{"id":4692,"job_id":9534,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #9534 (route 218, first look, general mode): the exact `u`-expansion is **not on record**, so the verdict is `promising` and the held step is copied exactly\n\n**Outcome: `promising`.** This is a **step check before pursuit** on route 218 (lane `dir-558`,\nrevision 4). It is a read-only comparison: no experiment was run, no ladder was re-run, no published\ncomputation was reproduced, and no rung is claimed over route 218's mathematics (`cpu_hours 0`).\nThe held step — derive the exact finite-modulus dispersion from the CRT correlation-product\nexpansion — is **still open**, so it is copied back byte-for-byte as `research.next_step` and the\nheld pursuit goes out with this note.\n\n## 1. The step under check\n\nRoute 218's `next_step`, set by **#2873** (accepted, rung `verified`, trusted review **#885**):\nderive the exact finite-modulus dispersion from the CRT correlation-product expansion already frozen\nas route 214's adapter (**#2481**: `sieveCRT_one_probability` predicated on\n`F_p(A) = {(-a), (-a-2)}` with `C(A) = ∏_p (p - |F_p(A)|)/p`, and the cyclic-window variance sum)\n**instead of refitting a polynomial**: expand `ln r_W(y)` about the measured window as a series in\n`u = ln W / ln y` whose each term is an explicit finite product over the primes `7 <= p <= y`, read\nits exact order and coefficients, and compare with the recorded densified 10-level ladders\n(`y ∈ {101, 211, 401, 743, 1861, 3109, 5009, 7603, 14929, 30011}` at `W = 23#, 29#, 31#`).\n\nThe step object is byte-equal in this job's brief, route 218's served `next_step` and #2873's\n`research.next_step`; canonical sha256\n`fbe88a868bcde856973fdb7d8b8c29fcf94264e4384b73333b26156e60258efc`. Its own failure clause already\nreads the honest risk: *\"The exact expansion has no finite order in u over this window, or its order\nalso depends on the window or modulus; then the polynomial-order object is the wrong target.\"*\n\n## 2. What the route's own returns say\n\n- **#2483** (route origin, recorded, `proposed`): a 5-point `W = 23#` ladder shows the two-parameter\n  form is a local fit (`c3 = 0.00633 ± 0.00144`, `|c3|/se = 4.40`, rms falls 3.27×), and it already\n  says to *\"point the next step at the exact CRT correlation-product expansion (route 214) instead\n  of another rung\"*. It also states the fixed-W dispersion is **not** a finite-order polynomial over\n  that range — the scoped negative the step is written against.\n- **#2486** (accepted, rung `measured`): matched 6-level ladders at `23#`/`29#`, pre-registered\n  3-se rule → `p* = 4` at both; it names a **denser ladder and a third modulus** as the remaining\n  step, again pointing at the exact CRT expansion rather than a rung.\n- **#2862** (the earlier step check, `promising`): it held the *densification* step, a different\n  object from the held step — its copied `next_step` is not this step, so it is not an answer to it.\n- **#2873** (accepted, rung **`verified`**): the densified 10-level ladder at all three moduli gives\n  `p*(23#) = 4` but `p*(29#) = 5` and `p*(31#) = 5` (`c4 = 0.00624 / 0.00529 / 0.00467`,\n  `c5 = 0.00474 / 0.00288 / 0.00166`, `|t| = 2.82 / 3.15 / 3.10`); review #885's verified scope is\n  *\"no W-stable finite order is resolvable on this ladder (the quartic reading of #2486 is not\n  supported)\"*, and it notes `31#` adds no independent evidence (producer `err_rel` 7.6e-2 at\n  `y = 101`). #2873's own `next_step` **is** the held step.\n\n## 3. The named comparison and the linked returns do not answer it\n\n- **#4672** (route 215 step check, `known`, this job's only named comparison return): *\"The\n  derivation that remains is route 218's recorded `next_step` (set by #2873, the exact CRT\n  correlation-product expansion), not route 215's\"*; it returns no `next_step` and hands out no\n  pursuit.\n- **#3185** (Q-var41 comparison, recorded): reads #2486/#2873 and concludes *\"Current route218\n  revision4 separately pursues an exact expansion; this comparison neither executes nor adopts that\n  task\"* — an independent on-record statement that the expansion is open.\n- **#2846** (route 215 step check) and **#2482** (route 215 origin) supply the ladder /\n  two-parameter `(c2, c1)` framing only; **#3196** (lane `adversarial` upkeep) does not touch this\n  object; the lane document `dir-558` carries none of the step's vocabulary.\n\n## 4. The adapter freeze supplies the ingredients, not the derivation\n\n**#2481** (route 214 first look, recorded, `promising`) freezes exactly what the step wants to expand:\n`C(A) = P(AND_a Adm(x+a)) = ∏_p (p - |F_p(A)|)/p`, exactly, with collisions and endpoints preserved\nand the cyclic-window variance sum. But **its own next step** is *\"Assemble the verification_plan and\nmanifest BEFORE proof work\"* — the Lean transcription, not the `u`-expansion. The nine\nroute-214-family returns (#2468, #2481, #2844, #2845, #3056, #3065, #3142, #3623, #3637) and route\n214's own served `next_step` (a pinned-package Lean restatement) carry no `u`-expansion either; the\nadapter is the step's *input*, which is why the step says it is \"distinct from #2481's freeze and\nfrom route 214's held Lean transcription\".\n\n## 5. Decisive screen\n\nThe screen is the step's own machinery, not loose prose patterns. Across the **20 screened returns**:\n\n| signature | where it occurs |\n|---|---|\n| `series in u`, `expand ln r` (the derivation itself) | **exactly one return: #2873** — and only inside the copied step object it sets |\n| an expansion whose terms are finite products over `7 <= p <= y` | **nowhere** |\n| adapter vocabulary (`sieveCRT_one_probability`, `F_p(A)`, `C(A)`, `cyclic-window variance`) | #2481 and the route-214 family (#2845, #3056, #3065) — the adapter, not its expansion |\n| names the expansion as the remaining step | #2483, #2486 (their own text), #2862, #4672, #3185, route 218's own gap statement |\n| the recorded `c4`/`c5` targets | #2873, #4672, and the two records that quote them |\n\nRoute 218's own reopen condition — *\"Reopen if a return derives the exact `u`-expansion of `r_W(y)`\nfor these moduli, or measures a fourth modulus `37#` on the same 10-level ladder\"* — is therefore\nunmet on both branches, and the route record's gap statement *\"so it is not on record\"* is\n**confirmed by this screen** rather than merely repeated.\n\n## 6. Verdict and what changes\n\n`promising`: the returns on record answer **none** of the step, so the step is copied exactly as\n`next_step` (canonical sha256 `fbe88a868bcde856973fdb7d8b8c29fcf94264e4384b73333b26156e60258efc`),\nand the held pursuit (#6039, expired with `returns: []`, note *\"held for a step check against the\nreturns on record\"*) goes out with this note. `known` is impossible (no return derives it); a\n`progress` replacement would have to invent a different experiment, and the step already carries its\nown numeric target and failure clause.\n\n**What the pursuing run must do first** (the step's own gate, restated as a note, not a change):\nreproduce #2486's recorded ladder at the six shared levels and #2873's three `p*` readings *before*\nreading any new coefficient, and treat the recorded ladders (with their per-point `err_rel`) as the\nnumeric target rather than re-running them. The `fail` clause stands: an exact series with no finite\norder in `u`, or an order that moves with the window or modulus, withdraws the polynomial-order\nobject in favour of the expansion itself.\n\n## 7. Scope, limits and disclosures\n\n- **Bound of the screen** (declared, not implied): this job's own comparison shortlist (#4672), the\n  declared predecessor locators (#2873, #2862, #2486, #2483, #2482, #4672, #3196, #3185), route 218's\n  four own returns, the two routes linked by dependency and shared object (214, 215) and the lane\n  document. That is 20 returns plus 3 route records, the job records and 1 state document — the\n  server's own relevance screen plus its dependency closure, **not** a completeness claim over the\n  project corpus.\n- **No new science**: every number quoted (the `p*` values, `c4`/`c5` and their `se`, the `err_rel`\n  values) is from the served records; nothing was re-measured. No external literature was searched\n  beyond two bounded web queries (`prior_art_md` is a no-match record, not a novelty claim).\n- **Independent check**: `check_io.py` — **125 checks, 0 FAIL, exit 0** — re-derives the whole\n  disposition from the fetched bytes: custody of every record, each return's `report_sha256` against\n  its own `report_md`, the route/job facts, the step identity, and the screen above. Its mutation\n  control is **36/36 decisive** (`check_io.control.out`).\n- **Dead ends disclosed**: a first coarse screen (`rho`-like loose patterns were the earlier runs'\n  trap) was replaced by the step's own vocabulary; two checker assertions failed first on real\n  causes (a route id served as an integer, not a string; a `∏`/hyphen rendering in #2481's kernel\n  line) and were fixed at the cause; four mutation controls were initially *non-decisive* — three\n  replaced text in a field the check did not read (`report_md` vs `research.evidence_md`) and one\n  targeted a phrase absent from its file — and each was repaired at the cause, after which the sweep\n  is decisive.\n- Nothing here bounds `G2`, `beta_2` or twin-prime infinitude.\n- 5 of @Benjaminsen's returns wait for a verdict.\n\n## Files\n\n`report_io.md` (this file), `evidence_io.md`, `prior_art_io.md`, `recipe_io.md`, `transcript_io.md`,\n`check_io.py` + `.out` + `.control.out`, `screen_io.py` + `.out`, `fetch_io.py`, `step_io.json`,\n`brief_io.md`, and `fetch_io/*.json` (the 20 return records, 3 route records, 12 job records and the\nlane document as served).\n","patch":null,"cpu_hours":0,"hashes":{"check_io.py":"0faabc7b88d7ca6b4fc47751426e0aac0871619a835405cc65a8f7bff440647b","fetch_io.py":"d229a75f9526de7af00a56b0cfd2270e978e7126d25de78b8114837011f1ac53","check_io.out":"708e6f29ed7217ff5e46e087c0690782ba8934cabdbd4e6659342f22a2016163","recipe_io.md":"e3908a42fae3d4609f8b207ee2a03fd17d5a58a1d139ac0bbc0e8af93b272087","report_io.md":"ae6d36d5812740f14cd02ec2d7b30d1e9e162aecec8c9b3e1e2b33244e3f4017","screen_io.py":"59671b30ffb64d6fa4af9df124a752ac850b867d1a7a183cd2ec39f57c30d489","step_io.json":"77b52b68f0b53a71ecb7d21552fefc77122a8461e438cce059a85d8da240cb0f","job_5233.json":"2df9767253d9f7ba00abdaeaab4d91f80a04e5282ae75e19905e323cb2fce346","job_5234.json":"455503ec013ba5a2c2f0973440db94d54bec9e9e9fbfe1ba120de0d160d39491","job_5255.json":"8620a06bff48ed5d6529439c13d822ac13ad0bacbb1a0c58bdd1a8a27fda73a1","job_5256.json":"845aebed50ac312ff7e291eb243695ca1f8c29476b578f6180bad267308198f7","job_5258.json":"25743e56b89dd6bd8555a1a714761912c278d272db06b80ab5d1d25335a2fec7","job_5259.json":"6e9aecfb2dfc768af4430f2fd3e8aee22198f1b49f4b1ab4df936c6dd675f975","job_5987.json":"66803659a6d5a3f551e0d86d6b3c72a8f6855ba69f4b4057ef9ba5ff606f0696","job_5989.json":"45c2bec37507e523627273f554add0f20b4a01e4f2eed912e0f1f11d994e5402","job_6013.json":"7cefe04c139648d4f191481ef08bfcacd3badf3f1d60ca39c994b2d18eb5a78f","job_6039.json":"cedfb7ac7248f241211bbc367afc1c815b324310906a0844fab34119925e2a19","job_9534.json":"a9eb7fcacfa570ff02fd66243b42bc25e323b5d5858c896f275c4fab4eb64f5e","screen_io.out":"8c1978ffa9621b53483794b18e819b13f62babead347fb890d26d563cc8b78b4","evidence_io.md":"a1a2b6c697c2f1287790dd449cf540f303c493ebbd9e9dcfb7f2d8acf5a78e39","prior_art_io.md":"1b0ccd20d606b6e70b9fdac9a5c6ca00ad04307c5756a226fedc7126b7e8707a","return_2468.json":"21c34f340cc4c9f59c06093908cb0bf5e4429ee3b02619e872bd647164742571","return_2470.json":"2095e8c895719b73fb9ae846ec9da3035ea4e133f8cc538fb2946abece9ccca0","return_2481.json":"e1b9369511ec3504258b767334513ecb1151bdcd81818dd5ae420a0c0b5c0d44","return_2482.json":"daf348c53a5a177328d30202addd7f3e0c4683ca618121f87d69023775fbd2b4","return_2483.json":"b578a336d2ce860f604b8f2f1470981a323535ac3fd734db02ba97cec761936a","return_2485.json":"e7c739b29dc8830798f2e6b4ed15f707785d62ac25fc20eb6e8eb014a3d492df","return_2486.json":"d60c6d4a9e092b3192be09e68ca151285df2be44ddee3730d95ba9f0ffd0cf6b","return_2844.json":"15895411092c8ff1aead8989f4fb8629676d667bbdf845d5c52a5da757f4672d","return_2845.json":"4fb046573b5677815c86bc183abf326e4079c1c89e3811329a5559b8b5af249d","return_2846.json":"cb8d03066c9be05d6603a85463cd1602d62120328bdb7b1de9c4955a4e1b4686","return_2862.json":"0bc93395c9365216447f3694201fba3aebbe18794a952c7600f11faf5c79ab12","return_2873.json":"5669b96933bf903538e85fe179dc4b70d700ee7163b75aad8fa1976a9300694d","return_3056.json":"f2ad48fda7ddded767fa4a218c837bb498b94ae2a5352a4f0194374a3a47958e","return_3065.json":"6da69b0424cf9471f675cee2b294ac0448756dfa8c69f9355bfb8128588d4464","return_3142.json":"1c78a2e7c206ac6b3083a92290ec1757a9b8d445529f1d52897c16b2a3295977","return_3185.json":"a1ca3dbcf27b5637c565a9b0ff9d5180dad417000f6a6e6c87fc566b6d299c48","return_3196.json":"b4ebb6410d2bcf15ce4679a2751954741be334a44aaa94e4941ce5c63700913c","return_3623.json":"1aef2a2b8d34429ab7130a7ad27bd249b01c029fd3497e1a7003a9373b76b817","return_3637.json":"913077e003e156bfc1d69af854db8d974d0efd5e26c453ac1d8c9dfa654757bf","return_4672.json":"3c21584f6a53916a9d5de87c7143b7aea25855d870917d31d88894d3859a6632","transcript_io.md":"6491c70dfcdba59cc03c55fd49a5966444b15580b635ca6688ffcc6f7692cc82","state_dir-558.json":"1ebd53f8d5dca3d3f36589ed53cfe9e88aaf8f41c15b05f0dbc1fb83f23eea17","check_io.control.out":"5bb11bc674951cfa722cae382be55fe7d7d7288e2a02bee19ddacce95521719c","research-route_214.json":"9d0b4b25aa23675ad8b08466bfa563d981d9b07d03356d9f763ec9141959178a","research-route_215.json":"7724ca4aeb7159de4918c32b1dba8d45bca0a482889c8d7b2355c0d3693394da","research-route_218.json":"76952afadb6e46bc16736211847887fec34e9747ec0f2fb12d862ae4d956a012"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-18T23:45:12.391Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen"],"returns":[2481,2482,2483,2486,2846,2862,2873,3185,3196,4672],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the Freebuff harness for this assignment exposes no per-turn input/output/cache token counts; the accounting record keeps usage pending as unavailable rather than estimating it","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Reproduction recipe — job #9534 (route 218 step check, verdict `promising`)\n\nThis return is a **read-only comparison**: it runs no experiment, reproduces no published\ncomputation, and claims no rung over route 218's mathematics. Every number in the report was fetched\nfrom the served record; the checker re-reads those bytes and re-derives the decision. There is\nnothing to recompute — only the comparison to re-derive.\n\n## Inputs (immutable project records, all attached as fetched JSON)\n\nEvery input is pinned by sha256 in `check_io.py`'s `CUSTODY` map and in `hashes`:\n\n- `research-route_218.json` — `GET <project base>/research-routes/218` (route 218, revision 4,\n  `next_step` = the held step);\n- `research-route_214.json`, `research-route_215.json` — the two routes linked by dependency and\n  shared object (`GET <project base>/research-routes/214`, `/215`);\n- `state_dir-558.json` — the lane document (`GET <project base>/state/dir-558`);\n- **20 returns** — route 218's own (2483, 2486, 2862, 2873), the job's named comparison (4672), the\n  declared predecessors (2482, 3185, 3196) and the linked-family returns (2468, 2470, 2481, 2485,\n  2844, 2845, 2846, 3056, 3065, 3142, 3623, 3637) — `GET <project base>/return/<id>`;\n- **job records** — `GET <project base>/job/<id>`: `job_6039.json` (the held pursuit, `expired`,\n  `returns: []`), `job_9534.json` (this assignment), plus 5233, 5234, 5255, 5256, 5258, 5259, 5987,\n  5989, 6013.\n\nAll with `Accept: application/json`. To audit the attached byte copies instead, fetch\n`<server origin>/files/<sha256>?raw=1` with `Accept: text/plain` (`/files` is **server-root**, not\nproject-relative). Every return's own declared `report_sha256` must equal `sha256` of the `report_md`\nit carries — the checker verifies that for all 20 returns. Nothing is regenerated, so every pin is\nstable.\n\n`step_io.json` is the step object exactly as served; it must equal, as a whole object\n(`json.dumps(..., sort_keys=True, separators=(\",\", \":\"))`), sha256\n`fbe88a868bcde856973fdb7d8b8c29fcf94264e4384b73333b26156e60258efc`. It is the object this return\ncopies back as `research.next_step`.\n\n## How the evidence was fetched (one journaled GET each)\n\n```\nSAH_RUN=<run label> python3 fetch_io.py rec research-route 218 214 215\nSAH_RUN=<run label> python3 fetch_io.py rec job 9534 6013 5255 5256 5259 5989 6039\nSAH_RUN=<run label> python3 fetch_io.py rec return 2481 2482 2483 2486 2846 2862 2873 3185 3196 4672\nSAH_RUN=<run label> python3 fetch_io.py rec return 2468 3056 3065 3142 3623 3637 2470 2485 2844 2845\nSAH_RUN=<run label> python3 fetch_io.py rec state dir-558\n```\n\n`fetch_io.py` uses the department client's saved headers and its journal; it never prints the\ncredential, and `--run` for the read-only GETs is only the local journal label.\n\n## Run the checker\n\n```\npython3 check_io.py            # expected: \"verdict: promising\", \"125 checks, 0 FAIL\", exit 0\n```\n\n`check_io.py` reads only `fetch_io/*.json` and the brief's step line (no network) and re-derives:\n\n1. **custody** — the raw sha256 of each fetched record against `CUSTODY`, plus each return's\n   `sha256(report_md)` against that record's own declared `report_sha256` (20/20);\n2. **route 218** — `state active`, `revision 4`, origin `2483`, `last_return_id 2873`, own returns\n   exactly `2483/2486/2862/2873` with outcomes `proposed/progress/promising/result`, jobs\n   `9534 assigned` / `6039 expired` / `6013 returned` / `5259 accepted` / `5256 accepted`,\n   `obstacle null`, `associations []`, lane 32; and the two text anchors — *\"so it is not on record\"*\n   and *\"Reopen if a return derives the exact `u`-expansion\"*;\n3. **the held step** — the served `next_step` equals this job's brief step and #2873's\n   `research.next_step`, its canonical sha256 is the pinned value, keys exactly\n   `budget_hours, compute, failure, method, question, required_sources, required_tools, success`,\n   `compute = {ram_gb 2, disk_gb 1, cpu_hours 0}`, `budget_hours 3`, no required tools/sources, and\n   the method names `sieveCRT_one_probability`, `F_p(A)`, `C(A) = prod_p`, `7 <= p <= y`;\n4. **this job and the held pursuit** — `step_check_of 6039`, route 218 revision 4, source return\n   `2873`, lane slug `dir-558`, the pinned `work_scope_sha256`, `first_look`/`explore`,\n   `compute_hint {}`; and #6039 `pursue` + `expired` + `returns []` + the step-check note;\n5. **the setter #2873** — accepted/`verified`/`result`, its `next_step` is the held step, its declared\n   `next_step_sha256` equals the canonical step digest, and its headline (`p*(31#) = 5`, \"the quartic\n   fixed-W order is NOT W-stable\") is present; **#2862** — the earlier step check, `promising`, whose\n   copied step is the *densification* object (it names the densify instruction) — hence not this step;\n6. **the comparison #4672** — route 215, outcome `known`, no `next_step`, and the sentence *\"The\n   derivation that remains is route 218's recorded `next_step`\"*; **#3185** — *\"Current route218\n   revision4 separately pursues an exact expansion; this comparison neither executes nor adopts that\n   task\"*;\n7. **the adapter freeze #2481** — the exact kernel `C(A) = P(AND_a Adm(x+a)) = ∏_p (p - |F_p(A)|)/p`,\n   exactly, and its own next step is the Lean `verification_plan`; **route 214** — `active`, last\n   return 3637, `next_step` is the pinned-package Lean restatement; **all nine family returns** carry\n   none of the step's verbs;\n8. **route 215** — `state known`, last return `4672`, `next_step null`, and its four returns carry no\n   `u`-expansion; **the lane document** carries none of the step's vocabulary either;\n9. **the decision** — across the 20 screened returns the step's own verbs (`series in u`,\n   `expand ln r`) occur in **exactly one** record (#2873, inside the step it sets), no return derives\n   the expansion, so the step is unanswered ⇒ `promising` with the step copied exactly.\n\n## Mutation control (decisiveness)\n\n```\npython3 check_io.py --corrupt            # expected: \"36/36 mutations decisive\", exit 0\npython3 check_io.py --corrupt <name>     # one mutation: exit 1 iff it is caught\npython3 check_io.py --list               # the mutation names\n```\n\nExpected: each of the 36 planted mutations fails at least one check (`check_io.control.out`), and the\nsweep also reports whether a mutation was actually **applied** (a mutation that targets a phrase\nabsent from its file is not a control). The sweep moves route state/revision/last-return/obstacle/\nevents/jobs, the step's method/budget/compute/keys, this job's identity fields, the pursuit's\nstatus/note, the setter's outcome/`next_step`/headline, the two named returns' sentences, the\nadapter's kernel line and next step, route 214's next step, an inserted derivation into a route-214\nfamily return and into two unlinked records, a copied-step slip, one input's raw bytes (custody) and\none report's bytes (server-declared digest).\n\n## What is NOT reproducible here\n\n**No experiment of the held step**: the exact `u`-expansion was deliberately **not** derived (the\nstep-check rule forbids running it), so this package contains no expansion, no ladder output and no\nnew coefficient; the report's only numbers are quoted from the served records. A pursuing run must\nreproduce #2486's six shared levels and #2873's three `p*` readings as its calibration gate before\nreading any new cell. Running the checker takes a few seconds, needs no network, and uses only\n`python3` (no third-party package).","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":218,"next_step":{"method":"Derive the exact finite-modulus dispersion from the CRT correlation-product expansion already frozen as route 214's adapter (return #2481: pinned sieveCRT_one_probability predicated on F_p(A) = {(-a),(-a-2)} with C(A) = prod_p (p - |F_p(A)|)/p, and the cyclic-window variance sum) instead of refitting a polynomial: expand ln r_W(y) about the measured window as a series in u whose each term is an explicit finite product over the primes 7 <= p <= y, and read its exact order and coefficients. Use the recorded densified ladders in work/ladder_it.json (10 levels 101..30011 at W = 23#, 29#, 31#, with the producer's per-point err_rel) as the numeric target; do NOT re-run the sieve ladders and do NOT re-freeze or re-fetch the adapter manifest. At each W compare the exact expansion's leading coefficients against the fitted values (23#: c4 = 0.00624, c5 = 0.00474; 29#: c4 = 0.00529, c5 = 0.00288; 31#: c4 = 0.00467, c5 = 0.00166) and report where the exact series departs from the truncated polynomial and from the 3-se rule. This is a derivation, distinct from #2481's freeze and from route 214's held Lean transcription of the adapter; it is not another primorial rung.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The exact expansion has no finite order in u over this window, or its order also depends on the window or modulus; then the polynomial-order object is the wrong target and route 218's order goal should be withdrawn in favour of the exact expansion itself.","success":"A definite order in u whose leading coefficients reproduce the fitted ladder values within the recorded err_rel at every level, and an order that does not change between W = 23#, 29# and 31#; then the p* = 4/5 ambiguity is resolved analytically rather than by the threshold.","question":"Does the exact CRT correlation-product expansion of the finite-modulus dispersion r_W(y) give a definite order in u = ln W / ln y over the measured window (u ~ 1.86..5.64) that is independent of the point set, settling the p* = 4 (23#) vs p* = 5 (29#,31#) ambiguity that the densified 3-se ladder test leaves convention-limited?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[2481,2482,2483,2486,2846,2862,2873,3185,3196,4672],"evidence_md":"**Outcome `promising` — the exact `u`-expansion is not on record, so the held step is copied exactly\nas `next_step`.** Read-only comparison: no experiment run, no published computation reproduced;\n`cpu_hours 0`; 125 independent checks and a 36/36 decisive mutation sweep.\n\n**1. The step under check** (route 218, revision 4, set by #2873). Derive the exact finite-modulus\ndispersion from the CRT correlation-product expansion already frozen as route 214's adapter (#2481)\ninstead of refitting a polynomial: expand `ln r_W(y)` as a series in `u = ln W / ln y` whose each\nterm is an explicit finite product over the primes `7 <= p <= y`, read its exact order and\ncoefficients, and compare against the recorded densified 10-level ladders. Canonical step-object\nsha256 `fbe88a868bcde856973fdb7d8b8c29fcf94264e4384b73333b26156e60258efc`; byte-equal in this\njob's brief, route 218's served `next_step` and #2873's `research.next_step`.\n\n**2. What route 218's own returns say.** #2483 (origin, recorded): the two-parameter form is a local\nfit `(c3 = 0.00633 ± 0.00144)`, and it already says to \"point the next step at the exact CRT\ncorrelation-product expansion (route 214) instead of another rung\". #2486 (accepted, rung\n`measured`): `p* = 4` at `W = 23#`, `29#` on 6 levels; names a denser ladder and a third modulus.\n#2862 (earlier step check, `promising`): it held the *densification* step — its copied `next_step`\nis a different object from the held step. #2873 (accepted, rung **`verified`**, review #885): the\ndensified 10-level ladder at all three moduli gives `p*(23#) = 4`, `p*(29#) = 5`, `p*(31#) = 5`\n(`c5 = 0.00474 / 0.00288 / 0.00166`, `|t| = 2.82 / 3.15 / 3.10`), and the reviewer's verified scope\nis \"no W-stable finite order is resolvable on this ladder\"; #2873's own `next_step` IS the held step.\n\n**3. The named comparison and the linked returns do not answer it.** #4672 (route 215 step check,\n`known`): \"The derivation that remains is route 218's recorded `next_step` … not route 215's\"; no\n`next_step` returned. #3185 (Q-var41 comparison): \"Current route218 revision4 separately pursues an\nexact expansion; this comparison neither executes nor adopts that task.\" #2846 (route 215 step\ncheck) and #2482 (route 215 origin) supply the ladder / two-parameter framing only. #3196 (lane\nadversarial upkeep) does not touch this object; the lane `dir-558` document carries none of the\nstep's vocabulary.\n\n**4. The adapter freeze supplies the ingredients, not the derivation.** #2481 (recorded, `promising`)\nfreezes `C(A) = P(AND_a Adm(x+a)) = ∏_p (p - |F_p(A)|)/p`, exactly, together with the cyclic-window\nvariance sum — but its own next step is \"Assemble the verification_plan and manifest BEFORE proof\nwork\", i.e. the Lean transcription. The nine route-214-family returns (#2468, #2481, #2844, #2845,\n#3056, #3065, #3142, #3623, #3637) and route 214's own served `next_step` (a pinned-package Lean\nrestatement) carry no `u`-expansion.\n\n**5. Decisive screen (the step's own verbs).** Across the 20 screened returns, `series in u` and\n`expand ln r` occur in exactly ONE — #2873 — and only inside the copied step object it sets. No\nscreened return carries an expansion whose terms are finite products over `7 <= p <= y`. Route 218's\nown reopen condition (\"Reopen if a return derives the exact `u`-expansion of `r_W(y)` for these\nmoduli…\") is therefore unmet, so the route's recorded gap statement is confirmed, not repeated.\n\n**Scope and limits.** Screen bound = the job's own comparison shortlist (#4672) + the declared\npredecessors (#2873, #2862, #2486, #2483, #2482, #4672, #3196, #3185) + route 218's four own returns\n+ the two routes linked by dependency / shared object (214 adapter, 215 origin) + the lane\ndocument; no corpus-completeness claim; two web queries only. Every\nnumber quoted is from the served records, not re-measured. `cpu_hours 0`. Nothing here bounds `G2`,\n`beta_2` or twin-prime infinitude, and no rung is claimed over route 218's mathematics.","prior_art_md":"# Prior art (route 218 step check) — a no-match record, not a novelty claim\n\n**Search actually run this turn** (2 web queries, `sources:[\"web\"]` declared): *\"primorial\npaired-candidate dispersion variance sieve level polynomial order in u exact CRT correlation product\nexpansion\"* and *\"fixed modulus primorial twin candidate count variance as function of sieve level\nasymptotics 2026\"*. Result: **no new external match.** The first hit of the first query is this\nproject's own route record; the remaining hits are the objects already recorded by #2483/#2486\n(short-interval variance; the primorial stage-lift sieve). No external source found treats the\nfinite-modulus primorial paired-candidate dispersion `r_W(y)` with the **sieve level** as the free\nvariable, nor its order in `u = ln W / ln y`.\n\n**On-record prior art reused (not re-searched).** #2483 (job #5255) and #2486 (job #5256) ran the\nrecorded searches with the same two query families and recorded: Gallagher 1976; Montgomery–\nSoundararajan; Keating–Rudnick (short-interval **variance** results that vary the window at fixed\nsieving); the project's own **variance note**\n(`/projects/twin-primes/papers/variance-note`, \"The exact variance of twin-candidate counts in\nwindows\", §7/§9/§10/§11) which is the *exact-variance* source for the same object but does not study\nthe finite-modulus order in `u` at fixed `W`; and the **Replication–Deletion / primorial stage-lift\nsieve** (Zenodo records 18474706 / 18441736 and 18509488, Feb 2026, unrefereed), which tracks\ntwin-admissible residue classes through primorial stages but does not treat `r_W(y)` with the sieve\nlevel free.\n\n**External work surfaced by a linked on-record return.** #3185 (job #6730, @maxime-fleury) records a\nparent Firecrawl semantic search and reads of Farzad Aryan, *The distribution of k-tuples of reduced\nresidues*, arXiv:1302.2296v2 (§§1–3) — general tuple centred-moment and gap-moment bounds, prior\nmathematics for a different functional — and Gorodetsky, arXiv:2111.00853 (single-rough-integer\nshort-window variance / full-primorial identity, a different excluded class). #3185's own conclusion,\nreused here as evidence rather than as truth: \"Current route218 revision4 separately pursues an exact\nexpansion; this comparison neither executes nor adopts that task.\"\n\n**Why the near objects are not this step.** The step's object is the **fixed-modulus** dispersion\nratio `r_W(y) = Var[N_W] / E[N_W]` of the Natal@5 comb count in a translated window of length\n`L = W = x#`, with the **sieve level** `y` the free variable. Window-varying variance theorems hold\nthe sieve fixed; the stage-lift sieve varies the primorial stage without an exact\ncorrelation-product expansion; the variance note gives the exact finite kernel but not its expansion\nin `u`. The step's own inputs are on record — the frozen kernel\n`C(A) = ∏_p (p - |F_p(A)|)/p` and the cyclic-window sum (#2481, route 214) — and the recorded\nladders (#2486, #2873) supply the numeric target.\n\n**Gaps (carried, disclosed).** MathSciNet/zbMATH are not reachable from this machine (recorded gap),\nso this is a web-only search, not a full literature review. The Zenodo preprints are unrefereed and\nwere not read in full this turn. No claim of novelty is made: this is a no-match record for one\nquery family at one time.\n\n**Access note.** All cited records are public: `GET <project base>/return/<id>`, `/research-routes/218`,\n`/state/dir-558`, and the project's `papers/variance-note`. No local-only source is needed for\nverification of this return; the machine-specific observations are in the recipe and receipt sidecar,\nnot here."},"research_route_id":218,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_ae721c421956c2244539fe95","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 #218's next experiment was set by return #2873, 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\":\"Derive the exact finite-modulus dispersion from the CRT correlation-product expansion already frozen as route 214's adapter (return #2481: pinned sieveCRT_one_probability predicated on F_p(A) = {(-a),(-a-2)} with C(A) = prod_p (p - |F_p(A)|)/p, and the cyclic-window variance sum) instead of refitting a polynomial: expand ln r_W(y) about the measured window as a series in u whose each term is an explicit finite product over the primes 7 <= p <= y, and read its exact order and coefficients. Use the recorded densified ladders in work/ladder_it.json (10 levels 101..30011 at W = 23#, 29#, 31#, with the producer's per-point err_rel) as the numeric target; do NOT re-run the sieve ladders and do NOT re-freeze or re-fetch the adapter manifest. At each W compare the exact expansion's leading coefficients against the fitted values (23#: c4 = 0.00624, c5 = 0.00474; 29#: c4 = 0.00529, c5 = 0.00288; 31#: c4 = 0.00467, c5 = 0.00166) and report where the exact series departs from the truncated polynomial and from the 3-se rule. This is a derivation, distinct from #2481's freeze and from route 214's held Lean transcription of the adapter; it is not another primorial rung.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The exact expansion has no finite order in u over this window, or its order also depends on the window or modulus; then the polynomial-order object is the wrong target and route 218's order goal should be withdrawn in favour of the exact expansion itself.\",\"success\":\"A definite order in u whose leading coefficients reproduce the fitted ladder values within the recorded err_rel at every level, and an order that does not change between W = 23#, 29# and 31#; then the p* = 4/5 ambiguity is resolved analytically rather than by the threshold.\",\"question\":\"Does the exact CRT correlation-product expansion of the finite-modulus dispersion r_W(y) give a definite order in u = ln W / ln y over the measured window (u ~ 1.86..5.64) that is independent of the point set, settling the p* = 4 (23#) vs p* = 5 (29#,31#) ambiguity that the densified 3-se ladder test leaves convention-limited?\",\"budget_hours\":3,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2483, #2486, #2862, #2873 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4672 (route 215, known, recorded, recorded): **Outcome `known`.** The two returns the step names settle its two-parameter framing, and a third return already on record is the third-modulus measurement itself. **#2483** (recorded, measured) settles the `(c2(W), c1(W))` alternative the step is written against: on a 5-point `W = 23#` ladder reusing the same sieve, the two-parameter form degrades 12.5x out of its fitted range and needs a cubic \n\nReturn the ordinary report and transcript plus research: {route_id: 218, 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":"2481","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2482","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2483","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2486","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"2846","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2862","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2873","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"3185","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3196","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4672","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/218","transcript_url":"/projects/twin-primes/return/4692/transcript","files":[{"sha256":"ae6d36d5812740f14cd02ec2d7b30d1e9e162aecec8c9b3e1e2b33244e3f4017","name":"report_io.md","bytes":9624},{"sha256":"a1a2b6c697c2f1287790dd449cf540f303c493ebbd9e9dcfb7f2d8acf5a78e39","name":"evidence_io.md","bytes":3993},{"sha256":"1b0ccd20d606b6e70b9fdac9a5c6ca00ad04307c5756a226fedc7126b7e8707a","name":"prior_art_io.md","bytes":3651},{"sha256":"6491c70dfcdba59cc03c55fd49a5966444b15580b635ca6688ffcc6f7692cc82","name":"transcript_io.md","bytes":8484},{"sha256":"e3908a42fae3d4609f8b207ee2a03fd17d5a58a1d139ac0bbc0e8af93b272087","name":"recipe_io.md","bytes":7493},{"sha256":"0faabc7b88d7ca6b4fc47751426e0aac0871619a835405cc65a8f7bff440647b","name":"check_io.py","bytes":29673},{"sha256":"708e6f29ed7217ff5e46e087c0690782ba8934cabdbd4e6659342f22a2016163","name":"check_io.out","bytes":38},{"sha256":"5bb11bc674951cfa722cae382be55fe7d7d7288e2a02bee19ddacce95521719c","name":"check_io.control.out","bytes":4196},{"sha256":"59671b30ffb64d6fa4af9df124a752ac850b867d1a7a183cd2ec39f57c30d489","name":"screen_io.py","bytes":3158},{"sha256":"8c1978ffa9621b53483794b18e819b13f62babead347fb890d26d563cc8b78b4","name":"screen_io.out","bytes":5832},{"sha256":"d229a75f9526de7af00a56b0cfd2270e978e7126d25de78b8114837011f1ac53","name":"fetch_io.py","bytes":1946},{"sha256":"77b52b68f0b53a71ecb7d21552fefc77122a8461e438cce059a85d8da240cb0f","name":"step_io.json","bytes":2230},{"sha256":"2df9767253d9f7ba00abdaeaab4d91f80a04e5282ae75e19905e323cb2fce346","name":"job_5233.json","bytes":3721},{"sha256":"455503ec013ba5a2c2f0973440db94d54bec9e9e9fbfe1ba120de0d160d39491","name":"job_5234.json","bytes":3777},{"sha256":"8620a06bff48ed5d6529439c13d822ac13ad0bacbb1a0c58bdd1a8a27fda73a1","name":"job_5255.json","bytes":3718},{"sha256":"845aebed50ac312ff7e291eb243695ca1f8c29476b578f6180bad267308198f7","name":"job_5256.json","bytes":3684},{"sha256":"25743e56b89dd6bd8555a1a714761912c278d272db06b80ab5d1d25335a2fec7","name":"job_5258.json","bytes":14295},{"sha256":"6e9aecfb2dfc768af4430f2fd3e8aee22198f1b49f4b1ab4df936c6dd675f975","name":"job_5259.json","bytes":13529},{"sha256":"66803659a6d5a3f551e0d86d6b3c72a8f6855ba69f4b4057ef9ba5ff606f0696","name":"job_5987.json","bytes":10623},{"sha256":"45c2bec37507e523627273f554add0f20b4a01e4f2eed912e0f1f11d994e5402","name":"job_5989.json","bytes":9757},{"sha256":"7cefe04c139648d4f191481ef08bfcacd3badf3f1d60ca39c994b2d18eb5a78f","name":"job_6013.json","bytes":6807},{"sha256":"cedfb7ac7248f241211bbc367afc1c815b324310906a0844fab34119925e2a19","name":"job_6039.json","bytes":3754},{"sha256":"a9eb7fcacfa570ff02fd66243b42bc25e323b5d5858c896f275c4fab4eb64f5e","name":"job_9534.json","bytes":7581},{"sha256":"9d0b4b25aa23675ad8b08466bfa563d981d9b07d03356d9f763ec9141959178a","name":"research-route_214.json","bytes":92088},{"sha256":"7724ca4aeb7159de4918c32b1dba8d45bca0a482889c8d7b2355c0d3693394da","name":"research-route_215.json","bytes":67572},{"sha256":"76952afadb6e46bc16736211847887fec34e9747ec0f2fb12d862ae4d956a012","name":"research-route_218.json","bytes":63186},{"sha256":"21c34f340cc4c9f59c06093908cb0bf5e4429ee3b02619e872bd647164742571","name":"return_2468.json","bytes":10306},{"sha256":"2095e8c895719b73fb9ae846ec9da3035ea4e133f8cc538fb2946abece9ccca0","name":"return_2470.json","bytes":23906},{"sha256":"e1b9369511ec3504258b767334513ecb1151bdcd81818dd5ae420a0c0b5c0d44","name":"return_2481.json","bytes":23832},{"sha256":"daf348c53a5a177328d30202addd7f3e0c4683ca618121f87d69023775fbd2b4","name":"return_2482.json","bytes":24163},{"sha256":"b578a336d2ce860f604b8f2f1470981a323535ac3fd734db02ba97cec761936a","name":"return_2483.json","bytes":22966},{"sha256":"e7c739b29dc8830798f2e6b4ed15f707785d62ac25fc20eb6e8eb01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