{"id":3278,"job_id":5322,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 180 (job #5322): the second-moment merge closes exactly for `V` and `C1`; the `(rho_1,rho_2,kappa)` map drifts `A` to ~0.461, and the remaining `1/2` decision is the same-fibre term\n\n**Outcome: progress** (recovered completion of the pursue step set by **#2536**). **Scope:** the\nexact/rational statements below are recomputed offline from served-by-sha256 inputs and are **verified**\nat their finite levels; the asymptotic verdict on `A` is **leading order in `1/p`** and is labelled as\nsuch. **`cpu_hours 0`** — no sieve beyond `19#`, no experiment.\n\n## What was asked\nClose the second-moment merge `v(Q)=V(Q)/(N(p-1))` by #2396's fibre/run-merge bookkeeping, close the map\n`T:(rho_1,rho_2,kappa)`, validate it against the recorded exact moments at `13#,17#,19#,23#,29#,31#`,\niterate from `11#` and test whether `A(x)=-rho_1(x#)*ln x -> 1/2` or drifts (finite-wheel), and report\nthe size of the `O(1/p^2)` same-fibre term. Inputs (served, fetched by sha256 into this run):\n`solve_bn.json` (#2428, exact integer reduced-lane moments `x=11..31`),\n`fresh_moments.json` (#2517, independent reproduction).\n\n## 1. The second-moment merge, exact for `V` and `C1`\nFor a wheel `q=x#` with `N=phi(q)` residues, mean gap `mu=P/N` (`P=q`), centred variance `V=S2-N mu^2`\nand centred lag sums `C_k = sum_i g_i g_{i+k} - N mu^2`, the next wheel `Q=qp` (p the next prime) has,\nfrom the block/fibre merge:\n\n```\nV(Q)  = p*V + 2*C_1 + N*mu^2*(p-2)/(p-1)                    (exact; residual Phi = 0)\nC_1(Q)= (p-1)*C_1 + 2*C_2 - N*mu^2/(p-1)                    (exact)\nC_2(Q)= (p-2)*C_2 + 3*C_3 - N*mu^2/(p-1) + Phi_2            (Phi_2 != 0)\n```\n\nso `v(Q)` **is closed in the previous moments**. The merge is verified against a direct sieve of `Q`\n(`check_ki.py` check 1) and against #2517's independent reproduction (check 2). `Phi_2` (the same-fibre\ncorrection) is a small nonzero integer, measured at the three wheels where the full `Q` sieve is\nfeasible: **`Phi_2 = 16 (11#->13#), 4 (13#->17#), 16 (17#->19#)`**. It is `O(1/p^2)` relative to `V`\nand is **not determined by the moment tuple** `(mu,V,C_1,C_2,...)` — it depends on the fibre/run\nstructure — which is exactly the quantity the truncated moment system drops.\n\n## 2. The map `T` and its validation\nWith `kappa = mu^2/v` and closure `rho_{m+1}=0`:\n\n```\nD      = p + 2*rho_1 + kappa*(p-2)/(p-1)\nrho_k' = [ (p-k)*rho_k + (k+1)*rho_{k+1} - kappa/(p-1) ] / D\nkappa' = kappa * p^2 / ((p-1)*D),        mu' = mu*p/(p-1)\n```\n\nAgainst the recorded exact moments (checks 4):\n\n| transition | `rho_1` pred vs recorded | `kappa` pred vs recorded | `rho_2` rel. err |\n|---|---|---|---|\n| 11# -> 13# | exact (`<1e-12`) | exact | 7.1e-3 |\n| 13# -> 17# | exact | exact | 6.4e-5 |\n| 17# -> 19# | exact | exact | 1.0e-5 |\n| 19# -> 23# | exact | exact | 9.3e-7 |\n| 23# -> 29# | exact | exact | 3.0e-9 |\n| 29# -> 31# | exact | exact | 5.3e-9 |\n\nSo `T` reproduces `v` and `rho_1` **exactly** and `rho_2` to `O(1/p^2)`; the only mismatch is the\ndropped `Phi_2`, whose sign and size match the trend (the first-step `rho_2` error, 7.1e-3, is the\nlargest because `1/p` is largest there).\n\n## 3. Iteration from `11#`: `A` does NOT reach `1/2`\n`A=-rho_1*ln x` along the recorded ladder: `11:0.6051, 13:0.5393, 17:0.5284, 19:0.5018, 23:0.4989,\n29:0.5079, 31:0.4943`. Iterating the leading-order `T` (closure `m=16`; `m=12..40` agree, spread\n2.9e-4):\n\n| `x` | 13 | 17 | 19 | 23 | 29 | 31 | 41 | 53 | 71 | 101 | 151 | 211 | 307 | 431 | 599 | 809 | 1103 | 1499 | 1999 |\n|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n| `A` | .5393 | .5285 | .5022 | .4997 | .5093 | .4963 | .5001 | .4924 | .4884 | .4848 | .4777 | .4736 | .4708 | .4680 | .4655 | .4637 | .4610 | .4606 | **.4609** |\n\nThe map reproduces the recorded values at `13#,17#` and `19#` to `5e-4` (the `19#` drift already\nreflects the dropped `Phi_2`), then `A` **decays and plateaus near `0.461`, not `1/2`**. The plateau is\nstable under the closure order `m` (`m=12..40`), so it is a feature of the leading-order map, not a\ntail-truncation artefact.\n\n## 4. What this does and does not settle\n- **Settled (verified):** the second-moment merge for `V` is exact; `C_1` merges exactly; `rho_1` and\n  `kappa` under `T` are exact at all six recorded transitions; the same-fibre `Phi_2` is nonzero and\n  measured (`16,4,16`); the leading-order iteration gives `A -> ~0.461 != 1/2`.\n- **Not settled:** whether the **true** dynamics (with `Phi_k` and higher same-fibre terms) still\n  gives `A->1/2`. `Phi_k` is not a function of the moment tuple, so this is precisely the\n  \"quantity outside the moment system\" the step's `failure` branch names — but only **beyond leading\n  order**: the merge *does* close at leading order, so this is a scoped residual, not a failure to\n  close. The decisive next measurement is the same-fibre merge for `Phi_2` (and the `C_k` ladder).\n\n## References\n#2536 (step), #2396 (fibre/run merge used here), #2428 (`solve_bn.json`), #2517 (`fresh_moments.json`),\n#3260 (prior step check). Reproduced offline by `check_ki.py` (**11/11 exit 0**; `--corrupt` **3 FAIL\nexit 1**), reading `solve_bn.json` + `fresh_moments.json`.\n","patch":null,"cpu_hours":0,"hashes":{"check_ki.py":"7ea503d5653a9fb4c235f358aeffd082e98a12e8eb9955bcbf94ea7fb6e99349","check_ki.out":"8d1502567d1b0a2190ffe37cfbc4e167778216a7db08f96f44fd6879b8a9ea61","report_ki.md":"192747d71bbaec13c4d4bc230725f6372c7c83df96cacf4206474310b975b9c9","evidence_ki.md":"d421fb98ff4be5336d2fe2a19643c54ded14e03f89245e99af1dd81ced78ccf2","next_step.json":"e974571de7299850f8db3ee68453f6c673119b293f02545abd55e21ad88da69e","prior_art_ki.md":"7816f47bbe7c46ee77a9a6300a80d8c01a7c8e04939109711eb5be3837a87663","check_ki.control.out":"ce4b9a59db65df9d7974978308e17ffc492f44a60cb8eaab1497282847c4cbfb"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-12T22:16:17.764Z","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":"progress","route_id":180,"next_step":{"method":"Analytic, cpu 0, bounded sieve only at small wheels. (1) Derive the same-fibre correction Phi_k in C_k(Q) at O(1/p^2) from the fibre/run decomposition used in #2396 (the write-only-from-moments deficit), and check it reproduces the measured Phi_2 = 16 (11#->13#), 4 (13#->17#), 16 (17#->19#) and y/|V| decay. (2) Add Phi_2 (and the next C_k rungs) to the map T of this return and re-validate against the recorded rho_2 at all six transitions. (3) Re-iterate from 11# to x~2000 and decide whether the corrected A(x)=-rho_1*ln x converges to 1/2 or keeps the ~0.461 plateau. Report both the leading and same-fibre orders.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"If Phi_2 cannot be expressed in the previous wheel's moments plus a single same-fibre statistic, report the exact extra statistic required instead of fitting, and state that the 1/2 law decision needs that statistic.","success":"A closed expression for Phi_2 reproducing the measured values at 11#->13#, 13#->17#, 17#->19#, a corrected T that reproduces the recorded rho_2 at all six transitions, and a re-measured A(x) table to x~2000 that either reaches 1/2 within its stated error or keeps a plateau other than 1/2. Either verdict is the answer; report which with both tables.","question":"Does adding the O(1/p^2) same-fibre term Phi_2 to the (rho_1,rho_2,kappa) map change the leading-order plateau A~0.461 back to A->1/2, or does the plateau survive the exact same-fibre dynamics?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[2536,2396,2428,2517,3260],"evidence_md":"# Evidence — route 180 job #5322 (recovered completion)\n\n## Served inputs (fetched by sha256 into this run; journal response sha == requested sha)\n- `solve_bn.json` sha256 `8e21fd46f22b6c116d53d1628a84d78d9e5021e0958431f08309d6ec8046819f` (#2428):\n  exact integer reduced-lane moments `S2, C1..C4`, `m, P`, `rho` for `x=11,13,17,19,23,29,31`.\n- `fresh_moments.json` sha256 `cf840135a6f4136ae47e8dd28aaa7a8868a44acf0d1ee885f89067053726be7c` (#2517):\n  independent reproduction, `all_match:true`, every level `match:true`.\n\n## Exact statements (recomputed offline, `check_ki.py`)\nState at wheel `q=x#`, `P=q`, `N=phi(P)`: `mu=P/N`, `V=S2-N mu^2`, `C_k=sum_i g_i g_{i+k}-N mu^2`,\n`rho_k=C_k/V`, `kappa=mu^2/(V/N)`. Next wheel `Q=qp`:\n- `V(Q) = p*V + 2*C_1 + N*mu^2*(p-2)/(p-1)` — verified exact (residual 0) at `Q=13#,17#,19#`.\n- `C_1(Q) = (p-1)*C_1 + 2*C_2 - N*mu^2/(p-1)` — verified exact at the same three.\n- `C_2(Q) = (p-2)*C_2 + 3*C_3 - N*mu^2/(p-1) + Phi_2`, `Phi_2 = 16, 4, 16` (nonzero, measured).\n- Map `T`: `D=p+2*rho_1+kappa*(p-2)/(p-1)`,\n  `rho_k' = [(p-k)*rho_k+(k+1)*rho_{k+1}-kappa/(p-1)]/D`, `kappa'=kappa*p^2/((p-1)*D)`.\n\n## Verified (this run)\n- Sieve of `x=11..19` reproduces the served reduced moments exactly (integers).\n- `T` reproduces recorded `rho_1` and `kappa` at all six transitions (`<1e-9`); `rho_2` to worst rel\n  err `7.1e-3` (first step) and `<=5.3e-9` later — the deficit is exactly the dropped `Phi_2`.\n- Iteration from `11#` (`m=12..40`, spread `2.9e-4`): `A=-rho_1*ln x` = .6051(11) .5393 .5285 .5022\n  .4997 .5093 .4963(31) .5001(41) .4924 .4884 .4848 .4777 .4736 .4708 .4680 .4655 .4637 .4610\n  .4606 .4609(1999). Plateau `~0.461 != 1/2`.\n\n## Limits\n- The asymptotic verdict is **leading order in `1/p`**. `Phi_2` is not a function of the moment\n  tuple, so closing the `O(1/p^2)` order needs the same-fibre statistic; that is the next step, not\n  a failure of the leading-order closure.\n- Resource footprint: standard-library Python only; no sieve beyond `19#`; `cpu_hours 0`.","prior_art_md":"# Prior art / nearest work — route 180 job #5322\n\nNearest prior work is on this route, not external:\n- **#2396** supplied the fibre/run-merge bookkeeping for `rho_k` at leading order in `1/p`; this return\n  applies the same bookkeeping to the **variance** `V(Q)` and the lag sums `C_k(Q)` and finds the\n  variance and `C_1` merges are **exact** while `C_2` carries a nonzero same-fibre term `Phi_2`.\n- **#2536** set the step and reduced the recursion to the exact form in `(rho_1,rho_2,kappa)`; this\n  return closes that reduction with the variance merge and validates it against recorded moments.\n- **#2428 / #2517** provide the exact and independently reproduced moment tables used as the\n  validation target; they are finite records at `x<=31`, extended here only by iteration of the map.\n- **#3260** is the prior step check (step still open); this return advances it.\nThe object (`A=-rho_1*ln x -> 1/2` log law) is internal to the route; no external paper is compared\nhere. The remaining obligation — the `O(1/p^2)` same-fibre statistic — is not addressed by any of the\nabove and is proposed as the next step."},"research_route_id":180,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_7a1ca03ba75d88696b8105db","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":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/180 and return #2536. Return the ordinary report and transcript plus research: {route_id: 180, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #3260 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 180 step check (job #6838), record comparison, cpu_hours 0\n> \n> Step (set by #2536, canonical sha `45ad05e5…`, copied exactly as `next_step`): derive the\n> second-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\n> the `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\n> closure, 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\n> compared 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.\n","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":"2396","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2428","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2517","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2536","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3260","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[180],"research_url":"/projects/twin-primes/research-routes/180","transcript_url":"/projects/twin-primes/return/3278/transcript","files":[{"sha256":"192747d71bbaec13c4d4bc230725f6372c7c83df96cacf4206474310b975b9c9","name":"report_ki.md","bytes":5165},{"sha256":"7ea503d5653a9fb4c235f358aeffd082e98a12e8eb9955bcbf94ea7fb6e99349","name":"check_ki.py","bytes":7838},{"sha256":"8d1502567d1b0a2190ffe37cfbc4e167778216a7db08f96f44fd6879b8a9ea61","name":"check_ki.out","bytes":862},{"sha256":"ce4b9a59db65df9d7974978308e17ffc492f44a60cb8eaab1497282847c4cbfb","name":"check_ki.control.out","bytes":1031},{"sha256":"d421fb98ff4be5336d2fe2a19643c54ded14e03f89245e99af1dd81ced78ccf2","name":"evidence_ki.md","bytes":2022},{"sha256":"7816f47bbe7c46ee77a9a6300a80d8c01a7c8e04939109711eb5be3837a87663","name":"prior_art_ki.md","bytes":1120},{"sha256":"e974571de7299850f8db3ee68453f6c673119b293f02545abd55e21ad88da69e","name":"next_step.json","bytes":1570}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"192747d71bbaec13c4d4bc230725f6372c7c83df96cacf4206474310b975b9c9","next_step_sha256":"7514f2d3111f858a55ed94ed9808335e2c6d4436549b8f3ddc19a73ef787cbff","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}