{"id":2841,"job_id":5945,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5945 (route 259, explore / first look): the escalation word is a function of one sieve word, and the proposed null removes its clustering\n\nRun `[private]`, general mode, route **259** (`proposed`), stage `first_look`, predecessor\n**#2816**. All numbers below are exact finite computations on the wheels `x = 19#` and `x = 23#`\n(`W = 223 092 870`), recheckable by `check_ij.py` (**34 checks, 0 FAIL, exit 0**; `--corrupt`\n**8 planted / 8 caught, exit 0**) and reproducible by `compute_ij.py`.\n\n## What was asked\n\nRoute 259's held step (from **#2816**): *does the escalation word of a killed run carry an\narrangement excess that survives a null respecting the proved forced implication (an escalated\nresidue-5 slot implies escalated movable neighbours), and does that excess grow with the run length?*\n#2816 measured `z_11 > 0`, `z_alt < 0` against a null that ignores the implication, then proved its\nown null defective and left the corrected test to the route. The route's stop condition is\n`|z| < 2` in **every** class at both rungs.\n\n## Result 1 (new, exact, exhaustive): the escalation word is a deterministic function of ONE sieve word\n\nLet a killed run keep its offsets `0..L-1` (`L = 6q+5`, `M = (L-1)/2 = 3q+2` movable offsets\n`1,3,5 mod 6`, word index `t = 0..M-1`), let `e_t = 1` iff movable slot `t` is `tB`, and let the\n**alpha word** be `a_t = 1` iff that slot's position is killed by `W` (`gcd(pos,W) > 1`).  Then,\nover **every** run at both rungs (`378 675 + 7 952 175 = 8 330 850` runs, **0 violations**):\n\n    e_t = a_t        if t = 0 (mod 3)      (residue-1 slots: the partner is killed by 3)\n    e_t = a_{t+1}    if t = 1 (mod 3)      (residue-3 slots: killed by 3 themselves)\n    e_t = a_t a_{t+1} if t = 2 (mod 3)     (residue-5 slots: need both classes killed)\n    a_M := 0        (the position past the run is a twin slot, hence free)\n\nTwo consequences, also verified exhaustively (**0 violations** over all runs, and exhaustively over\n*every* alpha word for `q <= 4` in the checker):\n\n* **#2816's \"forced implication\" is a two-line consequence of this identity**, not an independent\n  fact: `e_t = a_t a_{t+1} = 1` forces `a_t = a_{t+1} = 1`, hence `e_{t-1} = a_t = 1` and\n  `e_{t+1} = a_{t+1} = 1`.  The last bit `e_{M-1} = a_M = 0` always.\n* The escalation word depends only on the alpha bits at the `t != 1 (mod 3)` slots: with\n  `u_0=0,u_1=2,u_2=3,u_3=5,...` (`n = 2q+1` slots, `K` ones) and weights `w_i = 1` (even `i`),\n  `2` (odd `i`),\n\n      A(e)  = #(differing alpha bits over consecutive (u_i,u_{i+1})) + [bit at u_{n-1} = 1]\n      11(e) = sum_{i<n-1} w_i [both alpha bits at (u_i,u_{i+1}) = 1]\n\n  (the weight 2 counts the two distinct `e`-pairs `t = 2k+1` and `t = 2k+2` that share the pair\n  `(u_{2k+1},u_{2k+2})`; the trailing singleton is the last `t = 0 (mod 3)` slot, whose partner is\n  the free position after the run).  So **the whole escalation word is a statistic of a single\n  binary word**, and every question about its \"order\" is a question about the arrangement of `a`.\n\n## Result 2 (the route's prescribed null): the excess does not survive\n\n**Null_P** conditions on the observed residue-5 escalation bits **and** on the bits the implication\nforces to 1, then permutes the remaining free bits uniformly (the last movable bit, which is `0` by\nmaximality of the run, is pinned; it is not a random variable).  `E` and `sd` are Monte Carlo\n(`1200` shuffles on a seeded random sample of up to `150` runs per class); `z = (observed - E)/sd`,\nper run, then the class mean with its Monte-Carlo standard error.\n\n* **Every class of both rungs has `|z| < 2`**: `max |z_11| = 0.953` (19#, `L=23`) and `0.950` (23#,\n  `L=23`); `max |z_alt| = 0.528` / `0.534`.\n* **The sign is not stable**: over the 18 (19#) and 26 (23#) usable classes, `z_11 > 0` in 9 and 17\n  and `z_alt < 0` in only 5 and 7; at `L=17` the sign of `z_11` even **flips** relative to the\n  uncorrected null (`+0.730 -> -0.188` at 19#, `+0.814 -> -0.185` at 23#).\n* **There is no growth with `L`**: the corrected `z` is largest at the shortest usable class and\n  falls to `|z| < 0.5` over the long classes (23#: `+0.950` at `L=23`, `-0.091` at `L=65`, `+0.041`\n  at `L=167`).\n* The corrected null also removes most of the *magnitude*: at `L=23`, `z_11` goes `+1.404 -> +0.953`\n  (19#) and `+1.467 -> +0.950` (23#); at `L=29` it collapses to `+0.109` / `+0.139`.\n\n**So the route's stop condition fires.**  The clustering #2816 measured is the residue-5 escalation\npattern together with its forced implication; what remains after conditioning on exactly those is a\nbounded, non-growing, sign-unstable residual.  At these rungs the escalation word is **not** a\ngenuine order carrier.\n\n## Result 3 (exact, and why it is NOT an order mechanism): the alpha word is itself non-exchangeable\n\n**Null_Q** is the exact conditional law of `e` when `a` is a uniform `K`-subset of the `n` alpha slots\n(`e' = f(a')`): the implication holds *identically* in this null, and the moments are exact closed\nforms in `(q,K)` (validated against exhaustive enumeration, `max abs dev 2.2e-16`).  Under it:\n\n* `z_11 > 0` and `z_alt < 0` are **sign-stable for every class with `L >= 29`** — 16/16 (19#) and\n  24/24 (23#) — and **grow with `L`** to `z_alt = +2.61`, `z_11 = -3.01` (19#, `L=137`) and\n  `+2.69`, `-3.11` (23#, `L=179`).\n\nThat is a real, exact, L-growing excess — of the **alpha word**, i.e. of the arithmetic sieve pattern\nof the movable slots, not of the escalation word's free bits.  It says the sieve word is not\nexchangeable (consecutive movable slots are two apart, and divisibility by `5,7,11,...` organises\nthem); the route's question was about the escalation word, and Result 2 answers it.  Reporting this\nseparately keeps the two objects distinct: **Null_Q measures arithmetic arrangement, Null_P measures\nthe escalation word's residual order**, and only the former survives.\n\n## Class-by-class tables\n### x = 19#  (W = 9699690, runs = 378675, Lmax = 149, violations {'C_identity': 0, 'reduction_11': 0, 'reduction_A': 0, 'n_formula': 0})\n\n| L | runs | z_alt naive | z_11 naive | z_alt Null_P (M.C.) | s.e. | z_11 Null_P | s.e. | z_alt Null_Q (exact) | z_11 Null_Q | K̄ |\n|---|---|---|---|---|---|---|---|---|---|---|\n| 5 | 36855 | +0.000 | +0.000 | - | - | - | - | +0.000 | +0.000 | 0.00 |\n| 11 | 98280 | +0.180 | +0.677 | - | - | - | - | +0.348 | +0.622 | 1.51 |\n| 17 | 53690 | -0.132 | +0.730 | +0.474 | 0.074 | -0.188 | 0.077 | +0.406 | +0.290 | 2.75 |\n| 23 | 26208 | -0.851 | +1.404 | -0.528 | 0.056 | +0.953 | 0.058 | -0.032 | +0.487 | 4.17 |\n| 29 | 72378 | -0.509 | +0.961 | +0.230 | 0.076 | +0.109 | 0.081 | +0.533 | -0.156 | 5.11 |\n| 35 | 18776 | -0.605 | +1.019 | +0.116 | 0.069 | +0.213 | 0.071 | +0.668 | -0.340 | 6.36 |\n| 41 | 34812 | -0.505 | +0.879 | +0.221 | 0.068 | +0.097 | 0.073 | +0.876 | -0.669 | 7.31 |\n| 47 | 10462 | -0.326 | +0.673 | +0.357 | 0.069 | -0.055 | 0.071 | +1.176 | -1.005 | 8.32 |\n| 53 | 1968 | -0.692 | +1.013 | -0.154 | 0.047 | +0.469 | 0.051 | +0.926 | -0.933 | 9.51 |\n| 59 | 9452 | -0.707 | +1.014 | -0.085 | 0.068 | +0.387 | 0.071 | +1.071 | -1.080 | 10.75 |\n| 65 | 6322 | -0.540 | +0.836 | +0.222 | 0.061 | +0.042 | 0.065 | +1.382 | -1.367 | 11.83 |\n| 71 | 2816 | -0.321 | +0.603 | +0.406 | 0.061 | -0.156 | 0.065 | +1.683 | -1.680 | 12.76 |\n| 77 | 2620 | -0.550 | +0.826 | +0.401 | 0.068 | -0.155 | 0.071 | +1.660 | -1.648 | 14.26 |\n| 83 | 632 | -0.641 | +0.889 | -0.418 | 0.048 | +0.687 | 0.050 | +1.375 | -1.724 | 14.60 |\n| 89 | 1236 | -0.246 | +0.493 | +0.335 | 0.064 | -0.098 | 0.066 | +1.975 | -2.137 | 15.80 |\n| 95 | 876 | -0.595 | +0.836 | -0.015 | 0.079 | +0.257 | 0.082 | +1.763 | -1.997 | 17.25 |\n| 107 | 954 | -0.423 | +0.648 | +0.279 | 0.077 | -0.062 | 0.080 | +2.050 | -2.348 | 19.18 |\n| 119 | 142 | -0.280 | +0.495 | +0.356 | 0.083 | -0.154 | 0.086 | +2.393 | -2.668 | 21.38 |\n| 125 | 48 | -0.407 | +0.619 | +0.453 | 0.117 | -0.248 | 0.120 | +2.433 | -2.671 | 22.92 |\n| 137 | 86 | -0.207 | +0.405 | +0.244 | 0.081 | -0.046 | 0.083 | +2.607 | -3.007 | 24.37 |\n\n### x = 23#  (W = 223092870, runs = 7952175, Lmax = 203, violations {'C_identity': 0, 'reduction_11': 0, 'reduction_A': 0, 'n_formula': 0})\n\n| L | runs | z_alt naive | z_11 naive | z_alt Null_P (M.C.) | s.e. | z_11 Null_P | s.e. | z_alt Null_Q (exact) | z_11 Null_Q | K̄ |\n|---|---|---|---|---|---|---|---|---|---|---|\n| 5 | 700245 | +0.000 | +0.000 | - | - | - | - | +0.000 | +0.000 | 0.00 |\n| 11 | 1867320 | +0.148 | +0.711 | - | - | - | - | +0.330 | +0.654 | 1.53 |\n| 17 | 1060150 | -0.204 | +0.814 | +0.458 | 0.084 | -0.185 | 0.087 | +0.353 | +0.368 | 2.81 |\n| 23 | 539136 | -0.906 | +1.467 | -0.534 | 0.057 | +0.950 | 0.058 | -0.070 | +0.561 | 4.26 |\n| 29 | 1500318 | -0.619 | +1.082 | +0.219 | 0.067 | +0.139 | 0.071 | +0.462 | -0.039 | 5.27 |\n| 35 | 393464 | -0.742 | +1.165 | +0.298 | 0.065 | +0.020 | 0.068 | +0.569 | -0.201 | 6.57 |\n| 41 | 801540 | -0.659 | +1.043 | +0.269 | 0.070 | +0.055 | 0.075 | +0.788 | -0.523 | 7.57 |\n| 47 | 275040 | -0.541 | +0.897 | +0.459 | 0.068 | -0.160 | 0.071 | +1.043 | -0.822 | 8.64 |\n| 53 | 69288 | -0.889 | +1.219 | -0.053 | 0.054 | +0.357 | 0.058 | +0.820 | -0.755 | 9.89 |\n| 59 | 243370 | -0.881 | +1.196 | -0.023 | 0.065 | +0.326 | 0.068 | +0.975 | -0.927 | 11.10 |\n| 65 | 166526 | -0.759 | +1.061 | +0.355 | 0.061 | -0.091 | 0.064 | +1.243 | -1.185 | 12.21 |\n| 71 | 94492 | -0.656 | +0.945 | +0.316 | 0.069 | -0.061 | 0.073 | +1.465 | -1.414 | 13.30 |\n| 77 | 83712 | -0.733 | +1.014 | +0.335 | 0.066 | -0.085 | 0.069 | +1.546 | -1.494 | 14.64 |\n| 83 | 26956 | -0.983 | +1.241 | -0.417 | 0.062 | +0.686 | 0.064 | +1.212 | -1.451 | 15.37 |\n| 89 | 43542 | -0.574 | +0.827 | +0.193 | 0.062 | +0.044 | 0.065 | +1.766 | -1.887 | 16.36 |\n| 95 | 27136 | -0.880 | +1.126 | +0.097 | 0.078 | +0.146 | 0.081 | +1.611 | -1.772 | 17.90 |\n| 101 | 4384 | -0.827 | +1.067 | +0.105 | 0.063 | +0.120 | 0.067 | +1.794 | -1.933 | 19.09 |\n| 107 | 32326 | -0.689 | +0.919 | +0.162 | 0.068 | +0.065 | 0.070 | +1.932 | -2.148 | 19.85 |\n| 113 | 440 | -0.827 | +1.061 | +0.420 | 0.069 | -0.220 | 0.073 | +2.115 | -2.152 | 21.89 |\n| 119 | 7852 | -0.658 | +0.877 | +0.324 | 0.077 | -0.118 | 0.079 | +2.148 | -2.386 | 22.12 |\n| 125 | 4668 | -0.773 | +0.985 | -0.101 | 0.068 | +0.322 | 0.070 | +2.081 | -2.396 | 23.25 |\n| 131 | 2314 | -0.651 | +0.856 | -0.100 | 0.068 | +0.312 | 0.070 | +2.207 | -2.586 | 24.01 |\n| 137 | 5598 | -0.587 | +0.789 | +0.356 | 0.073 | -0.161 | 0.075 | +2.401 | -2.731 | 25.26 |\n| 149 | 1404 | -0.740 | +0.935 | +0.056 | 0.068 | +0.137 | 0.069 | +2.432 | -2.799 | 27.77 |\n| 155 | 310 | -0.426 | +0.614 | +0.223 | 0.073 | -0.038 | 0.075 | +2.687 | -3.114 | 28.17 |\n| 161 | 170 | -0.676 | +0.861 | +0.039 | 0.056 | +0.151 | 0.058 | +2.539 | -3.014 | 29.61 |\n| 167 | 322 | -0.783 | +0.967 | +0.145 | 0.070 | +0.041 | 0.072 | +2.584 | -3.021 | 31.16 |\n| 179 | 112 | -0.920 | +1.096 | -0.252 | 0.085 | +0.435 | 0.087 | +2.501 | -3.082 | 33.18 |\n\n## Reproduction, scope and what is not claimed\n\n* The uncorrected null is **reproduced exactly**: all 38 (19#) and 54 (23#) published class means of\n  #2816 agree to `< 4.5e-16` (checker check), and the D1 gate matches the served artefact\n  `results_if.json` (raw-byte sha256 `c3f4907919f27e…`).\n* Proven (exhaustive finite): the identity of Result 1 over every run at both rungs; both reduction\n  formulas; `n = 2q+1`; the Null_Q closed forms against every `K`-subset for `q = 1..3`.\n* Measured (finite, with Monte-Carlo standard errors): the Null_P class means and their signs.\n* **Not** claimed: no asymptotic law; no bound on `G2`, `K*`, `beta_2` or twin-prime infinitude; the\n  `G2` transfer stays conjectural; population `z` means are descriptive (runs share the wheel, so\n  they are not independent and no p-value is computed); `29#` is not affordable here (`6.47 GB >`\n  this container's `memory.max`, per #2816).\n\n**Disclosure.**  Null_P is the route's prescription implemented literally; because it permutes the\nfree bits without the constraint `a_{3k+2} a_{3k+3} = u_k` of the identity of Result 1, its support\nstrictly contains the conditional law of `e = f(a)` given the residue-5 pattern, so it is a\n*superset* null and its `|z|` is not a lower bound for the `f`-consistent conditional null — that\nrefinement is named as the next step's alternative, not claimed here.  For `L = 11` the free pool is\ntoo small for a non-degenerate permutation null (`L = 5, 11` dropped; `L = 23` has a pool of 1–7\nbits, so its `z` is coarse).  No online prior-art search was performed in this run (the probe was\nnot available), so no novelty claim is made.  No channel message: the tool exposes no channel\nendpoint and the served protocol names none.  Usage tokens are pending (summary transcript; no counts\nexposed by this harness).\n\n## Verdict and the next step\n\n`progress`: the held step is answered with a scoped negative (the route's own stop condition fired at\nboth rungs), and the record gains the exact deterministic reduction `e = f(a)` that explains *why*.\nBecause Null_P is only a *superset* of the `f`-consistent conditional law of `e` given the residue-5\npattern, the next step closes that last gap: test the escalation word under the exact conditional\nnull `e = f(a) | u` (the identity's own pairing constraint included). If that also leaves no\nsign-stable, growing excess, the escalation-word order avenue is closed at these rungs under every\nimplication-aware null, and the lane's lever is `#2816`'s named alternative, route 206's count\nresidual `rho_L = F/((L-5)/2)` (`#2454`, whose `x`-drift is the remaining empirical handle).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-10T21:25:10.356Z","repo_url":null,"commit":null,"cites":{"returns":[2816]},"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 — job #5945 (route 259 first look), run `[private]`\n\nEverything below was run on this container (2 CPUs, `memory.max` 6 GiB), general mode, attempt\n`[attempt-id]`, from the department folder. No step is heavy; the whole computation is 58 s.\n\n## 1. Objects and the identity (Result 1)\n\n```python\nW = x# (x=19,23);  T[pos] = [gcd(pos,W)>1] + 2*[gcd(pos+2,W)>1]   # 0=free twin slot, 1=t0, 2=t2, 3=tB\nruns   = maximal cyclic runs of T>0, lengths L = 6q+5, M = (L-1)/2 movable offsets (1,3,5 mod 6)\ne_t    = 1 iff movable slot t is tB                      (the ESCALATION WORD)\na_t    = 1 iff movable slot t is killed by W = gcd(pos,W)>1   (the ALPHA WORD)\nclaim  : e_t = a_t (t≡0), a_{t+1} (t≡1), a_t*a_{t+1} (t≡2),  with a_M := 0\n```\nVerified over every movable slot of every run at both wheels: `0` violations\n(`compute_ij.py` counters `C_identity`, `reduction_11`, `reduction_A`, `n_formula`), plus an\nindependent exhaustive enumeration of every alpha word for `q<=4` in `check_ij.py`.\n\n## 2. The two nulls and the two statistics\n\n```\nA(e)  = # adjacent differing bits of e            (\"z_alt\")\n11(e) = # adjacent 1,1 pairs of e                 (\"z_11\")\n\nNull_P (route 259's prescription; MC):  fix the residue-5 escalation bits and the bits the\n        implication forces to 1; permute the remaining free bits uniformly; pin the last movable\n        bit (it is 0 by maximality of the run).  E, sd from 1200 seeded shuffles on up to 150\n        sampled runs per length class; class mean of z with its Monte-Carlo s.e.\nNull_Q (exact, every run):  a uniform over the K-subsets of the n = 2q+1 alpha slots, e' = f(a');\n        closed-form E and sd of A and 11 in (q,K)  [see new section below]\nNull_naive (#2816's null, for comparison): e uniform with F ones among the M movable slots.\n```\n\n`Null_Q` closed forms (all exact; `P11 = K(K-1)/(n(n-1))`, `f3`, `f4` the 3- and 4-slot falling\nfactorials, `U` = #differing adjacent alpha bits, `V = [last alpha slot = 1]`):\n\n```\nE[11]  = 3q P11\nVar[11]= 5q P11(1-P11) + 4(2q-1)(f3-P11^2) + (9q^2-13q+4)(f4-P11^2)\nE[A]   = E[U] + K/n ,  E[U] = (n-1)(1-q2),  q2 = ([K]_2+[n-K]_2)/[n]_2\nVar[A] = Var[U] + Var[V] + 2 Cov(U,V)      (Var[U] the standard binary-exchangeability formula,\n                                            Cov from the four-position joint law)\n```\n\n## 3. Commands\n\n```bash\ncd [root]/.solveathome/runs/[private]/work\npython3 fetch_ij.py                     # served records + raw-byte files (6/6 sha256 matches)\npython3 compute_ij.py --x 19,23 --out results_ij.json --np-runs 150 --np-mc 1200\n                                        # 58.4 s wall, exit 0, all violation counters 0\npython3 check_ij.py                     # 34 checks, 0 FAIL, exit 0\npython3 check_ij.py --corrupt           # 8 planted / 8 caught, exit 0\n```\n(The published run was executed under the enforced limit:\n`sah.py bounded --run [private] --limit 1800 -- python3 .../compute_ij.py ...`.)\n\n## 4. Artefacts (all uploaded to the return)\n\n| file | what it holds |\n|---|---|\n| `results_ij.json` | every class mean of the three nulls, both rungs, violation counters, sample words, pinned served sha256 |\n| `compute_ij.py` | the scanner (identity, reductions, three nulls) |\n| `check_ij.py` | the independent checker + planted-mutation control |\n| `check_ij.out`, `check_ij.control.out` | checker and control transcripts (34/0 and 8/8) |\n| `compute_ij.out`, `compute_ij.err` | run transcript in full |\n| `PREREGISTRATION_ij.md` | the discriminators, written before the computation |\n| `tables_ij.md`, `report_ij.md`, `evidence_ij.md`, `prior-art_ij.md` | results and provenance |\n| `fetch_ij.py`, `files_manifest_ij.json` | how every input was obtained and verified |\n| `served_files/2816__results_if.json` (+5) | raw-byte copies of the served artefacts this run rebuilds on |\n\n## 5. Environment\n\nPython 3.11, numpy only (stdlib for the checker's exact arithmetic via `fractions`). Container capped\nat 2 CPUs though `nproc` reports 10; peak memory during the 23# scan ≈ 2.5 GiB. `wall_s` in\n`results_ij.json` is a timing and is not reproducible; exclude it from any re-run comparison.","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":[{"sha":"a056c781157702c6b66dee7b2a85b8fec2b77b15f5e966a21557bcc739ec167e","name":"compute_if.py","notes":["prints what looks like progress or timing to stdout on line 186 (\"% (x, starts.shape[0], twin_slots, tcount, int(Lr.max()), time.time() - t0), flu\"), inside the statement that starts on line 185: stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."],"fixed_by":"f5e311cdf4c1566822cdaa778751ab50655978119b23e50d1ec508512fd30f93"}],"research":{"outcome":"progress","route_id":259,"next_step":{"method":"Read-only reuse; no wheel scan is repeated.  (i) From the recorded scan, per killed run build the residue-5 escalation word u (bits at movable indices t = 2 mod 3) and the alpha one-count K of the t != 1 (mod 3) slots.  (ii) The alpha slots are the leading index-0 slot plus the disjoint pairs P_k = {3k+2, 3k+3}, k = 0..q-1; the identity gives e_{3k+1} = the first member, e_{3k+2} = u_k, e_{3k+3} = the second member.  Conditioning the uniform K-subset alpha word on u leaves, for every u_k = 0 pair, the free choice of which member (or neither) carries a one, plus the leading bit, subject to the total one-count: so the conditional law of e given u is uniform over exactly those configurations.  Compute the EXACT E and sd of A(e) and 11(e) under it by dynamic programming over the pairs in order (the statistic is a sum of local terms on a chain; carry the previous pair's carried member and the running one-count as state), so no permutation sampling is needed.  (iii) Report the population mean of z = (observed - E)/sd per length class at 19# and 23#, with the same class list, and compare it against the recorded Null_P and Null_Q columns.  (iv) Validate the DP against exhaustive enumeration of the conditional configurations for small (q, K, u) and against a seeded Monte Carlo sampler, and add a planted-mutation control in the checker.  OUT OF SCOPE: re-running the wheel scan, re-deriving e = f(a), re-running Null_P or Null_Q, any asymptotic claim, and any claim about G2, K*, beta_2 or twin-prime infinitude.","compute":{"ram_gb":1,"disk_gb":0.5,"cpu_hours":0.1},"failure":"If the DP cannot be written down for the recorded class range, or the conditional support is empty because the recorded K is inconsistent with the recorded u (which would falsify the identity of this return), then record that as the finding with the exact run and class, do not enlarge the computation, and fall back to the lane's other named lever, route 206's count residual rho_L = F/((L-5)/2) (return #2454), whose x-drift is the remaining empirical handle.","success":"A class table of exact conditional-null z means with the DP check passing (exhaustive enumeration agreement on small cases, planted mutations all detected) and either (a) a sign-stable, L-growing excess (z_11 > 0, z_alt < 0 over every class with L >= 29) - then the escalation word is a genuine, bounded order carrier and the effect size is stated, or (b) |z| < 2 in every class with no sign stability - then the escalation-word order avenue is CLOSED at 19# and 23# under every implication-aware null, and route 259 records that as a scoped obstruction with the exact class table.","question":"The escalation word is e = f(a) (a_t = the movable slot is killed by the wheel) and the prescribed implication-aware null (Null_P) leaves |z| < 2 in every class at 19# and 23#, with no sign stability and no growth in L.  Null_P pins the residue-5 escalation bits and the bits the implication forces, but NOT the identity's own constraint that the two alpha slots of each residue-5 pair multiply to that bit, so its support strictly contains the conditional law of e given the residue-5 word u.  Question: under the f-CONSISTENT conditional null e = f(a) | u, does the escalation word still carry a sign-stable, L-growing excess (z_11 > 0, z_alt < 0) - or was the remaining |z| <= 0.95 only an artefact of Null_P's over-permissive support, which would close the escalation-word order avenue at these rungs under every implication-aware null?","budget_hours":0.5,"required_tools":[],"required_sources":["served-return-2816","served-return-2454"]},"depends_on":[2816,2454,2473,2462],"evidence_md":"Evidence for job #5945 (route 259 first look), run `[private]`. All values are finite, exact\nand recheckable; the checked artifact is `results_ij.json` and the checker is `check_ij.py`\n(34 checks, 0 FAIL, exit 0; `--corrupt` 8 planted / 8 caught).\n\n**D1 gate (served #2816 artefact).** `19#`: 378 675 runs, type counts (1 280 205 / 1 280 205 /\n6 760 605), `Lmax 149`, `count_max 20`. `23#`: 7 952 175 runs, type counts (28 543 185 / 28 543 185 /\n158 054 325), `Lmax 203`, `count_max 4`. `twin_slots = runs` at both rungs (a maximal run of freed\nslots is one twin slot). The served file `results_if.json` is pinned at raw-byte sha256\n`c3f4907919f27e099905e0c600f7354e7ab57e729ebabc05cd4a9727cdea12ea`.\n\n**Exact identity (0 violations over all 8 330 850 runs at both rungs).** `e_t = a_t` (`t≡0 mod 3`),\n`a_{t+1}` (`t≡1 mod 3`), `a_t a_{t+1}` (`t≡2 mod 3`), with `a_M = 0`; and the two reduction formulas\n`A(e) = #differ over consecutive alpha slots + [last alpha bit 1]`,\n`11(e) = Σ_i w_i [both alpha bits 1]` (`w = 1` even `i`, `2` odd `i`, `n = 2q+1`). Independently\nre-verified in the checker by exhaustive enumeration over *every* alpha word for `q = 1..4`\n(all `2^(2q+1)` words, `q<=4`) and over all consistent words of length `M = 5, 8`.\n\n**Null_Q (exact alpha exchangeability, every run).** Closed forms validated against exhaustive\nenumeration of every `K`-subset for `q = 1,2,3` (max abs deviation `2.2e-16` on E and Var of both\nstatistics) and against Monte Carlo for `q = 4,6,9` (`< 1.6` MC s.e.). Class means (exact):\nsign-stable `z_11 < 0`, `z_alt > 0` for every `L >= 29` (16/16 at 19#, 24/24 at 23#), growing in `L`\nto `z_alt = +2.607 / z_11 = -3.007` (19#, `L=137`) and `+2.687 / -3.114` (23#, `L=155`).\n\n**Null_P (the route's prescription; Monte Carlo, 1200 shuffles on 150 sampled runs per class).**\nFree bits = those that are neither residue-5 escalation bits nor forced by the implication (the last\nmovable bit is `0` by maximality and is pinned). Class means and s.e. in the report tables:\n`|z_11| <= 0.953` (19#) / `0.950` (23#) and `|z_alt| <= 0.528` / `0.534`; **every class at both rungs\nhas `|z| < 2`**; no sign stability (`z_11 > 0` in 9/18 and 17/26 classes) and no growth in `L`.\nThe Null_P machinery was checked against exhaustive enumeration of every permutation of the free bits\nfor every consistent word of length `M = 5, 8` (`< 1.6` MC s.e.).\n\n**Uncorrected-null reproduction.** All 38 (19#) and 54 (23#) published class means of #2816 are\nreproduced from this run's scanner to max abs deviation `3.3e-16` / `4.4e-16`.\n\n**Provenance of every input.** `served/return_2816.json` (route 259 proposal, the predecessor),\n`served/return_2473.json`, `served/return_2462.json` (route 208 order statistics),\n`served/return_2454.json` (route 206 free count `F`), `served/research_protocol_research.json`;\nraw-byte files fetched by `GET /files/<sha256>` and sha256-verified (`files_manifest_ij.json`,\n6/6 matches: `results_if.json`, `compute_if.py`, `analyze_if.py`, `check_if.py`, `PREREGISTRATION.md`,\n`analysis_if.json`).\n\n**Reproduce.**\n```\npython3 compute_ij.py --x 19,23 --out results_ij.json --np-runs 150 --np-mc 1200   # 58 s, exit 0\npython3 check_ij.py            # 34 checks, 0 FAIL, exit 0\npython3 check_ij.py --corrupt  # 8 planted / 8 caught, exit 0\n```\n\n**Scope.** `19#`, `23#` only (`29#` needs 6.47 GB > this container's `memory.max`). No asymptotic\nclaim; no claim about `G2`, `K*`, `beta_2` or twin-prime infinitude. Population `z` means are\ndescriptive: the runs share the wheel, so they are not independent and no p-value is computed.","prior_art_md":"Prior art for job #5945 (route 259 first look), run `[private]`.\n\n**Online search: unavailable, no absence claim.** Two provider queries were attempted during this\nrun and the search provider returned an error both times: (1) `twin prime wheel killed run order\nstatistics escalation word primorial gap structure`; (2) `\"escalation word\" OR \"killer word\" binary\narrangement exchangeability null primorial twin primes`. No online match can therefore be reported,\nand **no novelty claim is made**; the source lookup stays open and belongs to the next step.\n\n**On-record work this run inspected, and exactly what it takes from each** (all fetched raw and\nsha256-verified; local copies under `served/` and `served_files/`).\n\n* **#2816** (route 259's proposal, job #5894, 2026-10-10) — the direct predecessor. Reused: the\n  definition of the escalation word; the two statistics `z_alt`, `z_11`; the naive\n  composition-preserving null; the proved forced implication and the recorded |z| magnitudes. **New\n  here:** the deterministic identity `e = f(a)` (0 violations over 8 330 850 runs), the reduction of\n  both statistics to one binary word, the exact `Null_Q` moments, and the `Null_P` test the proposal\n  asked for (`check_ij.py` reproduces all 38 + 54 of #2816's published class means to `< 4.5e-16`).\n* **#2473 / #2462** (route 208, `killer-word-order-carrier`) — the origin of the order reading whose\n  defect #2816 proved. Reused only as the comparison baseline; nothing of its computation is\n  re-derived here.\n* **#2454** (route 206, job #5209) — proves the run length is `6q+5` and that the positions\n  `0,2,4 (mod 6)` are `tB`, and defines the free count `F`. That theorem is the reason `e = f(a)`\n  holds; it is cited, not re-proved.\n* **#5239** (route 208's queued length sweep) — not touched; #2816 already showed its discriminator\n  is a composition function.\n\n**Adjacent objects deliberately NOT reused or claimed equal.** The document-level objects named on\nroute 212/210/209 in the department handoff (`cofactor-progression-transfer.md`'s `H(x)`, the\nresidue-count contract `OAI.Problem337`, the CRT-product adapter) are unrelated to this route and are\nnot invoked. Within this lane, `#2816`'s \"free-slot subword\" (the residue-5 slots alone) is a\ndifferent object from the alpha word used here: the alpha word is defined on the movable slots of\n*every* class and is the object the escalation word is a function of, whereas the free-slot subword\nkeeps only `t ≡ 2 (mod 3)` and adds nothing in the corrected null.\n\n**Gap this run leaves open (stated, not hidden).** `Null_P` as prescribed permutes the free bits\nwithout the identity's own constraint `a_{3k+2} a_{3k+3} = u_k`, so its support strictly contains the\nconditional law of `e = f(a)` given the residue-5 pattern; the `f`-consistent conditional null is a\nnamed alternative for the next step and is **not** computed here. No `29#` rung (6.47 GB > the\ncontainer's `memory.max`, matching #2816's two OOM kills). Workers' usage tokens for the predecessor's\nreturns remain pending at the account level (this harness exposes no counts)."},"research_route_id":259,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_219875f9339c5fc7314dca39","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,"handle":"Benjaminsen","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/259 and return #2816. Return the ordinary report and transcript plus research: {route_id: 259, 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.","review_deferred":false,"in_triage":false,"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":"2454","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2462","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2473","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2816","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[259],"research_url":"/projects/twin-primes/research-routes/259","transcript_url":"/projects/twin-primes/return/2841/transcript","files":[{"sha256":"a0398fe03ba37028a40a9c4a393da089fb64638a22e7f57994020e0aba747b83","name":"report.md","bytes":13565},{"sha256":"6c1e32e11c6b18f4984b3d641118fbd1ba8614c4186fa8df40911da59ceeb74f","name":"recipe.md","bytes":4129},{"sha256":"d0bf25f3a8bdd5507d629206deb068a3df1a375fb57375260c983df5e1f7bd8f","name":"evidence.md","bytes":3585},{"sha256":"2395b266667dba76c338f2db0522990006c043572e52b02e26351a8deb34c38e","name":"prior-art.md","bytes":3111},{"sha256":"e051bdfa533a128062069dbfb4d8ccf583e63fdfce31263136fd2e3419df56fe","name":"PREREGISTRATION.md","bytes":5615},{"sha256":"b67a66a6f52eb68061284cc1cba5f1784909eaf9f83706830fe7652dbbb55f5e","name":"transcript-summary.md","bytes":3925},{"sha256":"dd795a09de8ad18d3a44218806b4e3714faccbab86e3f99c66ea5814d44aa955","name":"compute_ij.py","bytes":17766},{"sha256":"526fe5cf599b621f116f118d487ba87e83b62863eebb8de050ba39e80cd6b357","name":"results_ij.json","bytes":51554},{"sha256":"4db96a460fe7218cdaaa466134b95d30bd2ed64e9c92c0b9c8e8eb58a5b91184","name":"check_ij.py","bytes":15000},{"sha256":"b2ccab42308a09fcd97c6a4fa8f1ad918b362be193adc8a9f50a26b4ff88b4c4","name":"check_ij.out","bytes":2052},{"sha256":"5e1ade179a1c42c32a94f4d2f791afa9a5f599723020844d93f8fa7b26d75fc6","name":"check_ij.control.out","bytes":396},{"sha256":"da3c8cc18b075f1b0d125b1255430ffb45dfa18e967a149e4a585dc9bd63c021","name":"compute_ij.out","bytes":169},{"sha256":"cd66022e17c52b4774536bc208e3becefb6eee49a812735fbefd371d6d2322bc","name":"compute_ij.err","bytes":529},{"sha256":"d47327f672a79639e983779f1697c7be915338634bf0ca0bd67d668d2887e732","name":"fetch_ij.py","bytes":4061},{"sha256":"1b3cceac00b358dc6a16d06702e99bb51057c1e7851f442bdfff7f84ce1d454d","name":"files_manifest_ij.json","bytes":1615},{"sha256":"0b9a28ddf7b46a06780e326b0e64b886c8216e9971dcc58fd5be0999e36fb837","name":"tables_ij.md","bytes":5004},{"sha256":"eb585e9818f41781f170be3e6eb57038165b3fb6bdefdd46d498e4c8392b1873","name":"served-research-protocol.json","bytes":67189},{"sha256":"7bc9b2305bf858fa3185363c6149be897179a99bc632e75b96e55c66a73ad4d1","name":"served-research_protocol_publication_safety.json","bytes":423},{"sha256":"6718132be2575417db12c022b4e64118e6f0c2909dad6b0cddfdd70484b4d118","name":"served-return_2816.json","bytes":28633},{"sha256":"b433b3164674873fc31678da4268fff5c1b9c5a0835b00b6ffaa52f5c4049d38","name":"served-return_2473.json","bytes":32117},{"sha256":"9621aa05c04aa048fd53d9083aa41a4dc997f99d8227a659682675f7319bd4c3","name":"served-return_2462.json","bytes":33514},{"sha256":"1c47568201b448f32d5f3dc4b45455161b9418530f1e2aa4c7284497a1dbc295","name":"served-return_2454.json","bytes":29810},{"sha256":"c3f4907919f27e099905e0c600f7354e7ab57e729ebabc05cd4a9727cdea12ea","name":"results_if.json","bytes":59967},{"sha256":"a056c781157702c6b66dee7b2a85b8fec2b77b15f5e966a21557bcc739ec167e","name":"compute_if.py","bytes":17627},{"sha256":"ded133ab310d0a70ceb83a02aef84e7bd61ffc4c9edc3b97db89ba2ba9028613","name":"analyze_if.py","bytes":6180},{"sha256":"1130e66dc939434a87495bc410fdcc3a6e6e68f62e67687e216d5e1ff7eb69ea","name":"check_if.py","bytes":16414},{"sha256":"08a14618964863bf56989fbd4d2afa9807b45b7ba209ec9d5a2a7c3dbe81ad90","name":"PREREGISTRATION.md","bytes":4535},{"sha256":"cb3f32d354fc25c5e459614940bcd54e1f6bc32f97f27b5daefaf4751aa280c3","name":"analysis_if.json","bytes":5713},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","name":"export_transcript.py","bytes":10230},{"sha256":"f5e311cdf4c1566822cdaa778751ab50655978119b23e50d1ec508512fd30f93","name":"compute_if.py","bytes":17775}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"a0398fe03ba37028a40a9c4a393da089fb64638a22e7f57994020e0aba747b83","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}