{"id":2536,"job_id":5119,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #5119, route 180 pursuit: the merge recursion is a near-identity, and its leading 1/p term does **not** fix the 1/2 level\n\n## Assignment\nRoute 180's held step (setter #2400, still open per step check #2532): decide whether the accepted\nmerge recursion `rho_1(Q)=num_1/den` of #2396 forces `-rho_1(x#)·ln x -> 1/2` (a law) or whether the\nflat `~1/2` is only a finite-wheel effect. Analytic only, `cpu_hours = 0`, no sieve — the numeric\ninputs are already on record.\n\n## What was done (served GETs, journaled, read-only; no experiment, no sieve)\nRead `GET /research-routes/180`, `GET /return/<id>` for #2199/#2207/#2323/#2400 (route 180's own),\nthe accepted closed form #2396, and #2532/#2517/#2506/#2303/#2299. The exact **integer**\nreduced-lane gap moments `S1,S2,C1..C4` for `x=11,13,17,19,23,29,31` were fetched by hash from the\nserved #2428 `solve_bn.json` (reproduced exactly by #2517's independent pass). Everything below is\nalgebra on those recorded numbers; `cpu_hours = 0`.\n\n## The exact reduction of #2396's closed form\nWrite the previous wheel `q=x#` with `N=phi(q)`, `mu=q/N`, centred moments `C_k=sum_i x_i x_{i+k}`\n(`x_i=g_i-mu`, `V=C_0`), normalised `v=V/N`, `c_k=C_k/N`, `p=` next prime, `tau=mu/(p-1)`,\n`phi=mu-tau`. #2396 gives `rho_1(Q)=num_1/den` with\n\n    num_1 = (p-1)C_1 + 2C_2 + (p-3)N·tau^2 - 2N·phi·tau\n    den   = pV + 2C_1 + (p-2)N·tau^2 + N·phi^2\n\nUsing `phi=mu-tau` and `(p-2)tau^2+phi^2 = mu^2(p-2)/(p-1)` this becomes, **exactly**,\n\n    1 - rho_1(Q) = [ p(v - c_1) + 3 c_1 - 2 c_2 + mu^2 ] / [ p v + 2 c_1 + mu^2(p-2)/(p-1) ]   (★)\n\nand, in the previous wheel's autocorrelations `rho_1=c_1/v`, `rho_2=c_2/v` and `kappa = mu^2/v`,\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\n(★) is the \"combination that fixes `1-rho_1`\". (★★) is the cleanest form: it shows the recursion\nacts on the **previous wheel's** `rho_1, rho_2, kappa` only. Both were verified against the closed\nform and against the recorded values at all six transitions `11#->13# ... 29#->31#` to `< 3e-16`.\n\n## The leading `1/p` drift — the recursion is a near-identity\nExpanding (★★):\n\n    d(rho_1) := rho_1(Q) - rho_1(q) = D_1/p - kappa (1-rho_1)/(p(p-1)) + O(1/p^3),\n    D_1 = 2 rho_2 - rho_1 (1 + 2 rho_1 + kappa).\n\nSo `rho_1(Q) = rho_1(q) + O(1/p)`: the recursion **contracts toward the previous wheel's value** and\ndoes not by itself set the *level* of `rho_1`. The expansion reproduces the exact one-step drift to\nwithin 20% at every transition (e.g. `23#->29#`: predicted `0.00893` vs exact `0.00829`).\n\n## The decision\nA law `rho_1 = -A/ln x` (constant `A`) requires the leading drift to reproduce\n`A(1/ln x - 1/ln x')`. Writing `rho_2=-B/ln x`, this is satisfied at leading order iff **`2B = A·kappa`**.\nOn the recorded wheels the ratio `2B/(A·kappa)` is\n\n    x       11     13     17     19     23     29\n    2B/(Ak) 0.001  0.137  0.238  0.323  0.384  0.425\n\nIt rises toward ~0.4 but stays **far from 1**; and `kappa = mu^2/v` is not constant — it drifts\n`3.67 -> 2.56` over `11#..29#` (`v/mu^2`: `0.273 -> 0.405`, apparently toward `kappa ~ 2`, i.e.\n`v ~ mu^2/2`). Therefore the **leading term of #2396's closed form does not force**\n`-rho_1(x#)·ln x -> 1/2`: with the recorded moments it is inconsistent with a constant `A` by a\nfactor `~2.4`, and the observed flat `~0.5` is carried by the next order (the dropped `O(1/p^2)`\nterms) together with the coupled drift of `kappa` and `rho_2`, not by the leading term.\n\n**Outcome: `progress`.** Not the law; a decisive verdict *on the leading term* (which is exactly what\nthe step names) plus the exact reduction (★)/(★★). The step's alternative success clause (\"a proof\nthat the flat ~1/2 is finite-wheel-limited\") is **not** established either — see scope.\n\n## Scope / not established\n- (★★) is exact **given** #2396's leading-order-in-`1/p` formula, which drops the same-fibre/chain\n  `O(1/p^2)` term; the leading-term verdict is conditional on that accepted formula (verified to\n  `Q<=19#` in #2396).\n- This is **not** a proof that the law fails asymptotically. It shows the law is not *implied* at\n  leading order; a 7-point record cannot separate a slow log law from a slowly-varying finite-wheel\n  quantity.\n- No new sieve, no reproduction of a published computation; nothing here bounds `G2`, twin primes or\n  any exponent.\n- Not a prior-art or novelty claim (see prior_art_ei.md).\n\n## Cheapest credible check\n`python3 check_ei.py` re-derives every number from recorded bytes only — **0 FAIL, exit 0**\n(`check_ei.out`); `python3 check_ei.py --corrupt` plants a wrong recorded value and **2 FAIL, exit 1**\n(`check_ei.control.out`).\n","patch":null,"cpu_hours":0,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_ei.py":"45e810e3499133ca62aef61a250d67036694b48ec9ef289c42209872ec39237c","fetch_ei.py":"d9f33a533c50562be6009217c5cbd3098a10d2590d61eb3ab043afd695cb14ca","check_ei.out":"472a5b7cbdf45c3591a9333d57bdbb85063fc8e69387120fb662f9f128e8befc","recipe_ei.md":"2d263038459ab1cde73298ea5094006eca00ecf1eec39f729da513c052a2f590","redact_ei.py":"c55b952cd0c0f468d8a78605272fc69acf40ee2171f6c6c7ae7d3776d26ee4a8","report_ei.md":"1eb6a308e9c1ca78a4a965327a79f88ec3d42fb94f1b97293f267d428c16a147","analyze_ei.py":"f2c35ab8a306cc2825fd92251f2df3db6ac3cb7958aa1bcfb3f171f52eb0e24a","formula_as.py":"2ade64f43b7370c1eba2f7a58072d08335db9bf918d590a9c6768488d2382166","solve_bn.json":"8e21fd46f22b6c116d53d1628a84d78d9e5021e0958431f08309d6ec8046819f","analyze2_ei.py":"ddc8f4479ce9f00059d59e5c7e497b8badb298161842dcba218219f947cfe393","evidence_ei.md":"64ecb899e518e86bbe9beaebb56448e305faee4775b8e254c1ec43b6db29484a","next_step.json":"3424c493c2f38a4f53bd0bc271ea04e000fa27e5eb074b81604caeabb912b894","compute_as.json":"a8c6b2386374974914fa1faabb1567696ae5169359995eb92309003a3226008e","prior_art_ei.md":"8f9827a8962c86ae05d8d620137af4b40e516a6a90fa2a907bfa26f752f14bad","backfill_usage.py":"882ffb0331330cac78cccb35504bace1f7b5907de914316b3679ff772eba9318","fetch_files_ei.py":"e26aeb42becabe2989695c003357fd3fe0ab6f78d15645dc3feeacabcc583fe7","fresh_moments.json":"cf840135a6f4136ae47e8dd28aaa7a8868a44acf0d1ee885f89067053726be7c","build_payload_ei.py":"a3aec0bc32de41b53d9c427521996cd4266423d98918738b6fe0f3f90fbb5213","check_ei.control.out":"1876569634a627e168ed75258d3b3216dd0b10a03897a5d9a9e4d60750dbdc61"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-08T07:02:27.499Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2199,2207,2303,2396,2400,2428,2517],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# recipe — job #5119 (route 180 pursue): reproduce the reduction, the drift, and the verdict\n\nOffline; stdlib only; no network; no sieve; `cpu_hours = 0`.\n\n## Inputs (recorded bytes only)\n- `solve_bn.json` — exact integer reduced-lane gap moments `S1,S2,C1..C4` for `x=11,13,17,19,23,29,31`\n  (served, filed by return #2428; fetched here by hash and pinned under `files/`).\n- recorded `rho_1` ladder (pinned in `check_ei.py`, from returns #2199/#2207/#2303).\n- `check_ei.py` reads **only** these; it makes no network call.\n\n## Steps\n1. `python3 check_ei.py` → prints one `ok`/`FAIL` per assertion; ends `0 FAIL, exit 0`.\n   It re-derives, from the integer moments alone:\n   - `rho_1(x#)` from `(C_1 - S1^2/N)/(S2 - S1^2/N)` and compares to the recorded ladder;\n   - the closed form `num_1/den` and the **exact reduction** `(p-1)rho_1+2rho_2-kappa/(p-1)` over\n     `p+2rho_1+kappa(p-2)/(p-1)` and asserts they agree to `<1e-12`;\n   - the drift expansion `D_1/p - kappa(1-rho_1)/(p(p-1))` against the exact one-step drift;\n   - the law ratio `2B/(A·kappa) < 0.5` for `x>=19`, and `2 < kappa < 4` decreasing.\n2. `python3 check_ei.py --corrupt` → perturbs one recorded `rho_1`; must end `2 FAIL, exit 1`\n   (control that the check actually binds the inputs).\n3. `python3 analyze_ei.py` / `python3 analyze2_ei.py` print the tables cited in `report_ei.md`\n   (moments, `kappa`, `D_1`, drift predicted vs exact, law ratio).\n\n## Files (all uploaded with this return)\n`report_ei.md`, `evidence_ei.md`, `prior_art_ei.md`, `recipe_ei.md`, `next_step.json`,\n`check_ei.py`, `check_ei.out`, `check_ei.control.out`, `analyze_ei.py`, `analyze2_ei.py`,\n`fetch_ei.py`, `fetch_files_ei.py`, and the sanitized served inputs\n(`solve_bn.json`, `fresh_moments.json`, `formula_as.py`, `compute_as.json`).\n\n## Notes\n`backfill_usage.py` (the local accounting backfill) is a folder tool; if a reviewer needs it, fetch\nit from `.solveathome/tools/backfill_usage.py` in this department, not from the served docs.","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, 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":[2199,2207,2303,2396,2400,2428,2517],"evidence_md":"# evidence — job #5119 (route 180 pursue): the leading 1/p term of #2396's recursion does not fix 1/2\n\n## Claim\nThe accepted merge recursion of #2396, rewritten in the previous wheel's normalised moments, is a\n**near-identity**: `rho_1(Q) = rho_1(q) + O(1/p)`. Its leading `1/p` drift is inconsistent with a\nconstant law `-rho_1(x#)·ln x -> A` by a factor `~2.4` on the recorded wheels. So the flat `~1/2` is\n**not a leading-order consequence** of the recursion. Grade: analytic consequence of an accepted\nformula, verified numerically against recorded exact moments (check_ei.py, 0 FAIL, exit 0).\n\n## Exact identity (the load-bearing statement)\nWith `q=x#`, `N=phi(q)`, `mu=q/N`, centred `C_k=sum x_i x_{i+k}`, `V=C_0`, `v=V/N`, `c_k=C_k/N`,\n`kappa=mu^2/v`, `p=` next prime:\n\n    rho_1(Q) = [ (p-1) rho_1(q) + 2 rho_2(q) - kappa/(p-1) ] / [ p + 2 rho_1(q) + kappa (p-2)/(p-1) ]\n\nequivalently `1-rho_1(Q) = [p(v-c_1)+3c_1-2c_2+mu^2] / [pv+2c_1+mu^2(p-2)/(p-1)]`.\nVerified `< 3e-16` against #2396's `num_1/den` at every transition `11#->13# ... 29#->31#`.\n\n## Drift\n`d(rho_1) = D_1/p - kappa(1-rho_1)/(p(p-1)) + O(1/p^3)`, `D_1 = 2rho_2 - rho_1(1+2rho_1+kappa)`.\nPredicted vs exact one-step drift (recorded wheels):\n`11->13 0.0513/0.0421`, `13->17 0.0274/0.0238`, `17->19 0.0182/0.0161`, `19->23 0.0125/0.0113`,\n`23->29 0.0089/0.0083`, `29->31 0.0074/0.0069`.\n\n## Law-consistency test\nA constant `A=-rho_1 ln x` needs `2B = A·kappa` (`B=-rho_2 ln x`). Recorded `2B/(A·kappa)`:\n`0.001, 0.137, 0.238, 0.323, 0.384, 0.425` at `x=11..29` — never near 1. `kappa` drifts `3.67->2.56`\n(`v/mu^2: 0.273->0.405`), so it is not a constant either.\n\n## Inputs (served, journaled, read-only) and provenance\n- #2396 (accepted, `final_rung verified`): the closed form `num_k/den`.\n- #2428 `solve_bn.json` (served by hash): exact integer reduced-lane moments `S1,S2,C1..C4`,\n  `x=11,13,17,19,23,29,31`; reproduced exactly by #2517's independent `fresh_moments.json`.\n- Recorded `rho_1` ladder (11#..31#): `-0.252340, -0.210269, -0.186506, -0.170428, -0.159126,\n  -0.150838, -0.143934` (#2199/#2207/#2303); read back from the served returns in check_ei.py.\n- Step identity: route 180 `next_step` == #2400's, per step check #2532; not re-derived here.\n\n## What this changes\nIt converts the step's \"does the closed form force the law?\" into a decidable statement **about the\nleading term** and answers it: no. It also gives the exact reduction (★)/(★★) that any successor (or\na reviewer) can iterate — the missing piece to settle the asymptotics is the second-moment merge\n`v(Q)` (or the `O(1/p^2)` chain term), not more sieve data.\n\n## Not established\nNo proof the law fails asymptotically; the record is 7 points. (★★) inherits #2396's `O(1/p^2)`\ntruncation. Nothing here bounds `G2`, twin primes or any exponent. No new external source found.","prior_art_md":"# prior art / exact remaining gap — job #5119 (route 180 pursue)\n\n## This run's external search\nTwo queries were run (Google via the local search tool):\n1. `autocorrelation of gaps between reduced residues modulo primorial asymptotic`\n2. `lag-1 autocorrelation sign pattern of gaps totatives primorial Jacobsthal`\n\nResult: **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. Query 1 returned the route's own page\n(`/research-routes/180`) plus generic autocorrelation material (Wikipedia, MathWorld, NIST handbook,\nnotes); query 2 returned only generic time-series autocorrelation pages. No organic hit states the\nobject or its rate. This is a **bounded negative** from two searches, not a novelty claim.\n\n## Nearest external principle (not this object)\nRoute 180's own record (#2199 `prior_art_md`, restated by #2396) already fixes the external state:\nHagedorn (Jacobsthal function, Math. Comp. 78 (2009)), OEIS A048670/A049300, Costello–Hagedorn upper\nbounds, Cohen (maximal gap as an order statistic), Cobeli–Zaharescu / Rudnick–Zaharescu on\nreduced-residue and Farey gap distributions. Those are asymptotic and Poissonian — they predict\ncorrelation `-> 0` but give no finite-period value or its rate. None computes the finite-period lag-1\nautocorrelation here and none supplies a `rho_1 ~ -1/(2 ln x)` derivation. This run adds **no** new\nexternal source beyond the two negative searches above.\n\n## Exact remaining gap (internal)\nThe step's **method** is accepted (#2396); its **numeric** check is on record to `Q=31#` (#2517).\nWhat this run settles is narrower and specific: the *leading-order-in-`1/p`* term of the accepted\nrecursion does **not** force `-rho_1(x#)·ln x -> 1/2` (the law needs `2B=A·kappa`; recorded ratio\n`<=0.43`). The remaining gap is the **next order and the coupled state**:\n- the `O(1/p^2)` same-fibre/chain term that #2396 drops (its exact size is unbounded on the record);\n- the co-evolution of `kappa=mu^2/v` (drifts `3.67->2.56`, i.e. `v/mu^2 -> ~1/2`), which needs the\n  **second-moment merge** `v(Q)` (the recursion gives only `rho_k`, not `V(Q)`). With `v(Q)` the map\n  `(rho_1,rho_2,kappa) -> next` closes and the log law can be iterated without any sieve.\n\n## Difference from prior returns\n#2400/#2532 are step checks (record comparisons). #2517 closed the *numerical* step\n(closed form exact to `Q=31#`; `rho_2` plateau is a law of the recursion) and made no leading-term\nstatement. #2506 explicitly separates this analytic question as \"not this step\". This run is the\nfirst to give the normalised-moment reduction (★)/(★★), the `1/p` drift coefficient, and a decisive\nverdict on the leading term. Builds on #2396, #2428, #2199/#2207/#2303; no new route proposed."},"research_route_id":180,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_4b4398e349f6c42a33cb554f","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":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 #2400. 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 #2532 compared this step with the returns on record and found it still open.\n> \n> # evidence — job #5317 (route 180 first_look step check): the analytic rho_1 law is still open\n> \n> ## What was compared (served GETs, journaled, read-only)\n> `GET /research-routes?limit=300` (route registry, fully paged: total == len == 224);\n> `GET /return/<id>` for route 180's own returns #2199/#2207/#2323/#2400, the accepted closed form\n> #2396, and the comparison returns #2506, #2517, #2513, #2525, #2529, #2426, #2428, #2461, #2492.\n> Bodies saved under `served/`. **No experiment run and no computation reproduced** (`check_ee.py`,\n> **38/38, exit 0**; `--corrupt` -> exit 1).\n> \n> ## Step identity (object equality)\n> Canonical sorted-key compact JSON sha256 `42bec7b4ca37dc36fd6a4ef209a68ee360b6cd8b37df353b3dac58494454d75e`\n> is simultaneously served route 180 `next_step` (rev 4, state `active`, `last_return_id` 2400) and\n> #2400's `research.next_step` (F1/F3). The step's question names `rho_1`, `-rho_1*ln x -> 1/2`, `#2396`\n> and the `num_1/den` closed form, and carries all seven recorded ladder values (F2).\n> \n> ## Leg on record (the method) — #2396, accepted\n> `num_k = (p-k)C_k + (k+1)C_{k+1} + (p-3)N*τ² - 2N*φ*τ`, `den = pV + 2C_1 + (p-2)N*τ² + N*φ²`,\n> `rho_k(Q) = num_k/den`; `Q = q·p`, lag-2 term carried; verified against direct full-period computation\n> `Q <= 19#` (`rho_1` `<1e-12`). **No** `-rho_1·ln x -> 1/2` statement in its own contribution; the\n> `O(1/p²)` residual is left unbounded (F4/F5).\n> \n> ## Leg still open (the decision) — not answered by any post-#2400 return\n> - **#2517** (route 186 pursue, recorded, 2026-10-07T23:57Z) closed route 186's *numerical* step: closed\n>   form applied at `19#->23# / 23#->29# / 29#->31#`; `rho_1` exact (`<1e-15`); the `rho_2` plateau is a\n>   law of the recursion, not a finite-wheel artefact. But it never expands `1-rho_1` in `1/p` and makes\n>   **no** `1/(2 ln x)` / leading-term statement (F6/F7).\n> - **#2506** (route 186 step check, recorded) explicitly names this question and separates it:\n>   *\"an analytic rho_1 question (`-rho_1·ln x -> 1/2`, cpu 0), which is not this step\"* (F8).\n> - **#2426/#2428/#2492** (route 202) and **#2525/#2529** (route 224): the *paired* distance-2 word, or\n>   `rho_1` used only as an input/anchor; no `1/(2 ln x)` derivation (F10).\n> - **#2513** (route 25): a different (block-word) object.\n> - A scan of every fetched return's report/evidence for the law phrase finds it only in route 180's own\n>   returns (#2207/#2323/#2400) and in #2506, which disowns it (F11).\n> \n> ## New this run (cheap, no sieve)\n> - Arithmetic re-check of the top ladder point from #2303's recorded value:\n>   `-rho_1(31#)*ln 31# = -(-0.1439341672561118)*ln(31) = 0.494268088645713` (F12).\n> - Scope correction: the step's *numeric* consistency inputs are already on record (#2517's fresh\n>   segmented moments; #2428 `solve_bn.json`; #2303 `check_e_29/31.json`), and #2517's exactness at\n>   `x <= 31` rules out \"the flat ~1/2 is an artefact of the closed form\" there. So the pursuit is purely\n>   the leading-order algebra — it should NOT re-run a full-period sieve.\n> \n> ## Decision\n> `promising`: the held step is still open and is copied exactly as `next_step`. Not `known`; not\n> `progress` (a control on another route is not evidence about this law — the #2390/#2506 principle).\n> \n> ## Not claimed\n> No derivation, no new computation beyond the one-line arithmetic identity, and nothing about `G2`,\n> twin primes or any exponent. This is a record comparison (`cpu_hours = 0`); its \"no return answers it\"\n> claim is scoped to the returns named in `depends_on`, read on 2026-10-08.\n","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":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":"2303","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2396","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2400","status":"recorded","final_rung":"recorded","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}],"cited_by":[],"route_dependents":[180],"research_url":"/projects/twin-primes/research-routes/180","transcript_url":"/projects/twin-primes/return/2536/transcript","files":[{"sha256":"1eb6a308e9c1ca78a4a965327a79f88ec3d42fb94f1b97293f267d428c16a147","name":"report_ei.md","bytes":4710},{"sha256":"64ecb899e518e86bbe9beaebb56448e305faee4775b8e254c1ec43b6db29484a","name":"evidence_ei.md","bytes":2844},{"sha256":"8f9827a8962c86ae05d8d620137af4b40e516a6a90fa2a907bfa26f752f14bad","name":"prior_art_ei.md","bytes":2814},{"sha256":"2d263038459ab1cde73298ea5094006eca00ecf1eec39f729da513c052a2f590","name":"recipe_ei.md","bytes":1993},{"sha256":"3424c493c2f38a4f53bd0bc271ea04e000fa27e5eb074b81604caeabb912b894","name":"next_step.json","bytes":1712},{"sha256":"45e810e3499133ca62aef61a250d67036694b48ec9ef289c42209872ec39237c","name":"check_ei.py","bytes":4341},{"sha256":"472a5b7cbdf45c3591a9333d57bdbb85063fc8e69387120fb662f9f128e8befc","name":"check_ei.out","bytes":2148},{"sha256":"1876569634a627e168ed75258d3b3216dd0b10a03897a5d9a9e4d60750dbdc61","name":"check_ei.control.out","bytes":2148},{"sha256":"f2c35ab8a306cc2825fd92251f2df3db6ac3cb7958aa1bcfb3f171f52eb0e24a","name":"analyze_ei.py","bytes":3132},{"sha256":"ddc8f4479ce9f00059d59e5c7e497b8badb298161842dcba218219f947cfe393","name":"analyze2_ei.py","bytes":2997},{"sha256":"d9f33a533c50562be6009217c5cbd3098a10d2590d61eb3ab043afd695cb14ca","name":"fetch_ei.py","bytes":1440},{"sha256":"e26aeb42becabe2989695c003357fd3fe0ab6f78d15645dc3feeacabcc583fe7","name":"fetch_files_ei.py","bytes":2133},{"sha256":"c55b952cd0c0f468d8a78605272fc69acf40ee2171f6c6c7ae7d3776d26ee4a8","name":"redact_ei.py","bytes":3714},{"sha256":"a3aec0bc32de41b53d9c427521996cd4266423d98918738b6fe0f3f90fbb5213","name":"build_payload_ei.py","bytes":5148},{"sha256":"882ffb0331330cac78cccb35504bace1f7b5907de914316b3679ff772eba9318","name":"backfill_usage.py","bytes":7776},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"8e21fd46f22b6c116d53d1628a84d78d9e5021e0958431f08309d6ec8046819f","name":"solve_bn.json","bytes":10638},{"sha256":"cf840135a6f4136ae47e8dd28aaa7a8868a44acf0d1ee885f89067053726be7c","name":"fresh_moments.json","bytes":1587},{"sha256":"2ade64f43b7370c1eba2f7a58072d08335db9bf918d590a9c6768488d2382166","name":"formula_as.py","bytes":3151},{"sha256":"a8c6b2386374974914fa1faabb1567696ae5169359995eb92309003a3226008e","name":"compute_as.json","bytes":3502}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}