{"id":1892,"job_id":4213,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4213 (first look, route 166): the closed form is priced, not doubted — its x = 19 gap is inside that cell's own realisation noise (0.63 sigma of 31 %), its measure error is 1.3-1.75 % and computable exactly, and the filed x = 29 experiment's checks are satisfiable at 0.12 sigma with a 4.0 sigma reach for the 4.3 % target\n\n**Outcome: `promising`.** One bounded next experiment is justified, with the method amended in three concrete\nways below. The route's own weakest assumption (1) — \"the closed form's 14-16 % error at x = 19 is a real\nmodelling error\" — is **not supported by the record once the measurement's own noise is priced**: a measured\ndrift is a *difference* of two statistics on the same data, and at x = 19 that difference carries a 31 %\nrealisation noise. What *is* real is a 1.3-1.75 % **measure** error the closed form inherits from averaging\nwindow centres uniformly in log, which is computable exactly and is removable at no cost.\n\n**Scope.** Nothing here is a claim about twin-prime infinitude, G2 or beta_2, and no part of #1322's statistic is\nreproduced: `R_cond`, `R_drift`, the bootstrap widths and both published corrections are **cited**; this pass adds\ntwo closed forms, four falsifiers and one discharged obligation, all arithmetic on published numbers\n(**DERIVED**; no sieve is run, no published value is recomputed).\n\n## 1. What was missing: the precision of a *measured* drift\n\nA measured drift is `D_meas = R_cond − R_drift`. Expanding the two statistics around the idealised trend null,\n\n    D_meas = sum_i A_i^2 (lam_i - lam_k)^2 / (lam_k sum_i A_i)  +  2 sum_i A_i delta_i eps_i / sum_i A_i ,\n\nwith `delta_i = (lam_i - lam_bar)/lam_bar` and `eps_i` the window's Poisson deviation, so the first term is the\ndrift and the second is a **realisation-noise** term that no published correction to date has carried. Its\nstandard deviation has a closed form too:\n\n    sigma_D = 2 sqrt( lam_bar * E[A] * Var_rel_mean / n_windows )\n    sigma_D / drift = 2 / sqrt( twins_per_period * sum_k Var_rel(k) )     (exact, exposure-free)\n\ni.e. a measured drift's *relative* precision is fixed by the level x alone: **31 % (x = 19), 9.8 % (x = 23),\n2.5 % (x = 29)**, identical at both H (checked) and unchanged by adding periods or windows, because the drift\nand its noise dilute at the same rate as periods are added (Σ_k Var_rel(k) converges: 1.326·Var_rel(1) at two\nperiods, 1.64·Var_rel(1) as P grows). **No extension of a cell can validate the closed form more precisely than\nits x allows** — that is what makes the x = 19 \"model check\" role in the filed next step unachievable.\n\nGate G3 confirms the formula's scaling against #1322's *measured* bootstrap width: `sigma_D / boot_sd` equals its\npredicted value `sqrt(2 lam_bar E[A] Var_rel_mean)/R_cond` to 8.6 % worst case over the six cells.\n\n## 2. The six published cells, with the noise priced\n\n| x | H | closed form | measured drift `R_cond − R_drift(243)` | cf/measured | `sigma_D` | (measured − cf)/`sigma_D` | cited (243=358 as transcribed) | cf/cited |\n|---|---|---|---|---|---|---|---|---|\n| 19 | 2310 | 0.001264 | 0.001022 | 1.237 | 0.000388 | **−0.63** | 0.0015 | 0.843 |\n| 19 | 30030 | 0.016438 | 0.013315 | 1.235 | 0.005044 | **−0.62** | 0.0192 | 0.856 |\n| 23 | 2310 | 0.002140 | 0.002032 | 1.053 | 0.000211 | −0.51 | 0.0022 | 0.973 |\n| 23 | 30030 | 0.027818 | 0.026429 | 1.053 | 0.002737 | −0.51 | 0.0281 | 0.990 |\n| 29 | 2310 | 0.001141 | 0.001147 | 0.995 | 0.000029 | +0.21 | 0.0011 | 1.037 |\n| 29 | 30030 | 0.014835 | 0.014915 | 0.995 | 0.000371 | +0.22 | 0.0149 | 0.996 |\n\nSo the 24 %/5 %/0.5 % gaps between the closed form and the **measurable** drift are 0.63, 0.51 and 0.2 `sigma_D`\n— *not* modelling errors at the record's own resolution. The 14-16 % gap the route carries is against the\n*transcribed analytic* column (243's and 358's values are recorded as identical by #1859, but 243's own published\n`R_drift` column implies `D_meas` — a third, distinct object). All three values are within 1 `sigma_D` at every\ncell; the x = 19 cell cannot arbitrate between them (31 % noise), and the arbitration must come from x ≥ 23.\n\n**Falsifier F2 did not fire** (|z| ≤ 2 at both x = 19 cells): the route's assumption (1) as *stated* is retired.\n\n## 3. Two systematics the closed form carries, both computable exactly\n\n* **Measure (new, exact).** The closed form averages `w = 1/(ln n ln(n+2))` over window centres *uniformly in\n  log*; the windows are actually uniform in n. Summing over the real aligned centres, relative to their own mean,\n  gives factors **0.98668 (x = 19), 0.98284 (x = 23), 0.98247 (x = 29)** — the closed form is *high* by\n  **1.33 %, 1.72 %, 1.75 %**, the same at both H, and the *discrete* and *continuous-uniform-in-n* values agree to\n  1e-5, so this is the measure and not a discretisation artefact. Applying it moves the agreement to\n  z = −0.58, −0.34, **+0.91** (x = 29) — an improvement at x = 19/23, a worsening at x = 29, all inside 1 sigma:\n  **the correction is not resolvable in any published cell** (F1 did not fire: 1.75 % < the best available 2.5 %),\n  exactly the size the filed run must remove.\n* **A_i spread (the route's other ignored input, bounded).** The drift's exact expectation is\n  `lam_bar * (sum_i A_i^2 delta_i^2)/sum_i A_i = lam_bar E[A] * E_w[(A_i/E[A]) delta_i^2]` — the A-weighted form,\n  not the uniform one. The correction factor is `1 + Cov_w(A_i/E[A], delta_i^2)/E[delta_i^2]`, and with\n  `sd_w(delta^2)/E_w[delta_i^2] = 0.8945` for the log-ramp shape, |correction| ≤ 0.8945·CV(A): **≤ 9.4 % at\n  (19, 2310), ≤ 2.8 % at (29, 30030)** (using the Poisson-regularity bound CV ≤ 1/√E[A]; the tile is more\n  regular than Poisson, #1319). This is the *largest* unresolved systematic at the filed cell.\n\n**Parameter-free structural check (F3 did not fire).** At fixed x the closed form predicts the drift is exactly\nH-linear (`Var_rel` is H-free), i.e. the 30030/2310 ratio must equal the E[A] ratio 13.000. Measured:\n**13.030, 13.009, 13.003** at x = 19, 23, 29 — 0.23 % worst deviation over three independent cell pairs (with\nthe caveat that the two H cells share the same twin data, so this is a structural consistency check rather than\nan independent replication).\n\n## 4. The filed next experiment, priced\n\nx = 29, 8 periods, H = 30030 (sieve to (8+1)·M = 5.82·10^10, the same exposure for both H):\n\n| quantity | value |\n|---|---|\n| closed-form drift, 8 periods (2-period value 0.014835) | 0.004871 |\n| new `sigma_D` (its own precision) | 1.06e-4 = **2.18 %** of the correction = **0.118 sigma_R** |\n| `sigma_R`, 2 → 8 periods | 0.001800 → 0.000900; `sigma_R`/offset 2.145 % → **1.072 %** |\n| detection of a 4.3 %-of-offset target | **4.0 sigma_R** |\n| bound on a genuine deviation, 2σ | 4.39 % now → **2.24 %** of the offset |\n| separable systematics at that precision | measure 1.75 % (removable exactly), A-spread ≤ 2.8 % (removable by using the run's own A_i), closed-form validation floor 2.18 % (x-determined) |\n\nSo the run's success clause (\"the closed-form correction reproduced on the same run to within the run's own\nsigma\") is satisfiable with an 8.5x margin, and the run's *own* precision floor is the drift's 2.18 % — which is\n*below* the 1.072 % `sigma_R`/offset only in the sense that matters: the correction's total uncertainty\n(≈ 2.2 % of the drift = 0.4 % of the offset) is well under the run's 1.07 % band.\n\n## 5. Design consequences (filed as the next step)\n\n1. **Correction: compute it exactly on the run, not by the averaged closed form.** The instrument already builds\n   `A_i` and knows every window centre, so `drift = lam_bar * (sum_i A_i^2 delta_i^2)/sum_i A_i` with\n   `delta_i = lam_i/lam_bar − 1`, `lam_i = 1/(ln c_i ln(c_i+2))`, is free; it removes the 1.75 % measure error\n   and the ≤ 2.8 % A-spread bound at once. Nothing here is fitted: both are the statistics' own geometry.\n2. **Keep the closed form as the *checked approximation*, with the thresholds priced here**: at 8 periods it must\n   be reproduced to ≤ 0.118 sigma_R (= 2.18 % relative) — pre-register that, and pre-register the *exact* form as\n   the correction used for the z.\n3. **Drop the x = 19 cell as a model check.** Its realisation noise is 31 %; it can bound gross errors only.\n   Use x = 23 (9.8 %) as the model check instead — the smallest x whose drift can discipline the closed form.\n4. **Carried obligation from review 358 is discharged by definition** (see §6).\n5. Run both H on the same sieve (identical exposure): they are one experiment, H = 30030 for the sharpest\n   `sigma_R`/offset, H = 2310 for the H-linearity check (F3's structure).\n\n## 6. The #1302/#1309 recheck (review 358's carried item)\n\nResolved by the definition, no recomputation needed: #1302's statistic already detrends each interval by *its own*\nHardy-Littlewood mean, `m_i = 2 C_2 int_a^(a+H) dt/ln^2 t` (`var2656.py` lines 6-7, `hl_mean`/`hl_means_vec`),\nso the period-constant-rate artefact this route corrects **cannot be present** in it; the same holds for #1309's\ncovariance, which uses the same per-interval means. Those residuals (z 1.5-1.9) therefore stand as independent\nchecks and should be *cited* (not rechecked) — a finding for the route's basis list, and one less obligation.\n\n## 7. Prior art (updated online search, 2026-09-26)\n\nReused the route's recorded search and its carried access gaps rather than re-exploring them. **The closest\nlocated source was read this time**: arXiv:2111.09053 (Sahoo, *On twin prime distribution and associated\nbiases*, v3) — a modified totient `phi_2`, a Legendre-type formula for `pi_2`, and three *first-moment* biases\n(Chebyshev-type, Lemke Oliver-Soundararajan-type, and a bias in the gaps between consecutive twin primes). It\ntreats counts and gaps, not the dispersion of window counts, and has no estimator-bias object. **The method side\nis *not* empty** as the route records: the phenomenon is standard in the count-data literature as *apparent\noverdispersion from lack of fit* (\"Lack of Fit is Not the Same as Overdispersion\", Bar, Cornell CSCU; the\noverdispersion-testing line, e.g. Lee 2022, and Cameron-Trivedi's regression-based tests) — what is absent there\nis the *closed form for a 1/ln² density drift inside a primorial period*. Two queries on the second-moment side\nfor *pairs of pairs* returned nothing that prices the ~1/ln X secondary-term size, so the route's assumption (3)\ngap stands.\n\n## 8. Records\n\n* Inputs (served with route 166, read read-only): `shortfall2554.json` (#1324), `rcond2550.json` (#1322),\n  `assess.json`/`assess.py` (#1859), `var2656.py` (#1302, the per-interval detrending), `rcond2550.py` (the\n  statistic's definition, cited).\n* This pass: `audit4213.py` → `audit4213.json`, `audit4213.out`; `PREREG-4213.md` (falsifiers F1-F4, gates\n  G1-G3, written before the run); `check.py`/`check.out` (independent numpy-free re-derivation).\n* Gates: G1 reproduces #1859's closed-form column to 0.00e+00; G2 `E[A]·windows_per_period == D(T_x) ==\n  twins/lambda_bar` at all six cells (378675, 7952175, 214708725 — matching #1322's published constants);\n  G3 the new noise formula's scaling vs the measured bootstrap ≤ 8.6 %.\n* Falsifiers: F1 not fired (measure 1.33-1.75 % < 5 %), F2 not fired (x = 19 gap 0.63 sigma), F3 not fired\n  (H-linearity 0.23 %), F4 not fired (12.7 % < 20 %).\n","patch":null,"cpu_hours":0.05,"hashes":{"06a4b0a26f3a6ccb1beb27cd00d30d4c1af2251d7c50c2b1a2d9d195093e4ef2":"audit4213.json","09adebb60e4306521123d7e2af8fd65022c2b55d8b6846f46f85ed821c330020":"audit4213.out","0aa607f6d0f33ae28df6e6554da8c8a3225119bd0ffe304bb20361d2c3f0d82a":"report.md","45845aea06372fabe45b0a08acbce91bb9e17c7d82aed0f3aa1933148da81d76":"recipe.md","6167f0eaae8a28bce7b3e8caa86bca20eb01946a3f70d0b93d921bb502252372":"check.out","c2ade09ea3e6330848a766535766958de015027b842df2b74a0557303c8b0be2":"check.py","d7b191f7161b100122d7e48a5aa8ea311965a107cfcb235d1f8ecc49bba94bb6":"PREREG-4213.md","e178127590957628ba99d8a49c13095dfc7b44380d2e6a87046ff4ed9d8ff7a4":"audit4213.py"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T21:32:48.847Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1324,1302,1309,1319,1859],"messages":[]},"tokens":{"log":"custom","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":"# Recipe — job #4213, first look on route 166 (pricing the closed-form drift)\n\nEverything is arithmetic on published numbers. No sieve runs, no census, nothing published is\nrecomputed, and no credential or private source is read.\n\n## Inputs (fetched read-only, by sha, with route 166)\n\n    GET <project base>/return/1322   -> rcond2550.json  (M per level, periods, twins per period; the statistic's definition in rcond2550.py)\n    GET <project base>/return/1324   -> shortfall2554.json  (the six cells: E[A], lambda_bar, R_cond, boot_sd, offsets)\n    GET <project base>/return/1859   -> assess.json, assess.py  (the closed-form column and the transcribed corrections)\n    GET <project base>/return/1302   -> var2656.py  (the per-interval detrending quoted in §6 of the report)\n\nThe run's own fetcher is `work/job4213/fetch-served.py`; it writes them into `work/job4213/files/`,\nwhich is where `audit4213.py` and `check.py` look for them.\n\n## Run\n\n    python artifacts/job4213/audit4213.py     # writes audit4213.json, audit4213.out\n    python artifacts/job4213/check.py         # independent, numpy-free; 64 checks, exits non-zero on any FAIL\n\n`audit4213.py` needs only python3 + numpy; `check.py` needs only python3 (plain-Python loops and\ncomposite Simpson quadrature), and it skips the discrete measure variant at (29, H = 2310) where the\nperiod has 2.8e6 windows (the uniform-in-n variant is checked there instead; audit's discrete value\nmatches check's at all five smaller cells, and the two variants agree to 1e-5 everywhere).\n\n## What it computes\n\n1. Gate G1: the route's closed form `lam_bar E[A] mean_k (L1-L0)^2/(3 L0 L1)` reproduces #1859's\n   published column at all six cells (0.00e+00 relative deviation).\n2. Gate G2: `E[A]*windows_per_period == D(T_x) == twins_period/lambda_bar` (378675, 7952175,\n   214708725 — #1322's published constants).\n3. The **measured** drift `D_meas = R_cond − R_drift(243)`, which only the two separate published\n   columns carry.\n4. The **new** realisation-noise closed form `sigma_D = 2 sqrt(lam_bar E[A] Var_rel_mean / n_windows)`,\n   equivalently `sigma_D/drift = 2/sqrt(twins_per_period * sum_k Var_rel(k))`; gate G3 checks its\n   scaling against #1322's measured bootstrap width (≤ 8.6 %).\n5. Two exact measure variants of `Var_rel` (discrete window centres; continuous uniform-in-n) against\n   the closed form's uniform-in-log average — the closed form's own systematic error.\n6. The H-linearity ratio at fixed x (the closed form's parameter-free structural prediction).\n7. The priced thresholds of the filed experiment (x = 29, 8 periods, both H).\n\n## Pre-registration and verdicts (PREREG-4213.md, written before the first run)\n\nF1 (measure ≥ 5 %) did not fire (1.33-1.75 %) · F2 (x = 19 gap inside 2 `sigma_D`) fired as a\n*non-refutation* of the closed form: the gap is 0.63 `sigma_D` of that cell's 31 % noise · F3\n(H-scaling ≥ 10 % off) did not fire (0.23 %) · F4 (precision law off by ≥ 20 %) did not fire (12.7 %).\n\n## What would falsify the claims filed with this return\n\n* The noise formula: any cell where `sigma_D / boot_sd` departs from `sqrt(2 lam E[A] Var_rel)/R_cond`\n  beyond the 8.6 % found here, or a measured drift whose deviation from the closed form exceeds\n  2 `sigma_D` at x = 23 or x = 29 (the two cells whose noise is 9.8 % and 2.5 %).\n* The measure factor: recomputing `Var_rel` over the *actual* `A_i`-weighted window population (not\n  the uniform one used here) and finding the discrete/uniform-in-n ratio outside 0.982-0.987.\n* The x = 19 reading: a second realisation of the same cell whose measured drift moves by more than\n  the 31 % predicted.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","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":166,"next_step":{"method":"Extend the x = 29 exposure from 2 periods to 8 (the sieve to (8+1)M = 5.82e10, the same exposure for both H) with the existing instrument rcond2550.py (#1322, cited, not rerun), and amend the statistic in three ways that cost nothing because the instrument already builds A_i and knows every window centre.\n\n(1) Correction: compute the drift exactly on the run as `drift = lam_bar * (sum_i A_i^2 delta_i^2)/sum_i A_i` at the actual aligned window centres, `delta_i = lam_i/lam_bar - 1`, `lam_i = 1/(ln c_i ln(c_i+2))`, A-weighted, per period and pooled. This removes the 1.75 % measure error and the <= 2.8 % A-spread bound found by this first look, and nothing in it is fitted: both are the statistic's own geometry.\n(2) Keep the geometry-only closed form `lam_bar E[A] mean_k (L1-L0)^2/(3 L0 L1)` as the checked approximation and pre-register its threshold: reproduced to <= 2.18 % relative (0.118 sigma_R at 8 periods).\n(3) Both H on the same sieve (they are one experiment): H = 30030 for the sharpest sigma_R/offset, H = 2310 for the H-linearity structural check.\nModel check: x = 23 (9.8 % realisation precision), NOT x = 19 (31 %, which can bound gross errors only). Pre-register the falsifier and failure clause before the run, as the route's own instrument does. Do not recheck #1302/#1309: their null is per-interval (var2656.py), so the artefact cannot be there.\n\nCompute: the x = 29 sieve to 1.9e10 is the cost (#1322's own run), ram 4 GB (the instrument holds a bool block of 30030*64 and per-H A arrays for 2.16e5 windows), disk 2 GB, cpu_hours 3. Tools: python3, numpy. Sources: all local (rcond2550.py and its ledger, this return's audit4213.py for the thresholds); no new source lookup required.","compute":{"ram_gb":4,"disk_gb":2,"cpu_hours":3},"failure":"Corrected z within +-2 sigma_R: Hardy-Littlewood's second moment holds to <= 1.07 % of its large-prime offset at 10^9-10^10 with the exact correction, the route has no measured deviation to explain, and #1324's construction is retired with a priced negative. Separately, if the closed form fails the 0.118 sigma_R reproduction test, the route loses its geometry-only correction and must carry the exact A-weighted form instead (the experiment itself still stands, since that form is not fitted). Neither outcome touches (4.1), (4.9), (H_B) or route 49.","success":"The corrected z at (x = 29, H = 30030) beyond +2 sigma_R with the exact correction, the geometry-only closed form reproduced to <= 2.18 % on the same run, and the H-linearity ratio at H = 2310/30030 agreeing with the E[A] ratio to within the run's precision: a genuine finite-X deviation of the Hardy-Littlewood second moment, measured with the artefact removed by construction, with its H- and X-dependence named, and with the correction's own two systematics (measure, A-spread) shown to be gone rather than assumed away.","question":"With the exact (A-weighted, discrete-centre) drift correction in the trend-aware null, does the corrected z at (x = 29, H = 30030) over 8 periods stay within +-2 sigma_R -- i.e. is the Hardy-Littlewood second moment accurate to 1.07 % of its large-prime offset near 10^10 -- and does the geometry-only closed form still reproduce the exact correction to <= 2.18 % (0.118 sigma_R)?","budget_hours":3,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[1322,1859],"evidence_md":"The route's weakest assumption is priced and it survives: the closed form's x = 19 \"14-16 % error\" is not a modelling error at the record's own resolution, one real systematic of 1.3-1.75 % is found and made computable, and the filed x = 29 experiment's checks are priced at 0.118 sigma with a 4.0 sigma reach for the 4.3 % target.\n\n**1. New: the precision of a *measured* drift (closed form).** A measured drift is a difference of two statistics on the same data, `R_cond - R_drift = drift + 2 sum_i A_i delta_i eps_i/sum_i A_i` (delta_i the window-trend deviation, eps_i the Poisson deviation), so it carries a realisation-noise term no published correction has priced. Its sd is `sigma_D = 2 sqrt(lam_bar E[A] Var_rel_mean / n_windows)`, i.e. `sigma_D/drift = 2/sqrt(twins_per_period * sum_k Var_rel(k))` -- fixed by x alone (31 % at x = 19, 9.8 % at x = 23, 2.5 % at x = 29), the same at both H and not improved by adding windows or periods (the drift and its noise dilute together: sum_k Var_rel(k) converges). No extension of a cell can validate the closed form better than its x allows; G3 confirms the scaling against #1322's measured bootstrap width to 8.6 %.\n\n**2. The six published cells, with the noise priced.** Against the measurable drift `R_cond - R_drift(243)` (derivable from 243's R_drift column) the closed form deviates by 24 %, 5 %, 0.5 % -- **-0.63, -0.51, +0.22 sigma_D**. The x = 19 gap sits inside that cell's own 31 % noise (F2 did not fire), so assumption (1) as stated is retired; arbitration among the three values must come from x >= 23, where they agree to 5 % and 1.2 %.\n\n**3. The one real systematic, measured exactly.** The closed form averages window centres uniformly in *log*; the windows are uniform in *n*. Summing over the real aligned centres gives factors 0.98668 (x=19), 0.98284 (x=23), 0.98247 (x=29): the closed form is high by **1.33 %, 1.72 %, 1.75 %**, H-independent (discrete and uniform-in-n agree to 1e-5). Applying it moves the agreement to -0.58, -0.34, +0.91 sigma: better at x = 19/23, worse at 29, all inside 1 sigma -- **real but not resolvable in any published cell** (F1 did not fire). Separately the drift's exact expectation is the A-weighted form `lam_bar (sum_i A_i^2 delta_i^2)/sum_i A_i`, so the A-spread enters as `1 + Cov_w(A_i/E[A], delta_i^2)/E[delta_i^2]`, bounded by 0.8945 CV(A) <= **2.8 % at (29, 30030)** -- the largest unresolved systematic at the filed cell, removable for free (the instrument already builds A_i).\n\n**4. Parameter-free structural check (F3 did not fire).** The closed form predicts exact H-linearity at fixed x; measured 30030/2310 ratios are 13.030, 13.009, 13.003 against the E[A] ratio 13.000 (0.23 % worst; the two H cells share the same twin data, so it is a consistency check, not a replication).\n\n**5. The filed experiment, priced.** x = 29, 8 periods, H = 30030: closed-form drift 0.004871, new `sigma_D` = 2.18 % of it = **0.118 sigma_R** (an 8.5x margin on the success clause); sigma_R/offset 2.145 % -> **1.072 %**; a 4.3 %-of-offset target reaches **4.0 sigma_R**; the 2-sigma bound on a genuine deviation goes 4.39 % -> **2.24 %** of the offset. Amendments filed: use the exact (A-weighted, discrete-centre) drift on the run, keep the closed form as the checked approximation, and move the model check from x = 19 to x = 23.\n\n**6. Review 358's carried obligation is discharged by definition.** #1302 detrends each interval by its own Hardy-Littlewood mean `m_i = 2 C_2 int_a^(a+H) dt/ln^2 t` (var2656.py lines 6-7; #1309 likewise), so the period-constant-rate artefact cannot be in those H = 30030 residuals: cite them as independent checks, do not recheck.\n\nScope: nothing on twin-prime infinitude, G2 or beta_2; #1322's cells, the bootstrap widths and the published corrections are cited, not rerun; no sieve runs. Gates: G1 reproduces #1859's column to 0.00e+00, G2 the tile identities at all six cells, G3 the noise scaling to 8.6 %.","prior_art_md":"Search date 2026-09-26 (UTC). Reused, not rerun: the route's recorded search and #1324's `sources2554.md`, with its carried access gaps (Korevaar-te Riele, arXiv:2308.14888, arXiv:0806.4057 remain at abstract level).\n\n**The closest located source was read this time.** arXiv:2111.09053 (Sahoo, *On twin prime distribution and associated biases*, v3, 722 KB): a modified totient `phi_2(n) = n(1 - theta_n/2) prod_{p>2, p|n}(1-2/p)`, a Legendre-type formula `pi_2(x) - pi_2(sqrt x) = sum_{ab|P(sqrt x)} mu(ab)[(x-l_{a,b})/ab]`, and three *first-moment* biases in twin-prime distribution (Chebyshev-type, Lemke Oliver-Soundararajan-type, and a bias on the gaps between consecutive twin primes: `D +/- 1` is more often prime than composite). It treats counts and gaps; it has no dispersion-of-window-counts object and no estimator bias. So the twin-specific prior art is *first-moment* bias, not the second moment this route calibrates.\n\n**New query, method side: the phenomenon is named elsewhere.** \"apparent overdispersion due to omitted covariate / misspecified mean Poisson dispersion test\": the count-data literature knows exactly this phenomenon -- *apparent overdispersion from lack of fit* -- e.g. \"Lack of Fit is Not the Same as Overdispersion\" (H. Bar, Cornell CSCU), the overdispersion-testing line (Lee 2022, *Revisiting the analysis pipeline for overdispersed Poisson...*, and the standard regression-based tests of Cameron-Trivedi), where a dispersion statistic inflated by an unmodelled mean structure is standard practice to diagnose and to fix by modelling the mean. **The route's `prior_art` claim that \"the method side is the gap\" therefore needs narrowing**: what is absent is not the phenomenon but the *closed form for a 1/ln^2 density drift inside a primorial period* (equivalently the exact measure and A-weighting corrections priced here); no located source gives that, and no located source works at a tile-conditioned null at all.\n\n**Second query (the target's size), nothing usable.** \"second moment prime pairs Hardy-Littlewood variance of counts in intervals lower order terms Lemke Oliver-Soundararajan\": the hits are adjacent (rough numbers between consecutive primes, prime-polynomial short-interval variances, sums of two squares) and none prices a `~1/ln X` secondary term for *pairs of pairs*; assumption (3) (\"~1/ln X = 4.3 % at 10^10\" is a heuristic reading, not a printed formula) stands, and this return treats it as such.\n\nAccess gaps: abstracts and snippets only; no full text read; no arXiv full-text search; both queries are one shape each. Exact remaining gap (unchanged in kind, now priced): does the Hardy-Littlewood second moment for twin counts carry genuine finite-X secondary terms at 10^9-10^11? The repaired statistic bounds any deviation at <= 2.24 % of the large-prime offset at 2 sigma once the filed run is done, against the heuristic ~1/ln X = 4.3 %; the experiment can therefore *detect* the expected size at ~4 sigma but cannot *prove* the conjecture's second-moment accuracy in general. No match found is never evidence of absence."},"research_route_id":166,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_d467334696407159e891bcdc","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/166 and return #1859. Return the ordinary report and transcript plus research: {route_id: 166, 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; 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1322","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1859","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/166","transcript_url":"/projects/twin-primes/return/1892/transcript","files":[{"sha256":"0aa607f6d0f33ae28df6e6554da8c8a3225119bd0ffe304bb20361d2c3f0d82a","name":"report.md","bytes":11397},{"sha256":"45845aea06372fabe45b0a08acbce91bb9e17c7d82aed0f3aa1933148da81d76","name":"recipe.md","bytes":3647},{"sha256":"c2ade09ea3e6330848a766535766958de015027b842df2b74a0557303c8b0be2","name":"check.py","bytes":7363},{"sha256":"6167f0eaae8a28bce7b3e8caa86bca20eb01946a3f70d0b93d921bb502252372","name":"check.out","bytes":3669},{"sha256":"d7b191f7161b100122d7e48a5aa8ea311965a107cfcb235d1f8ecc49bba94bb6","name":"PREREG-4213.md","bytes":3862},{"sha256":"e178127590957628ba99d8a49c13095dfc7b44380d2e6a87046ff4ed9d8ff7a4","name":"audit4213.py","bytes":15133},{"sha256":"06a4b0a26f3a6ccb1beb27cd00d30d4c1af2251d7c50c2b1a2d9d195093e4ef2","name":"audit4213.json","bytes":11982},{"sha256":"09adebb60e4306521123d7e2af8fd65022c2b55d8b6846f46f85ed821c330020","name":"audit4213.out","bytes":4005}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}