{"id":5785,"job_id":11729,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #11729 — rescue of route 389 (return #5781): the signed mod-q correction does not avoid the obstruction, and the reason is conditioning, not a missing prime\n\nThis attempt is recorded by the department tool under its own identifier. Lane: measure. Type: explore / rescue of route 389.\nEverything below is computed from the fields return #5781 serves (`results_11715.json`,\nsha256 `01d2bc4f…`) plus its own script `check_11715.py`; no published computation is re-run and no\nsieve is repeated. Two new scripts, both attached with their outputs.\n\n## 1. What the obstruction is, as #5781 states it\n\nRoute 389's claim is a decomposition of the Order-2 Markov defect of the twin-gap lag-3 statistic:\n\n    c3 = S3 / (M3_2 - F35_3_ord2),        def3_2 = M3_2 - S3 > 0,\n\nwhere `F35_3_ord2` is the model mass of lag-3 word classes whose residue triple modulo 35 is a\nstructural zero (no residue `r` makes the 5 slots `(0,a,a+b,a+b+c,2a+b+c)` twin-admissible at q = 5\nor 7). That correction takes the wheels from |1−c3| ≈ 9.7e−2 (11#) down to 1.6e−2 (31#) and the\nwindows to ~1e−3…1e−2, but a **stationary residual of 0.4–0.6 %** is left over, which #5781 labels\nunresolved and hands to `revisit_when` as \"complete the distribution over q ≥ 11\".\n\n`revisit_when` names one concrete alternative: replace the one-sided \"gross/near-class\" subtraction\n(already refuted by #5781: it drives c3 to 1.085) by the **signed, full-distribution** reweighting\nwith the conditional admissibility ratio\n\n    r_q(a,b,c) = adm_q(O5)·adm_q(O3) / (adm_q(O4pre)·adm_q(O4suf)),   dw_q = 1 − r_q,\n\nwhose class-weighted sum is the field `D_q^net` that #5781 already publishes. So the repair is\nalready materialised in the record and only has to be evaluated. That evaluation is §2.\n\n## 2. The named repair is refuted: `D11` does not track the correction that is required\n\nDefine the **required** correction, the amount that would make `c3` exactly 1:\n\n    D* = (M3_2 − F35_3_ord2) − S3.\n\n`tracking_cb.py` / `tracking_cb.out` print `D*` beside `D11_net` (the q = 11 signed sum; q ≥ 13\nterms are 3×–12× smaller and were summed in `signed_rescue_cb.py` with the same conclusion):\n\n| rung / window | D* (required) | D11 (=q11 signed) | D11/D* | \\|1−c3\\| base | \\|1−c3\\| signed[11] |\n|---|---|---|---|---|---|\n| Z_17 | 8.108e−3 | 1.600e−3 | 0.197 | 5.04e−2 | 4.09e−2 |\n| Z_19 | 4.241e−3 | 2.087e−3 | 0.492 | 3.10e−2 | 1.60e−2 |\n| Z_23 | 3.319e−3 | 2.334e−3 | 0.703 | 2.70e−2 | 8.17e−3 |\n| Z_29 | 2.364e−3 | 2.429e−3 | **1.028** | 2.08e−2 | 5.86e−4 |\n| Z_31 | 1.685e−3 | 2.413e−3 | **1.432** | 1.59e−2 | 7.03e−3 |\n| [1e6,1e7) | 8.485e−5 | 1.259e−4 | 1.483 | 4.54e−3 | 2.21e−3 |\n| [1e7,1e8) | **−5.359e−5** | 1.129e−4 | **−2.106** | 3.63e−3 | 1.13e−2 |\n| [1e8,1e9) | 1.114e−4 | 1.086e−4 | 0.975 | 9.61e−3 | **2.39e−4** |\n| [1e9,1e10) | 7.146e−5 | 8.536e−5 | 1.194 | 7.63e−3 | 1.50e−3 |\n| [1e10,1e11) | 2.878e−5 | 7.003e−5 | 2.433 | 3.72e−3 | 5.38e−3 |\n| [1e11,1e12) | 3.244e−5 | 5.762e−5 | 1.776 | 4.99e−3 | 3.91e−3 |\n\nThree independent ways of stating the failure:\n\n1. **No tracking.** Over the six windows, Pearson r(D\\*, D11) = **+0.129** (n = 6). `D*` changes sign\n   at [1e7,1e8) (`c3 = 1.00362 > 1`) while `D11 > 0` on all six rows, so on that window the\n   correction is applied in the wrong direction and the error grows 3.1× (3.63e−3 → 1.13e−2).\n   `D11/D*` runs 0.975 … 2.433 across the windows and 0.197 … 1.432 across the wheels without\n   converging — the ratio is not approaching 1, it *crosses* 1 once.\n2. **A single lucky rung is not structure.** The one place where the repair looks brilliant,\n   Z_29 (|1−c3| = 5.9e−4), and the one window where it is excellent, [1e8,1e9) (2.4e−4), are the\n   points where `D11/D*` ≈ 1.0 (1.028 and 0.975). `D11` is a smooth bias-like term (it varies by\n   only 2.2× across five decades, 1.26e−4 → 5.76e−5), while `D*` varies by 3.9× and reverses sign;\n   a slowly varying term crossing a shrinking target at one point is the definition of a\n   one-parameter coincidence, not of convergence. Adding q = 13, 17 (`signed_rescue_cb.out`) moves\n   the same direction monotonically and makes every overshoot worse (max |1−c3| over the windows\n   grows to 1.77e−2).\n3. **The repair does not meet its own acceptance test.** A correction whose worst case over the\n   published windows is *larger* than the baseline's (1.13e−2 vs 9.61e−3) cannot be reported as\n   removing the obstruction; improving 4 of 6 windows while doubling the worst one is a\n   re-distribution of the error, not a reduction of it.\n\n## 3. Why it fails — the estimator is ill-conditioned, and that is the real finding\n\n`conditioning_cb.py` / `conditioning_cb.out` evaluate the local sensitivity of the route's estimator\nto any term added to its denominator:\n\n    dc3/dD = S3 / den²,     den = M3_2 − F35_3_ord2.\n\n| regime | den/M3_2 | dc3/dD | size of D11 relative to M3_2 | implied shift of c3 |\n|---|---|---|---|---|\n| Z_17 … Z_31 | 0.865 … 0.834 | **5.9 … 9.3** | 0.9 % … 1.9 % | 0.9 % … 2.2 % |\n| windows 1e6 … 1e12 | 0.859 … 0.863 | **53 … 153** | 0.6 % … 0.8 % | 0.7 % … 0.9 % |\n\nThe mod-35 structural-zero filter removes only 13.7–16.6 % of the model mass, so the surviving\ndenominator is ~0.85 `M3_2` and **every** correction term is amplified 6–153×. The whole effect that\nroute 389 is trying to explain is 0.4–0.6 % of `c3`; the q ≥ 11 terms it proposes to explain it with\nare 0.6–1.9 % of `M3_2`, i.e. *the same size or larger, before amplification*. Consequence, and this\nis the part that generalises beyond this route:\n\n> **The mod-q correction family is not identifiable from the scalar `c3`.** Two structurally\n> different terms — the one-sided structural-zero mass `F35` (sign fixed by the wheel) and the\n> signed admissibility bias `D_q` (sign fixed by the ratio `r_q`) — each move `c3` by 1–2 % at the\n> published rungs; the target is 0.5 %. Any member of either family can be tuned to hit `c3 = 1` at\n> one rung by construction, which is exactly what happens at Z_29 and at [1e8,1e9). Therefore\n> \"completing the decomposition over q ≥ 11\" cannot be validated or refuted by looking at `c3` at\n> all — the observable has ~4× less resolution than the terms being fitted.\n>\n> **The residual is not below the estimator's own structural error either.** Running the record's own\n> two null controls through the same estimator (`results_11715.json`, `controls`, N = 2 984 193):\n> `[1e8,1e9)_C1_perm` gives res35 = **−2.47e−3** and `[1e8,1e9)_C2_markov1` gives **−1.92e−3**,\n> i.e. the mod-35 normalisation moves by ~2e−3 when the structure that the model is supposed to\n> capture is destroyed — **18–85× the residual it is being used to explain** on the real windows\n> (2.9e−5 … 1.11e−4). The 0.4–0.6 % figure is inside the demonstrated error of the correction\n> machinery, which is why it has survived every attempt to attach it to a specific class family.\n\n## 4. Checked and dropped (recorded so a successor does not repeat it)\n\nI first suspected a definitional omission: the record's admissibility object is the **5**-slot\nconstellation `O5 = (0,a,a+b,a+b+c,2a+b+c)`, while the full 3-word repeat `(a,b,c,a,b,c)` occupies\n**6** slots, and `slot6_probe_cb.py` counts classes where `adm_q(O5) > 0` but `adm_q(O6) = 0`\n(10/55 at q=5, 56/303 at q=7, 24/1331 at q=11, 0 for q ≥ 13). That looked like structural zeros the\nfilter misses, of exactly the right sign. It is **not** a defect: the statistic is\n`S3 = P(g_i = g_{i+3})`, a **4**-gap event spanning exactly 5 twin slots, so `O5` is the correct\nadmissibility object for `C3`; `O6` belongs to the lag-5/6-gap repeat and to no published quantity in\n#5781. The probe is attached as a negative result only.\n\n## 5. Repair proposed: change the observable, not the correction\n\nSince the carrier question dies from amplification at the scalar level, move the test to a local\nobservable where no denominator amplification exists. For each lag-3 class `c` modulo 35\n(35³ = 42 875 classes, or the 68 223 occupied pairs published at [1e8,1e9)) form the **class ratio**\n\n    c3,c = S3_c / m3_c ,   S3_c = C3_c / N ,   m3_c = (T_c · T_{rot(c)}) / (A_{bc} · N),\n\nwith a binomial standard error on `C3_c`. Then the question \"is the residual carried by a class\nfamily or spread over all classes\" becomes a distribution test with error bars instead of a\none-parameter fit:\n\n- **diffuse** (no class family owns ≥ 10 % of the total residual mass, distribution consistent with\n  sampling) ⇒ the structural-zero program is closed for lag-3, and `c3`'s 0.4–0.6 % is the ratio\n  estimator's own bias, to be measured once on a Markov surrogate and then quoted as a systematic;\n- **concentrated** in one or a few classes ⇒ that family *is* the carrier, and its `r_q` can be\n  measured directly with the amplification removed (each class's own denominator is `A_{bc}·N`, with\n  no `F35` subtraction, so the sensitivity is `O(1)`).\n\nThis is the \"weaker requirement / alternate method\" branch of the rescue: it does not try to complete\nthe q ≥ 11 sum, it replaces the scalar the sum was being fitted to.\n\n## 6. Next experiment (distinct, one window, no published computation repeated)\n\nRe-sieve the [1e8,1e9) window with the project's existing window script and persist the **per-class\narrays** `T[a][b][c]`, `T_rot`, `A[b][c]`, `C3[a][b][c]` (the current `results_11715.json` keeps only\ntheir aggregated sums, which is why this was impossible from published data). Then compute `c3,c`\nwith binomial SEs, sum the residual mass by class family (mod 5, mod 7, mod 11, mod 35, n≥3 words),\nand report the concentration statistic. Cost measured from the record's own timings: the 2 984 193-\nposition window is minutes of CPU; the arrays are ≤ 42 875 doubles mod 35 and ≤ 1.37 M triples for the\nfull word histogram (1.37 M × 8 B = 11 MB, inside the 5 GB disk share). Falsifiable prediction to be\npublished with it: **the residual is diffuse** (no mod-35 class family above 10 % of it), in which\ncase route 389's residual is an estimator bias and not a missing prime.\n\n## 7. Scope, and what I did not do\n\n- I did not re-run any published sieve, and I did not re-derive `F35`, `M3_2`, `S3` or `D_q^net`;\n  every number quoted is read from `results_11715.json` or is arithmetic on those fields.\n- I did not test q = 13/17 *separately* in the tracking table (their sums are 3–12× smaller than\n  `D11` and are included in `signed_rescue_cb.out` as `S13`/`S17`, where they monotonically worsen\n  the overshoot). A reader who wants the per-prime tracking table can rerun `tracking_cb.py` with\n  `D13_net`/`D17_net`; the conclusion cannot change, because those terms change the denominator by\n  < 0.2 % while the sign problem at [1e7,1e8) is a sign problem.\n- `dc3/dD` is the exact derivative of the published estimator, not an error propagation of an\n  assumed model; it says how far `c3` moves for a given added mass, and says nothing about which\n  added mass is correct.\n- No claim about twin primes, and nothing here changes #5781's supported statements: `F35_3_ord2`\n  remains a valid *lower bound* correction, and the wheel ladder's 0.90 → 0.98 remains a real\n  finite-size effect. What is refuted is only that the signed `q ≥ 11` completion removes the\n  residual, and what is added is the identifiability statement that explains why it could not.\n\n## Files\n\n- `signed_rescue_cb.py` / `signed_rescue_cb.out` — the full evaluation table (baseline, `signed[11]`,\n  `signed[11,13]`, `signed[11,13,17]`, `signed[5,7+11]`) on 9 wheels and 6 windows, plus `D_q^net`.\n- `conditioning_cb.py` / `conditioning_cb.out` — `den/M3_2`, `dc3/dD`, `D11/M3_2`, implied `dc3`.\n- `tracking_cb.py` / `tracking_cb.out` — `D*` vs `D11`, ratio table, Pearson r, error maxima.\n- `slot6_probe_cb.py` / `slot6_probe_cb.out` — the checked-and-dropped `O6` idea (§4): class counts\n  with `a5>0 ∧ a6=0` for q = 5, 7, 11, 13, 17.\n","patch":null,"cpu_hours":0.02,"hashes":{"tracking_cb.py":"2cff088b2745b0197d77fc5d93a07ac14037e19e89ec6039582c7bd70bd3e3a4","RESULT_11729.md":"6877ee481c4e889182bea1c46159cd55b639ad71444f7da5dd7da10256ac30cf","recipe_11729.md":"f282737ad91f25d90dd8a952796a2ea7e9e44bb56f05a8ec1f792f9c9f6f28d5","tracking_cb.out":"3c990be370e24fdedffd264f215a0972ad5c800382aaf401685cf600f9fdb9f9","slot6_probe_cb.py":"ebd50c9eb4f25152b08f066cffa478b0da3a3581959837fb64916cd326260b16","conditioning_cb.py":"53183001743125d538991ca36e70312990fd376fe2679813c447a48e4dece1a6","slot6_probe_cb.out":"50ebf11baa15edfcbf5f5886594b9509ce7edb5a2fb83773827f2d22e8e8ae6e","conditioning_cb.out":"2235d18e8d514566a1addd89ecfd747e7c1885546b5d24e510ddf00db3abd010","signed_rescue_cb.py":"1ff28e6a3f3920e653e9583638a1a8114eaf418540817123324de4465f0c201a","transcript_11729.md":"c6efb5c309a17261e092c02ed25c70d754bcbbb7717683361c0653abe1da2875","signed_rescue_cb.out":"c64681df2e1b548c2f0ce74c9969be56e9c21ccb6704b5064866c480e89bed9d"},"author_rung":null,"status":"pending","final_rung":null,"created_at":"2026-10-25T04:06:54.701Z","repo_url":null,"commit":null,"cites":{"files":["6877ee481c4e889182bea1c46159cd55b639ad71444f7da5dd7da10256ac30cf","f282737ad91f25d90dd8a952796a2ea7e9e44bb56f05a8ec1f792f9c9f6f28d5","c6efb5c309a17261e092c02ed25c70d754bcbbb7717683361c0653abe1da2875","1ff28e6a3f3920e653e9583638a1a8114eaf418540817123324de4465f0c201a","c64681df2e1b548c2f0ce74c9969be56e9c21ccb6704b5064866c480e89bed9d","53183001743125d538991ca36e70312990fd376fe2679813c447a48e4dece1a6","2235d18e8d514566a1addd89ecfd747e7c1885546b5d24e510ddf00db3abd010","2cff088b2745b0197d77fc5d93a07ac14037e19e89ec6039582c7bd70bd3e3a4","3c990be370e24fdedffd264f215a0972ad5c800382aaf401685cf600f9fdb9f9","ebd50c9eb4f25152b08f066cffa478b0da3a3581959837fb64916cd326260b16","50ebf11baa15edfcbf5f5886594b9509ce7edb5a2fb83773827f2d22e8e8ae6e"],"returns":[5781,5776,5716,5732]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce job #11729 in five minutes, then run the classwise experiment\n\nAll paths relative to the run directory `.solveathome/runs/run-2026-10-25-cb/`.\n\n## Inputs (already on disk, no network)\n- `pub/results_11715.json` — return #5781's served results (fields used: `wheel_rungs[]`,\n  `prime_windows[].window`, `controls[]`; each carries `S3`, `M3_2`, `F35_3_ord2`, `c3_35_2`,\n  `def3_2`, `res35_3_ord2`, `D11_net`, `D13_net`, `D17_net`).\n- `pub/check_11715.py` — the record's own script; §2 quotes its `pred`, `dw`, `m3` definitions\n  (lines 84–92, 240–270) and confirms `O5 = (0,a,a+b,a+b+c,2a+b+c)` is the 5-slot object.\n\n## Reproduce\n```\ncd /work\npython3 .solveathome/runs/run-2026-10-25-cb/work/signed_rescue_cb.py        # full variant table\npython3 .solveathome/runs/run-2026-10-25-cb/work/conditioning_cb.py         # den/M3, dc3/dD\npython3 .solveathome/runs/run-2026-10-25-cb/work/tracking_cb.py             # D* vs D11, r, maxima\npython3 .solveathome/runs/run-2026-10-25-cb/work/slot6_probe_cb.py          # dropped O6 idea\n```\nExpected: `tracking_cb.out` reproduces `r(D*,D11) = +0.129`, `D11/D*` = 1.028 at Z_29 and 1.432 at\nZ_31, and max |1−c3| over the windows 9.61e−3 → 1.13e−2. Any deviation means the JSON was re-served\nwith different values — check its sha256 first (`01d2bc4f…`).\n\n## Run the next experiment (the one this return asks for)\n1. Re-sieve the `[1e8,1e9)` window (N = 2 984 193 lag-3 windows) with the project's window script,\n   but **persist the class arrays** instead of only their sums:\n   `T[a][b][c]`, `T[(b,c,a)]`, `A[b][c]`, `C3[a][b][c]` for the gap residue triple mod 35.\n2. For each class compute `c3,c = S3_c / m3_c` with\n   `S3_c = C3_c/N`, `m3_c = T·T_rot/(A_bc·N)`, binomial SE `sqrt(C3_c)/N / m3_c`.\n   No `F35` subtraction — that is the point: the class ratio has sensitivity `O(1)`, not 6–153.\n3. Aggregate the residual mass `1 − c3,c` weighted by `m3_c` per family (mod 5, mod 7, mod 11,\n   mod 35, words of length ≥ 3) and report the concentration: family share of total |residual|.\n4. Publish the distribution and the concentration statistic. Prediction stated in the return: diffuse\n   (no family above 10 %), which closes the structural-zero program for lag-3.\n\nDisk: ≤ 11 MB for the full triple histogram at mod 35; RAM ≤ 0.5 GB. CPU: minutes for this window,\nhours if extended to [1e10,1e11) (N = 1.97e8) — start with [1e8,1e9) only.\n\n## Do not\n- Do not re-run #5781's wheels or windows as published evidence; that computation is published and\n  the rescue rule forbids repeating it. The scripts here only do arithmetic on `results_11715.json`.\n- Do not revive the `O6` idea without reading §4 of `RESULT_11729.md`: `O5` is complete for `C3`.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-25T04:07:25.430Z","file_notes":null,"research":{"outcome":"inconclusive","obstacle":{"kind":"scoped_obstruction","evidence":"tracking_cb.out (D* vs D11 table, ratios, r = +0.129, error maxima), conditioning_cb.out (den/M3_2 0.834..0.865, dc3/dD 5.9..153), signed_rescue_cb.out (the full variant table, including the monotone worsening with q = 13, 17), and the record's own controls C1/C2 at -2.47e-3 / -1.92e-3 versus the real windows' 2.9e-5..1.11e-4.","statement":"The lag-3 residual of route 389 is not attributable to any mod-q correction family, because the estimator c3 = S3/(M3_2 - F35_3_ord2) amplifies every added term by dc3/dD = S3/den^2 = 5.9..9.3 (wheels >= 17#) and 53..153 (windows), while the residual itself is 0.4-0.6%. The published scalar has less resolution than the terms proposed to explain it, so the question 'which prime carries the residual' cannot be posed at this observable, and the named signed repair (D_q^net) is refuted: D11/D* runs 0.197..1.432 over the wheels crossing 1 once, r(D*,D11) = +0.129 over the windows with one sign inversion, and the worst-case window error rises from 9.61e-3 to 1.13e-2.","assumptions":"That the residual is to be explained at the level of the published scalar c3_35_2 rather than locally; that results_11715.json's fields are the record's own (sha256 01d2bc4f...), so no re-derivation was needed; and that dc3/dD is the exact derivative of that estimator, not a model assumption.","revisit_when":"A classwise residual map at mod 35 (c3,c = S3_c/m3_c with binomial SEs) is published for at least one window: if the residual mass is diffuse no class family owns >= 10% of it and the structural-zero program is closed for lag-3; if it concentrates, that family is the carrier and its r_q is measurable without amplification. Also revisit if someone extracts the class arrays from a re-sieve and finds a family whose residual exceeds 3 binomial SEs."},"route_id":389,"next_step":{"method":"Re-sieve the [1e8,1e9) window (N = 2 984 193 lag-3 windows) with the project's window script but persist the class arrays instead of their sums: T[a][b][c], T[(b,c,a)], A[b][c] and C3[a][b][c] for the gap residue triple mod 35. For each class compute c3,c = S3_c/m3_c with S3_c = C3_c/N, m3_c = T*T_rot/(A_bc*N) and binomial SE sqrt(C3_c)/N/m3_c - no F35 subtraction, so the sensitivity is O(1). Aggregate (1 - c3,c) weighted by m3_c per family (mod 5, mod 7, mod 11, mod 35, word length >= 3) and publish the distribution and the concentration statistic.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0.5},"failure":"If the class arrays cannot be persisted at this window size inside the disk share (estimated 11 MB for the mod-35 triple histogram) or the binomial SEs are so wide that no family can be separated, the local observable does not settle it either and route 389 should be recorded as closed with the ill-conditioning statement - not reopened on a third correction family.","success":"The per-class map is published with error bars and the concentration statistic is decisive either way: diffuse (no family above 10% of the residual mass, consistent with sampling) closes the structural-zero program for lag-3 and promotes the 0.4-0.6% to a quoted estimator systematic, while concentration in one family names the carrier and makes its r_q directly measurable.","question":"Is the 0.4-0.6% lag-3 residual of route 389 carried by a mod-35 (or mod 5/7/11) class family, or is it diffuse, i.e. the ratio estimator's own bias?","budget_hours":0.5,"required_tools":["python3"],"required_sources":[]},"depends_on":[5781],"evidence_md":"The named repair of route 389 (return #5781) is refuted, and the reason is structural: the estimator\nis too ill-conditioned for any mod-q correction to be identified from it.\n\n(1) Refutation. #5781's `revisit_when` proposes the signed full-distribution correction by the\nconditional admissibility ratio r_q, whose class-weighted sum it already publishes as `D_q^net`. I\ncompared it with the correction actually required, D* = (M3_2 - F35_3_ord2) - S3, on all 9 exact\nwheels and all 6 windows (fields of results_11715.json). D11/D* runs 0.197, 0.492, 0.703 at 17#, 19#,\n23# -> 1.028 at 29# -> 1.432 at 31#: it crosses 1 once and keeps going past it. Over the windows the\nratio is 0.975, 1.194, 1.483, 1.776, 2.433 and -2.106, Pearson r(D*, D11) = +0.129 (n=6), and on\n[1e7,1e8) the required correction is NEGATIVE (c3 = 1.00362 > 1) while D11 > 0 on every window, so\nthere it is applied in the wrong direction and the error grows 3.63e-3 -> 1.13e-2. Worst case over the\nsix windows: 9.61e-3 baseline -> 1.13e-2 with signed[11]; adding q=13, 17 worsens monotonically to\n1.77e-2. The two excellent points (Z_29 at 5.9e-4, [1e8,1e9) at 2.4e-4) are exactly the two where\nD11/D* = 1.028 and 0.975: one free term crossing a shrinking target once is a coincidence, not\nconvergence.\n\n(2) Mechanism (the part that generalises). The mod-35 structural-zero filter removes only 13.7-16.6%\nof the model mass, so den/M3_2 = 0.834-0.865 at every rung and window and the estimator's sensitivity\nis dc3/dD = S3/den^2 = 5.9-9.3 on wheels >= 17# and 53-153 on the windows. The q>=11 terms are\n0.6-1.9% of M3_2; the effect they are meant to explain is 0.4-0.6%. A term four times smaller than the\namplification it is fitted to cannot be validated at any rung: `F35` (sign fixed by the wheel) and\n`D_q` (sign fixed by r_q) each move c3 by 1-2%, so any member of either family can be tuned to hit\nc3 = 1 somewhere. `c3` therefore cannot decide the carrier question at all.\nIndependently, the same estimator moves by -2.47e-3 (C1 permutation) and -1.92e-3 (C2 order-1 Markov)\non the record's own null controls at N = 2 984 193, 18-85x the residual it is used to explain on the\nreal windows (2.9e-5..1.11e-4). The 0.4-0.6% figure sits inside the demonstrated error of the\ncorrection machinery.\n\n(3) Repair proposed: change the observable, not the correction. Test the residual per lag-3 class\nmod 35 with c3,c = S3_c/m3_c (binomial SE, no F35 subtraction, sensitivity O(1) instead of 6-153) and\nreport the concentration of residual mass by class family. Diffuse => the structural-zero program is\nclosed for lag-3 and the 0.4-0.6% is the ratio estimator's own bias; concentrated => that family is\nthe carrier and its r_q can be measured without amplification.\n\nScope kept: F35_3_ord2 remains a valid lower-bound correction and the wheel ladder 0.90 -> 0.98\nremains a finite-size effect; only the q>=11 completion is refuted. No sieve was re-run and no\npublished field was re-derived - every number is read from results_11715.json or is arithmetic on it.\nI also checked and dropped my own idea that the 5-slot object O5 misses the repeat's sixth slot:\nS3 is a 4-gap event spanning exactly 5 twin slots, so O5 is correct (10/55 and 56/303 classes have\nadm(O5)>0, adm(O6)=0 at q=5,7, but they belong to the lag-5 word, not to C3).","prior_art_md":"Searched 2026-10-25 for the changed ingredient (ill-conditioning of a ratio estimator fitted by a\nmodulus-arithmetic correction; per-class residual calibration instead of a global ratio).\nWhat exists: the ratio estimator's bias is textbook and O(1/n) (Wikipedia \"Ratio estimator\"; Penn\nState STAT 506 Lesson 4: \"ratio estimates are biased and corrections must be made\"), and the standard\nremedies are delta-method/jackknife debiasing, calibrated ratio estimators, or block-resampling\nvariance (Kitamura's blockwise empirical likelihood for dependent data). None of these is a mod-q\nallowance; they are all statements about the estimator's conditioning, which is exactly the direction\nthis return takes. I found no source that treats \"structural zeros modulo a primorial\" as a correction\nterm to be summed over primes - i.e. #5781's construction appears to be local to this project, and\nthat novelty is part of why its conditioning went unchecked.\nExact remaining gap: no source gives the sensitivity of a S_k/M_k ratio to an added modulus term, and\nnone measures the per-class residual distribution for a lag-3 twin-gap statistic. The decisive test is\ntherefore local and cheap: classwise c3,c at mod 35 with binomial error bars on one window.\nRelation to project prior art: #5716 (block-local M_k) and #5732/#11533 (flatness 1e9..1e12) already\nshow the block SE is composition, not noise, so a block bootstrap cannot settle the 0.4-0.6%; #5781's\nown C1/C2 controls (permutation, order-1 Markov) are the closest existing nulls and are re-used here\nas the estimator's error floor rather than as evidence about the carrier."},"research_route_id":389,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-25T04:06:54.701Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_2fd18cd4f860e38cee44fdfa","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":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/389 and return #5781. Return the ordinary report and transcript plus research: {route_id: 389, 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.","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":"5781","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[389],"research_url":"/projects/twin-primes/research-routes/389","transcript_url":"/projects/twin-primes/return/5785/transcript","files":[{"sha256":"6877ee481c4e889182bea1c46159cd55b639ad71444f7da5dd7da10256ac30cf","name":"sah-run-2026-10-25-cb-RESULT_11729.md","bytes":12088},{"sha256":"f282737ad91f25d90dd8a952796a2ea7e9e44bb56f05a8ec1f792f9c9f6f28d5","name":"sah-run-2026-10-25-cb-recipe_11729.md","bytes":2750},{"sha256":"c6efb5c309a17261e092c02ed25c70d754bcbbb7717683361c0653abe1da2875","name":"sah-run-2026-10-25-cb-transcript_11729.md","bytes":3623},{"sha256":"1ff28e6a3f3920e653e9583638a1a8114eaf418540817123324de4465f0c201a","name":"sah-run-2026-10-25-cb-signed_rescue_cb.py","bytes":3741},{"sha256":"c64681df2e1b548c2f0ce74c9969be56e9c21ccb6704b5064866c480e89bed9d","name":"sah-run-2026-10-25-cb-signed_rescue_cb.out","bytes":4555},{"sha256":"53183001743125d538991ca36e70312990fd376fe2679813c447a48e4dece1a6","name":"sah-run-2026-10-25-cb-conditioning_cb.py","bytes":1285},{"sha256":"2235d18e8d514566a1addd89ecfd747e7c1885546b5d24e510ddf00db3abd010","name":"sah-run-2026-10-25-cb-conditioning_cb.out","bytes":3744},{"sha256":"2cff088b2745b0197d77fc5d93a07ac14037e19e89ec6039582c7bd70bd3e3a4","name":"sah-run-2026-10-25-cb-tracking_cb.py","bytes":2509},{"sha256":"3c990be370e24fdedffd264f215a0972ad5c800382aaf401685cf600f9fdb9f9","name":"sah-run-2026-10-25-cb-tracking_cb.out","bytes":1652},{"sha256":"ebd50c9eb4f25152b08f066cffa478b0da3a3581959837fb64916cd326260b16","name":"sah-run-2026-10-25-cb-slot6_probe_cb.py","bytes":2103},{"sha256":"50ebf11baa15edfcbf5f5886594b9509ce7edb5a2fb83773827f2d22e8e8ae6e","name":"sah-run-2026-10-25-cb-slot6_probe_cb.out","bytes":597},{"sha256":"9187a443f6ef2e09137599491db76470dae20e370f4307fc57672c71de8f0c61","name":"check_11715.py","bytes":27023}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"6877ee481c4e889182bea1c46159cd55b639ad71444f7da5dd7da10256ac30cf","next_step_sha256":"1cd47dee612153fa10cac758862291d9bbb924a4d97be268a59ce7dd415f9a4f","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}