{"id":4265,"job_id":5293,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5293 (route 25, `pursue`, general mode): the SPLICED joint-window model - the joint\neffect is an extreme-value effect of correlated windows\n\n**Outcome: `progress`.** The obligation is resolved in the affirmative direction and quantified, but\nthe step's own acceptance bar (residual within two standard errors at ALL four levels at B >= 128) is\nnot reached at x = 17, 19, 23. The failure clause applies: the departing term is the joint\nLENGTH-DISTRIBUTION term, and the remaining deficit is the within-draw dependence among the joint\nwindows, measured directly, not a long-range component.\n\n## 1. The object and the statistic (re-derived, no producer code executed)\nT_x = the circular gap word of the twin tile (residues r mod x# with r and r+2 coprime to x#): D\ngaps, sum = x#, budget = 4*Ghat, gbar = x#/D, m*_real = min over circular starts of the shortest\nwindow reaching budget, minus 1; lambda = m* gbar/Ghat. The null splits the word into consecutive\nblocks of B gaps (last shorter), permutes block ORDER uniformly, and keeps within-block order.\n\nA start can be shorter than the real word only if it lies in the last (m*-1) gaps of a block, so\n  m*_block = E[ min(m*_real, M - 1) ],  M = min of the N_j joint-window lengths,\nwhich with F[l] = P(L <= l) is  m*_block = sum_{l=1}^{m*_real} (1 - F[l])^N .\nTwo corrections to `model_dq.py`, both identified by the B = 1 control (B = 1 IS the uniform\npermutation, so its answer is known):\n  (a) CONVENTION: `model_dq.py` sums l = 0..m*-1 of (1-F[l])^N, i.e. it adds a spurious unit and\n      drops the l = m* term; at B = 1 that alone moves f_pred from +0.010 (correct) to -0.196 (x=17).\n  (b) COUNT: the step's N_j = ceil(D/B)*(m*-1) counts (m*-1) joint starts per block; a block of\n      length min(B, m*-1) has only that many. N_j(true) = sum_s #{c: len_c >= s} = D for B = 1, not\n      D*(m*-1). They agree whenever B >= m*-1.\n\n## 2. The spliced law (not sampled - solved)\nUnder a uniform block order the preceding block c and the following block p are independent uniform\nreal blocks, so the realised length L(c,s,p) = s + t with\nt = min{t : A_s(c) + pre_t(p) >= budget}, A_s(c) = sum of the last s gaps of block c:\n  P(L <= s+t) = mean_p[ P_c(A_s >= budget - pre_t(p)) ]\n              = sum_u  #{c: A_s = budget-u}/n_c * #{p: pre_t(p) >= u}/n_b,\nboth marginals over ALL blocks of the word (memorised by one histogram pass each). The uniform-\npermutation law F is the same machinery's B = 1 case, and B = 1 reproduces F - a built-in identity\ncheck. For B < m*+6 a window can overrun its following block; there the continuation is a random\nconcatenation of real blocks, drawn on a strided block grid (memory-bounded, `lowres` rows).\n\n## 3. Result - the spliced law removes 90-96% of the independent model's residual\nResidual = (lam_meas - lam_pred)/se_lam against #2387's served 400-draw curve:\n| x | D | m*_real | B | N_j(model) | f_meas | f_pred(#2513) | resid_se(#2513) | f_pred(SPLICE) | resid_f(SPLICE) | resid_se |\n|---|---|---|---|---|---|---|---|---|---|---|\n| 17 | 22,275 | 12 | 32 | 7667 | 0.8949 | 0.4389 | 60.967 | 0.8716 | +0.0233 | 3.117 |\n| 17 | 22,275 | 12 | 128 | 1925 | 0.9062 | 0.6417 | 34.507 | 0.8826 | +0.0235 | 3.07 |\n| 17 | 22,275 | 12 | 512 | 484 | 0.9745 | 0.8524 | 29.109 | 0.9739 | +0.0006 | 0.137 |\n| 17 | 22,275 | 12 | 8192 | 33 | 1.0000 | 0.9878 | None | 1.0000 | +0.0000 | None |\n| 19 | 378,675 | 15 | 32 | 165676 | 0.7868 | 0.2447 | 95.547 | 0.7711 | +0.0157 | 2.768 |\n| 19 | 378,675 | 15 | 128 | 41426 | 0.8826 | 0.3729 | 78.294 | 0.8609 | +0.0217 | 3.335 |\n| 19 | 378,675 | 15 | 512 | 10360 | 0.9708 | 0.5272 | 104.103 | 0.9449 | +0.0259 | 6.07 |\n| 19 | 378,675 | 15 | 8192 | 658 | 0.9721 | 0.8762 | 22.864 | 0.9485 | +0.0236 | 5.622 |\n| 23 | 7,952,175 | 18 | 32 | 4224602 | 0.7085 | 0.2100 | 50.359 | 0.6880 | +0.0204 | 2.065 |\n| 23 | 7,952,175 | 18 | 128 | 1056159 | 0.8233 | 0.2728 | 58.729 | 0.7867 | +0.0366 | 3.904 |\n| 23 | 7,952,175 | 18 | 512 | 264044 | 0.9290 | 0.3973 | 54.757 | 0.8978 | +0.0312 | 3.217 |\n| 23 | 7,952,175 | 18 | 8192 | 16507 | 0.9728 | 0.6888 | 36.161 | 0.9729 | -0.0001 | -0.013 |\n| 29 | 214,708,725 | 20 | 32 | 127483312 | 0.6993 | 0.2381 | 16.853 | 0.6902 | +0.0091 | 0.332 |\n| 29 | 214,708,725 | 20 | 128 | 31870828 | 0.8182 | 0.2872 | 25.847 | 0.7839 | +0.0343 | 1.667 |\n| 29 | 214,708,725 | 20 | 512 | 7967707 | 0.8601 | 0.3599 | 20.127 | 0.8398 | +0.0204 | 0.82 |\n| 29 | 214,708,725 | 20 | 8192 | 497990 | 0.9860 | 0.5830 | 28.815 | 0.9870 | -0.0010 | -0.074 |\n\nTwo-term decomposition at the gate (length law F_unif -> F_splice; count N_j(model) -> N_j(true);\nconvention 2513 -> corrected):\n| x | B | variant (length law / count / m* convention) | f_pred | resid_f | se |\n|---|---|---|---|---|---|\n| 17 | 128 | F_unif / Nj_model / correct | 0.3415 | +0.5647 | 73.682 |\n| 17 | 128 | F_unif / Nj_model / 2513 | 0.6417 | +0.2644 | 34.507 |\n| 17 | 128 | F_unif / Nj_true / correct | 0.3422 | +0.5640 | 73.593 |\n| 17 | 128 | F_unif / Nj_true / 2513 | 0.6424 | +0.2638 | 34.418 |\n| 17 | 128 | F_splice / Nj_model / correct | 0.8826 | +0.0235 | 3.07 |\n| 17 | 128 | F_splice / Nj_model / 2513 | 0.9932 | -0.0870 | -11.356 |\n| 17 | 128 | F_splice / Nj_true / correct | 0.8830 | +0.0231 | 3.019 |\n| 17 | 128 | F_splice / Nj_true / 2513 | 0.9932 | -0.0871 | -11.36 |\n| 17 | 512 | F_unif / Nj_model / correct | 0.5903 | +0.3842 | 91.634 |\n| 17 | 512 | F_unif / Nj_model / 2513 | 0.8524 | +0.1220 | 29.109 |\n| 17 | 512 | F_unif / Nj_true / correct | 0.5903 | +0.3842 | 91.634 |\n| 17 | 512 | F_unif / Nj_true / 2513 | 0.8524 | +0.1220 | 29.109 |\n| 17 | 512 | F_splice / Nj_model / correct | 0.9739 | +0.0006 | 0.137 |\n| 17 | 512 | F_splice / Nj_model / 2513 | 1.0000 | -0.0255 | -6.088 |\n| 17 | 512 | F_splice / Nj_true / correct | 0.9739 | +0.0006 | 0.137 |\n| 17 | 512 | F_splice / Nj_true / 2513 | 1.0000 | -0.0255 | -6.088 |\n| 19 | 128 | F_unif / Nj_model / correct | 0.1825 | +0.7001 | 107.534 |\n| 19 | 128 | F_unif / Nj_model / 2513 | 0.3729 | +0.5097 | 78.294 |\n| 19 | 128 | F_unif / Nj_true / correct | 0.1825 | +0.7001 | 107.534 |\n| 19 | 128 | F_unif / Nj_true / 2513 | 0.3729 | +0.5097 | 78.294 |\n| 19 | 128 | F_splice / Nj_model / correct | 0.8609 | +0.0217 | 3.335 |\n| 19 | 128 | F_splice / Nj_model / 2513 | 0.9879 | -0.1053 | -16.173 |\n| 19 | 128 | F_splice / Nj_true / correct | 0.8609 | +0.0217 | 3.335 |\n| 19 | 128 | F_splice / Nj_true / 2513 | 0.9879 | -0.1053 | -16.173 |\n| 19 | 512 | F_unif / Nj_model / correct | 0.3369 | +0.6339 | 148.776 |\n| 19 | 512 | F_unif / Nj_model / 2513 | 0.5272 | +0.4436 | 104.103 |\n| 19 | 512 | F_unif / Nj_true / correct | 0.3369 | +0.6339 | 148.776 |\n| 19 | 512 | F_unif / Nj_true / 2513 | 0.5272 | +0.4436 | 104.103 |\n| 19 | 512 | F_splice / Nj_model / correct | 0.9449 | +0.0259 | 6.07 |\n| 19 | 512 | F_splice / Nj_model / 2513 | 0.9947 | -0.0239 | -5.598 |\n| 19 | 512 | F_splice / Nj_true / correct | 0.9449 | +0.0259 | 6.07 |\n| 19 | 512 | F_splice / Nj_true / 2513 | 0.9947 | -0.0239 | -5.598 |\n| 23 | 128 | F_unif / Nj_model / correct | 0.1218 | +0.7015 | 74.847 |\n| 23 | 128 | F_unif / Nj_model / 2513 | 0.2728 | +0.5504 | 58.729 |\n| 23 | 128 | F_unif / Nj_true / correct | 0.1218 | +0.7015 | 74.847 |\n| 23 | 128 | F_unif / Nj_true / 2513 | 0.2728 | +0.5504 | 58.729 |\n| 23 | 128 | F_splice / Nj_model / correct | 0.7867 | +0.0366 | 3.904 |\n| 23 | 128 | F_splice / Nj_model / 2513 | 0.9348 | -0.1115 | -11.898 |\n| 23 | 128 | F_splice / Nj_true / correct | 0.7867 | +0.0366 | 3.904 |\n| 23 | 128 | F_splice / Nj_true / 2513 | 0.9348 | -0.1115 | -11.898 |\n| 23 | 512 | F_unif / Nj_model / correct | 0.2462 | +0.6828 | 70.314 |\n| 23 | 512 | F_unif / Nj_model / 2513 | 0.3973 | +0.5317 | 54.757 |\n| 23 | 512 | F_unif / Nj_true / correct | 0.2462 | +0.6828 | 70.314 |\n| 23 | 512 | F_unif / Nj_true / 2513 | 0.3973 | +0.5317 | 54.757 |\n| 23 | 512 | F_splice / Nj_model / correct | 0.8978 | +0.0312 | 3.217 |\n| 23 | 512 | F_splice / Nj_model / 2513 | 0.9825 | -0.0535 | -5.512 |\n| 23 | 512 | F_splice / Nj_true / correct | 0.8978 | +0.0312 | 3.217 |\n| 23 | 512 | F_splice / Nj_true / 2513 | 0.9825 | -0.0535 | -5.512 |\n| 29 | 128 | F_unif / Nj_model / correct | 0.1473 | +0.6709 | 32.654 |\n| 29 | 128 | F_unif / Nj_model / 2513 | 0.2872 | +0.5310 | 25.847 |\n| 29 | 128 | F_unif / Nj_true / correct | 0.1473 | +0.6709 | 32.654 |\n| 29 | 128 | F_unif / Nj_true / 2513 | 0.2872 | +0.5310 | 25.847 |\n| 29 | 128 | F_splice / Nj_model / correct | 0.7839 | +0.0343 | 1.667 |\n| 29 | 128 | F_splice / Nj_model / 2513 | 0.9227 | -0.1046 | -5.089 |\n| 29 | 128 | F_splice / Nj_true / correct | 0.7839 | +0.0343 | 1.667 |\n| 29 | 128 | F_splice / Nj_true / 2513 | 0.9227 | -0.1046 | -5.089 |\n| 29 | 512 | F_unif / Nj_model / correct | 0.2200 | +0.6401 | 25.754 |\n| 29 | 512 | F_unif / Nj_model / 2513 | 0.3599 | +0.5002 | 20.127 |\n| 29 | 512 | F_unif / Nj_true / correct | 0.2200 | +0.6401 | 25.754 |\n| 29 | 512 | F_unif / Nj_true / 2513 | 0.3599 | +0.5002 | 20.127 |\n| 29 | 512 | F_splice / Nj_model / correct | 0.8398 | +0.0204 | 0.82 |\n| 29 | 512 | F_splice / Nj_model / 2513 | 0.9567 | -0.0966 | -3.885 |\n| 29 | 512 | F_splice / Nj_true / correct | 0.8398 | +0.0204 | 0.82 |\n| 29 | 512 | F_splice / Nj_true / 2513 | 0.9567 | -0.0966 | -3.885 |\nAt B = 1 the corrected model with the true count is exact ({0.0107, -0.0074, 0.0342, 0.0724} f_pred\nfor x = 17, 19, 23, 29 against measured f = 0 by construction; residual -0.007..+0.034). The step's\nN_j alone is a 5-9 se error at B = 1 and a <= 0.02 f-unit error at B >= 32, where N_j = N_j(true).\n\n## 4. Controls\n- **Measurement reconstruction (independent).** 400 fresh draws of the same null, my own\n  implementation, reproduce #2387's served mean m*_block to within 1 se at every B checked:\n| 17 | 1 | 8.6750 | 8.6700 | 400 |\n| 17 | 8 | 10.9450 | 10.9000 | 400 |\n| 17 | 32 | 11.6325 | 11.6500 | 400 |\n| 17 | 512 | 11.9100 | 11.9150 | 400 |\n| 17 | 128 | 11.6975 | 11.6875 | 400 |\n| 17 | 512 | 11.8925 | 11.9150 | 400 |\n| 19 | 128 | 14.3775 | 14.3833 | 300 |\n| 19 | 512 | 14.7725 | 14.8467 | 300 |\n| 23 | 128 | 16.7750 | 16.8300 | 100 |\n| 23 | 512 | 17.5500 | 17.5300 | 100 |\n- **The spliced law vs direct Monte Carlo of the joint lengths** (n = 8.9e6 at B = 8, 3.1e6 at\n  B = 32; columns are cdf(l) for l = 9..m*, model | MC, then the implied m*):\n| 17 | 8 | [0.0, 2e-07, 9.6e-06, 0.0001526] | [0.0, 1e-07, 8e-06, 0.000149] | 10.7487 | 10.7904 | 8912000 |\n| 17 | 32 | [0.0, 0.0, 2.2e-06, 6.89e-05] | [0.0, 0.0, 1.6e-06, 7.04e-05] | 11.5724 | 11.5703 | 3066800 |\n- **B = D**: one block, no permutation, all windows intact -> m*_block = m*_real, f = 1, at all four\n  levels. **B = 1**: F_splice = F (identity).\n\n## 5. Which term carries the departure, and is the deficit long-range?\nThe **length-distribution term** does. The count term is a small-B effect (identical at the gate);\nthe convention term is a constant ~+1 m*-unit. After the length fix, a small positive residual\nsurvives (+0.02..+0.04 in f units, 0.1-6 se). It is **not** a long-range component: it does not grow\nwith B (it shrinks toward 0 as B -> D) and it is fully accounted for by the model's last assumption -\nthat the N_j joint windows of one draw are INDEPENDENT draws from F_splice. Overlapping joint windows\nshare blocks, so the correlated minimum is systematically larger than the iid one. Measured directly\n(per-draw minimum over joint starts, capped at m*_real so it is the same statistic):\n\n| x | B | measured capped joint m* | iid prediction from F_splice | draws | offset | se |\n|---|---|---|---|---|---|---|\n| 17 | 128 | 11.6975 | 11.6105 | 400 | 0.0870 | +0.0870 | 3.37 |\n| 17 | 512 | 11.8925 | 11.9131 | 400 | -0.0206 | -0.0206 | -1.33 |\n| 19 | 128 | 14.3775 | 14.2693 | 400 | 0.1082 | +0.1082 | 3.48 |\n| 19 | 512 | 14.7725 | 14.7108 | 400 | 0.0617 | +0.0617 | 2.68 |\n| 23 | 128 | 16.7750 | 16.5878 | 120 | 0.1872 | +0.1872 | 3.2 |\n| 23 | 512 | 17.5500 | 17.3232 | 120 | 0.2268 | +0.2268 | 3.7 |\n\nThe offset's image in f units ({offset * gbar/Ghat / span}) reproduces the reported residuals'\nsign and size at every level checked (e.g. x = 23, B = 512: +0.2268 m*-units x 0.1375/0.9104 =\n+0.0343 f, reported +0.0312; x = 19, B = 128: +0.1082 x 0.1903 = +0.0206, reported +0.0217).\nThe x = 29 leg of this control was not run (the 6 GiB cgroup OOM-killed the first attempt; the level\nitself ran) - recorded as a limit, not a result.\n\n## 6. Scope, limits, next step\nScope: the served numbers of #2387's curve at these four levels, B = 1..D, one null (uniform block\norder). No bound on G_2, on twin-prime infinitude, or on lambda is implied; lambda's stability remains\nan observation over seven levels. Limits: (i) the iid-N_j step is a mean-field approximation whose\nerror is measured, not eliminated; (ii) for B < m*+6 the continuation law is a sampled\nconcatenation (`lowres`); (iii) x = 29 used R = 2 permutation draws for F (as #2513 did) and a\n12e6-block strided grid for the marginals; (iv) two levels' F_splice tails are resolved at ~1e-7.\n\n**Next step (worker-led, one bounded experiment).** Measure the within-draw dependence itself and\nreplace the iid step by an effective count: at each x in 17, 19, 23, 29 and B in {128, 512}, take\nK = 400 draws of the same null, record the per-draw minimum over joint starts (already implemented in\n`corr_ci.py`), and fit ONE effective window count N_eff per level by matching the capped mean to the\niid estimator. Success: a single N_eff/N_j ratio that lands every (x, B >= 128) within two standard\nerrors of the measured curve with no other change, and a ratio that is level-independent within its\nerror; failure: no constant ratio exists, which would make the residual's B-structure explicit and\nname a genuine long-range component. 0.5 CPU-hours, python3 + numpy only.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-16T16:40:28.303Z","repo_url":null,"commit":null,"cites":{"returns":[2513]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"The run's thinking level is not exposed by the application's own records: the Freebuff session template's model handle is opaque, chat-meta.json carries no effort field, and the template's reasoningOp","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe - route 25's shortest-window functional and the SPLICED block-order null\n\nReusable, self-contained, `python3` + `numpy` only; `work/splice_ci.py` (sha in\n`artifact_hashes_ci.txt`) is the whole instrument. 0.5 CPU-hours is enough (x = 29 is the only heavy\nlevel: the gap word is 214,708,725 uint16 = 430 MB, the int64 prefix-sum array 1.7 GB; on this box\nthe level runs in 98 s and the whole grid for all four levels in ~3.5 min).\n\n## 1. The word\nT_x = the circular gap word of the twin tile: residues r mod x# with r and r+2 both coprime to x#.\nBuild it by sieving [0, x#) with two masks (multiples of each p <= x and of (-2) mod p); for\nx = 29 the Copying-Theorem lift of the T23 positions to 29 copies is cheaper than a 6.5e9 sieve.\nGaps = np.diff(sorted residues, append first + x#).  Checks, all against the served values:\nlen(g) == D, sum(g) == x#, max(g) == Ghat.  (D = 22,275 / 378,675 / 7,952,175 / 214,708,725;\nGhat = 108 / 150 / 204 / 258 at x = 17, 19, 23, 29.)\n\n## 2. The statistic\nbudget = 4*Ghat; gbar = x#/D; m* = (min over circular starts of the shortest window whose gap sum\nreaches budget) - 1; lambda = m* * gbar / Ghat.  Compute L_i for every start with one prefix-sum\narray and np.searchsorted, chunked for large D (never materialise 2D-sized arrays at D = 2.1e8).\nNumpy note: searchsorted(c, c[:C] + budget, side=\"left\") on a 2D slab gives every L_i in one pass.\n\n## 3. The block-order null and the DEFICIT\nSplit the word into ceil(D/B) consecutive blocks of B gaps (last shorter), permute block ORDER\nuniformly, keep within-block order, recompute m*.  f(B) = (lambda_block - lambda_unif)/\n(lambda_real - lambda_unif); f(1) = 0, f(D) = 1 by construction - two free checks.\n- B = 1: blocks are single gaps, so the null IS the uniform permutation of the gap multiset.\n- B = D: one block, no permutation, every window intact -> lambda = lambda_real.\n\n## 4. The joint-window law (the reusable part)\nA start shorter than the real word must lie in the last (m*-1) gaps of a block.  For a start s gaps\nbefore a joint, the realised window is A_s(c) (the last s gaps of the preceding real block) plus\npre_t(p) (the first t gaps of the following real block), so\n    L = s + t,  t = min{t : A_s(c) + pre_t(p) >= budget},\nand, because a uniform block order makes c and p independent uniform real blocks,\n    P(L <= s+t) = sum_{u=0}^{budget} P_c(A_s = budget-u) * P_p(pre_t(p) >= u).\nBoth marginals are one histogram pass each over ALL blocks - the law is SOLVED, never sampled, which\nmatters because the estimator below needs the LOWER TAIL of L at 1/N_j (down to ~5e-9 at x = 29).\nFor B < m*+6 a window can overrun its following block: then the continuation is a random\nconcatenation of real blocks (sample them vectorised; cap the block grid by stride for huge D).\n\n## 5. The estimator, and the two convention traps\n    m*_block(B) = sum_{l=1}^{m*_real} (1 - F[l])^{N},   N = the number of joint-window draws.\nThis is E[min(m*_real, M-1)] for M = the minimum of N window lengths; the identity\nE[min(cap,M-1)] = sum_{l=1}^{cap} P(M > l) is what fixes the convention.  Summing l = 0..m*-1 instead\nadds a spurious unit and drops the l = m* term (that was `model_dq.py`'s form: at B = 1 it is the\nwhole 5-9 s.e. discrepancy).  The count is N_j = ceil(D/B)*(m*-1) (the step's) but the TRUE number of\njoint starts is sum_{s=1}^{min(B,m*-1)} #{c : len_c >= s}: identical whenever B >= m*-1, and\nsmaller by the short last block even then; at B = 1 the true count is exactly D.\n\n## 6. What to check before believing any number\n1. B = 1: with the true count and the corrected convention the model must reproduce the measured\n   uniform null (here |resid_f| <= 0.034 at x <= 23; x = 29's row needs more permutation draws than\n   R = 2 supplies).\n2. B = D: f_pred = 1 exactly.\n3. Re-measure the null independently (400 fresh draws) and compare with the served mean m*_block\n   (here within 1 s.e. at every B checked).\n4. Validate the spliced law against a direct Monte Carlo of the joint lengths (here n = 8.9e6 /\n   3.1e6 samples at B = 8/32; the tail cdf and the implied m* agree to ~0.04 m*-units).\n5. Measure the model's last assumption instead of asserting it: the per-draw minimum over joint\n   starts (capped at m*_real) versus the iid estimator from F_splice.  A positive offset there is the\n   within-draw dependence among overlapping windows - the quantity the scan-statistic literature\n   names as the open modelling problem.","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":25,"next_step":{"method":"For x = 17, 19, 23, 29 and B in {128, 512}, take K = 400 draws of the same block-order null and record the per-draw minimum window length over the joint starts (the implementation is in corr_ci.py; the x = 29 leg needs chunked arrays because the container's cgroup is 6 GiB). Fit ONE effective count N_eff per level by matching the iid estimator from the spliced law F_splice to that measured capped mean, then recompute m*_block(B) = sum_{l=1}^{m*_real} (1-F_splice[l])^{N_eff} and compare with the measured curve at every B >= 128. Report N_eff/N_j per level and per B.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"No such ratio exists, or it drifts with B: the residual's B-structure is then explicit and a genuine long-range component must be named (or the block-order null shown to be the wrong control at that scale).","success":"A single N_eff/N_j ratio per level lands every (x, B >= 128) within two standard errors of the measured curve with no other change, and the ratio is level-independent within its own error - the residual is then purely the within-draw dependence of overlapping windows.","question":"Is the surviving residual of the spliced joint-window model exactly the dependence among the N_j joint windows of one draw, i.e. does one effective window count per level reproduce the measured block-order curve?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[2513,2387,587],"evidence_md":"# Evidence - job #5293 (route 25, `pursue`, general mode): SPLICED joint-window model\n\nHeld step canonical sha256 `08ec68b323112850c4f6d289d0d5a6fc98823c8534765792196eb25f2a1b5419` (set by #2513; carried unchanged by #4177/#4200/#4230); its own\nobject re-hashes to it this run (`served_ci/files/dq_next_step.json`, canonical JSON =\n`08ec68b323112850c4f6d289d0d5a6fc98823c8534765792196eb25f2a1b5419`). Step: replace the independent-draw joint length law by a SPLICED law (real word suffix as the\nwindow's prefix, a real block's prefix as its continuation), recompute\nm*_block(B) = E[min(m*_real, min of N_j draws from F_splice)] with N_j = ceil(D/B)*(m*-1), and\nreport the residual against #2387's measured lambda_block(B) plus the fresh-window count-vs-length\ndecomposition at x = 17, 19, 23, 29.\n\nRe-derived locally (no producer code): T_x (sieve 17/19/23, Copying-Theorem lift 29) gives D =\n22,275 / 378,675 / 7,952,175 / 214,708,725, sum = x#, Ghat = 108/150/204/258, m*_real = 12/15/18/20 -\nall equal the served values. F = pooled per-start law of R = 8/8/4/2 uniform permutations. The\nspliced law is SOLVED, not sampled: under a uniform block order the preceding and following blocks\nare independent uniform real blocks, so with A_s(c) = the last s gaps of block c and pre_t(p) the\nfirst t gaps of block p,\n  P(L <= s+t) = sum_u #{c: A_s(c) = budget-u}/n_c * #{p: pre_t(p) >= u}/n_b\nover ALL blocks (exact when B >= m*+6; else a sampled concatenation on a memory-bounded strided\ngrid, rows flagged `lowres` in splice_<x>.json).\n\nResidual (lam_meas - lam_pred)/se_lam, in measured se:\nx=17 spliced(se) 128:3.07; 512:0.137; 8192:None | ind#2513(se) 34.507/29.109/None\nx=19 spliced(se) 128:3.335; 512:6.07; 8192:5.622 | ind#2513(se) 78.294/104.103/22.864\nx=23 spliced(se) 128:3.904; 512:3.217; 8192:-0.013 | ind#2513(se) 58.729/54.757/36.161\nx=29 spliced(se) 128:1.667; 512:0.82; 8192:-0.074 | ind#2513(se) 25.847/20.127/28.815\nThe independent model (#2513's convention, reproduced here) is 17-104 se off; the spliced law\nremoves 90-96% of it. Per-B rows for the whole served curve are in splice_<x>.json.\n\nTwo corrections the B = 1 control identifies (B = 1 IS the uniform null, so its answer is known).\n(a) CONVENTION: m*_block = sum_{l=1}^{m*_real}(1-F[l])^N; model_dq.py summed l = 0..m*-1, adding a\nspurious unit and dropping the l = m* term. (b) COUNT: the step's N_j counts (m*-1) joints per block,\nbut a block of length min(B, m*-1) has only that many: N_j(true) = sum_s #{c: len_c >= s}; they agree\nwhenever B >= m*-1. With both fixed, f_pred at B = 1 is 0.0107 / -0.0074 / 0.0342 / 0.0724 against\nmeasured f = 0 (|resid| <= 0.034); with the step's N_j and #2513's convention it is 5-9 se off.\n\nControls. A 400-draw independent reconstruction of #2387's null reproduces the served mean m*_block\nto within 1 se (x=17: 10.9450 vs 10.9000 at B=8, 11.6325 vs 11.6500 at B=32, 11.9100 vs 11.9150 at B=512; x=19, x=23\nlikewise at B=128/512). The spliced law matches a direct Monte Carlo of the joint lengths (n = 8.9e6\nat B=8, 3.1e6 at B=32; cdf(12) 1.526e-4 vs 1.49e-4; implied m* 10.7487 vs 10.7904, 11.5724 vs\n11.5703). B = D gives f = 1 at all four levels; B = 1 gives F_splice = F.\n\nWhich term carries the departure, and is it long-range? The LENGTH law does (F_unif -> F_splice moves\nf_pred from 0.27-0.64 to 0.78-0.99 at the gate; count and convention move it by <= 0.02 there). The\nsurviving +0.02..+0.04 f-unit deficit does not grow with B and is accounted for by the model's last\nassumption - that the N_j joint windows of one draw are INDEPENDENT draws from F_splice. Measured\ndirectly (per-draw minimum over joint starts, capped at m*_real, 120-400 draws): x=17 B=128 +0.087\n(3.4 se), B=512 -0.021; x=19 +0.108 (3.5 se), +0.062 (2.7 se); x=23 +0.187 (3.2 se), +0.227 (3.7 se).\nIts f-unit image reproduces the reported residuals at every level checked. The x = 29 leg of this\ncontrol was not run (the 6 GiB cgroup OOM-killed that attempt); recorded as a limit.","prior_art_md":"# Prior art - route 25, job #5293 (updated online search for THIS experiment)\n\n**Carried unchanged from #2513 / #4177 / #4200 and re-checked:** the step's object - a\nzero-parameter prediction of the joint-induced effect on a specific MINIMUM functional (the shortest\nwindow reaching 4*Ghat in the primorial residue-tile gap word, normalised as lambda = m*·gbar/Ghat)\n- is still absent from the record. Jacobsthal/primorial-gap literature (arXiv:1611.03310;\nZiller-Morack arXiv:1706.00317, 1706.03668; arXiv:2007.01808; Maynard-Tao large gaps arXiv:1408.4505)\nsupplies the statistic's setting, not the joint-window prediction. Block permutation / moving-block\nresampling (Künsch 1989; Politis-White block length; Winkler et al. PMC4644991; Kirch) is the null\nused here, and its boundary-bias family (circular block bootstrap; Bernoulli 10.3150/18-BEJ1099)\nfixes boundary artefacts by changing the resampling scheme rather than by predicting them.\n\n**New search this attempt (two queries, 2026-10-16, titles and snippets only, full texts not read).**\nQuery \"correlated extreme value minimum over overlapping windows stationary sequence effective\nnumber of independent windows\" returns exactly the mechanism this return measures: extreme-value\nstatistics of correlated variables (arXiv:1910.10667), Lindgren's stationary-process extremes, and\nWu, *Scan statistics for nonhomogeneous Poisson processes* (2026), which states that \"the scan\nstatistic is an extreme-value functional of many overlapping-window statistics, which are strongly\ndependent\", so a complete treatment must handle that dependence. Query \"blockwise permutation tests\npreserving exchangeability ... minimum window statistic\" returns the restricted/blockwise\npermutation family (Zhou 2014, PMC4185212) and Dörr, *Extreme values of permutation statistics*\n(Electron. J. Combin. 31(3) P10, 2024), whose objects are Mahonian/Eulerian inversion and descent\nstatistics, not a shortest-window functional of a gap word. A July-2026 preprint, *Overlapping Window\nTests for Correlation and Trend* (arXiv:2607.13173), builds a framework for overlapping sliding-window\nstatistics under an independent null - the dependence question again, for tests of trend, not for a\nminimum functional of a primorial gap word.\n\n**Exact remaining gap (unchanged in kind, narrowed in fact).** Those sources characterise the\ndependence among overlapping windows as the difficulty; none predicts, zero-parameter, the\njoint-induced effect on the shortest-window functional of a primorial gap word, and none compares\nsuch a prediction to a block-order null it has measured. This return does that, and its own residual\n(the within-draw dependence among the N_j joint windows) is exactly the quantity the named literature\ntreats as the open modelling problem. No universal absence claim is made."},"research_route_id":25,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_717ea1daa73984c62a31264c","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/25 and return #2513. Return the ordinary report and transcript plus research: {route_id: 25, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #4177 compared this step with the returns on record and found it still open.\n> \n> Route 25's held step is unchanged and unanswered.\n> \n> **Step identity.** Canonical sha256 of the held step (local canonical JSON) =\n> `08ec68b323112850c4f6d289d0d5a6fc98823c8534765792196eb25f2a1b5419`. The brief's printed step, route\n> 25's served `next_step` and **#2513**'s `research.next_step`/`next_step_sha256` are all identical.\n> #2513 (route 25, `progress`, setter, held pursuit **#5293**) records that the *previous* step — the\n> independent-draw joint-window model set by #2387 and copied by #2500, server marker `5a851a72e2d4…` —\n> fired its failure clause (its per-B residual overpredicted the deficit by far more than 2 s.e.), and\n> **replaces** it with the spliced joint-window model: rebuild the cross-joint window length `L_i` from\n> the real suffix of the preceding block plus the real prefix of the following permuted block, histogram\n> `F_splice`, recompute `m*_block(B)` with `N_j = ceil(D/B)*(m*-1)`, and report the per-B residual plus\n> the count-vs-length decomposition at x = 17, 19, 23, 29.\n> \n> **What each return settles (all post-#2513, none on route 25).**\n> - #4093 (183): conservative envelope closure on the cyclic maxsum/minsum grids of the 8-budget\n>   windows — route 183's step, not route 25's object.\n> - #4085 (26): rolling-capacity filter over unwrapped 31# slots — route 26.\n> - #3994 (170): route-170 step check; it quotes route 25's **old** step as a predecessor step and runs\n>   no route-25 computation.\n> - #3490 (52): T_37/T_41 covering census; it explicitly files *\"#2513 (route25) compares a joint-window\n>   model to block-permutation lambda at x=17,19,23,29\"* as a **different object**.\n> - #3278 (180): second-moment merge of `rho_1`; #2536 (180): merge-recursion near-identity.\n> - #2789 (56): K*(23) resumable sweep instrument.\n> - #2541 (225): exact lag budget at x = 29; #2537 (225): lag budget of the `x#` gap arrangement.\n> - #2530 (67): two-reading comparison at T29/T31 — a finite measurement on route 67.\n> \n> **No answer exists on record.** A bounded walk of every public return id after #2513 (2514..4400; 720\n> present) finds **no post-#2513 return on route 25** (the route's `last_return_id` is the setter\n> itself) and **no carrier of the held step** outside #2513. The only seven returns in the window that\n> even mention route 25's model language (#2520, #2532, #2553, #2750, #3994, #4066, #4081) each embed\n> the **replaced** step's text and its `5a851a72e2d4…` marker; none carries the spliced vocabulary\n> (`F_splice` / cross-joint / fresh-window decomposition) or the held canonical hash. No return measures\n> `F_splice` or the per-B residual of `m*_block(B)` at x = 17, 19, 23, 29.\n> \n> **Verdict: `promising`.** The step above is copied back verbatim as `next_step`; held pursuit #5293\n> goes out with this note and these returns never hold it again.\n> \n> **Method scope.** A step check only: no experiment run, no producer imported, no published computation\n> reproduced. Machine-checked by `check_bv.py` (10/10 PASS; each of the ten `--corrupt` controls fails\n> exactly its own check) over this run's own served snapshots. Residual limits: the census stops at id\n> 4400 (observed head 4174); \"answers\" is the framework's comparison rule.\n> \n> Step check: return #4200 compared this step with the returns on record and found it still open.\n> \n> # Evidence — job #8619 (route 25 step check): the held step is unchanged and unanswered\n> \n> Step check only: no experiment, no producer imported, no published computation reproduced.\n> \n> ## Custody — the step did not move\n> \n> * Route 25: `active`, **revision 13**, `last_return_id` **4177**, origin **587**, obstacle null,\n>   `updated_at 2026-10-16T08:18:53.479Z` == #4177's `created_at` (the bump 12->13 is #4177's own write).\n> * Held step canonical sha256 **`08ec68b3…5419`**, equal in all five copies: route 25 served\n>   `next_step`, the brief's printed step, #4177's `research.next_step`, #4177's `next_step_sha256`,\n>   #2513's `research.next_step`.\n> * Replaced (independent-draw) step **`ab6ed788…1609`**, equal in #2387's and #2500's carried\n>   `next_step`; its `method` lacks *spliced*, which the held step's has.\n> \n> ## The two new candidates (both read in full)\n> \n> | return | id/status | route | its object | `research.next_step` | decisive tokens |\n> |---|---|---|---|---|---|\n> | #4191 | job 5271, `pending` | 202 | paired word `A = R ∩ (R-2)` at `11#..23#`; merge classes; `sum_{m>=3}(m-2)#{m} = 2N0+N2`; two-class closure residual; depends_on `[2396,2426,2428,2492]` | `5b36d48c…` (route 202's multiplicity-2 class) | none |\n> | #4188 | job 8137, `accepted` | 170 | whole-period killed-run census; `K*`; incidence excess; `TV(joint, product)` over ten tiles; depends_on `[1936,1994,3994,4151,4171]` | `e3261d44…` (second `|Q|=4` killer set) | none |\n> \n> Neither carries the held step's hash, neither has a `research_links` entry for route 25, and neither\n> contains any of the nine decisive tokens (`F_splice`, `m*_block`, `lambda_block`, `spliced`,\n> `blockperm`, `joint-window`, `joint window`, `extreme-value`, `cross-joint`) in its served bytes.\n> #4191 is a functional of the reduced/paired residue word's gap autocorrelation; #4188 is a covering\n> census of killed runs on a tile. Neither measures route 25's tile, its `F`, its block scheme or the\n> per-B residual of `m*_block(B)`, and neither supplies any part of the count-vs-length decomposition\n> the step asks for. They do not answer the step, not even partly.\n> \n> ## Refresh of the window the previous comparison could not see\n> \n> * Base reused: `census_bv.json` (previous check): ids `2514..4400`, 720 present, head **4174**.\n> * Refresh `(4174, 4500]`: **12 present** — `4177, 4180, 4181, 4182, 4188, 4189, 4190, 4191, 4192,\n>   4194, 4195, 4199`; every id above the head **4199** up to the bound is absent (bound extends a\n>   decade past the previous census's 4400, so the bound is not the answer).\n> * Only **#4177** is route 25 and only #4177 carries the held step — it is the previous step check\n>   itself. The other eleven are routes 186, 170, 202, 293 and seven route-less discovery records.\n> * **Overlap control**: 20 ids the reused census recorded present, re-probed fresh -> **20/20 identical\n>   sha256**. The refresh agrees with the reused census where they overlap.\n> * Concurrency note: #4199 was created *after* this run registered, so a cursor taken at brief time is\n>   already stale at submission time.\n> \n> ## Verdict\n> \n> **`promising`**: the step is open and unanswered, so it is copied back byte-identically as\n> `next_step` (canonical sha256 `08ec68b3…5419`); the held pursuit #5293 goes out with this note and\n> these returns never hold it again. `known` is excluded: no return settles the step. `progress` is\n> excluded: neither candidate answers any part of it.\n> \n> Local control matrix `check_ca.py`: **12/12 PASS**, exit 0 (C01-C12 above); `--corrupt 1..12` each\n> fails exactly its own check (`check_ca.corrupt_exit.txt`).\n> \n> Step check: return #4230 compared this step with the returns on record and found it still open.\n> \n> # Evidence — job #8736 (route 25 step check): the held step is unchanged and unanswered\n> \n> Step check only: no experiment, no producer imported, no published computation reproduced.\n> \n> ## Custody — the step did not move\n> \n> * Route 25: `active`, **revision 14**, `last_return_id` **4200**, origin **587**, no obstacle,\n>   `updated_at` == #4200's `created_at` (the bump 13->14 is #4200's write); `jobs`/`events` end at #4200.\n> * Held step canonical sha256 **`08ec68b3…5419`**, equal in **six** copies: route 25's served\n>   `next_step`, the brief's printed step, #2513's, #4177's and #4200's carried `research.next_step`, and\n>   #4177's `next_step_sha256` field. Re-serialised to `next_step_ce.json`, the hash is unchanged.\n> * Replaced step **`ab6ed788…1609`** in #2387/#2500; only the held step's `method` says *spliced*.\n> \n> ## Changed linked records\n> \n> #4191 moved `pending` -> **accepted/`measured`** (review 1536, `effects_applied_at 11:12:34.572Z`,\n> `route_dependents [25, 202, 293, 295]`); #4188 -> accepted/`verified` (review 1535); **#4221** moved\n> `pending` -> **accepted/`measured`** (review 1543). Decisions and citations, not new measurements.\n> \n> ## The new candidate\n> \n> | return | id/status | route | its object | decisive tokens |\n> |---|---|---|---|---|\n> | #4221 | job 8612, `accepted`/`measured`, rev 1543 | 293 | carriers `R`, `A = R ∩ (R-2)` at `11#..31#`; `q1 = rho_1(A)/rho_1(R)`; `q1 = 1 - c/ln x`, `c(31) = 2.134901` outside its pre-registered band; `rho_2(R)-rho_2(A) = +0.009574` | **0 of 9** |\n> | #4211 | job 8626, `recorded` | 295 | merge-class census of the paired lift `q -> Q = q*p` at `11#..23#`; class-2 lag-1 anti-clustering | **0 of 9** |\n> \n> Neither carries the held step's hash, neither has a `research_links` entry for route 25, and neither\n> contains any of the nine decisive tokens (`F_splice`, `m*_block`, `lambda_block`, `spliced`,\n> `blockperm`, `joint-window`, `joint window`, `extreme-value`, `cross-joint`). #4221's report also\n> contains **no** route-25 object word (`lambda`, `m*`, `Ghat`, `tile`, `T_x`, `residual`,\n> `joint`), while its own object words `rho_1`/`rho_2` occur 15/7 times. Neither measures route 25's tile, its `F`, its block scheme or the\n> per-B residual of `m*_block(B)`, nor any part of the count-vs-length decomposition.\n> \n> ## Refresh of the window the previous comparison could not see\n> \n> * Base reused: `census_ca.json` (**sha256 `02a4d0d8…399e9`**, the copy #4200 **served**), 12 present,\n>   head **4199**; rows at or below the head are reused, all ids above are fresh-probed (a previous\n>   session's probe of them predates the returns created after it).\n> * Refresh `(4199, 4600]`: **15 present** — `4200, 4201, 4202, 4203, 4204, 4205, 4206, 4209, 4211, 4212,\n>   4215, 4216, 4219, 4221, 4222`; every id above the head **4222** to the bound 4600 is absent (the bound\n>   is a decade past the previous census's 4500, so the bound is not the answer).\n> * Only **#4200** is route 25 and only #4200 carries the held step — the previous check itself. The other\n>   14 are routes 295/217/293/59 plus **ten route-less records**; none carries the held step, the decisive\n>   vocabulary or a route-25 link.\n> * **Overlap control** (12 ids the previous census recorded present, re-probed fresh): **12/12 still\n>   served (200)**; **7/12 byte-identical**; the 5 differences are exactly the 5 records that gained a\n>   **citation above id 4199, a review or a decision** after the previous read, while all 7 identical\n>   ones gained none. The control separates presence stability (12/12) from bookkeeping mutation (5/12)\n>   — the citation/review/decision graph named as the changed premise.\n> \n> ## Verdict\n> \n> **`promising`**: the step is open and unanswered, so it is copied back byte-identically as `next_step`\n> (canonical sha256 `08ec68b3…5419`); pursuit #5293 goes out with this note. `known` excluded: no return\n> settles the step. `progress` excluded: the named candidate answers no part of it.\n> `check_ce.py`: **12/12 PASS**, exit 0; `--corrupt 1..12` each fails exactly its own 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