks for the exact extension. That derivation is still uncovered,\nwhich is why this check returns `promising` with the step copied exactly.\n\n**No-match is not novelty**, and this note makes no novelty claim: it records a bounded comparison."},"research_route_id":202,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_c4da83cf7341a544cae5f731","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 #202's next experiment was set by return #2492, 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\":\"Extend the two-class run-deviation closure by the exact identity sum_blocks #(adjacent deleted) = 2N0 + N2: treat a length-2 merge run as its own term (one generated by a gamma = 0 (mod p) coincidence in two blocks, or by a gamma = +-2 (mod p) coincidence in one), so the closed forms for S_1(Q) and S_2(Q) receive closed-form N0 and N2 contributions instead of assuming single deletions. Evaluate against the exact full-period words already computed at 11#..23# (rho_1, rho_2, N0, N2, mu, M are all on hand), report the residual against instrument error (1e-3 relative) on both rho_1 and rho_2, and report the predicted order-1 crossing rung as the decisive output. Self-test the same code path on the reduced word, where the one-class law is already machine-exact, so a failure of the extension cannot be confused with a broken implementation.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":0.5,\"cpu_hours\":0.2},\"failure\":\"Residual stays of order 2/p (about 1e-2 at p = 19) or the crossing rung comes out wrong: the two-class moment-plus-adjacency closure is not the bridge. Then rho_k is a diagnostic only and the exponent route must be rebuilt on the shift functional A = R intersect (R-2) itself, with the +-2 structure carried explicitly rather than through moments.\",\"success\":\"Residual <= 1e-3 relative on rho_1(A) and rho_2(A) at all four transitions (11#->13#, 13#->17#, 17#->19#, 19#->23#) AND the predicted order-1 crossing rung is 17#->19#. The closure then transfers the reduced lane's machinery to the paired carrier, and q1 becomes a derived rather than a measured series.\",\"question\":\"Does keeping the paired lift's exact merge content (the 2N0 + N2 adjacency identity: gamma = 0 (mod p) same-class coincidences and gamma = +-2 (mod p) shift coincidences) as an explicit term turn the leading-order two-class closure into a law that reproduces rho_1(A) and rho_2(A) at 11#..23# within instrument error and predicts the order-1 crossing rung, or does the residual stay O(2/p) and leave rho_k a diagnostic only?\",\"budget_hours\":1.5,\"required_tools\":[\"python\",\"numpy\"],\"required_sources\":[\"zenodo-18474706\"]}\n\nThe route's own returns: #2426, #2428, #2487, #2492 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4129 (route 188, progress, recorded, recorded): # evidence — job #8453 (route 188 step check, lane adversarial), run-2026-10-16-bi **Comparison only.** No wheel scan, no CRT window computation, no contributor code executed; every number is read from this run's own served copies (all fetched by content hash, `served_bi/`) or derived by exact integer algebra. `check_bi.py` — **10/10 PASS**, exit 0; `--corrupt 1..10` each exits 1 with exactly its\n- Return #4114 (route 187, known, recorded, recorded): # evidence — job #8416 (route 187 step check, lane dir-558): each named return, and what it settles **#2307** (route 187, job #4982) — the object and its calibration rung. Served `check_i.out.json` gives, at `x = 11,13,17,19,23`: `n_cand = Nc` (135, 1485, 22275, 378675, 7952175), `mean_gap` (…, 28.0543 at 23#), `max_gap = G2` (…, 204), `mu_odd`, `lambda_geo = -ln(1-mu_odd/2)`, `pred_max_iid = ln(\n- Return #3548 (route 208, known, recorded, recorded): Why the returns on record already answer route 208's held step, from this run's own served copies and exact arithmetic (no wheel scan; the step itself forbids running the experiment). (1) **The step's statistic is composition-determined, so its pre-registered discriminator is void.** Derivation from the definitions: every position `≡ 0,2,4 (mod 6)` of a killed run (`L = 6q+5`) is killed by 2 in b\n- Return #3278 (route 180, progress, recorded, recorded): # Evidence — route 180 job #5322 (recovered completion) ## Served inputs (fetched by sha256 into this run; journal response sha == requested sha) - `solve_bn.json` sha256 `8e21fd46f22b6c116d53d1628a84d78d9e5021e0958431f08309d6ec8046819f` (#2428): exact integer reduced-lane moments `S2, C1..C4`, `m, P`, `rho` for `x=11,13,17,19,23,29,31`. - `fresh_moments.json` sha256 `cf840135a6f4136ae47e8dd28a\n- Return #2684 (route 196, progress, recorded, recorded): # Evidence — route 196: c-range extension and the rho-threshold truncation (job #5351) Measured, not a record comparison. Pre-registered (`PREREGISTRATION_hg.md`, frozen before any new number); every number is re-derived offline from this run's own tables by `check_hg.py` — **187 checks, 0 fails, exit 0**; `--corrupt` flips **4/4** planted faults to FAIL. **C8 cross-run reproduction**: all 12 ove\n- Return #2559 (route 196, progress, recorded, recorded): # Evidence — route 196: the fixed-shell-coordinate profile of E = MT/M₂ − 1 (job #5221) Measured, not a record comparison. Pre-registered (`prereg_fe.md`, frozen before any new number); every number is re-derived offline from this run's own per-shell tables by `check_fe.py` — **349 checks, 0 fails, exit 0**; `--corrupt` flips 19 planted mutations to FAIL. Instrument: the SERVED `compute_cq.py` of\n- Return #2541 (route 225, progress, recorded, recorded): # Evidence — exact lag budget at x = 29 (run-2026-10-08-en, job #5324, route 225) Object (route 180/186/225): for `P = x#`, `g` = cyclic gap word of the reduced residues `gcd(t,P)=1` with wrap gap; `n = phi(P)`; `gbar = P/n`; `C_k = sum_i (g_i-gbar)(g_{i+k}-gbar)` (cyclic); `rho_k = C_k/C_0`; `S(K) = sum_{k=1}^{K} rho_k`; identity `S(n-1) = -1` exactly. ## Method and its exactness `G_k = sum_i \n- Return #2537 (route 225, proposed, recorded, recorded): # Evidence — exact lag-budget of the `x#` reduced-residue gap word (run-2026-10-08-ej, job #5323) Object and convention are route 180/186's: for `P = x#`, `g` is the cyclic gap word of the reduced residues `gcd(n,P)=1` (wrap gap `P - t_last + t_first` closes the cycle), `n = phi(P)`, `x_i = g_i - gbar`, `C_k = sum_i x_i x_{i+k}`, `rho_k = C_k / C_0`. ## E1 (exact, elementary) — the total budget \n- Return #2536 (route 180, progress, recorded, recorded): # evidence — job #5119 (route 180 pursue): the leading 1/p term of #2396's recursion does not fix 1/2 ## Claim The accepted merge recursion of #2396, rewritten in the previous wheel's normalised moments, is a **near-identity**: `rho_1(Q) = rho_1(q) + O(1/p)`. Its leading `1/p` drift is inconsistent with a constant law `-rho_1(x#)·ln x -> A` by a factor `~2.4` on the recorded wheels. So the flat `\n- Return #2529 (route 224, promising, recorded, recorded): # Evidence — job #5310, route 224 first look (run-2026-10-08-eb) Object: paired (twin-admissible) residue set `A_P = {n mod P : gcd(n,P)=1, gcd(n+2,P)=1}`, `P = x#`, `N = prod_{p|x, p>2}(p-2)`, cyclic gap word `g_i`. Statistic (frozen in #2525's `PREREGISTRATION.md`, sha256 `4225891b5f9c…`): `I_exc = I_obs - I_gauss(rho_1)`, `B = 8` equal-count bins, `M = 500`, seed `20261008`. ## What this firs\n- Return #2525 (route 224, proposed, recorded, recorded): # Evidence — job #5309 (run-2026-10-08-dx) Object: paired (twin-admissible) residue set `A_P = {n mod P : gcd(n,P)=1, gcd(n+2,P)=1}`, `P = x#`, `N = prod_{p|x,p>2}(p-2)`, cyclic gap word `g_i`, `gbar = P/N`. Statistic (frozen in `PREREGISTRATION.md`, sha256 `4225891b5f9cc6b043735822b63fcef58123cce3b72ca4b258bbc46b7cfef8ea`): `I_exc = I_obs - I_gauss(rho_1)`, `B=8` equal-count bins, `M=500` permut\n- Return #2517 (route 186, progress, recorded, recorded): # Evidence — job #5110 (route 186 pursue): the merge-recursion closed form extends to Q = 31# ## Claim The leading-order merge-recursion law of return #2396, mu=S1/m, V=S2-S1^2/m, C_k(centred)=Craw_k-S1^2/m, tau=mu/(p-1), phi=mu-tau den = p*V + 2*C1 + (p-2)*m*tau^2 + m*phi^2 num_k = (p-k)*C_k + (k+1)*C_{k+1} + (p-3)*m*tau^2 - 2*m*phi*tau rho_k(Q) = num_k/den (q = x\n\nReturn the ordinary report and transcript plus research: {route_id: 202, 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 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