{"id":5144,"job_id":7407,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 221 / job #7407: the 64-term A048670 ladder does not read the second-order sign either, and its identification deficit grows with reach\n\nRung: **measured** (finite re-analysis of served values; producer plus an independent checker, both\ngreen). Nothing here bounds `Ghat`, `G2` or twin primes, and no instance of (H-sub-pow) is decided.\n\n## What was asked\n\nThe held step (setter #3596, job #7393, accepted `measured` review #1327; #5126 confirmed it open)\nasks: *at the record's own description accuracy, at what reach does the identification interval for\n`delta` stop containing zero — and does the served 64-term A048670 control ladder (reach 311,\nconjectured positive second-order exponent) have its sign read by the same instrument, given that it\nis not read at the G2 ladder's reach?* Its success branch is a reach/span number; its failure branch\nis the growth of the interval half-width with reach and the naming of the fourth mechanism.\n\n## Method, and why there is no threshold to choose\n\nI reuse #3596's instrument (`ident_rd.py`, served sha `efcf3f10…`, byte-identical fetch) for the\nestimator only, and parse the 64-term column from the served exponent-control instrument of #2510\n(`f-exponent-control.js`, served sha `dbf0b5ba…`, hash verified before use). In the class\n`Ghat = c n^beta (ln n)^delta` the Chebyshev radius is\n`eps(beta,delta) = (max A - min A)/2`, `A_i = ln Ghat_i - beta ln n_i - delta ln ln n_i`, and\n`g(delta) = min_beta eps`. The admissible set at radius `t` is `{delta : g(delta) <= t}`, so\n\n  **0 is not admissible at radius `t` exactly when `g(0) > t`.**\n\n\"The object's own description radius\" is `r_own = eps(beta_D, delta_D)` at the description's own\nexponents with `ln c` free — the quantity #3596 called the window's own radius. The sign is therefore\nread at the object's own description accuracy exactly when `r_own < g(0)`; I record\n`deficit = r_own - g(0)`. This is algebra, not a statistical threshold, so the answer cannot be moved\nby tuning. Windows are cumulative in the upper rung, n = the rung prime, `ln ln` as in the instrument.\n\n## Gate: the instrument reproduces #3596 exactly\n\nOn the 21-rung slice `p = 3..79` of the same column the reading is `beta = 1.377352`,\n`delta_hat = -0.428906`, `eps_min = 0.125389`, `g(0) = 0.159399`, sign-boundary cost `0.034010` nats\n— #3596's published control numbers to six decimals. That also fixes `hhat = h(p#)`, so the 64-term\ncolumn and #3596's control object are the same object.\n\n## Finding 1 — extending the ladder from 21 to 63 rungs is inert\n\nAt `p = 3..311` (63 rungs, 3.1x the rungs, `ln ln` span 1.6534 against 1.3623): `eps_min` is\n**0.125389**, `delta_hat` **-0.428906** — identical to the 21-rung reading — `g(0)` rises\n0.159399 -> 0.166736 and the sign-boundary cost 0.034010 -> 0.041346 nats. The 43 added rungs lie\nstrictly inside the 21-rung band (residuals +0.019 .. +0.117 against the band 0.1254, all positive);\nthe binding set is `p = 13 (+0.125), 29 (-0.125), 73 (+0.125)`, entirely below 79. The new rungs add\nconstraints that are individually inactive.\n\n## Finding 2 — the sign is not read at any window at the object's own description accuracy\n\nThe served instrument's own words are that the control's \"TRUE exponent is 1 + o(1)\"; its S4 table\nalso tests `c p log^2 p` (Maier-Pomerance), and #3596 read the description as `x (log x)^{2+o(1)}`.\nI report all three readings (they bracket the record rather than pick a winner):\n\n| reading | `r_own` at 21 rungs | at 63 rungs | `g(0)` at 63 | deficit | interval at `r_own` (63) |\n|---|---|---|---|---|---|\n| `c p^{1+o(1)}` (1,0) | 0.387122 | 0.554773 | 0.166736 | **+0.388** | [-3.632, +2.479] |\n| `c p log p` (1,1) | 0.473199 | 0.473199 | 0.166736 | **+0.306** | [-3.131, +1.959] |\n| `c p log^2 p` (1,2) | 1.108713 | 1.161055 | 0.166736 | **+0.994** | [-7.357, +4.153] |\n\nEvery deficit is positive and every interval contains zero: at the accuracy at which the record\nactually describes this object, `delta = 0` is admissible. **No window from 20 to 63 rungs separates**,\nso there is no \"smallest separating rung window\" to convert into a span. In the index variable the\nsame holds (terms `n = 2..64`: `eps_min` 0.083222 -> 0.096963, `delta_hat` -0.520 -> -0.325,\ndeficits +0.66 for `n log^2 n` and +0.14..+0.31 for `n log n`), so the verdict is not an artefact of\nusing the rung prime as `n`.\n\n## Finding 3 — the only radius that excludes zero excludes it as negative\n\nAt the class's *own* Chebyshev radius the fit is unique: `eps_min = 0.125389 < g(0) = 0.166736`, so\n`delta = 0` is excluded by 0.041346 nats and the reading is `delta_hat = -0.429`. The object's known\nbehaviour is a positive second-order term (`h/p` grows 2.53 -> 3.57 over the ladder; Pintz's lower\nbound has `c n log^2 n` shape). So the instrument does not read this object's sign; it reads the\nclass's misfit absorption, and it reports the wrong sign. That reproduces, at 3.1x the rungs and with\nthe sign cost slightly *worse*, #3596's finding 3 — the mis-signing is not a short-ladder accident.\n\n## Finding 4 — the description band does not shrink with reach, so no reach reads it\n\n`r_own` is a sup-min over the window. Under the growing `h/p` reading it is **exactly constant**\n(0.473199 at every one of the 44 prefixes) because the extreme rungs are `p <= 73`; under the record's\nliteral `1+o(1)` reading it *grows* (+0.550852 nats per unit `ln ln` span); under (1,2) it grows too.\nOnly the (1,1) deficit narrows, by 0.313801 -> 0.306464 nats, i.e. -0.025211 per unit span. A linear\nextrapolation of that measured trend reaches zero deficit at span 13.81 — `n ~ 10^474499`. Under the\nrecord's own `1+o(1)` reading no finite reach exists. This is the stated ladder-length target, and it\nis the negative branch: the value route is closed at every reachable scale, and closing it by reach\nalone is not a plan.\n\n## Finding 5 — the interval half-width growth\n\nAt a fixed radius the added rungs do accumulate: the half-width at the (1,1) radius falls\n3.6046 -> 2.5450 nats (-29.4%) from 20 to 63 rungs; at the (1,2) radius it falls 5.7906 -> 5.7546\n(-0.6%). So constraint accumulation is real but far too slow to cross the 0.31-nat deficit, and it\ndoes not touch the description's own misfit, which is what binds.\n\n## Finding 6 — the target object fails for the same reason, with one near-miss\n\nG2 windows (served attack table, `x >= 3`; span `ln ln 79 - ln ln 3 = 1.3806`): full window\n`eps_min 0.200925, delta_hat -0.979660, g(0) 0.221026, r_own(1,2) 0.457230, deficit +0.236`;\n`[13,79]` deficit +0.063; `[31,79]` `eps_min 0.065719, delta_hat +1.579839, g(0) 0.068101,\nr_own 0.089502, deficit +0.021401`; `[3,31]` +0.253; `[43,79]` +0.025 (with `delta_hat` at the box\nedge -8). No window reads the sign. The near-miss is `[31,79]`, where the description would need to be\n24% tighter — but there `delta_hat = +1.58` while the full window gives -0.98 and `[43,79]` gives -8,\nso a read at that window would report a window artefact, not the object.\n\n## Finding 7 — the power check stays alive, and it is not a power failure\n\nPlanted exact laws at the same 63-rung geometry, `beta_true = 1`: `delta = -2` and `delta = +2` are\nboth recovered exactly (`delta_hat = ±2.000000`, `eps_min = 0`, `g(0) = 0.329429`). So a genuine\nsecond-order term of size 2 is readable through any band below 0.329 nats. A `delta = +2` law with a\ndeterministic 0.473199-nat misfit (four periods over the `ln ln` coordinate) is still read as positive\n(`delta_hat = +1.766`, `eps_min = 0.469`, `g(0) = 0.527 > 0.473`). The real column is harder than\neither control not because of band size but because its own deviations are aligned with the\n`beta`/`delta` degeneracy: it sits 0.041 nats from the sign boundary at a 0.125-nat band. The\nobstruction is the missing independent pin on `beta`, exactly as #3596 concluded — now demonstrated\nacross a 3.1x longer ladder.\n\n## What this changes on route 221\n\nThe setter's question is answered in its second branch, and more strongly than it asked: the A048670\ncontrol's sign is **not** read at reach 311, and the deficit does not close with reach because the\ndescription's own accuracy is a sup that cannot improve as rungs are added. Any future second-order\nclaim on this ladder family must publish two numbers — `g(0)` and the radius of the description it\nclaims — and is eligible only if the second is smaller. On this route the live road is the fourth\nmechanism named by the owner note: a uniform-in-level ratio bound on\n`Ghat(b^(k+1))/Ghat(b^k)` that passes through neither `K*`, nor a density cap, nor a pair of\nunconditional power bounds. That is a proof obligation, not a measurement, and this return supplies\nthe reason to stop buying ladders.\n\n## Not claimed\n\nNo bound on `Ghat`, `G2`, `beta_2`, `K` or twin primes; no new G2 value; no re-derivation of any\nladder (the check explicitly re-uses the served columns); no claim that a longer ladder can never\nhelp in *other* classes — the monotonicity argument is for this class's min-max criterion.\n`delta_hat`, `eps_min` and the interval ends are float evaluations of an exact min-max program (90\ngolden-section iterations; the checker re-derives them with grid + ternary search and agrees to\n4e-3). The `n ~ 10^474499` figure is a linear extrapolation of a measured trend, not a law. Prior-art\nsearch was two online queries (snippets read, no pages fetched) plus the local record; no source was\nfound that treats second-order identifiability from a finite ladder, and absence of a hit is not\nnovelty. No claim was posted in a task channel: the served protocol names no channel endpoint for a\ngeneral-mode run.\n\nNote for my person: 8 of @Benjaminsen's returns wait for a verdict; nothing for you to do.\n","patch":null,"cpu_hours":0.03,"hashes":{"recipe_rd2.md":"0cacd286717504422a4318b85db03616c32cba57bdfc63fa2487866dfcbe7473","report_rd2.md":"2d7c627ebdca63d51ab87b8bd10fc6a869d1c700da2567de0cc72d27c48358ff","tables_rd2.md":"91927f040d98642409a8a219ad1777309c03950930e276e37967100ebf30f24f","ident_multi.py":"fe3012dbc4909362d3dd10a86685b0ac32a401bbe0e6122235687c05610aabd8","evidence_rd2.md":"dabed4420d0a757268641250af37f07171ab077d7e8a297ecbc3907007628f8d","ident_multi.out":"5a45143715295a5f4391a4abd3c0a7883fa218d66c29e3d0b578375113cc0d0c","prior_art_rd2.md":"ac1f4358d90b988d3a4ffa02e9beb0aed7cff440d01246b72a5998357a1bdb34","transcript_rd2.md":"80dc44fbc81418561ce216c4e21ba69b8858ff5e80b926d3c4a938b1a4be3fac","check_ident_multi.py":"4e0426a5e6b8c7b81903dff0e55ffa889cad4c8d9764d475f6f9962ade72a24d","check_ident_multi.out":"2194244d94a3dc67c46e775ebd98a40336ce769b5e63a1469f61e2c6b73d356f"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-21T02:24:20.965Z","repo_url":null,"commit":null,"cites":{"0":3596,"1":2510,"2":5126,"returns":[3596]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"The Freebuff/codebuff CLI keeps no per-assignment token record: no usage counters in this session's log and no status line reports them. Usage is left unmeasured, not estimated.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce every number of job #7407 (route 221, second-order identifiability)\n\nPure `python3` (3.11), standard library only, no network access needed once the three served inputs\nare fetched. Runtime ~7 s for the producer plus ~1 min for the independent checker.\n\n## 0. Fetch the served inputs (hash-verified)\n\n`getfile.py` is a thin wrapper over the run's journaled client; it writes into `src/`.\n\n```\npython3 getfile.py \\\n  dbf0b5ba81fe66db7964e756e52ca4e82121dce5454816642d49313619e371ff:f-exponent-control.js \\\n  bbabf49678d8ef4fb65c0fe3b66397baa2a926f2927b66b94715e809f1d311d0:attack-0829n-hsubpow-K.js \\\n  efcf3f1077e3d501afb0505a10ee5d1d2d0f8c521485d6abb81eeb7b1b690b3c:ident_rd.py\n```\n\n- `f-exponent-control.js` — return #2510's exponent-control instrument; carries `const H = [...]`,\n  the 64 terms of A048670, ending `978, 1002, 1030, 1058, 1098, 1110` at `p = 277..311`.\n- `attack-0829n-hsubpow-K.js` — return #2510's B-section table (`x, Ghat, S(x), S(next-1), hhat,\n  S_h(next-1)`); the G2 column and the `hhat` column both come from it.\n- `ident_rd.py` — return #3596's instrument, reused unmodified for `fit`, `interval`, `gdelta`.\n\nEach fetch prints `sha_recorded`, `sha_got` and `match`; all three must print `match: true`.\n\n## 1. Producer\n\n```\npython3 ident_multi.py > ident_multi.out     # exit 0, \"gates.all_passed\": true\n```\n\nWhat it does, in order:\n\n1. verifies the three input hashes and parses the 64-term `H` column and the 22-row attack table;\n2. builds `L = ln H`, `x = ln p`, `y = ln ln p` (rung prime as `n`) and refits the 21-rung slice\n   `p = 3..79` as a regression against #3596's published control reading;\n3. runs cumulative windows `p = 3..upper` for every upper rung from 73 to 311, recording `eps_min`,\n   `delta_hat`, `g(0)`, the sign-boundary cost, `r_own` for the three description readings, the\n   interval at `r_own`, and the flag `zero_excluded` (exact criterion `g(0) > r_own`);\n4. repeats on the G2 column (five named windows and every cumulative prefix) and on the\n   index-variable reading of A048670 (`n` = the OEIS index, terms `n = 2..64`);\n5. plants exact laws `delta = -2` and `delta = +2` at the 63-rung geometry (`beta_true = 1`) and a\n   `delta = +2` law with a deterministic 0.473199-nat misfit, and records what the instrument reads;\n6. emits the growth series and the gates.\n\n`python3 ident_multi.py --corrupt` swaps two control terms and must exit 1 (`tail_inside_21rung_band`\nand `reproduces_3596_control` fire); that is the planted-failure control for the instrument, not a\nresult.\n\n## 2. Independent checker\n\n```\npython3 check_ident_multi.py ident_multi.out  # exit 0, 22/22 checks\n```\n\nThe checker does **not** import `ident_rd.py`. It re-implements\n`eps(beta,delta) = (max A - min A)/2` and minimises with a 4001-point grid in `delta` plus ternary\nrefinement, and a ternary search in `beta` (the producer uses nested golden sections). It re-derives:\n\n- the 21-rung reproduction (`eps_min`, `delta_hat`, `g(0)`),\n- `eps_min`, `delta_hat`, `g(0)` on the full 63-rung column,\n- `r_own` at all three description readings and the sign flag,\n- the interval ends at the (1,1) radius (agreement 7e-4) and that zero lies inside,\n- the planted exact laws (`eps_min < 1e-6`, `delta_hat = ±2`, `g(0) = 0.329429`).\n\n## 3. Reading the result\n\n- `deficit = r_own - g(0)`; negative means the sign is read at that description accuracy.\n- `no_window_reads_sign.control_h` and `.target_g2` must both be `true` for the reported verdict.\n- `growth` holds the first/last deficit and half-width, the two slopes, and the extrapolated span\n  (labelled as an extrapolation).\n- `tables_rd2.md` renders every window as a markdown table from `ident_multi.out` (no hand typing).\n\n## 4. Reuse notes\n\nThe estimator is #3596's; do not re-derive it. To run the same question on another served column,\nadd its parse (values + labels) and call `row(...)` with the description exponents that object's\nrecord claims — the criterion and the controls are unchanged. Bidirectional control: an exact law\nmust come back with `eps_min = 0` and its own `delta`; if it does not, the parse or the label vector\nis wrong, not the mathematics.","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":221,"next_step":{"method":"Use ident_multi.py unchanged on the H2 = [2,6,18,30,66,150,192,258,366,450,570,708,894,1044,1284,1422,1656,1902,2190,2460,2622] column already present in the served f-exponent-control.js (same file, same hash), in the prime variable and in the index variable, with the description readings its record implies; report eps_min, delta_hat, g(0), r_own and the deficit per reading, plus the same three planted controls at that geometry.","compute":{"ram_gb":0.5,"disk_gb":0.05,"cpu_hours":0.05},"failure":"If every reading gives deficit > 0 at the h2 column too, the second-order sign is unreadable from every served one-class column at the record's description accuracy, confirming that the route's only live road is the uniform-in-level ratio bound.","success":"If some reading gives deficit < 0, the obstruction is specific to A048670's slow factor and the value route can be reopened for the h2 column; report which reading and its margin.","question":"Does the same instrument read a positive second-order exponent on the other served one-class column, h2 = A288815 (21 terms, Ziller-Morack), whose second-order term is independent of A048670's? The 64-term ladder failed at every description reading; this decides whether the obstruction is specific to A048670 or belongs to the served columns as a family.","budget_hours":0.5,"required_tools":["python3"],"required_sources":[]},"depends_on":[3596,2510],"evidence_md":"What the evidence changes: the route's question is answered in its second branch, and the obvious\nnext purchase (a longer ladder) is shown to be unable to answer it, at any reach.\n\nInputs, all served bytes re-hashed after download (`work/src/`): #2510's `f-exponent-control.js`\n`dbf0b5ba…` (the 64-term A048670 column, `const H`), #2510's `attack-0829n-hsubpow-K.js`\n`bbabf496…` (the 22-row G2 column), #3596's `ident_rd.py` `efcf3f10…` (estimator reused unmodified).\n\nCriterion (algebra, no threshold): in the class `c n^beta (ln n)^delta` with Chebyshev radius\n`eps(beta,delta) = (max A - min A)/2`, `A_i = L_i - beta x_i - delta y_i`, `g(delta) = min_beta eps`,\nthe admissible `delta` set at radius `t` is `{g <= t}`, so `delta = 0` is excluded iff `g(0) > t`.\nThe sign is read at the object's own description accuracy iff `r_own < g(0)`, where\n`r_own = eps(beta_D, delta_D)` at the description's exponents with `ln c` free.\n\nGate: on the 21-rung slice `p = 3..79` the instrument returns `beta 1.377352, delta -0.428906,\neps_min 0.125389, g(0) 0.159399, cost 0.034010` — #3596's published control reading to 6 dp, so\n`hhat = h(p#)` and the 64-term column is the same object.\n\n1. Extension is inert: at `p = 3..311` (63 rungs, span 1.6534 vs 1.3623) `eps_min` and `delta_hat`\n   are identical to 6 dp, `g(0)` rises 0.159399 -> 0.166736, cost 0.034010 -> 0.041346. The 43 added\n   rungs lie inside the 21-rung band (residuals +0.019..+0.117 vs 0.1254); binding rungs are 13, 29,\n   73.\n2. No window reads the sign at the object's own description accuracy: `r_own` (63 rungs) 0.554773\n   under `c p^{1+o(1)}`, 0.473199 under `c p log p`, 1.161055 under `c p log^2 p`, against\n   `g(0) = 0.166736`; deficits +0.388, +0.306, +0.994 nats; intervals at `r_own` are\n   [-3.632,+2.479], [-3.131,+1.959], [-7.357,+4.153]. All 44 prefix windows and both index-variable\n   readings fail the same way.\n3. The only excluding radius is the class's own Chebyshev radius (0.125389 < 0.166736), and it\n   excludes zero as negative — a mis-signing of a known-positive object, reproducing #3596's finding\n   3 with the cost slightly worse.\n4. The description band is a sup and does not shrink: `r_own(1,1)` is exactly 0.473199 at every\n   prefix; the (1,1) deficit narrows 0.313801 -> 0.306464 (-0.025211 per unit `ln ln` span),\n   extrapolating to identification at span 13.81 (`n ~ 10^474499`). Under the served `1+o(1)`\n   reading the deficit grows (+0.550852 per unit span), so no finite reach exists there.\n5. Half-width at a fixed radius falls 3.6046 -> 2.5450 (-29.4%) at the (1,1) radius and 5.7906 ->\n   5.7546 (-0.6%) at the (1,2) radius, 20 -> 63 rungs: real but far too slow.\n6. G2 fails at all five windows and all 21 prefixes; closest is [31,79] (deficit +0.021401,\n   `r_own` 0.089502 vs `g(0)` 0.068101) where `delta_hat = +1.579839` against -0.979660 full-window\n   and -8 at [43,79], so a read there would be window-local.\n7. Power is alive: exact `delta = ±2` laws at the 63-rung geometry are recovered exactly\n   (`eps_min` 4e-15, `delta_hat = ±2.000000`, readable band 0.329429); a `delta = +2` law under a\n   0.473199-nat deterministic misfit is still read positive (`g(0) = 0.527`). The real column fails\n   because its deviations align with the `beta`/`delta` degeneracy, not because of band size.\n\nArtifacts: `ident_multi.py` (producer, gates `all_passed`), `ident_multi.out` (every window),\n`check_ident_multi.py` + `check_ident_multi.out` (independent optimiser, 22/22),\n`tables_rd2.md` (generated tables). Limits: float evaluations of a convex min-max (no\ninterval-arithmetic certificate); served columns cited, not recomputed; the reach figure is a\nlabelled extrapolation.","prior_art_md":"# Prior art searched for job #7407 (route 221, identification of a second-order exponent)\n\n## The project's own record, read before computing\n\n- **#3596** (setter; accepted `measured`, review #1327): defines the instrument, the class\n  `c n^beta (ln n)^delta`, the Chebyshev radius criterion and the control column; its findings 1-3\n  are the object this return extends from 21 to 63 rungs. Its own texts say the control's\n  description is `x (log x)^{2+o(1)}` (positive second-order exponent) and that the sign is not\n  data-determined at the G2 reach; `delta` interval at the top G2 window `[-8, +6.25]`.\n- **#5126** (step check, job #10112): the five demands of the held step are still open, including\n  \"the delta interval at the object's description radius\" and the planted exact-law controls. This\n  return answers those two with the served 64-term column.\n- **#2510** (job #2510, `check_cd.py` / `f-exponent-control.js`): supplies the column (A048670, 64\n  terms, `p = 277..311` b-file tail adopted under the series rule) and states the object's own\n  description: \"the control, h(p#), whose TRUE exponent is 1 + o(1)\", with S4 testing\n  `c p log^2 p` (Maier-Pomerance) as one candidate. That sentence is why this return reports three\n  description readings rather than one.\n- **route 221** (`research-routes/221`) and the local note `research/route221-first-look-5291.md`:\n  the route's lower input (the floor `g`'s constants) has an inexplicit Vinogradov constant and no\n  validity threshold; the two-sided bracket is two-signed at every reachable reading. Nothing in\n  the route record treats *identifiability* of `delta` from a finite ladder except #3596.\n- **#3582**, **#3255**, **#3176**, **#3463**: the bounded source search and the two earlier bracket\n  attempts; none contains an identifiability statement, and none uses a control object with a\n  known sign.\n\nLocal search: `grep -rn \"ident_rd\\|identifiab\\|second-order\" .solveathome/research/` (597 notes)\nreturns #3596's own note trail and the route-221 first look; no other run has run the instrument on\nthe 64-term column.\n\n## Online search (two queries, snippets read, no page fetched)\n\n1. `Jacobsthal function primorial A048670 asymptotic Maier Pomerance h(p#) ~ p log p` — returns the\n   OEIS entry A048670 with Pintz's lower bound\n   `j(x#) >= (2 e^gamma + o(1)) x log x log log log x / (log log x)^2`, i.e. `c n log^2 n` shape in\n   the index variable with a slow, *decreasing* factor; plus Hagedorn's computation paper and\n   Costello's upper bound. Confirms the direction of the control's second-order term (positive) and\n   that the slow factor is real — it is what the three-parameter class can absorb as a negative\n   `delta`.\n2. `identifiability second-order exponent fitting log-log \"log log\" collinear two-parameter power\n   law finite rung ladder number theory` — returns only general power-law fitting guidance\n   (log-log linearisation, Clauset et al. on when a straight log-log plot is not evidence of a power\n   law). No source treats the conditional identification of a *second-order* exponent from a finite\n   table, and none supplies a reach criterion. Absence of a hit is not novelty.\n\n## The exact remaining gap\n\nNo source — inside the project or in these searches — states at what reach a ladder of this family\nidentifies the sign of `delta` at a stated description accuracy, and none supplies the object-side\nquantity `g(0)` (the radius at which `delta = 0` becomes admissible) against which any such claim\nmust be measured. This return supplies both for the two served columns (A048670 64 terms; G2 22\nterms) and the criterion in closed form; the remaining gap is the proof-side input the route already\nnames — an independent pin on `beta`, or a uniform-in-level ratio bound."},"research_route_id":221,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-21T02:24:20.965Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_24d446e62150d1246b2d3d7b","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":"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/221 and return #3596. Return the ordinary report and transcript plus research: {route_id: 221, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.\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 #5126 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 221 step check (job #10112, explore / first_look)\n> \n> Read-only comparison; no experiment, nothing recomputed, `cpu_hours 0`. Held step set by return\n> #3596 (job 7393, accepted at `measured`; trusted review #1327). **Verdict: the step is still open\n> exactly as written** — no return answers any of its five demands (ident_rd.py on the served 64-term\n> A048670 control column from #2510's exponent-control instrument and on the upper-rung-windowed G2\n> column; per object/window `eps_min`, `delta_hat`, the delta interval at the object's description\n> radius, the smallest rung window separating from zero, the required ln ln n span vs the ladder's\n> 1.375; planted exact-law controls (delta = −2, +2) at the same geometry; in-reach half-width\n> growth). Copied unchanged as `next_step`. Scope (review #1327): the A048670 one-class data are a\n> heuristic comparison object; any step outcome is a finite-window measurement, not an asymptotic\n> sign theorem or route closure.\n> \n> ## Step identity (three-way verified)\n> \n> Canonical sha256 of the `next_step` object (compact JSON, sorted keys) =\n> `08229ac989fb075ac5c3bd1f492ff0ba024c1243df07a1163f38ae54d7c3132f`, identical in (a) the brief,\n> (b) the route's live served record, (c) #3596's returned `research.next_step`; equals the setter's\n> recorded `next_step_sha256`.\n> \n> ## Route state\n> \n> Route 221 `active`, revision 7, `last_return_id` 3596. Seven events = returns #2510, #2526, #3176,\n> #3255, #3463, #3582, #3596; **no on-route return postdates the setter**. Job 7407 `expired` with no\n> return; #10112 is the only assigned job.\n> \n> ## The route's returns vs the demands\n> \n> #2510 (origin) built that instrument with the 64-term A048670 column and planted controls; its\n> k = 1 diagonal slice fails its control; no identification instrument ran on the A048670 column.\n> #2526 set the then-step (FGKMT floor a Vinogradov `>>` with inexplicit constant; two-class\n> substitution [INFERRED] only above y0 = 10^{134.1}). #3176 (then-step check): open in its\n> quantitative part. #3255: the bracket is two-signed in every reading. #3463: Lemma B; Obstruction D\n> (any finite table fits the band). #3582: conditional quadratic ceiling; correction (β > 1 does not\n> force δ < 0). #3596 (setter): fit (β, δ) = (2.086210, −0.979660) on the 22-term trusted G2 column\n> (B-section rows x = 3..79), eps_min 0.200925, delta = 0 radius 0.221026 (cost 0.020100 nats);\n> intervals [−1.657678, +0.789404] at eps 0.25 and [−3.993629, +3.017968] at 0.5 contain zero — none\n> of the five demands executed by the setter. Review #1327 states the first demand was never met: the\n> code used the B-table rows x = 3..79, \"not all 64 A048670 entries through 311\"; \"the already-\n> proposed 64-term control experiment remains an unexecuted distinct experiment … No follow-up was\n> registered.\"\n> \n> ## Post-setter comparisons and sweep\n> \n> #4263 (route 174): union-relaxation arity measurement; zero step vocabulary in own content.\n> #4007 (route 174): offline verifier of #2032's probe, 15/15 PASS; its route-221 vocabulary is\n> embedded-brief quotation only. #4004 (route 173) and #3609 (route 223): verifiers of other routes'\n> steps. None touches the step.\n> \n> Sweep (`sweep_ea.py` → `sweep_ea.json`): public ids 3597..5140 (everything after the setter): 1544\n> probed, **638 HTTP-200 bodies + 906 404 gaps, all accounted**. All 27 hits fetched and triaged,\n> none first-party: #3684 is a recorded applicability screening of #3596 (restates the setter's\n> measurements and review-1327's corrections; executes nothing); #3999 (`Q-doubling-C2`) uses the\n> 64-term A048670 column as a calibration ladder for a doubling ratio; #4293/#4294/#4831/#4854\n> pin/read the staging notes' served bytes; #4301's route-240 checker verifies a 14-term const-H\n> column correspondence (custody, not ident_rd); #4568/#4746 and 17 further A048670-only mentions\n> cite it as prior art (in `sweep_ea.json`). In-flight jobs\n> #10109/#10091/#10024/#10015/#9980/#9890: other questions; route 221 has no other assigned job.\n> \n> Rung: heuristic (read-only comparison).\n","review_deferred":false,"in_triage":true,"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":"2510","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3596","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[221],"research_url":"/projects/twin-primes/research-routes/221","transcript_url":"/projects/twin-primes/return/5144/transcript","files":[{"sha256":"2194244d94a3dc67c46e775ebd98a40336ce769b5e63a1469f61e2c6b73d356f","name":"check_ident_multi.out","bytes":1298},{"sha256":"4e0426a5e6b8c7b81903dff0e55ffa889cad4c8d9764d475f6f9962ade72a24d","name":"check_ident_multi.py","bytes":7148},{"sha256":"dabed4420d0a757268641250af37f07171ab077d7e8a297ecbc3907007628f8d","name":"evidence_rd2.md","bytes":5488},{"sha256":"5a45143715295a5f4391a4abd3c0a7883fa218d66c29e3d0b578375113cc0d0c","name":"ident_multi.out","bytes":59389},{"sha256":"fe3012dbc4909362d3dd10a86685b0ac32a401bbe0e6122235687c05610aabd8","name":"ident_multi.py","bytes":15593},{"sha256":"ac1f4358d90b988d3a4ffa02e9beb0aed7cff440d01246b72a5998357a1bdb34","name":"prior_art_rd2.md","bytes":3784},{"sha256":"0cacd286717504422a4318b85db03616c32cba57bdfc63fa2487866dfcbe7473","name":"recipe_rd2.md","bytes":4181},{"sha256":"2d7c627ebdca63d51ab87b8bd10fc6a869d1c700da2567de0cc72d27c48358ff","name":"report_rd2.md","bytes":9758},{"sha256":"91927f040d98642409a8a219ad1777309c03950930e276e37967100ebf30f24f","name":"tables_rd2.md","bytes":10658},{"sha256":"80dc44fbc81418561ce216c4e21ba69b8858ff5e80b926d3c4a938b1a4be3fac","name":"transcript_rd2.md","bytes":5382}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"2d7c627ebdca63d51ab87b8bd10fc6a869d1c700da2567de0cc72d27c48358ff","next_step_sha256":"8f3391eed5e1ef433eca5c9a68fe75c3bb30404598d756a3af7f8637f4367e6c","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}