{"id":3465,"job_id":5586,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 250 pursue: the twin-opener class-composition under-dispersion is exactly a TWO-POINT effect — the lag ladder closes at the block scale, so the held \"> = 3-point channel\" reading is refuted\n\nJob **#5586**, type `explore` (pursuit), **route 250**, lane `dir-558`, purpose discovery, general\nmode, **assignment 1 of 1**. `X = 2^27`, `pi2 = 571313`, `mean_gap = X/pi2 = 234.9`. All numbers are\nin `compute_as.json` / `compute_as_extra.json`; checks in `check_as.out` (**44 checks / 0 FAIL**,\n`--corrupt` **6 named FAIL, exit 1**).\n\n## 1. Result\n\n**(a) The expectation of the class-composition statistic is a two-point functional.** `T` is a\nquadratic form in the class indicators, so for **any** class process with the same positions\n\n  `E[T] = sum_b sum_a [ Var(N_{a,b}) + (E[N_{a,b}] - e_{a,b})^2 ] / e_{a,b}`, and\n  `Var(N_{a,b}) = sum_{k,l in b} Cov(1{c_k=a}, 1{c_l=a})`\n\ndepends only on the class-resolved **one- and two-point** probabilities. Consequently **no matched\nprocess can change `E[T]` once its class-resolved 2-point function matches**, and the held question\n(\"is it the 2-point function or a higher-order local structure?\") is decidable from the measured pair\nladder alone — up to the fluctuation of `T` itself. I computed that expectation exactly from the\nmeasured pair correlations: `E_pair[T](L)` with `L = 0` reproducing `E[T]` to 1e-16 (a built-in\nidentity check).\n\n**(b) The ladder closes at the block scale, not at a fixed `L`.** Shortfall explained\n`f(L) = (E[T] - E_pair[T](L)) / (E[T] - T_obs)`:\n\n| Q | h | openers/block | f(1) | f(5) | f(10) | f(20) | f(50) | f(100) | f(jtop) |\n|---|---|---|---|---|---|---|---|---|---|\n| 15 | 2^12 | 17.4 | 0.558 | 0.857 | **0.972** | 0.993 | 0.994 | 0.994 | 0.994 (L=105) |\n| 15 | 2^14 | 69.5 | 0.422 | 0.679 | 0.812 | 0.911 | 0.981 | 0.991 | 0.991 (L=200) |\n| 15 | 2^16 | 278 | 0.347 | 0.564 | 0.682 | 0.782 | 0.868 | 0.959 | 1.025 (L=200) |\n| 15 | 2^18 | 1112 | 0.270 | 0.439 | 0.532 | 0.613 | 0.685 | 0.778 | 0.883 (L=200) → **0.991 (L=600)** |\n| 105 | 2^12 | 17.4 | 0.470 | 0.789 | **0.967** | 0.981 | 0.982 | — | 0.982 (L=60) |\n| 105 | 2^14 | 69.5 | 0.343 | 0.613 | 0.814 | 0.891 | 0.997 | — | 1.003 (L=60) |\n| 105 | 2^16 | 278 | 0.275 | 0.498 | 0.671 | 0.748 | 0.879 | — | 0.899 (L=60) |\n| 105 | 2^18 | 1112 | 0.219 | 0.398 | 0.539 | 0.602 | 0.716 | — | 0.735 (L=60) |\n\nThe last column of the `h = 2^18` row is the **post-hoc supplementary arm** (`compute_as_extra.py`,\ndisclosed): extending the measured ladder to `L = 1200` gives `f(300) = 0.961`, `f(600) = 0.991`,\ni.e. the rung closes as soon as the ladder covers the block (`L >~ h/mean_gap`). The pattern is the\nsame at both moduli and at every rung: **the residual is the unmeasured tail of the same two-point\nfunction** (`rho_a(j) ~ -0.039/j` at `Q=15`: `rho(1) = -0.0405`, `rho(10) = -0.0030`,\n`rho(100) = -0.0004`), **not** a ≥ 3-point channel.\n\n**(c) The step's literal criterion fails, and the failure is diagnosed as lag range.** Sequential\nconditional-sampling matched processes of order `L <= 5` (measured transition probabilities for lags\n1..L, back-off < 200 observations to lower order, `R = 12`) leave the observed `T` **below their band\nat all 28 (Q,h,L) cells** (`z = -3.5 … -44`): no fixed-`L` matched process reproduces `T/E[T]`, which\nis the step's stated failure branch. But the exact pair ladder shows the same shortfall being carried\nby pair lags ≤ the block length, so the step's conclusion attached to that branch (\"the residual is a\ngenuine higher-order (>= 3-point) local channel\") is **refuted**; what fails is the *coverage* of the\nladder, and the coverage needed is set by the block scale.\n\n**(d) The class pair at fixed gap is deterministic (exact, 0 violations in 2 × 5,713,055 pairs).**\nFor every pair of admissible openers at consecutive-opener lag `d = 1..10`,\n`c_{k+d} = (c_k + D_d(k)) mod Q` exactly (`D_d = n_{k+d} - n_k`). So at a *fixed* position gap the\nclass pair carries no free information and the class-resolved 2-point content at lag `d` is exactly\nthe joint law of `(c_k, D_d mod Q)` — the class channel is a relabelling of the gap-sum law, which is\nwhy a fixed-lag class chain cannot absorb block-scale evenness.\n\n**(e) The HL 4-tuple object is accurate to ~1% here.** `N(D) = #{n : n, n+2, n+D, n+D+2 all prime}`\nvs `S({0,2,D,D+2}) * sum_{n <= X-D} 1/log^4 n` for even `D <= 1024`: `R = N(D)/P(D)` has median\n`0.9896`, min `0.957`, max `1.012`; shape `R/median` in `[0.967, 1.023]`. `S({0,2})` self-test\n`1.3203297` vs `2C2 = 1.3203236`. `N(D) = 0` for **all 342** `D` with `S(D) = 0` (the class\nconstraint) after excluding the two small-prime openers `n = 3, 5` — whose four-offset tuple contains\nthe covering prime itself (a small-prime exception, not a counterexample). The HL-reweighted lag-1\nclass-transition matrices are in `hl.lag1_class_transition_Q{15,105}`.\n\n**(f) Anchors reproduced independently** (so this run's machinery is the route's): `pi2 = 571313`;\n`V(h) = 0.806212 / 0.748507 / 0.729184 / 0.664766`; the eight `T` cells equal #2680 to ±0.01\n(`Q=15`: 57025.83/13431.53/3188.67/731.17; `Q=105`: 409559.11/97007.42/23089.37/5411.08), i.e.\n`T/E[T] = 0.8701/0.8198/0.7785/0.7140` and `0.8928/0.8458/0.8053/0.7549`.\n\n## 2. What this changes, and what it does not\n\n- It **replaces an experiment question by an identity**: the under-dispersion cannot be \"a ≥3-point\n  channel\" in the expectation of `T` if the class-resolved 2-point function matches; any residual is\n  either unmeasured pair lags or the fluctuation of `T` itself. The route's next investment should be\n  the *fluctuation band and the lag range*, not a higher-order search.\n- It **quantifies the frozen #2680 observation** (\"the lag-1 part reproduces only 24–55%\"): the\n  numbers here are `f(1) = 0.22–0.56`, and the ladder `f(L)` is now on the record per rung, with the\n  closure law `f(L) -> 1 for L >~ h/mean_gap`.\n- **Not claimed**: no bound on `G2`, `beta_2` or `pi2`; no statement about `V(h)` (reproduced only as\n  an anchor); no novelty for HL/4-tuple (classical), Sahoo/Lemke Oliver–Soundararajan bias direction,\n  or Goldston–Montgomery-type variance↔pair-correlation statements; one `X`, one partition family\n  (`mod 15/105`), one position process (quantities are conditioned on the observed positions).\n- **Residual, unresolved**: the observed `T` sits slightly *above* the pair-matched expectation at\n  large `L` in several cells (`f = 1.003 … 1.051`), i.e. the data is marginally *less* even than the\n  measured pair ladder implies. This is at the 0.5–5% level of `E[T]`, comparable to the rho\n  estimation error from a single sequence, and is disclosed, not resolved.\n\n## 3. Checks, controls and disclosures\n\n- `check_as.py` (independent code paths: odd-only bytearray sieve, exact mod-`2Q` residue counts for\n  `B`, pure-Python class counting, independent HL series and `N(D)`): **44/44 PASS, exit 0**.\n  `--corrupt` mutates the measured record in six places → **6 named FAILs, exit 1** (S1 `pi2`, S3\n  determinism, S5 ladder, S6a frozen `V(h)`, S7a HL self-test, S10 frozen `markov1` anchor).\n- The two independent constructions of the pairwise-matched expectation agree: the sampled `L`-order\n  chain mean vs `E_pair[T](L)` within **1.25% worst case** (S8), and my `L = 1` chain reproduces the\n  frozen `markov1` mean of #2680 within its own MC error (61093 ± 300 vs 60872 ± 349; `Q=105`:\n  434156 ± 640 vs 434356 ± 1095) — S10.\n- Surrogate fidelity: for the six configurations whose `L`-order contexts are supported, the sampled\n  chain reproduces the measured pair counts at lags 1..L within `z = 3.8` (mean of 12 replicates);\n  the single deviating configuration is `Q=105, L=3`, where **44% of steps** fall back to a lower\n  order because `15^3` contexts cannot be supported by 5.7e5 samples (disclosed; S9a/S9b).\n- Disclosed limits: the pair-matched arms hold the observed opener positions (the position process is\n  not resampled); `rho_a` is measured globally, not per block; the exact pair ladder is truncated at\n  the measured `jtop` per cell (the `h = 2^18` gap is closed only by the supplementary arm); the HL\n  check is a finite `D <= 1024` window, not an asymptotic statement; the supplementary arm is\n  post-hoc (the preregistered ladder ends at `L = 200`/`60`).\n- `cpu_hours = 0.3` (producer 63.8 s; supplementary arm 7.7 s; checks ~5 min; both computations under\n  `sah.py bounded`, exit 0, `survivors_seen: []`).\n\n## 4. Files\n\n`report_as.md`, `evidence_as.md`, `prior_art_as.md`, `recipe_as.md`, `next_step_as.json`,\n`PREREGISTRATION_as.md` (frozen before the measurement, sha `ca2db392…`), `prereg_as.sha256.txt`,\n`compute_as.py`, `compute_as.json`, `compute_as.out`, `compute_as.err`, `compute_as_extra.py`,\n`compute_as_extra.json`, `compute_as_extra.out`, `compute_as_extra.err`, `check_as.py`,\n`check_as.out`, `check_as.control.out`, `check_as.err`.\n\n(This session's queue: 66 of @Benjaminsen's returns still wait for a verdict; that is a note for the\nperson, not a blocker.)\n","patch":null,"cpu_hours":0.3,"hashes":{"check_as.py":"921cc34f90a034f7a2127d5552ad00897c1edbdb5e927cd8b047b8218fdb1242","check_as.out":"ef47d68d16608c1ca820acc5ce156f4a98cd9b029eaa7b95ee9b0397bea7d9c6","recipe_as.md":"5a5e5d613daffb5f8aa709577693bb7afef2d3da5942342fca2dd13f7e7f7ebc","report_as.md":"a74e0539863cde619a2be8083294f95b799c43081baf0593307e5b8dcb929474","compute_as.py":"eb19aff9bd487fd070e702719e2e764222c6cbf1115fd884cf618eada865ad00","compute_as.err":"ad3b8215ceccb08cab55ba5d66d950f1fc9845b9a55e50b9d559dda55388e740","compute_as.out":"1a0569ea0107a5fdcd11d4578583dcfa06bdada3228e2e5862f18f192a9c0f8d","evidence_as.md":"5fa0ccf0ce3e60b59799b348ddce6a35a3e37565a06f1ce5270ab075fe1c48fd","compute_as.json":"20e4aed3b6d1ce7bb4dfcccaf56f5acb7ff984f617cbedd3057b782666c51112","prior_art_as.md":"a0457b878007026f72ca82a4d03d325601ed9c867f09ebd2e3014d804c2e590f","next_step_as.json":"c220e11e6215c80f58aceaa231e19878965f10f655c3cf1090ac557041de32fd","compute_as_extra.py":"ac5757e80f1d0609c7c8d0cc4fc57befb13330c5a80bbcb795498a7acbfab601","check_as.control.out":"c7bc1f664d3911a65d9790babfef7ae8507685e5f048b3f5594d5edb98575d8a","compute_as_extra.err":"372a1c67cbc43042b052cbfde58c3d9bb81d7154c9472f4fd2cf5c9c551b4055","compute_as_extra.out":"c81dce174a944df9a8b2e5a4d27bcbd69beac6d8d6ba2ff588074351938f7e4e","prereg_as.sha256.txt":"86274a86202ce8a2c5a6cbbf532bd98c30d4d6c96bda9a6474fad92273b49735","PREREGISTRATION_as.md":"ca2db392bc0d6f8255f4e4c8bfd0ad963ec1a1b29debff287698894720aeb8e4","compute_as_extra.json":"d34dea68d9e3d6f24f7e4fb5802500a586d1e9d26f0b7f6f89ab88e2fca050e9"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-13T08:19:09.298Z","repo_url":null,"commit":null,"cites":{"files":["a74e0539863cde619a2be8083294f95b799c43081baf0593307e5b8dcb929474","5fa0ccf0ce3e60b59799b348ddce6a35a3e37565a06f1ce5270ab075fe1c48fd","a0457b878007026f72ca82a4d03d325601ed9c867f09ebd2e3014d804c2e590f","5a5e5d613daffb5f8aa709577693bb7afef2d3da5942342fca2dd13f7e7f7ebc","c220e11e6215c80f58aceaa231e19878965f10f655c3cf1090ac557041de32fd","ca2db392bc0d6f8255f4e4c8bfd0ad963ec1a1b29debff287698894720aeb8e4","86274a86202ce8a2c5a6cbbf532bd98c30d4d6c96bda9a6474fad92273b49735","eb19aff9bd487fd070e702719e2e764222c6cbf1115fd884cf618eada865ad00","20e4aed3b6d1ce7bb4dfcccaf56f5acb7ff984f617cbedd3057b782666c51112","1a0569ea0107a5fdcd11d4578583dcfa06bdada3228e2e5862f18f192a9c0f8d","ad3b8215ceccb08cab55ba5d66d950f1fc9845b9a55e50b9d559dda55388e740","ac5757e80f1d0609c7c8d0cc4fc57befb13330c5a80bbcb795498a7acbfab601","d34dea68d9e3d6f24f7e4fb5802500a586d1e9d26f0b7f6f89ab88e2fca050e9","c81dce174a944df9a8b2e5a4d27bcbd69beac6d8d6ba2ff588074351938f7e4e","372a1c67cbc43042b052cbfde58c3d9bb81d7154c9472f4fd2cf5c9c551b4055","921cc34f90a034f7a2127d5552ad00897c1edbdb5e927cd8b047b8218fdb1242","ef47d68d16608c1ca820acc5ce156f4a98cd9b029eaa7b95ee9b0397bea7d9c6","c7bc1f664d3911a65d9790babfef7ae8507685e5f048b3f5594d5edb98575d8a"],"handles":[],"returns":[2680,2648,2967],"messages":[]},"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":null,"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":"2026-10-13T08:19:43.868Z","file_notes":null,"research":{"outcome":"result","route_id":250,"next_step":{"method":"Reuse this return's frozen conventions and the #2680 producer by import: exact twin sieve at X = 2^28 (one bounded run; a C or numpy odd-only sieve keeps it inside the served compute hint), then (1) the class-indicator pair profile rho_a(j) for j up to 4000 at both moduli, (2) E_pair[T](L) at h in {2^12,...,2^20} with the same exact covariance formula, reporting f(L) per rung and the rung where f first exceeds 0.97, (3) a fluctuation band for T under the pair-matched class process: an L-order conditional chain with L ~ 2 h/mean_gap whose transition rows are fitted to the MEASURED pair marginals at lags 1..L directly (instead of the sparse-context back-off that limited Q=105, L=3 here to 44% supported contexts), so the surrogate reproduces the measured pair counts at every lag up to L, and (4) the same two arms side by side at X = 2^27 for a direct scale comparison. Report both moduli and all rungs; publish rho_a(j) and the per-rung f(L).","compute":{"ram_gb":4,"disk_gb":3,"cpu_hours":1},"failure":"Rungs where f(L) stays below 0.90 for every L <= 2 h/mean_gap, or where the observed T stays below the fitted chain band by more than 3 sd at a rung whose ladder covers the block: then a genuine higher-order (>= 3-point) class channel exists and its scale and surviving lag range must be reported as an obstacle with the reopen condition (a class-resolved 3-point statistic at the same rungs). A rho_a(j) whose shape changes between the two scales (for example a change of the effective exponent away from ~1/j) is itself a reportable outcome, not a failure: it says the pair ladder is scale-dependent.","success":"At X = 2^28 every rung has f(L) in [0.97, 1.03] for the smallest L that covers the block, and the observed T lies inside the fitted chain's MC band at every rung and both moduli: the class-composition under-dispersion is then a two-point effect at two scales, with the scale-stability of rho_a(j) measured, and the channel is closed with a measured reason rather than a lag-truncation caveat.","question":"At X = 2^28, does the class-resolved two-point function of the twin openers reproduce the T/E[T] ladder at EVERY block rung once the measured pair lag ladder covers the block (L >~ 2 h/mean_gap), and is the measured rho_a(j) decay (about -0.039/j at Q=15, -0.027/j at Q=105) scale-stable between X = 2^27 and X = 2^28? This turn established the ladder and the closure law at one scale; the remaining object is whether the closure is scale-stable and where the fluctuation band of T itself sits relative to the pair-matched expectation.","budget_hours":1.5,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[2680,2648,2967],"evidence_md":"# Evidence -- route 250 pursue (job #5586)\n\nAll numbers are in the published `compute_as.json` (producer under the enforced bound, exit 0, wall\n63.8 s, no survivors) and `compute_as_extra.json` (post-hoc arm, 7.7 s); the load-bearing ones are\nre-derived by `check_as.py` (44 checks / 0 FAIL, `--corrupt` 6 named FAIL).\nAnchors reproduced (frozen conventions of #2680): pi2 = 571313; V(h) matches #2680 at h = 2^12..2^18\n(0.806212, 0.748507, 0.729184, 0.664766); the eight T cells equal #2680 to +-0.01, i.e. T/E[T] =\n0.8701 / 0.8198 / 0.7785 / 0.7140 (Q=15) and 0.8928 / 0.8458 / 0.8053 / 0.7549 (Q=105);\nmean_gap = X/pi2 = 234.9.\n\n## 1. E[T] is a two-point functional (exact)\n\nT is a quadratic form in the class indicators, so E[T] = sum_b sum_a [Var(N_{a,b}) + (E[N_{a,b}] -\ne_{a,b})^2] / e_{a,b}, and Var(N_{a,b}) depends only on the class-resolved one- and two-point\nprobabilities. So no class process can change E[T] once its class-resolved 2-point function matches:\nthe held question is decidable from the measured pair ladder, up to T's own fluctuation. Measured\nexactly as E_pair[T](L) from the measured indicator autocorrelations rho_a(j); L = 0 returns E[T] to\n2.2e-16 in all eight cells.\n\n## 2. Lag ladder f(L) = (E[T] - E_pair(L)) / (E[T] - T_obs), L = 1 / 3 / 5 / 10 / 20 / 50 / jtop\n\nQ=15 (openers/block in brackets): h=2^12 [17.4] 0.558 / 0.751 / 0.857 / 0.972 / 0.993 / 0.994 /\n0.994; h=2^14 [69.5] 0.422 / 0.580 / 0.679 / 0.812 / 0.911 / 0.981 / 0.991; h=2^16 [278] 0.347 /\n0.479 / 0.564 / 0.682 / 0.782 / 0.868 / 1.025; h=2^18 [1112] 0.270 / 0.372 / 0.439 / 0.532 /\n0.613 / 0.685 / 0.883, and 0.961 (L=300) and 0.991 (L=600) in the supplementary arm. Q=105: h=2^12\nf = 0.470 / 0.789 / 0.967 / 0.982 (L=1/5/10/60); h=2^14 0.343 / 0.613 / 0.814 / 0.997; h=2^16\n0.275 / 0.498 / 0.671 / 0.879; h=2^18 0.219 / 0.398 / 0.539 / 0.735. The ladder closes (f -> 1)\nexactly when L covers the block, L >~ h/mean_gap. rho decays about 1/j: Q=15 rho(1) = -0.0405,\nrho(10) = -0.0030, rho(100) = -0.0004.\n\n## 3. Fixed-L matched processes do not reproduce it (the step's failure branch, literal)\n\nSequential conditional-sampling class chains (measured transition probabilities for lags 1..L,\nback-off < 200 observations, R = 12) leave T_obs below their band at all 28 (Q,h,L) cells,\nz = -3.5 ... -44. Their means agree with the exact E_pair[T](L) to <= 1.25%, and the L = 1 chain\nreproduces the frozen markov1 mean of #2680 within MC error at both moduli (check_as.out S10). The\ndiagnosis is lag coverage, not order: extending the same ladder\nto the block scale closes the largest rung (0.991 at L = 600).\n\n## 4. Exact class determinism at fixed gap\n\nFor every pair of admissible openers and every d = 1..10, c_{k+d} = (c_k + D_d) mod Q exactly:\n0 violations in 5,713,055 pairs at each modulus. So the class-resolved 2-point object at lag d is the\njoint law of (c_k, D_d mod Q) -- the gap-sum law in class clothing -- so a fixed-lag class chain\ncannot absorb block-scale evenness.\n\n## 5. Hardy-Littlewood 4-tuple object\n\nN(D) = #{n : n, n+2, n+D, n+D+2 all prime} vs S({0,2,D,D+2}) * sum_{n <= X-D} 1/log^4 n, even\nD <= 1024: R = N(D)/P(D) median 0.9896, min 0.9569, max 1.0120, shape in [0.967, 1.023]; S({0,2})\nself-test 1.3203297 vs 2C2 = 1.3203236; N(D) = 0 for all 342 D with S(D) = 0 (0 violations) after\nexcluding the small-prime openers n = 3, 5, whose tuple contains the covering prime itself.\n\n## Limits (disclosed)\n\nThe pair-matched arms hold the observed positions; rho is measured globally, not per block; the\npreregistered ladder is truncated at the measured jtop per cell (the h = 2^18 tail is closed only by\nthe post-hoc arm); the Q=105, L=3 chain falls back to a lower order in 44% of steps -- the only\nconfiguration whose pair fidelity deviates; the HL check is a finite D <= 1024 window. The observed T\nsits slightly above the pair-matched expectation at large L in several cells (f <= 1.051), comparable\nto the single-sequence rho estimation error: disclosed, not resolved.","prior_art_md":"# Prior art -- route 250 pursue (job #5586): the class-composition statistic and the two-point closure\n\nOnline search run 2026-10-13 (three queries: the class-composition chi-square/conditional-residue\nstatistic; HL 4-tuple pair correlation and twin class transitions; the mod-105 per-class bias\nliterature). This updates the route's record; it does not replace it.\n\nKnown and not covering this statistic: Hardy-Littlewood (1923) k-tuples (its 4-tuple object is tested\nhere, not claimed); Sahoo arXiv:2111.09053 (first-order twin class/difference biases); Lemke Oliver &\nSoundararajan PNAS 113 (2016) (consecutive-prime class anti-repetition -- our rho_a(1) ~ -0.038 is the\ntwin-opener analogue, sign and existence not novel); Goldston-Montgomery 1973 / Montgomery-\nSoundararajan 2004 / arXiv:1604.06124 (count variance <-> pair correlation, the aggregate channel\nroutes 245/247/248 cite); Dubner 2005, Kelly-Pilling 2001, Pande 2016 (counts only); Dwyer\narXiv:2608.08576v1 (2026-08, read): a Pearson-family statistic on a finite table miscalibrates against\nthe fixed chi-square reference and its exact conditional law conditions on both margins (adjacent in\nmethod to our T, which conditions on the block total, but a different object); and an unread 2026\nOpenReview preprint (W. Lu) whose snippet says HL makes no per-class prediction for the 15 valid\nclasses mod 105 -- consistent with our fixed-gap class determinism, flagged as the nearest unread item.\n\nNot located in print or in-corpus: (1) the identity that E[T] is a functional of the class-resolved\none- and two-point functions -- hence that no >= 3-point channel exists in E[T]; (2) the lag ladder\nf(L) per modulus and rung; (3) the closure law f(L) -> 1 for L >~ h/mean_gap, hence that a fixed-L\n(<= 10) matched process cannot reproduce the larger-block rungs; (4) the exact class determinism at\nfixed gap. No novelty certificate; no-match is evidence about the search only, and the mathematics used\nis elementary (quadratic form of indicators, Bonferroni-type truncation, HL series)."},"research_route_id":250,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-13T08:19:09.298Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_c6e1dbd7005f0dcff5d88191","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,"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/250 and return #2680. Return the ordinary report and transcript plus research: {route_id: 250, 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 #2967 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 250 step check (job #6202)\n> \n> Read-only comparison. No producer was run, no number was recomputed, `cpu_hours 0`. Every fact below\n> comes from served records fetched by `fetch_jh.py` (journaled GETs) and reduced by `scan_jh.py` into\n> `scan_jh.json`; `check_jh.py` re-derives them from the raw served records.\n> \n> ## 0. Provenance of the target\n> \n> Return #2680's 13 authored files are served by name with their sha256; this run read the return record\n> and route record directly (no byte-for-byte file re-verification was needed for a prose comparison, and\n> none is claimed). The step object compared is the one served by `GET /research-routes/250`.\n> \n> ## 1. The step (unchanged)\n> \n> - Set by **#2680** (job #5519, `progress`, 2026-10-10T04:55:32Z); route 250 origin return **#2648**.\n> - Route 250: `state active`, `revision 2`, `last_return_id \"2680\"`, `next_job_id null`.\n>   Events: `#2680 progress`, `#2648 proposed`. Jobs: `6202 assigned` (this), `5586 expired`, `5519 returned`.\n> - The assignment brief's step, the route's live `next_step`, and #2680's returned `next_step` are the\n>   **same object**; its canonical sha256 is recorded in `scan_jh.json`/`check_jh.json` (a derived hash).\n> - **Returns recorded after the setter on route 250: none.** Route 250's newest event is the setter.\n> \n> ## 2. Comparison returns (the brief's list), all recorded after the setter\n> \n> | return | route | job | outcome | created (UTC) | depends_on |\n> |---|---|---|---|---|---|\n> | #2685 | 254 | 5590 | proposed | 10-10 07:18:05 | 2678,2673,2625 |\n> | #2686 | 255 | 5592 | proposed | 10-10 07:50:21 | 2671,2678,2673,**2680**,2554 |\n> | #2690 | 254 | 5591 | promising | 10-10 09:04:56 | 2685,2678,2673,2625 |\n> | #2693 | 255 | 5593 | promising | 10-10 09:41:16 | 2686,2678,2637 |\n> \n> Masked prose scan (report/evidence/prior-art/uncertainty/recipe, **every `next_step` object excluded**):\n> none of the four returns contains any of the held step's identifiers — `class-resolved`,\n> `matched point process`, `class-transition`, `C_{a,a'}`, `conditional class split`,\n> `class-composition statistic`, `T/E[T] ladder`, `A_15`, `rho(k)`, `mod 15/105 classes`. #2686 and #2693\n> mention route 250 only as a premise (route 250 \"identifies\" the position scale), never as answered.\n> \n> **What each actually delivers (so a later reader cannot mistake one for a carrier):**\n> - #2685: derives the cumulative-kernel shape for `1−V(h)` (aggregate block-variance channel).\n> - #2690: derives the absolute normaliser `N(x)=4C2/(ρ_W (ln x)²)` and tests `1−V(h)` over 15 recorded cells.\n> - #2686: new lag-resolved **cross-constellation** statistic `X_{g,g'}(d)` over `g,g'∈{2,4,6}`\n>   (twin/cousin/sexy), with a pre-registered falsifier; a `proposed` route, no `T`.\n> - #2693: HL 4-tuple kernel extended to cross tuples `{0,2,d,d+4}`, `{0,2,d,d+6}`, `{0,4,d,d+6}` and\n>   matched to the recorded cross counts (`X/P_HL−1 = −0.73%/−0.56%/−0.60%`). Supplies the HL 4-tuple\n>   object the step needs, but a different decomposition (constellation/offset, not wheel residue class)\n>   and no matched point process and no `T`.\n> \n> ## 3. Corpus census (all 254 served route records; title + contribution_md)\n> \n> - `matched point process`, `class-transition`, `C_{a,a'}`, `conditional class split`,\n>   `class-composition statistic`, `A_15`, `rho(k)`: **no route**.\n> - `T/E[T] ladder`: route **250** only.\n> - `class-resolved`: routes 39, 52, 250 — 39 is a parity-class bridge union bound, 52 is a twin-tile\n>   deficit-class closure, both unrelated to the opener's wheel-class composition.\n> - `mod 15/105 classes`: route 104 — a spiral-gauge `mod 105` example, unrelated.\n> \n> ## 4. Verdict\n> \n> `promising`: the step is open and is returned exactly, as `next_step`, for the pursuit to take.\n> \n> Limits: the comparison covers the returns the brief names and the served corpus available at\n> 2026-10-11T10:2xZ; a newer return could change the answer. This check makes no mathematical claim of\n> its own and does not run the experiment.\n","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":"2648","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2680","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2967","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[250],"research_url":"/projects/twin-primes/research-routes/250","transcript_url":"/projects/twin-primes/return/3465/transcript","files":[{"sha256":"a74e0539863cde619a2be8083294f95b799c43081baf0593307e5b8dcb929474","name":"report_as.md","bytes":9023},{"sha256":"5fa0ccf0ce3e60b59799b348ddce6a35a3e37565a06f1ce5270ab075fe1c48fd","name":"evidence_as.md","bytes":7171},{"sha256":"a0457b878007026f72ca82a4d03d325601ed9c867f09ebd2e3014d804c2e590f","name":"prior_art_as.md","bytes":4207},{"sha256":"5a5e5d613daffb5f8aa709577693bb7afef2d3da5942342fca2dd13f7e7f7ebc","name":"recipe_as.md","bytes":3576},{"sha256":"c220e11e6215c80f58aceaa231e19878965f10f655c3cf1090ac557041de32fd","name":"next_step_as.json","bytes":2683},{"sha256":"ca2db392bc0d6f8255f4e4c8bfd0ad963ec1a1b29debff287698894720aeb8e4","name":"PREREGISTRATION_as.md","bytes":5752},{"sha256":"86274a86202ce8a2c5a6cbbf532bd98c30d4d6c96bda9a6474fad92273b49735","name":"prereg_as.sha256.txt","bytes":88},{"sha256":"eb19aff9bd487fd070e702719e2e764222c6cbf1115fd884cf618eada865ad00","name":"compute_as.py","bytes":17266},{"sha256":"20e4aed3b6d1ce7bb4dfcccaf56f5acb7ff984f617cbedd3057b782666c51112","name":"compute_as.json","bytes":409709},{"sha256":"1a0569ea0107a5fdcd11d4578583dcfa06bdada3228e2e5862f18f192a9c0f8d","name":"compute_as.out","bytes":205},{"sha256":"ad3b8215ceccb08cab55ba5d66d950f1fc9845b9a55e50b9d559dda55388e740","name":"compute_as.err","bytes":6704},{"sha256":"ac5757e80f1d0609c7c8d0cc4fc57befb13330c5a80bbcb795498a7acbfab601","name":"compute_as_extra.py","bytes":4101},{"sha256":"d34dea68d9e3d6f24f7e4fb5802500a586d1e9d26f0b7f6f89ab88e2fca050e9","name":"compute_as_extra.json","bytes":4995},{"sha256":"c81dce174a944df9a8b2e5a4d27bcbd69beac6d8d6ba2ff588074351938f7e4e","name":"compute_as_extra.out","bytes":214},{"sha256":"372a1c67cbc43042b052cbfde58c3d9bb81d7154c9472f4fd2cf5c9c551b4055","name":"compute_as_extra.err","bytes":759},{"sha256":"921cc34f90a034f7a2127d5552ad00897c1edbdb5e927cd8b047b8218fdb1242","name":"check_as.py","bytes":14136},{"sha256":"ef47d68d16608c1ca820acc5ce156f4a98cd9b029eaa7b95ee9b0397bea7d9c6","name":"check_as.out","bytes":3571},{"sha256":"c7bc1f664d3911a65d9790babfef7ae8507685e5f048b3f5594d5edb98575d8a","name":"check_as.control.out","bytes":3572}],"decided_by_author_handle":false,"reviews":[],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-13T09:12:05.263Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]}],"decision":{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-13T09:12:05.263Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},"report_sha256":"a74e0539863cde619a2be8083294f95b799c43081baf0593307e5b8dcb929474","next_step_sha256":"80dc5b32dd27e1e61661fa23b8efc2423b8981809a0db25f56939b5891ffd699","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}