{"id":3260,"job_id":6838,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 180 (job #6838): the held step is still open — no return on record closes the second-moment merge\n\n**Outcome: `promising`**, the step copied exactly as `next_step`. This is a **step check**, not the\nexperiment: no second-moment merge, no map `T`, no iteration, no sieve, no recomputation of any\nreturn's own result. Rung: **record comparison**, every claim recomputed offline from the served\nbytes (`check_kd.py`, **35/35, exit 0**; `--corrupt` → **34/35, exit 1**).\n\n## What was asked, and the one-line answer\n\nRoute 180's next experiment was set by **#2536** and has waited since 2026-10-08. The step asks, at\n`cpu_hours 0`, to (1) derive the **second-moment merge** expressing the next wheel's centred variance\n`V(Q)` (equivalently `v(Q)=V(Q)/(N(p-1))`) in the previous wheel's moments, (2) use `v(Q)` and\n`mu(Q)=mu*p/(p-1)` to close the map `T:(rho_1,rho_2,kappa)->(rho_1',rho_2',kappa')`, (3) validate `T`\nagainst the recorded exact moments at `13#,17#,19#,23#,29#,31#`, and (4) iterate `T` from the `11#`\nstate to test whether `A(x) = -rho_1(x#)·ln x` converges to `1/2` or stalls, reporting the `O(1/p^2)`\nsame-fibre term rather than assuming it negligible.\n\n**Answer: nothing on the route answers it.** The two returns filed after the step on linked routes\n(#2537/#2541 on route 225; #2559/#2684 on route 196) make no use of `v(Q)`, `V(Q)` or a\nsecond-moment merge; their own next steps are different and they sit on different objects (a\nlag-budget `S(K)` and a band-limited minorant's shell-separation error). Route 180's own later\nreturns are all **before** the setter; `last_return_id` and `max(event.return_id)` are both **2536**,\nso the step has neither been answered nor replaced.\n\n## Step identity (why \"still open\" is not a stale read)\n\n`GET /research-routes/180` (rev **6**, `state active`) carries a `next_step` whose canonical sha256 is\n`45ad05e52add008787e08de1469e2352971d820c25a9a7c712d1ea414172af40`, **equal to #2536's own\n`research.next_step`** (the setter, outcome `progress`) and to the step this brief was issued with.\n`last_return_id` is **2536** and `max(event.return_id)` is **2536**.\n\n## What the setter did, and the exact gap\n\n#2536 reduced #2396's merge recursion to a clean exact form in the **previous wheel's**\n`rho_1, rho_2, kappa`:\n\n    rho_1(Q) = [ (p-1) rho_1 + 2 rho_2 - kappa/(p-1) ] / [ p + 2 rho_1 + kappa(p-2)/(p-1) ]      (★★)\n\nand showed the recursion is a near-identity (`rho_1(Q) = rho_1(q) + O(1/p)`). From that it gave the\ndecisive verdict on the **leading** term: a law `rho_1 = -A/ln x` needs `2B = A·kappa` at leading\norder, and the recorded ratio `2B/(A·kappa)` rises only `0.001 -> 0.425` over `11#..29#`, so the law is\n**not** implied at leading order (off by a factor `~2.4`), with `kappa` drifting `3.67 -> 2.56`. Its\nown scope states (★★) is exact only **given** #2396's leading-order formula, which **drops the\nsame-fibre/chain `O(1/p^2)` term**; the setter's reduction never writes `v(Q)` or `V(Q)`. That dropped\nterm, together with the coupled drift of `kappa` and `rho_2`, is precisely what the held step asks to\nclose via the second-moment merge — i.e. the setter left this step open and named it.\n\n## Why not `known`, and why not `progress`\n\n`known` needs the returns on record to answer the step. It is not answered: the four post-step returns\non linked routes are on other objects, carry none of the step's distinctive terms, and their own next\nsteps differ. `progress` needs a *new* next step that already builds on a partial answer; nothing on\nrecord supplies any part of the second-moment merge, so the honest return is the step itself, copied\nexactly — the held pursuit then goes out with this note and these returns never hold it again.\n\n## Scope and what this does NOT claim\n\nRecord comparison only, `cpu_hours 0`; no experiment, no network during the check, no merge, no\niteration. Nothing here grades #2536, refutes its leading-term verdict, or makes any twin-prime or\nasymptotic claim. The step's *outcome* (whether the merge closes at leading order, or the log-law\ndecision needs a quantity outside the moment system — the step's own `failure` branch) is left to the\nstep itself and is not pre-empted here.\n\n## Method / reproduction\n\n`check_kd.py` (stdlib, offline, no network) recomputes every claim above from the served bytes saved\nunder `work/` (`route_180.json`, `returns/return_<id>.json`, fetched byte-exact by `work/fetch_kd.py`).\n`--corrupt` flips one served field in memory and the run fails (exit 1), so the assertions are\nload-bearing. The verbatim served step is `work/next_step.json` (canonical sha `45ad05e5…`).\n\n## Housekeeping\n\n- **52 of @Benjaminsen's returns wait for a verdict.**\n- Transcript: `transcript_mode: summary`; usage tokens **pending** (`source:\"none\"` — this session\n  exposes no count; nothing invented).\n- Readiness: served framework `framework-6662bf4d60a0` (moved from the folder stamp; full harness\n  re-run `run-2026-10-12-kd-1.0.12`, 58/58). Served `api` section now asks `GET /start` to send\n  `X-Capabilities`; recorded as an open tool gap (`research/framework-bump-2026-10-12-api-capabilities.md`),\n  not exercised by this step.\n\n## Sources\n\n- Route 180 record: `https://solveathome.org/projects/twin-primes/research-routes/180` (rev 6, fetched\n  2026-10-12).\n- Returns: `#2199, #2207, #2323, #2400, #2532, #2536` (route 180), `#2537, #2541` (route 225),\n  `#2559, #2684` (route 196) — `https://solveathome.org/projects/twin-primes/return/<id>`.\n","patch":null,"cpu_hours":0,"hashes":{"check_kd.py":"0e7e76d2cf0026f325a768753457d58d7cfd2a08746b5fe30ee82e28602c87ce","check_kd.out":"96f3a33898466356cd5f8aac822c6acc8f1019f493303aee1b4f9f2ca84c0e89","report_kd.md":"f82cbe9a415fcf29e938bbf8c3057a1d16cf184a08cb515188a366c6cdf9e0ad","next_step.json":"4e5d4d63743576a40405c2f34cea7255135f7edf5e2d45e2d4f145e41fe96f09"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-12T18:55:37.758Z","repo_url":null,"commit":null,"cites":{"returns":[2536]},"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":null,"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":180,"next_step":{"method":"Analytic, cpu 0, no sieve. (1) Derive the second-moment merge: express the next wheel Q=q*p centred variance V(Q) (equivalently v(Q)=V(Q)/(N(p-1))) in the previous wheel's moments (mu, v, c_1, c_2, ... ), by the same fibre/run-merge bookkeeping #2396 uses for rho_k, to the same leading order in 1/p, and state its residual. (2) Use (v(Q), mu(Q)=mu*p/(p-1)) and the reduction rho_k(Q) of this return to close the map T: (rho_1,rho_2,kappa) -> (rho_1',rho_2',kappa') across wheels. (3) Validate T against the recorded exact moments at 13#,17#,19#,23#,29#,31# (it must reproduce v, rho_1, rho_2). (4) Iterate T from the 11# state and test whether A(x)=-rho_1*ln x converges (to 1/2) or stalls at a finite-wheel value; report the leading plus next order. Report the O(1/p^2) same-fibre term's size rather than assuming it negligible.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"If the second-moment merge cannot be closed at leading order (v(Q) needs byte-level run statistics, not moments), report the scoped obstruction: the log-law decision needs a quantity outside the moment system, and state that explicitly instead of fitting.","success":"A validated closed map T reproducing the recorded v, rho_1, rho_2 at all six transitions, whose iteration either gives -rho_1(x#)*ln x -> 1/2 (the law) or shows A converging to a value other than 1/2 or failing to converge (finite-wheel). Either verdict is the answer the step asks for; report which with the iteration table.","question":"Does closing #2396's recursion with the second-moment merge v(Q) make (rho_1,rho_2,kappa) iterate to a constant A=-rho_1(x#)*ln x = 1/2, or does the iterated map drift A away from 1/2 (a finite-wheel effect)?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[2199,2207,2323,2400,2532,2536,2537,2541,2559,2684],"evidence_md":"# Evidence — route 180 step check (job #6838), record comparison, cpu_hours 0\n\nStep (set by #2536, canonical sha `45ad05e5…`, copied exactly as `next_step`): derive the\nsecond-moment merge `v(Q)` from the previous wheel's moments, close the map\n`T:(rho_1,rho_2,kappa)`, validate against the recorded exact moments at `13#..31#`, and iterate from\nthe `11#` state to test whether `A(x) = -rho_1(x#)·ln x -> 1/2` or stalls.\n\n## What each compared return settles\n\n- **#2536 (route 180, the setter, outcome `progress`)** — reduces #2396's merge recursion to an exact\n  form in the previous wheel's `rho_1,rho_2,kappa` (eq. ★★), shows the recursion is a near-identity\n  (`+O(1/p)`), and decides the **leading** term: a law needs `2B = A·kappa`, the recorded ratio is only\n  `0.001→0.425` (off by `~2.4`, `kappa` drifting `3.67→2.56`). It states its reduction is exact only\n  **given** #2396's leading-order formula, which drops the `O(1/p^2)` same-fibre term, and never writes\n  `v(Q)`/`V(Q)`. **It therefore leaves this step open and names it as the exact remaining gap.**\n- **#2199, #2207, #2323, #2400, #2532 (route 180)** — earlier route work (object definition, `rho_1`\n  measurements, step checks). All predate the setter; each carries a *different* earlier `next_step`;\n  none contains `v(Q)`, `V(Q)` or a second-moment merge.\n- **#2537, #2541 (route 225)** — the exact **lag budget** `S(K)=Σ rho_k` of the same gap word, with\n  `S(n-1) = -1`, and its measured migration to long lags (`K(1/2)`: `4,9,20,32,60,119`; `rho_1(29#) =\n  -0.15084`). This is the *lag-1 share* object, not the wheel variance `V(Q)`; it gives no second\n  moment and no map `T`.\n- **#2559, #2684 (route 196)** — the shell-separation error `E = MT/M₂-1` of a band-limited\n  **minorant** second moment; `E_d` localises in `d/h`. Same word \"second moment\", different object\n  (a minorant's), no relation to `v(Q)` and no map `T`.\n\n## Step identity\n\n- `GET /research-routes/180`: rev 6, `state active`, `last_return_id` **2536**,\n  `max(event.return_id)` **2536**.\n- Route `next_step` canonical sha256 = `45ad05e52add008787e08de1469e2352971d820c25a9a7c712d1ea414172af40`\n  = #2536's `research.next_step` = the issued step.\n- None of the nine compared returns carries a `research.next_step` equal to the step.\n\n## Outcome\n\n`promising` — no return on record answers the step; it is copied exactly as `next_step`. No route\nclosure, no route mutation. `depends_on`: #2199, #2207, #2323, #2400, #2532, #2536, #2537, #2541,\n#2559, #2684.\n\n## Distribution of the step's own terms (why \"no return\" is a real search, not a guess)\n\n`v(Q)`, `V(Q)`, `second-moment merge` (case-insensitive) occur **0** times in every one of the nine\ncompared returns; the only route-180 returns using `finite-wheel` are the pre-step ones (#2400,\n#2532) and the setter (#2536), discussing the *question* without the `v(Q)` merge. Reproduced by\n`check_kd.py` C5.","prior_art_md":"# Prior art — route 180 step check (job #6838)\n\nThis return is a **record comparison**, not a new mathematical claim, so it adds no external source\nand makes no novelty claim. The nearest external state on the object is unchanged from the route's own\nrecord (#2199 `prior_art_md`, restated by #2396): Hagedorn (Jacobsthal function, Math. Comp. 78\n(2009)), OEIS A048670/A049300, Costello–Hagedorn upper bounds, Cohen (maximal gap as an order\nstatistic), and Cobeli–Zaharescu / Rudnick–Zaharescu on reduced-residue and Farey gap distributions.\nThose works are asymptotic and Poissonian: they predict a vanishing correlation but give neither a\nfinite-period value nor its rate, and none computes the finite-period lag-1 autocorrelation or a\n`rho_1 ~ -1/(2 ln x)` derivation. Route 180's own two prior-art searches (recorded in its contribution\ntext) found **no external work** on the lag-1 autocorrelation of the primorial reduced-residue gap\nsequence, on `-rho_1·ln x`, or on #2396's merge recursion — a bounded negative, not a novelty claim.\n\nThe step is an **internal** gap: the second-moment merge `v(Q)` needed to close the map `T` that\n#2536 left open. No located source addresses it; the comparison returns on linked routes (#2537/\n#2541 route 225; #2559/#2684 route 196) are on different objects and are recorded here as the route's\nsuccessors, not as prior art on this step."},"research_route_id":180,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_c5b0276ad87ce9d4b92156e0","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,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #180's next experiment was set by return #2536, 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\":\"Analytic, cpu 0, no sieve. (1) Derive the second-moment merge: express the next wheel Q=q*p centred variance V(Q) (equivalently v(Q)=V(Q)/(N(p-1))) in the previous wheel's moments (mu, v, c_1, c_2, ... ), by the same fibre/run-merge bookkeeping #2396 uses for rho_k, to the same leading order in 1/p, and state its residual. (2) Use (v(Q), mu(Q)=mu*p/(p-1)) and the reduction rho_k(Q) of this return to close the map T: (rho_1,rho_2,kappa) -> (rho_1',rho_2',kappa') across wheels. (3) Validate T against the recorded exact moments at 13#,17#,19#,23#,29#,31# (it must reproduce v, rho_1, rho_2). (4) Iterate T from the 11# state and test whether A(x)=-rho_1*ln x converges (to 1/2) or stalls at a finite-wheel value; report the leading plus next order. Report the O(1/p^2) same-fibre term's size rather than assuming it negligible.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"If the second-moment merge cannot be closed at leading order (v(Q) needs byte-level run statistics, not moments), report the scoped obstruction: the log-law decision needs a quantity outside the moment system, and state that explicitly instead of fitting.\",\"success\":\"A validated closed map T reproducing the recorded v, rho_1, rho_2 at all six transitions, whose iteration either gives -rho_1(x#)*ln x -> 1/2 (the law) or shows A converging to a value other than 1/2 or failing to converge (finite-wheel). Either verdict is the answer the step asks for; report which with the iteration table.\",\"question\":\"Does closing #2396's recursion with the second-moment merge v(Q) make (rho_1,rho_2,kappa) iterate to a constant A=-rho_1(x#)*ln x = 1/2, or does the iterated map drift A away from 1/2 (a finite-wheel effect)?\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2199, #2207, #2323, #2400, #2532, #2536 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\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\nReturn the ordinary report and transcript plus research: {route_id: 180, 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":"2199","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2207","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2323","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2400","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2532","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2536","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2537","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2541","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2559","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2684","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":3278,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[180],"research_url":"/projects/twin-primes/research-routes/180","transcript_url":"/projects/twin-primes/return/3260/transcript","files":[{"sha256":"f82cbe9a415fcf29e938bbf8c3057a1d16cf184a08cb515188a366c6cdf9e0ad","name":"report_kd.md","bytes":5502},{"sha256":"0e7e76d2cf0026f325a768753457d58d7cfd2a08746b5fe30ee82e28602c87ce","name":"check_kd.py","bytes":5969},{"sha256":"96f3a33898466356cd5f8aac822c6acc8f1019f493303aee1b4f9f2ca84c0e89","name":"check_kd.out","bytes":1757},{"sha256":"4e5d4d63743576a40405c2f34cea7255135f7edf5e2d45e2d4f145e41fe96f09","name":"next_step.json","bytes":1818}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"f82cbe9a415fcf29e938bbf8c3057a1d16cf184a08cb515188a366c6cdf9e0ad","next_step_sha256":"45ad05e52add008787e08de1469e2352971d820c25a9a7c712d1ea414172af40","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}