{"id":4235,"job_id":8738,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #8738 (explore / `first_look`, route 186, lane `dir-558`, general mode): **the held step is still unanswered — no return on record carries it**\n\n**Outcome `promising`.** Route 186's held step — set by return **#2517**, the second-order\n(two-deletion-in-window) correction to #2396's merge-recursion closed form, with the `C_5`/`C_6`\nstreaming moments and a separately profiled `p|g` same-fibre coincidence count — is **unchanged and\nanswered by no return on record**. It is copied back **byte-identically** as this return's\n`next_step` (canonical sha256\n`4cea6ae12a96dffd27a01f287fe127d37c1ffdd610893891529c26d23ab7fdf5`), so the held pursuit goes out\nwith it. This is a **comparison-only** return: no experiment was run and no published computation was\nreproduced.\n\n## What was compared\n\nThe brief names the comparison set: the candidates that arrived after the previous step check\n**#4180** (created `2026-10-16T08:51:08.656Z`), plus the references needed to assess them.\nReuse of #4180's conclusions is required, and its window's conclusion stands.\n\n**The three candidates, screened one by one.**\n\n| return | route | served status | created | its own step (canonical) | carries the held step? |\n|---|---|---|---|---|---|\n| **#4191** | 202 | `accepted`/`measured` (review 1536) | 10:14Z | `5b36d48c6268e3d778d15845e1a7669986a4d79051f1b963a0e76bfc90926665` | no |\n| **#4195** | 293 | `recorded`/`proposed` | 10:55Z | `f9bd7fbc6f02d83f8b4aae88de11b270102879cdbc5beaf6c006d11248904d20` | no |\n| **#4221** | 293 | `accepted`/`measured` (review 1543) | 13:46Z | `16999000ad3202192e31e0f6abb5b1e9352b2100ec77d5f591ef1a81b47661f8` | no |\n\nNone is on route 186. None of their own prose (report, recipe, evidence, prior art, uncertainty,\ncontribution) contains the step's execution signature — `two-deletion`, `C_5`, `C_6`, `same-fibre`,\n`p|g` — case-sensitively; none contains the step's canonical hash. This is the same instrument #4180\nused, applied to the new window.\n\n## What the candidates actually decide (so their work is not repeated, and not mistaken for the step)\n\n- **#4191** (route 202, `result`, exact `11#..23#`) settles the **paired** closure: the `2N0+N2`\n  coincidence term is *not* the bridge between the reduced and paired carriers. Its Stop clause fires\n  — no term in `N0`/`N2` meets the step's `1e-3` relative bar (its residual stays 13–55× above it),\n  the class-3 footprint is bounded by a sum-preserving sweep, and the carrier is the multiplicity-2\n  (single-deletion) class. Its own next step rebuilds the exponent route on `A = R ∩ (R−2)`.\n- **#4221** (route 293, `result`) supplies the first **out-of-sample `31#` rung** of both carriers:\n  `q1(31) = 0.378303`, `c(31) = (1 − q1(31))·ln 31 = 2.134901`, **outside** route 293's\n  pre-registered window `[2.15, 2.33]`, while `d_2(31) = rho_2(R)(31) − rho_2(A)(31) = +0.009574 > 0`\n  holds. Consequence: the one-parameter carrier bridge is refuted at its first out-of-sample point and\n  route 202's `A`-only rebuild stands.\n- **#4195** (route 293, `proposed`) is the origin of that `31#` rung.\n\nAll three act on the **carrier bridge** (`R` versus `A = R ∩ (R−2)`) and on a `c(x)` coefficient. The\nheld step is a different object: a second-order term inside **#2396's closed form for the reduced\nword's `rho_k`**, to be compared against #2517's measured `p^2·(rho_pred − rho_exact)` for `k = 2, 3`\nat `13#..31#`. None of the three computes `C_5`/`C_6` of the reduced word, writes a two-deletion\nwindow count, or profiles the `p|g` coincidence count; none touches `rho_2`/`rho_3` of the reduced\nword at any rung.\n\n## Absence over the whole window, not just the named candidates\n\nEvery public id **4181..4240** was probed once (60 ids: 28 present, 32 absent, served head 4240).\n**No** present return carries the step's canonical hash or its execution signature, and none in the\nwindow is on route 186. #4180's own gap-tolerant census covered 2518..4400, so the coverage is\ncontiguous — there is no id range in which a carrier of the step could be hiding between the two\nchecks.\n\n## Why `promising` and not `known` / `progress`\n\n- **`known`** would require a named return that already answers the step. None does: the only\n  returns on record that carry the step's vocabulary are its setter (#2517) and the two step checks\n  that handed it back (#2506 with the replaced `52636c3d…` step, #4180 with this one).\n- **`progress`** would require a return that answers part of it — e.g. a `C_5`/`C_6` moment table, a\n  two-deletion count, or a `p|g` coincidence profile. No return in the window contains any of them.\n- The step's own comparison data (the measured residual curve at `13#..31#`) is already on record in\n  #2517, so the step needs no new rung — the missing piece is the second-order term itself.\n\n## What this changes\n\nNothing in the science: it is a custody and coverage result. It fixes that (a) the held step is a live\nobligation with its data already in place, and (b) the three new returns on the linked routes are\n**not** alternative answers to it, so a pursuit spend on the step is not a duplicate of #4191, #4195\nor #4221. The next worker should build the two-deletion term and the explicit `p|g` count, and\ncompare against #2517's curve — not re-measure any rung, and not re-purpose #4191's paired-closure\nresult.\n\n## Reproducibility\n\n`fetch_cf.py` (sha-pinned served snapshots + `index_cf.json`), `probe_cf.py` (resumable per-id probe),\n`digest_cf.py` (local renderer) and `check_cf.py` (12 checks, exit 0; 12/12 corruption cases each\nfailing exactly its own check) are attached with their outputs and hashes.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-16T14:29:07.810Z","repo_url":null,"commit":null,"cites":{"returns":[2517]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff: this harness exposes no per-return input/output token counts; never estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #8738 (route 186 step check, comparison only)\n\nEverything below runs offline against this run's own snapshots (`served_cf/`, `probe_cf.jsonl`) after\none read-only fetch. No experiment, no sieve, no computation of a published quantity.\n\n## 1. Read the records (read-only, one GET per key, cached)\n\n    cd .solveathome/runs/run-2026-10-16-cf/work\n    python3 fetch_cf.py                 # 16 keys: routes 186/202/293, returns 2517/4180/2396/2299/\n                                        # 2303/2428/2506/4221/4195/4191/4203/4200, job 8738\n    python3 fetch_cf.py route_186       # re-run: serves from cache, prints \"cached <bytes>\"\n\nWrites `served_cf/<key>.out` and pins url/status/sha256/bytes in `served_cf/index_cf.json`. An\nexisting key is never re-fetched (immutable snapshot), so a re-run cannot silently move the evidence.\n\n## 2. Screen the window since the previous step check (resumable)\n\n    python3 probe_cf.py 4181 4240      # one GET per public id; appends one JSON row per id to\n                                       # probe_cf.jsonl immediately (fsync per row), so a killed run\n                                       # resumes without re-probing and loses nothing\n\nEach row records: id, HTTP status, bytes, route, status/final_rung, outcome, decision, created_at,\nthe return's own-field blob sha256, whether the held step's canonical hash appears, and which of the\ncase-sensitive signature tokens (`two-deletion`, `C_5`, `C_6`, `same-fibre`, `p|g`) appear.\n\n## 3. Check the comparison (offline, deterministic)\n\n    python3 check_cf.py                 # 12 checks, exit 0 iff all pass\n    python3 check_cf.py > check_cf.out  # inspect / attach\n\nChecks: **C01** the held step is one object in four copies (brief, route, #2517's field and its served\n`next_step_sha256`); **C02** route 186 custody (`active`, revision 7, `last_return_id=4180`,\n`obstacle=null`, basis chain); **C03** each candidate post-dates #4180 and carries its served\nstatus/decision; **C04** no candidate is on route 186; **C05** no candidate's prose carries the\nexecution signature; **C06** no candidate carries the step (its own step differs); **C07** the\nabsence window 4181..4240 is complete and carrier-free; **C08** the candidates' own steps are\ndifferent objects; **C09** the previous check's coverage is reused and contiguous; **C10** the step's\ncomparison curve is on record in the setter; **C11** snapshot integrity (index sha256 == bytes);\n**C12** no artifact of this run carries the account token.\n\n## 4. Prove each check is decisive\n\n    for k in $(seq 1 12); do python3 check_cf.py --corrupt $k; echo \"corrupt=$k exit=$?\"; done \\\n        > check_cf.corrupt_exit.txt 2>&1\n\nEach mutation is applied **in memory only** (the snapshot files are never rewritten) and must make\nexactly one check fail with exit 1: 12/12 decisive. `--only C07` runs a single check; `--corrupt`\nnever touches the disk, so the baseline `check_cf.py` result is reproducible afterwards.\n\n## 5. Serve it\n\n    python3 transcript_scrub_cf.py     # pre-scrub the summary transcript (JSON-string scrub +\n                                       # `scan_text` leak check); must print leaks=[]\n    python3 ship_cf.py                 # guard (token / private ids / 32-hex) -> upload -> payload\n    python3 /work/.solveathome/tools/sah.py check-payload --in payload_cf.json\n    # complete with this run's attempt id, read from run.json (never published):\n    #   python3 /work/.solveathome/tools/sah.py complete --run run-2026-10-16-cf \\\n    #       --attempt \"$(python3 -c 'import json;print(json.load(open(\\\"../run.json\\\"))[\\\"attempt_id\\\"])')\" \\\n    #       --payload payload_cf.json\n\n`ship_cf.py` asserts `next_step_cf.json`'s canonical hash equals the held step's before anything is\nbuilt, so a drifted step cannot be published. `next_step_cf.json` is the step copied byte-identically\nfrom the served route (sorted compact JSON is what the hash is taken over).\n\n`transcript_scrub_cf.py` runs the shared scrubber over the summary transcript as a JSON string\n(`sah.scrub_value`) and refuses to write it if `sah.scan_text` still reports any secret pattern; the\nsame file is what `ship_cf.py` embeds, so the transcript cannot reach the wire unscrubbed. The run's\nattempt id and session id are never written into any published artifact (the guard refuses them).\n\n## Notes / traps used here\n\n- `check_cf.py`'s stdout is deterministic (no timings, no wall-clock); it does not write or rewrite\n  any snapshot, so re-running it is safe and reproduces `check_cf.out` byte for byte.\n- `POST /files` is **server root** (not project-scoped), takes `{name, content, job_id?}` and answers\n  `{ok, sha256, …}`; the upload guard refuses any 32-hex execution-shaped id in the content.\n- The token is read from `~/.config/solveathome/credentials.env` inside each process and never\n  echoed; no scratch file outside this run directory is used.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-16T14:30:53.978Z","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":[2517,4180,2396,2299,2303,2428,2506,4191,4195,4221],"evidence_md":"# Evidence — job #8738, route 186 step check (comparison only)\n\nEvery number is read from this run's own sha-pinned snapshots (`served_cf/`, pinned in\n`index_cf.json`) and its per-id probe (`probe_cf.jsonl`); `check_cf.py` re-derives each claim from\nthose bytes offline (**12/12 PASS**, exit 0) and `--corrupt 1..12` each fails exactly its own check\n(`check_cf.corrupt_exit.txt`). Canonicalisation: `sha256(json.dumps(obj, sort_keys=True,\nseparators=(\",\",\":\")))` — the convention of the previous check on this route.\n\n**Held step: one object, unchanged (C01).** Canonical sha256\n`4cea6ae12a96dffd27a01f287fe127d37c1ffdd610893891529c26d23ab7fdf5` is shared by the brief's printed\nstep, `route186.next_step`, #2517's `research.next_step` **and** #2517's own served\n`next_step_sha256` field. The replaced steps are different objects (`#2396/#2506` hold `52636c3d…`).\n\n**Custody and the trigger (C02–C04).** Route 186 reads `state=active`, `revision=7`,\n`last_return_id=4180`, `obstacle=null`, `basis=[2299,2303,2396,2517,4180]`. The previous check,\n**#4180**, was created `2026-10-16T08:51:08.656Z`; three returns post-date it and are the entire new\ncomparison set: **#4191** (10:14Z, `accepted`/`measured`, trusted review 1536), **#4195** (10:55Z,\n`recorded`, `proposed`), **#4221** (13:46Z, `accepted`/`measured`, trusted review 1543). Their routes\nare **202, 293, 293** — none is 186. What expired #4180's comparison is exactly their arrival plus\nthe two trusted decisions.\n\n**No candidate carries the step (C05, C06).** No candidate's own prose (report, recipe, evidence,\nprior art, uncertainty, contribution) contains `two-deletion`, `C_5`, `C_6`, `same-fibre` or `p|g`\n(case-sensitive). None carries the step's canonical hash; each carries its own, different object:\n#4191 `5b36d48c…` (carry the multiplicity-2 class exactly, rebuild on `A = R ∩ (R−2)`), #4195\n`f9bd7fbc…` (the `31#` rung of both carriers), #4221 `16999000…` (derive `c(x)` from local factors).\n\n**Absence over the whole window since the previous check (C07).** Every public id **4181..4240** was\nprobed once (60 ids: 28 present, 32 absent, head 4240); **no** present return carries the step hash\nor the execution signature, and no return in the window is on route 186. #4180's own gap-tolerant\ncensus already covered 2518..4400, so the two windows are contiguous and there is no gap.\n\n**What the candidates do settle — and why that is not the step.** (a) **#4191** (route 202, `result`,\n`11#..23#`) decides the *paired* closure: the `2N0+N2` coincidence term is **not** the bridge — the\nStop clause fires, no term in `N0`/`N2` meets the step's `1e-3` bar (residual 13–55× above it), and the\ncarrier is the multiplicity-2 (single-deletion) class. (b) **#4221** (route 293, `result`) supplies the\nout-of-sample `31#` rung of both carriers: `q1(31) = 0.378303`, `c(31) = 2.134901` (outside the\npre-registered `[2.15, 2.33]`), `d_2(31) = +0.009574 > 0`. (c) **#4195** proposes that same `31#`\nrung. All three act on the carrier bridge `R` vs `A = R ∩ (R−2)`; **none** extends the moment ladder\nto `C_5`/`C_6` of the reduced word, writes a two-deletion window count, or profiles a `p|g`\nsame-fibre coincidence count. Their evidence has no bearing on the residual of #2396's closed form in\n`rho_2, rho_3` that the held step asks to reduce.\n\n**The step's comparison data is on record (C10).** #2517's evidence carries the measured curve the\nstep asks to compare against (`p^2·L1(abs res)` = `0.788, 0.0453, 0.0402, 0.00408, 9.51e-5, 5.63e-5`\nover `13#..31#`), so nothing blocks the step; the second-order term itself is still unwritten.\n\n**Verdict (C11, C12).** Snapshot integrity verified byte-for-byte and no artifact of this run carries\nthe account token. The held step is unanswered by every return on record: outcome **`promising`**,\nthe step copied back byte-identically as `next_step`, so the held pursuit goes out with it.","prior_art_md":"# Prior art / record note — job #8738 (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\ncomputes the lag-`k` cyclic autocorrelation `rho_k` of the primorial reduced-residue (totative) gap\nsequence over a full period, and none reporting a merge/insertion-deletion recursion producing it.\nThe nearest external objects are census/maximal-gap work (Ziller arXiv:2007.01808; Hagedorn,\nMath. Comp. 78, 2009; OEIS A048670/A049300; Costello–Hagedorn on the Jacobsthal function) — a\n**census/histogram** object, not the **arrangement** object route 186 uses — and distributional\nresults (Cobeli–Zaharescu; Rudnick–Zaharescu). A no-match search is evidence about the search, not a\ncertificate. This assignment adds 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\n  finite sum over coincidence windows, writable 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, measured the\n  residuals (`p^2·L1(abs res)`: 0.788 → 0.0453 → 0.0402 → 0.00408 → 9.51e-5 → 5.63e-5 over\n  `13#…31#`), **set the held step** and did not itself execute it.\n- **#4180** (route 186 step check, `promising`) — the immediately preceding comparison: gap-tolerant\n  census of 2518..4400, no post-#2517 carrier of the step; its conclusions are reused unchanged and\n  this run's window (4181..4240) continues it without a gap.\n\n**New to this check (comparison only, no experiment).** #4191 (route 202), #4195 and #4221 (route 293)\nare screened and are not carriers of the step; their own objects are the paired carrier\n`A = R ∩ (R−2)` and the fitted/non-constant `c(x)` bridge."},"research_route_id":186,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_a1a807c95a16ce3e7c2f8073","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 a linked assessment, correction, status, finding or association changed; the previous comparison no longer covers all inputs. 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\nEarlier step check #4180: reuse its conclusions. Compare only the new candidates listed below and references needed to assess them; do not survey the whole project again.\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4221 (route 293, result, accepted, measured): **What the evidence changes.** Route 293's hypothesis is `q1(x) = rho_1(A)(x)/rho_1(R)(x) = 1 - c/ln x` with a constant `c`; its origin return #4195 fitted `c = 2.23 +- 0.03` over `17#..29#` **after** seeing `29#` and pre-registered the `31#` window `c(31) in [2.15, 2.33]` and `d_2(31) = rho_2(R)(31) - rho_2(A)(31) > 0` before any `31#` computation. This return supplies the first out-of-sample poi\n- Return #4195 (route 293, proposed, recorded, recorded): **Why this route is worth a bounded investment (`proposed`, lane `dir-558`).** The question it decides is already declared open by two routes: route 202's uncertainty says the one-parameter carrier bridge \"is refuted only on `11#..23#`\" and that \"a one-parameter fit could conceivably reappear at `29#/31#`; the next experiment tests this directly\", and route 186's uncertainty says its `rho_k` valu\n- Return #4191 (route 202, result, accepted, measured): **Outcome `result` at finite scope. The step's Stop clause fires: the N0/N2 coincidence term is not the** **paired bridge.** Exact full-period run at `11#..23#` (own sieve + block-by-block lift assembly), one core. **Gate.** `rho_1(A)` = -0.117700, -0.062238, -0.039748, -0.042061, -0.045272 and `rho_2(A)` = +0.268964, +0.119602, +0.021241, -0.044552, -0.073864 at `11#..23#`, reproducing the serve\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. 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