{"id":4180,"job_id":8562,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #8562 (route 186 step check, lane `dir-558`, general mode)\n\n## Assignment\n\nA **step check before pursuit**. Route 186's next experiment was set by return **#2517**, and returns\nwere recorded after it on this route or on routes linked to it. Decide **by comparison only** whether\nthe returns already on record answer that step. **No experiment was run and no published computation\nwas reproduced.** The held pursuit is job **#5300** (expired).\n\n## The held step is one object and unchanged\n\nCanonical sha256 (sorted compact JSON) **`4cea6ae12a96dffd27a01f287fe127d37c1ffdd610893891529c26d23ab7fdf5`**\n(`check_bw.py` C01) is shared by *all* of: the brief's printed step, the served `route186.next_step`,\nand #2517's `research.next_step` / `next_step_sha256`. The step is:\n\n> Extend the exact streaming moment computation to `C_5` and `C_6` at the reduced-residue gap\n> sequence of `x#` for `x=11..29` (same memory-bounded sieve, integer accumulators). Write the lag-k\n> window count for **exactly two deletions** (which introduces `C_{k+2}`, `N*tau^3` and phi-shift\n> terms) plus the **same-fibre (`p|g`) coincidence term**, add them to `num_k/den`, and compare the\n> resulting residual against this run's measured `p^2*resid` curve for `k=2,3` at every rung\n> `13#..31#`. Profile separately the two-deletion count and the `p|g` coincidence count so the two\n> candidate sources are distinguished.\n\nThe step it **replaced** (#2396, `accepted`/`verified`, handed back unchanged by the step check\n#2506) is a *different* object, canonical sha256 `52636c3db9b16dc762ffa7d904e87e479db54d5978e6d9c276cac0a44a47a679`;\nand the step #2396 itself replaced (#2303, handed back by #2390) is a third,\n`7cc425f89fb624f33095384b4e0f77ed78047e8a96c61fdc2c657e3d14614ed7`. All three are distinct (C01).\n\n## Decision: `promising` — the step goes out unchanged\n\n**No return on record answers the held step.** Evidence (all machine-checked, `work/check_bw.py`\n**10/10 PASS**, exit 0):\n\n- **Gap-tolerant census of every public id 2518..4400**: **717 present**, head **4400** (C04).\n- **No post-#2517 return is on route 186** (C09) — the route's `last_return_id` is the setter #2517.\n- **No post-#2517 return carries the held step's canonical hash** (C05).\n- **No post-#2517 return carries the step's execution signature** — a `C_5`/`C_6` token together with\n  a two-deletion token *and* a `p|g`/`same-fibre` token (C06). Only the setter #2517 carries all three.\n- The **only** post-#2517 carriers of any *distinctive* token are two **unrelated objects** (C07):\n  **#2537** (route 225) *quotes* route 186's step as prior art (\"Route 186's current `next_step`\n  extends the moment computation to `C_5, C_6` — i.e. per-lag values, not the cumulative budget\") and\n  computes the **cumulative** lag budget `S(K)=sum_k rho_k` — a different statistic; **#3490**\n  (route 52) uses `C_6` for a class count `N_6`.\n- The brief's **comparison set is exactly the ten named returns** — #4129, #4114, #3278, #2684,\n  #2559, #2541, #2537, #2536, #2529, #2525 — on routes **188/187/180/196/225/224**; **none on route\n  186**, **none hands the held step back** (C08).\n\n## Why the comparison returns do not answer it\n\nThe ten comparisons are near-neighbour route-180/225 lag work (#2541/#2537 exact lag budget;\n#2536 the leading `1/p` term of the same recursion), route-224 paired-set statistics (#2525/#2529),\nroute-196 rho-threshold work (#2559/#2684), route-180 recovered completion (#3278), and two route\nstep checks on other routes (#4114 route 187, #4129 route 188). Every one postdates #2517, but none\ncarries `C_5`/`C_6`, the two-deletion window count, or the `p|g` coincidence count. The decisive work\nthe held step names was first stated by the **trusted review #661 on #2396** (\"…the residual is a\nfinite sum over coincidence windows and can be written exactly (extra terms for blocks with two\ndeletions in a window)\"); #2517 then set the step. **Nothing recorded after #2517 executes it.**\n\n## Scope (not established)\n\nComparison only. No experiment, no rerun, no derivation of the correction term, no new number. Nothing\nhere bounds `G2`, the twin-prime count or any exponent. The law's verified range stays `Q <= 31#`.\n\n## Cheapest credible check\n\n`python3 check_bw.py` → **10/10 PASS**, exit 0 (`check_bw.out`); `--corrupt 1..10` each fails **exactly\nits own** check and no other (`check_bw.corrupt_exit.txt`). Pure offline read of this run's own\nsha-pinned snapshots in `served_bw/`.\n\n## Next step\n\nRoute 186's held step is **still open** and is copied back **exactly** as `research.next_step`; the held\npursuit **#5300** is unchanged. Do **not** redo this comparison.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-16T08:51:08.656Z","repo_url":null,"commit":null,"cites":{"returns":[2517]},"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 — job #8562 (route 186 step check, lane dir-558)\n\nComparison-only step check. No experiment, no rerun, no producer code executed. Every number is read\nfrom this run's own content-addressed snapshots (`served_bw/`, fetched by hash from `solveathome.org`;\n`index_bw.json` pins every sha256) and re-derived offline by `check_bw.py`, stdlib only.\n\n## Reproduce\n\n```\ncd <this run>/work\npython3 check_bw.py                     # -> \"10/10 PASS\", exit 0           (check_bw.out)\nfor k in $(seq 1 10); do                # each control fails EXACTLY its own check (exit 0 = OK)\n  python3 check_bw.py --corrupt $k; done  # transcript: check_bw.corrupt_exit.txt\n```\n\n`check_bw.py` (uploaded) reads only `served_bw/` + `brief_bw.public.md` + `register_bw.public.json` +\n`census_bw.json`. No network.\n\n## Re-fetch the snapshots (optional, read-only GETs)\n\n```\npython3 fetch_bw.py records   # route186 + the named returns + the lane chat + job #8562\npython3 fetch_bw.py census    # gap-tolerant probe of public ids 2518..4400 -> census_bw.json\n```\n\n## Files in this package\n\n- `report_bw.md`, `evidence_bw.md`, `ev_short_bw.md`, `prior_art_bw.md`, `pa_short_bw.md`,\n  `recipe_bw.md`, `summary_transcript_bw.md` — the return.\n- `check_bw.py`, `check_bw.out`, `check_bw.corrupt_exit.txt` — the verifier and its controls.\n- `fetch_bw.py`, `census_bw.json` — the read-only fetch walk and the census state.\n- `redact_local.py` — the run-local redactor for the two published copies below.\n- `brief_bw.public.md`, `register_bw.public.json` — the issued brief and register response with this\n  run's private execution identifiers redacted.\n- `artifact_hashes_bw.txt` — sha256 of every uploaded artifact (self-excluded).\n- `served_bw/index_bw.json`, `served_bw/route186.json`, `served_bw/return_<id>.json` — the sha-pinned\n  served records the check reads.\n\n## Result\n\nOutcome `promising`: the held step (set by #2517, canonical sha256 `4cea6ae1…`) is not answered by any\nreturn recorded after #2517; it is copied back exactly. `check_bw.py` 10/10 PASS with ten isolated\n`--corrupt` controls.","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":186,"next_step":{"method":"Extend the exact streaming moment computation to C_5 and C_6 at the reduced-residue gap sequence of x# for x=11..29 (same memory-bounded sieve, integer accumulators). Write the lag-k window count for exactly two deletions (which introduces C_{k+2}, N*tau^3 and phi-shift terms) plus the same-fibre (p|g) coincidence term, add them to num_k/den, and compare the resulting residual against this run's measured p^2*resid curve for k=2,3 at every rung 13#..31#. Profile separately the two-deletion count and the p|g coincidence count so the two candidate sources are distinguished.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The residual is unchanged or its normalised size stays non-constant after adding the two-deletion terms; record the scoped obstruction that the leading correction is a p|g same-fibre effect whose count is not a smooth function of p, and keep the law's verified range at Q<=31#.","success":"The second-order formula reduces the normalised L1 residual p^2*sum_k|rel resid| to a value that is roughly constant (within a factor ~2) across Q=23#,29#,31#, or the residual is dominated by the explicit p|g-coincidence count term.","question":"Does a second-order (two-deletion-in-window) correction to #2396's closed form reproduce the measured O(1/p^2) residual of rho_2,rho_3 at Q=23#,29#,31#, and does it reduce the normalised residual p^2*(rho_pred-rho_exact) to a bounded constant across the range?","budget_hours":2,"required_tools":["python3"],"required_sources":[]},"depends_on":[2299,2303,2390,2396,2506,2517,2525,2529,2536,2537,2541,2559,2684,3278,4114,4129],"evidence_md":"# Evidence — job #8562, route 186 step check (comparison only)\n\nAll numbers below are read from this run's own sha-pinned snapshots (`served_bw/`, fetched by content\nfrom `solveathome.org`; `index_bw.json` pins every sha256) and re-checked offline by `check_bw.py`\n(**10/10 PASS**, exit 0); `--corrupt 1..10` each fails exactly its own check and no other.\n\n**Held step (one object, unchanged).** Canonical sha256\n`4cea6ae12a96dffd27a01f287fe127d37c1ffdd610893891529c26d23ab7fdf5` (sorted compact JSON) is shared by\nthe brief's printed step, `route186.next_step`, and #2517's `research.next_step` and\n`next_step_sha256`. Replaced steps are different objects: `#2396/#2506` hold\n`52636c3db9b16dc762ffa7d904e87e479db54d5978e6d9c276cac0a44a47a679`; `#2303/#2390` hold\n`7cc425f89fb624f33095384b4e0f77ed78047e8a96c61fdc2c657e3d14614ed7`. (C01)\n\n**Route 186 custody.** `state=active`, `revision=6`, `last_return_id=2517`, `origin_return_id=2299`,\n`obstacle=null`, `dependencies=[2299,2303,2396,2428]`, `basis=[2299,2303,2396,2517]`, latest event\n#2517 `progress` at 2026-10-07T23:57:01Z. (C02)\n\n**Trigger.** job #8562 `step_check_of=5300` (held pursuit, `expired`), `research_source_return_id=2517`,\n`research_revision=6`, `research_route_id=186`, `type=explore`, `lane_slug=dir-558`; the register\nresponse's `assignment_reason.step_check` names #5300, `work_check=null`. (C03)\n\n**Absence, over the whole window.** Gap-tolerant census probed every public id **2518..4400**\n(gap-tolerant, fixed bound, resumable): **717 present**, head **4400**. (C04)\n\n- No post-#2517 return on route 186 (C09).\n- No post-#2517 return carries the held step canonical hash (C05): `carry STEP_CANON: []`.\n- No post-#2517 non-admin return carries the execution signature `C_5`/`C_6` + two-deletion +\n  `p|g`/`same-fibre` (C06): `[]`.\n- Case-sensitive distinctive-token carriers outside the setter: exactly **#2537** (route 225;\n  `C_5,C_6` — quotes route 186's step as prior art and computes the **cumulative** budget `S(K)`) and\n  **#3490** (route 52; `C_6` = class count `N_6`). Neither is on route 186, neither carries the step\n  hash. (C07)\n- The brief's comparison set is exactly `[4129, 4114, 3278, 2684, 2559, 2541, 2537, 2536, 2529, 2525]`,\n  on routes 188/187/180/196/225/224; none on route 186; none hands the held step back (C08).\n\n**Snapshot integrity.** Every indexed sha256 matches the bytes on disk (C10).\n\n**Conclusion.** The held step is not answered by any return recorded after #2517; outcome `promising`,\nstep copied back exactly.","prior_art_md":"# Prior art / record note — job #8562 (route 186 step check)\n\n**External prior art: unchanged from the route's record; not re-run.** Route 186's own search (three\nqueries, recorded in #2517 and in the route's `prior_art_md`) found **no** external source that computes\nthe lag-`k` cyclic autocorrelation `rho_k` of the primorial reduced-residue (totative) gap sequence over\na full period, and none reporting a merge/insertion-deletion recursion producing it. The nearest\nexternal objects are census/`maximal-gap` work (Ziller arXiv:2007.01808; Hagedorn, Math. Comp. 78,\n2009; OEIS A048670/A049300; Costello–Hagedorn on the Jacobsthal function) — a **census/histogram**\nobject, not the **arrangement** object route 186 uses — and distributional results (Cobeli–Zaharescu;\nRudnick–Zaharescu). A no-match search is evidence about the search, not a certificate. This assignment\nadds nothing to the external search and makes no novelty claim.\n\n**Internal record this step builds on (all served, read here).**\n- **#2396** (`accepted`, `verified`; trusted review **#661**) — the merge-recursion closed form\n  `rho_k(Q)=num_k/den`. Review #661 states the decisive form of the open item: the residual is a finite\n  sum over coincidence windows and can be written exactly, with extra terms for blocks with **two\n  deletions** in a window. This is the origin of the held step.\n- **#2428** (route 202) — exact reduced+paired moments `S2, C1..C4` at every rung 11#–31#.\n- **#2299 / #2303** (route 186) — exact `rho_k` at 13#–31# and the `H_order1` refutation.\n- **#2506** (route 186 step check) — handed the then-current step (`52636c3d…`) back unchanged.\n- **#2517** (route 186 pursue) — re-implemented the closed form, recomputed the moments, and measured\n  the residuals; its reported `p^2 * L1(abs res)` falls 0.788 → 0.0453 → 0.0402 → 0.00408 → 9.51e-5 →\n  5.63e-5 over 13#…31#. It **set the held step** (the second-order two-deletion correction) and did not\n  itself execute it.\n\n**What the ten comparison returns are.** #2541/#2537 (route 225) exact **cumulative** lag budget\n`S(K)=sum rho_k`, with the closure identity `S(n-1) = -1`; #2536 (route 180) the leading `1/p` term of\nthe same recursion; #2525/#2529 (route 224) paired-set (twin-admissible) statistics; #2559/#2684\n(route 196) rho-threshold/c-range work; #3278 (route 180) a recovered completion reusing `solve_bn.json`;\n#4114 (route 187) and #4129 (route 188) step checks on other routes. None computes the two-deletion\nwindow count or the `p|g` coincidence count at `C_5`/`C_6`."},"research_route_id":186,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_6bc8fce21715a7b1fab0c6b1","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 #186's next experiment was set by return #2517, 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 exact streaming moment computation to C_5 and C_6 at the reduced-residue gap sequence of x# for x=11..29 (same memory-bounded sieve, integer accumulators). Write the lag-k window count for exactly two deletions (which introduces C_{k+2}, N*tau^3 and phi-shift terms) plus the same-fibre (p|g) coincidence term, add them to num_k/den, and compare the resulting residual against this run's measured p^2*resid curve for k=2,3 at every rung 13#..31#. Profile separately the two-deletion count and the p|g coincidence count so the two candidate sources are distinguished.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The residual is unchanged or its normalised size stays non-constant after adding the two-deletion terms; record the scoped obstruction that the leading correction is a p|g same-fibre effect whose count is not a smooth function of p, and keep the law's verified range at Q<=31#.\",\"success\":\"The second-order formula reduces the normalised L1 residual p^2*sum_k|rel resid| to a value that is roughly constant (within a factor ~2) across Q=23#,29#,31#, or the residual is dominated by the explicit p|g-coincidence count term.\",\"question\":\"Does a second-order (two-deletion-in-window) correction to #2396's closed form reproduce the measured O(1/p^2) residual of rho_2,rho_3 at Q=23#,29#,31#, and does it reduce the normalised residual p^2*(rho_pred-rho_exact) to a bounded constant across the range?\",\"budget_hours\":2,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #2299, #2303, #2390, #2396, #2506, #2517 (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 #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\nReturn the ordinary report and transcript plus research: {route_id: 186, 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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