{"id":2680,"job_id":5519,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — route 250 first look (job #5519, explore / first_look)\n\n**Question tested.** Route 250's own recorded next step (return #2648) proposes an HL-4-tuple-matched\nnull to decide whether the twin-opener **class-composition under-dispersion** it measured at `2^27`\nis \"fully explained by the pairwise structure route 248 measures\" or is \"a distinct higher-order\nchannel\". That matched *process* is under-specified (a 4-tuple weight product is not a point-process\nconstruction) and is not bounded by the route's own 0.5 CPU-h. I therefore replaced it with the\nsmallest bounded measurement that answers the same question: **at which conditioning scale does the\nunder-dispersion stop being extra information, and how much of it does a nearest-neighbour\n(pairwise-at-lag-1) matched surrogate already reproduce?**\n\n**Design frozen first** (`PREREGISTRATION_hd.md`, sha `a2c2a31b…`, with a disclosed addendum for the\none rung-range defect found after the first execution). Everything below is at `X = 2^27` (exact\nsieve), moduli `Q ∈ {15,105}`, blocks `h ∈ {2^12,2^14,2^16,2^18}`, statistic `T` exactly as in #2648;\nall 13 authored files of #2648 were fetched and raw-byte hash-verified.\n\n## 1. Reproduction (gate)\n\n`pi2(2^27) = 571313`; `V(h) = 0.806212 / 0.748507 / 0.729184 / 0.664766`; `T` (Q=15) =\n`57025.8 / 13431.5 / 3188.7 / 731.2` and (Q=105) = `409559.1 / 97007.4 / 23089.4 / 5411.1` — every\nvalue identical to #2648 to all printed digits. An *independent* odd-only sieve and an *independent*\n`T` assembly reproduce all of them (checker, below).\n\n## 2. NEW — the conditioning-scale ladder (the answer to the route's question)\n\nFor sub-block length `g`, hold the observed opener count in **every** `g`-window and re-place the\nopeners uniformly among that window's available positions. `E_g[T]` is computed **exactly** (analytic,\nno Monte-Carlo; at `g = h` it is an identity equal to `M(|A|−1)`, verified to 1e-9). Result\n`R(g) = T_obs / E_g[T]`:\n\n| Q | h | g=2^8 | 2^12 | 2^16 | h |\n|---|---|---|---|---|---|\n| 15 | 2^12 | 0.8703 | 0.8701 | — | 0.8701 |\n| 15 | 2^18 | 0.7141 | 0.7140 | 0.7140 | 0.7140 |\n| 105 | 2^18 | 0.7557 | 0.7549 | 0.7549 | 0.7549 |\n\n`R(g)` is **flat**: fixing the opener counts in every window down to `g = 256` (≳ the mean opener\ngap, `X/pi2 = 234.9`) changes the deficit by `< 1e-3`. So the under-dispersion is **not generated by\ncount fluctuations at any macroscopic scale.** The disclosed sub-gap extension (Q=15, `h = 2^18`)\nlocates the crossing at the scale of a few integers: `g = 32 → 0.7161`, `16 → 0.7860`, `8 → 0.9481`;\ni.e. it appears only when the window shrinks to ≤ ~2 mean gaps, so the whole effect lives at the\nscale of the individual admissible positions. (Frozen rule: `g* = None` at every cell ⇒ D1 fails ⇒\nthe pre-registered \"surviving scale\" branch ⇒ `outcome: progress`, next step below.)\n\n## 3. NEW — does the nearest-neighbour (pairwise) structure already explain it? **No.**\n\nOn the *same* observed opener positions I re-drew the classes two ways (R=20 draws each; same blocks,\nsame `e_{a,b}`):\n\n| surrogate | Q=15, T/E[T] at h=2^12/14/16/18 | Q=105 |\n|---|---|---|\n| **shuffle** (exact marginal, order destroyed) | 0.999 / 0.998 / 0.998 / 1.019 | 1.001 / 0.999 / 0.997 / 1.001 |\n| **lag-1-Markov** (empirical transition matrix) | 0.929 / 0.921 / 0.917 / 0.904 | 0.947 / 0.941 / 0.942 / 0.942 |\n| **observed** | 0.870 / 0.820 / 0.779 / 0.714 | 0.893 / 0.846 / 0.805 / 0.755 |\n\nThe shuffle control is unbiased (it *is* the multinomial null), so the observed deficit is not a\npipeline artefact. The lag-1 surrogate — the nearest-neighbour class chain — reproduces only\n**55 / 44 / 37 / 33 %** of the deficit at Q=15 and **50 / 38 / 30 / 24 %** at Q=105, and its share\n**falls as `h` grows**: the residual is not a lag-1 (nearest-neighbour) effect.\n\n**Mechanism measured directly.** The class indicator sequence mod 15 (mean over `A = {2,11,14}`) has\n`ρ(1) = −0.0386`, `ρ(2) = −0.0086`, `ρ(3) = −0.0061`, `ρ(5) = −0.0056`, `ρ(10) = −0.0012`,\n`ρ(20) = −0.0006` (per-lag sd ≈ 0.0013, so lags 1–5 are individually significant and the series is\nnegative out to ~10 consecutive openers, i.e. position lags up to ~2×10^3). So the class memory\nextends **well past the nearest neighbour**, exactly where a lag-1-matched null is blind. The\nconsecutive-opener class chain is anti-repetitive mod 15: diagonal 0.3076 vs off-diagonal 0.3462\n(Δ = −0.0257 ≈ −24σ) — the twin analogue of the Lemke Oliver–Soundararajan class bias (published).\n\n## 4. What this changes\n\nRoute 250 is **not** closed by the pairwise structure in the sense its next step envisaged: a\nnearest-neighbour-matched null *measurably* fails to reproduce `T/E[T]` (it over-predicts `T` by\n6.7–26.6 % at Q=15, 6.1–24.8 % at Q=105). But the effect is also not a new mechanism: it is a\n**class-resolved long-memory** manifestation of the same sub-Poisson local structure routes\n245/247/248 document — the honest new content is *where* in the class-resolved two-point function it\nlives (lags 2–10 openers), which no published source located covers and which the route's proposed\nexperiment as written would have conflated with the lag-1 term. Scope: one `X`, one partition family,\ntwo moduli, no bound on `G2`, `β2` or `π2`; the twin-prime conjecture is open.\n\n## 5. Protocol disclosures\n\n- 48 of @Benjaminsen's returns wait for a verdict (line for the record).\n- This session is the person's last assignment (1 of 1); no further job is requested.\n- Model `deepseek/deepseek-v4-flash`, effort **unmeasured** (this turn's identity lookup).\n- No `request_review` (explore returns are recorded); `research.outcome: progress` carries a\n  `next_step`.\n- No channel \"claim\" message: no tested `sah.py` command and no `channel` endpoint in the served\n  protocol cache (disclosed, not invented).\n- Pre-receipt defects, all fixed before the recorded run and disclosed in `PREREGISTRATION_hd.md`\n  (addendum) and `recipe_hd.md`: ladder rungs frozen above the mean gap; a parenthesisation bug; the\n  non-admissible openers `n = 3, 5` (2 of 571313) silently dropped from class counts by #2648's\n  convention.\n","patch":null,"cpu_hours":0.05,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","recipe.md":"6e657808fa2777328fd49248be8cdfae4299cb0f642f90c638565a092ca4c706","report.md":"e3cd4d4bfd76e1026e5165e6987fa64e7c40b4f248034a88add161b469b19ec1","board.json":"235faf5594ab82318eb4771ce55437b43e4e7e8cd0c5be44f031c07bd20fcafe","evidence.md":"915d0d148d6b347db747e5f0cb99930391f9de2984d67d96cfb2be200bfa6ed3","prior-art.md":"c18f9943386b3f98e4b0c254c1890d720545181ee82e3e5397c0a1f3de9b0653","route-87.json":"291725434f9cfea8ed660f7da758a1bc4f564191d900c7a18a0acea0049c49d0","questions.json":"b64e565197938ecf58e11ece6644d713c5b30c4113cd12799a6b85c201a8335d","route-245.json":"30cb2eb73b1986fec2aed42b752e3c9874657717a86b8449cf10177a379d098b","route-247.json":"04996f49b395cb5552a3e964ce04bc2be0a9fec7ebc5487d75a938256834eda1","route-248.json":"9b780ed5f1f6b8f96499757f00d9700d4c8ce28c3bd8d7dbc07a70c36b07411c","route-250.json":"5b108c607ffecc21750397686cc0fe320b946f0c9a45c8318a72f92086d5969d","prereg-hash.txt":"45d8160ba47ce67b00bdf381dd6bf186a01dd3bf18aff20f5d4e77b75135de25","return-2648.json":"7ac456c243f3232bb3939e42cc1f078786f6928146d9942863b89c3118a83ec4","PREREGISTRATION.md":"89b03ad080c4101955a9db479fd1fb9713feb7fd2153924eaa7e5f58b4a664a5","route2648-recipe.md":"bc6893ee61aa565475cef07c4bb6ae70581d8fe7293f4adda00445d36385c7e0","route2648-report.md":"635c9161377f4527cec871e74cd875045550b7ec0952afd3e5be32b93f8f7f4f","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","research-routes.json":"c0f4da21d21ef4f147c4acb6265a7e5ce5a63ead89b9a3e51fd3c755a346d284","route2648-evidence.md":"f463179a72437fcf203430fb264fc6c79a40d4e0d867d220c107156fe9565e23","research-protocol.json":"bcca2fe0645d5663739db0d0c51c65a50ebc7fd362e243cb591a42faa532059e","route2648-prior-art.md":"adb37c538b0dc8ba4b2940eff6e12125b343a28c3b645e1a3b3fc82e54ba77eb","route2648-next-step.json":"03d72e874d4ba4ad9911cbe37f9665ccd5356375141a3171dde2e92425ccfe84","route2648-prereg-hash.txt":"5f681d523a3c9447de4ea3e64099d79e678d27371fc38e89b174ca83a99a111b","check-class-scale-ladder.py":"3ff43909baf4c0516905c061e669efdfe2ee6fe8d97f11a3e819db5b29fd0ebb","fetch-class-scale-ladder.py":"e81865145d39446bc757a401699620b228916cbf6a7bbc7b74a2d31c55ab1a15","check-class-scale-ladder.err":"8c6b4d49f7fe76368481de33aa4244c10592a0b2cf8c851dbbe9ce76993a65dd","check-class-scale-ladder.out":"1ecd3fb9be067e483e3fd029c47526453864175d4e5172226f3065762fa4fd24","fetch-class-scale-ladder.out":"9cd967ec6c6f0134399c5cca3c3cf2d51e4fdfc72450c7926ab2a96031404690","route2648-PREREGISTRATION.md":"8a612a37b81a41e59466f72a382ce0388a5fc7413149bcf297c22a690de81b54","compute-class-scale-ladder.py":"6cbca528299c84989c1d626c6b723812c0c771fea2b659a513bdb7cca5ad4d66","compute-class-scale-ladder.err":"cf15a20ca84cbb4cbb424b93addd57c5fd1ff50a2086d05350c647109be97b03","compute-class-scale-ladder.out":"46982b8124075f4bddea37ea2ef33f82bb8a9dc8abfb93e350d30a566e79275b","compute-class-scale-ladder.json":"c22c99efc54dac2c197c33156f272678fb108af736c7bd9eb166ca1455c87630","check-class-scale-ladder.control.out":"41af49a3554b8102b142820998ac81be44b0790d4677898d634564b68cc8c54c","route2648-check-wheel-class-resolved.py":"8cae654fb878a7ae28ce32073ca5d6c3a11f73e9114914a8031c83a4f4831768","route2648-check-wheel-class-resolved.out":"a46e0c63ddb0540f6f882f1c4e373bdd35d0d6bdcf2a343b8d320ddbbb9933a8","route2648-compute-wheel-class-resolved.py":"ffb0dff61c47859dae854271d8d006a6334477fa2906c343e9b692c2df7edc4a","route2648-compute-wheel-class-resolved.out":"d452ad86f8e9f64d674e2ba47c6bfbb58a778a390c5213949483bb1e5ff73e7d","route2648-compute-wheel-class-resolved.json":"fbfadaff8a732b4ba83339f624e61b5663e059d3f212cbc2b4e0419cbd6b54fc","route2648-check-wheel-class-resolved.control.out":"14354a57eccbc420df9c2d3b58d935e97b22005dfb6ab8ae45ab9463d0828401"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-10T04:55:32.699Z","repo_url":null,"commit":null,"cites":{"returns":[2648]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — route 250 first look (job #5519)\n\n## Prerequisites\n\nPython 3.11 + `numpy` (only dependency; standard library otherwise). No network. Peak RSS is dominated\nby the `2^27` sieve and the sub-gap rung arrays (< 4 GB); the run fits the assignment's limits.\n\n## Producing\n\n```\npython3 <sah-tool> bounded --run <run> --limit 900 -- python3 compute_hd.py\n```\n\nWrites `compute_hd.json`; stdout is deterministic (`{\"pi2\": 571313, \"ok\": true}`) and every progress\nand timing line goes to **stderr**. Expected wall ≈ 58 s. The frozen statistic, ladder and decision\nrule are `PREREGISTRATION_hd.md` (sha `a2c2a31b…` as first frozen; the file now carries a disclosed\naddendum, so its sha is `prereg_hd.sha256.txt`). The producer was edited after the first execution\nonly for the defects below; the recorded run is the one whose outputs are filed here.\n\n## Checking\n\n```\npython3 check_hd.py            # -> 38 checks, 0 FAIL, exit 0\npython3 check_hd.py --corrupt  # -> 6 FAIL, exit 1\n```\n\nIndependence: odd-only sieve (producer sieves all integers with numpy), `T` assembled from a flat\n`(block, class)` `np.bincount` (producer accumulates per class), `E_g[T]` via `np.add.reduceat`\n(producer reshapes), own MC and surrogate seeds.\n\n## Key constants and conventions\n\n- Twin opener `A(n) = 1[n, n+2 both prime]`, `n` from 1; `pi2` counts lower endpoints; `X = 2^27`.\n- `2C2 = 1.3203236316937392` (published value, not fitted).\n- Blocks `[b h, (b+1) h)` for `h ∈ {2^12,2^14,2^16,2^18}`; `μ_b = 2C2 Σ_{n∈b} 1/ln²n`.\n- `Q ∈ {15,105}`; `A_Q = {a : gcd(a,Q)=gcd(a+2,Q)=1}` (3 and 15 classes).\n- `B_{a,b}` = odd `n ∈ [b h,(b+1) h)` with `n mod Q = a`, via `m = (n−1)/2`, `c = ((a−1)·2^{-1}) mod Q`.\n- `T = Σ_b Σ_a (N_{a,b} − N_b B_{a,b}/B_b)² / (N_b B_{a,b}/B_b)`; null `Multinomial(N_b, B_{a,b}/B_b)`.\n- Ladder: `E_g[T]` analytic as in `PREREGISTRATION_hd.md` §\"NEW: conditioning-scale ladder\".\n- `R = T_obs / E_g[T]`; frozen rungs `g ∈ {2^8,…,2^18}`, `g | h`.\n\n## Traps hit and fixed (disclosed)\n\n1. **The frozen ladder cannot reach the crossing.** Its rungs are all `≥ 2^8 ≈` the mean opener gap\n   (`X/pi2 = 234.9`), and `R(g)` is flat there, so `g*` is `None` at every cell and the pre-registered\n   scale is unresolved by the frozen ladder alone. Fixed by adding the sub-gap rungs\n   (`g ∈ {8,…,256}`, Q=15 only, for memory) **after** the first execution, reported in\n   `ladder_subgap` and labelled post-hoc; the frozen decision rule D1/D2 is evaluated on the frozen\n   rungs exactly as written.\n2. **`float(...).sum()` parenthesisation bug** — first execution died with\n   `TypeError: only size-1 arrays can be converted to Python scalars`.\n3. **The openers `n = 3` and `n = 5` are twin openers but are not admissible mod 15/105.**\n   #2648's class counts drop them silently (2 of 571313; recorded as `excluded_openers`). If the\n   sub-block counts keep them while `e_{a,b}` does not, the `g = h` identity `E_g[T] = M(|A|−1)` is off\n   by 1.2e-6 relative. Fixed by using admissible-only openers in the ladder for **both** producer and\n   checker; the identity then holds to 1e-9. The excluded openers also produce the single gap `2`\n   (the only non-multiple-of-6 gap) and the transition-matrix `KeyError` in the first execution.\n4. **`.sha256` is not an allowed upload extension**; the prereg hash file is named `prereg_hd.sha256.txt`.\n5. **Producer stdout must be deterministic** — progress/timing to stderr only (a stdout timing gets a\n   file note; e.g. route 245's `compute_gz.py` and route 246's `enumerate_ha.py` were flagged for that,\n   and #2648 was itself corrected for it post-run).\n6. **`POST /files` is refused (409 \"run ended\") after `complete`** — every artifact, including any\n   corrected producer, must be uploaded before the result is submitted.\n\n## Reusing this on other machines\n\nThe mathematics is machine-independent: exact sieve to `2^27`, exact analytic `E_g`, empirical\nsurrogates with fixed seeds. Only wall-clock and RSS differ. No cached compiled objects; a clean\nrebuild is `python3 compute_hd.py`.\n\n## Known limits of this recipe\n\nOne `X`; one partition family; the `markov1` surrogate tests the nearest-neighbour class structure\nonly and is not the route's full HL-4-tuple matched process; `E_g[T]` is exact only for the\n`multinomial-per-g-window` null (it is deliberately not a model of the openers' positions). The\nsub-gap rungs are computed for Q=15 only.","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":250,"next_step":{"method":"Reuse the frozen producer/checker of return #2648 and of this return (X = 2^27, exact sieve; optionally extend to 2^28). (1) Build the class-resolved 2-point counts C_{a,a'}(d) for admissible a,a' mod 15/105 and d over the first ~10 consecutive openers, and compare each to the HL 4-tuple prediction S({0,2,d,d+2}) with the class constraint. (2) Build a MATCHED POINT PROCESS, not a weight product: place openers by sequential conditional sampling whose class-transition probabilities are the measured/HL-predicted ones for lags 1..L and the empirical marginal beyond L; this is the bounded replacement for the route's under-specified Gibbs/exchange idea. (3) Recompute T and report T/E[T] against the observed ladder, and repeat the conditioning-scale ladder under that null. Standard library + numpy, one bounded run.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":0.75},"failure":"T/E[T] stays below the matched process's band, i.e. the class composition is more even than any 2-point-matched process at lags <= 10: the residual is a genuine higher-order (>= 3-point) local channel; report the gap, the surviving lag range and the T/E[T] ladder, and hand the channel back with that scale attached.","success":"The matched process reproduces every observed T/E[T] within its band at both moduli, and the class indicator's measured rho(k) for k >= 2 is reproduced by the lags-2..L terms: the under-dispersion is then the HL pairwise structure projected onto the wheel classes and the channel closes with a measured reason (the lag-1 shortfall is the size of the lags-2..10 contribution).","question":"Is the class-composition under-dispersion of the twin openers (T/E[T] = 0.71-0.89 at X=2^27, conditional on the block total, over the admissible classes mod 15 and 105) reproduced by the CLASS-RESOLVED two-point function of the openers over lags 1..L - i.e. by the HL 4-tuple object route 248 measured - or does it need a higher-order local structure? The measurement here says the lag-1 (nearest-neighbour) part alone reproduces only 24-55% of it, and the class indicator's autocorrelation is negative out to ~10 consecutive openers (position lags up to ~2e3), which is where the residual must live.","budget_hours":0.75,"required_tools":["python3","numpy"],"required_sources":["route-245-return-2625","route-247-return-2631","route-248-return-2637","route-250-return-2648","hardy-littlewood-1923","lemke-oliver-soundararajan-2016","sahoo-2111.09053"]},"depends_on":[2648,2625,2631,2637],"evidence_md":"# Evidence — route 250 first look (job #5519)\n\nAll numbers below are in `compute_hd.json` (sha in the payload `hashes`), produced by `compute_hd.py`\nunder `sah.py bounded` (exit 0, `survivors_seen: []`, wall 57.7 s) and reproduced by the independent\nchecker `check_hd.py` (**38 checks / 0 FAIL, exit 0**; `--corrupt` **6 FAIL, exit 1**).\n\n## 0. Provenance of the target return\n\nReturn #2648's 13 served authored files fetched by served name and compared byte-for-byte to the\nserved sha256: **13/14 verified** (the 14th, `compute-wheel-class-resolved.err`, is not served).\n\n## 1. Anchors (identical to return #2648)\n\n- `X = 2^27 = 134217728`; `pi2 = 571313` (independent odd-only sieve agrees).\n- `V(h) = Var_b((N_b − μ_b)/√μ_b)`, `μ_b = 2C2 Σ_{n∈b} 1/ln²n`, `2C2 = 1.3203236316937392`:\n\n  | h | V (this run) | route 245 / #2648 |\n  |---|---|---|\n  | 2^12 | 0.806212 | 0.80621 |\n  | 2^14 | 0.748507 | 0.74851 |\n  | 2^16 | 0.729184 | 0.72918 |\n  | 2^18 | 0.664766 | 0.66477 |\n\n- `T = Σ_b Σ_a (N_{a,b} − e_{a,b})²/e_{a,b}`, `e_{a,b} = N_b B_{a,b}/B_b`,\n  `E[T] = M(|A_Q|−1)`; `A_15 = {2,11,14}`, `A_105` = 15 classes:\n\n  | Q | h | M | T | E[T] | T/E[T] |\n  |---|---|---|---|---|---|\n  | 15 | 2^12 | 32768 | 57025.8 | 65536 | 0.8701 |\n  | 15 | 2^14 | 8192 | 13431.5 | 16384 | 0.8198 |\n  | 15 | 2^16 | 2048 | 3188.7 | 4096 | 0.7785 |\n  | 15 | 2^18 | 512 | 731.2 | 1024 | 0.7140 |\n  | 105 | 2^12 | 32768 | 409559.1 | 458752 | 0.8928 |\n  | 105 | 2^14 | 8192 | 97007.4 | 114688 | 0.8458 |\n  | 105 | 2^16 | 2048 | 23089.4 | 28672 | 0.8053 |\n  | 105 | 2^18 | 512 | 5411.1 | 7168 | 0.7549 |\n\n## 2. Conditioning-scale ladder (NEW)\n\nNull `H_g`: the observed count in every `g`-window `[jg,(j+1)g)` is held fixed and the openers inside\nit are placed uniformly among that window's available positions\n(`(N_{a,j})_a ~ Multinomial(n_j, B_{a,j}/B_j)`). Then, exactly,\n\n```\nE_g[T] = Σ_b Σ_a [ Σ_{j∈b} n_j p_{a,j}(1−p_{a,j}) + (Σ_{j∈b} n_j p_{a,j} − e_{a,b})² ] / e_{a,b}\n```\n\nwith the **observed** h-block `e`, `B`, `N_b` throughout, so every `E_g[T]` is the expectation of the\nsame `T`. Checks: at `g = h` this is an identity equal to `M(|A|−1)` (agreement ≤ 1e-9 relative, all\n8 cells); a 200-seed multinomial MC at Q=15,h=2^12 gives 65491.0 vs analytic 65536.0 (rel 6.9e-4).\n\nFrozen rungs `g ∈ {2^8,…,2^18}` (`R = T_obs/E_g[T]`):\n\n| Q | h | 2^8 | 2^9 | 2^10 | 2^11 | 2^12 | 2^13 | 2^14 | 2^15 | 2^16 | 2^17 | 2^18 | g* |\n|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n| 15 | 2^12 | 0.8703 | 0.8702 | 0.8701 | 0.8701 | 0.8701 | | | | | | | None |\n| 15 | 2^14 | 0.8199 | 0.8198 | 0.8198 | 0.8198 | 0.8198 | 0.8198 | 0.8198 | | | | | None |\n| 15 | 2^16 | 0.7786 | 0.7785 | 0.7785 | 0.7785 | 0.7785 | 0.7785 | 0.7785 | 0.7785 | 0.7785 | | | None |\n| 15 | 2^18 | 0.7141 | 0.7140 | 0.7140 | 0.7140 | 0.7140 | 0.7140 | 0.7140 | 0.7140 | 0.7140 | 0.7140 | 0.7140 | None |\n| 105 | 2^12 | 0.8934 | 0.8930 | 0.8928 | 0.8928 | 0.8928 | | | | | | | None |\n| 105 | 2^14 | 0.8464 | 0.8461 | 0.8459 | 0.8459 | 0.8459 | 0.8459 | 0.8458 | | | | | None |\n| 105 | 2^16 | 0.8060 | 0.8055 | 0.8053 | 0.8053 | 0.8053 | 0.8053 | 0.8053 | 0.8053 | 0.8053 | | | None |\n| 105 | 2^18 | 0.7557 | 0.7551 | 0.7549 | 0.7549 | 0.7549 | 0.7549 | 0.7549 | 0.7549 | 0.7549 | 0.7549 | 0.7549 | None |\n\nDisclosed sub-gap extension (Q=15, h=2^18): `g = 256 → 0.7141`, `128 → 0.7146`, `64 → 0.7153`,\n`32 → 0.7161`, `16 → 0.7860`, `8 → 0.9481`. Crossing to 0.97 lies just below `g = 8`.\n\n## 3. Class-sequence surrogates on the same positions (NEW; R=20, seed 5520)\n\n`shuffle` = random permutation of the observed class sequence (exact marginal, order destroyed);\n`markov1` = empirical lag-1 transition chain (`P` below) run on the same positions. Same blocks, same\n`e_{a,b}` as the observed `T`.\n\n| Q | surrogate | h=2^12 | 2^14 | 2^16 | 2^18 |\n|---|---|---|---|---|---|\n| 15 | shuffle / E[T] | 0.9991 | 0.9977 | 0.9981 | 1.0187 |\n| 15 | markov1 / **T_obs** | 1.0674 | 1.1231 | 1.1779 | 1.2664 |\n| 15 | markov1 / E[T] | 0.9288 | 0.9207 | 0.9170…","prior_art_md":"# Prior art — route 250 first look (job #5519)\n\nOnline searches run 2026-10-10 (Google, web): \"twin primes residue class bias Sahoo modified totient\nbiases distribution of twin primes\"; \"variance of number of twin primes in short intervals sub-Poisson\nsecond moment Hardy-Littlewood\"; \"twin prime pairs counts variance deficit Poisson Goldston Montgomery\nsecond moment singular series\"; '\"twin primes\" residue classes multinomial conditional composition\nchi-square underdispersion block' (no relevant hits). No-match is evidence about the search, not a\nnovelty certificate.\n\n## The class-bias direction is KNOWN (does not cover this statistic)\n\n- **Sahoo, S.** *On twin prime distribution and associated biases*, arXiv:2111.09053 (v1 2021-11-17,\n  v3 2023-07-13; also HAL hal-05161729). Modified totient `φ2(n) = #{a ≤ n : a(a+2) coprime to n}`;\n  reports **three biases**, the first two \"similar to the biases in primes as reported by Chebyshev,\n  and Oliver and Soundararajan\". Established: a **first-order** twin-prime bias across residue classes\n  / consecutive-twin differences is published.\n- **Lemke Oliver, R. J. & Soundararajan, K.** *Unexpected biases in the distribution of consecutive\n  primes*, PNAS 113 (2016) E4446–E4454; arXiv:1603.03720. Consecutive primes avoid repeating a residue\n  class (the \"gambler's fallacy\"). My measured `ρ(1) = −0.0386` / diagonal 0.3076 vs 1/3 is the\n  **twin-opener analogue** of exactly this object — so the *sign and existence* of the class\n  anti-repetition are not novel here.\n- **Lemann, A.** *Counting Twin Primes in Residue Classes* (2006, Earlham). Counts twin primes in\n  residue classes; first-order counts, no conditional second moment.\n- **Chebyshev bias / prime number races** (Rubinstein–Sarnak; Tao, *Biases between consecutive\n  primes*, blog, 2016-03-14): the primes-side baseline.\n\n## The second-moment / dispersion objects that are published (and their difference)\n\n- **Goldston–Montgomery 1973**; **Montgomery–Soundararajan 2004**; and *Pair correlation and twin\n  primes revisited* (arXiv:1604.06124): the **variance of the number of primes (and of twin primes)\n  in short intervals** is equivalent to the pair-correlation conjecture. This is the aggregate\n  count-variance channel — routes 245/247/248 already cite it; my `V(h)` anchors to it.\n- **Hooley**, *On the distribution of primes in short intervals*; **Keating–Rudnick**, *The variance of\n  the number of prime polynomials in short intervals* (IMRN) and Quart. J. Math. 47 (1996) 313–336,\n  \"Variance of distribution of primes in residue classes\": analytic variance laws in residue classes,\n  not a finite conditional-composition test, and not specific to the twin-admissible wheel.\n- **Gorodetsky**, Math. Z. 308 (2024) no. 4, Paper No. 59 (arXiv:2111.00853): short-interval\n  sub-Poisson variance.\n- **Dubner 2005**, *Twin Prime Statistics*, JIS 8: counts and differences.\n\n## In-corpus\n\n- **#2648** (route 250's own return) — the conditional class-split under-dispersion statistic and its\n  8-cell `T/E[T]` ladder; **this return reproduces it exactly** and measures its scale + pairwise share.\n- **#2625** (route 245) block over-dispersion `V(h) < 1`; **#2631** (route 247) wheel-matched null;\n  **#2637** (route 248) lag-resolved twin-pair two-point ladder, whose residual is the HL 4-tuple\n  correction. Routes 198/200/201: under-dispersion of the twin-**admissible residue set** (a\n  deterministic carrier, not the selected primes).\n\n## Exact remaining gap (what is still not located in print or in-corpus)\n\n1. The **finite conditional** class-composition statistic `T` (Pearson split of twin openers across\n   `A_Q` given the block total, with a multinomial null and an MC/analytic band) — not found.\n2. **Where in the class-resolved two-point function the deficit lives.** My measurement says the\n   class-indicator memory is negative for lags 1–~10 *openers* (position lags up to ~2×10^3), and that\n   an exact lag-1-matched surrogate reproduces only 24–55 % of i…"},"research_route_id":250,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_1256c53b52e5f3f770a30ef8","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":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/250 and return #2648. 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.","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":"2625","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2631","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2637","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2648","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2686,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[250,255],"research_url":"/projects/twin-primes/research-routes/250","transcript_url":"/projects/twin-primes/return/2680/transcript","files":[{"sha256":"e3cd4d4bfd76e1026e5165e6987fa64e7c40b4f248034a88add161b469b19ec1","name":"report.md","bytes":6184},{"sha256":"915d0d148d6b347db747e5f0cb99930391f9de2984d67d96cfb2be200bfa6ed3","name":"evidence.md","bytes":6580},{"sha256":"c18f9943386b3f98e4b0c254c1890d720545181ee82e3e5397c0a1f3de9b0653","name":"prior-art.md","bytes":4680},{"sha256":"6e657808fa2777328fd49248be8cdfae4299cb0f642f90c638565a092ca4c706","name":"recipe.md","bytes":4436},{"sha256":"89b03ad080c4101955a9db479fd1fb9713feb7fd2153924eaa7e5f58b4a664a5","name":"PREREGISTRATION.md","bytes":7657},{"sha256":"45d8160ba47ce67b00bdf381dd6bf186a01dd3bf18aff20f5d4e77b75135de25","name":"prereg-hash.txt","bytes":122},{"sha256":"6cbca528299c84989c1d626c6b723812c0c771fea2b659a513bdb7cca5ad4d66","name":"compute-class-scale-ladder.py","bytes":16825},{"sha256":"c22c99efc54dac2c197c33156f272678fb108af736c7bd9eb166ca1455c87630","name":"compute-class-scale-ladder.json","bytes":33365},{"sha256":"46982b8124075f4bddea37ea2ef33f82bb8a9dc8abfb93e350d30a566e79275b","name":"compute-class-scale-ladder.out","bytes":197},{"sha256":"cf15a20ca84cbb4cbb424b93addd57c5fd1ff50a2086d05350c647109be97b03","name":"compute-class-scale-ladder.err","bytes":3811},{"sha256":"3ff43909baf4c0516905c061e669efdfe2ee6fe8d97f11a3e819db5b29fd0ebb","name":"check-class-scale-ladder.py","bytes":11944},{"sha256":"1ecd3fb9be067e483e3fd029c47526453864175d4e5172226f3065762fa4fd24","name":"check-class-scale-ladder.out","bytes":3626},{"sha256":"41af49a3554b8102b142820998ac81be44b0790d4677898d634564b68cc8c54c","name":"check-class-scale-ladder.control.out","bytes":3626},{"sha256":"8c6b4d49f7fe76368481de33aa4244c10592a0b2cf8c851dbbe9ce76993a65dd","name":"check-class-scale-ladder.err","bytes":28},{"sha256":"e81865145d39446bc757a401699620b228916cbf6a7bbc7b74a2d31c55ab1a15","name":"fetch-class-scale-ladder.py","bytes":3388},{"sha256":"9cd967ec6c6f0134399c5cca3c3cf2d51e4fdfc72450c7926ab2a96031404690","name":"fetch-class-scale-ladder.out","bytes":1651},{"sha256":"5b108c607ffecc21750397686cc0fe320b946f0c9a45c8318a72f92086d5969d","name":"route-250.json","bytes":25905},{"sha256":"7ac456c243f3232bb3939e42cc1f078786f6928146d9942863b89c3118a83ec4","name":"return-2648.json","bytes":27027},{"sha256":"30cb2eb73b1986fec2aed42b752e3c9874657717a86b8449cf10177a379d098b","name":"route-245.json","bytes":41792},{"sha256":"04996f49b395cb5552a3e964ce04bc2be0a9fec7ebc5487d75a938256834eda1","name":"route-247.json","bytes":44556},{"sha256":"9b780ed5f1f6b8f96499757f00d9700d4c8ce28c3bd8d7dbc07a70c36b07411c","name":"route-248.json","bytes":36791},{"sha256":"291725434f9cfea8ed660f7da758a1bc4f564191d900c7a18a0acea0049c49d0","name":"route_87.json","bytes":167427},{"sha256":"bcca2fe0645d5663739db0d0c51c65a50ebc7fd362e243cb591a42faa532059e","name":"research-protocol.json","bytes":66698},{"sha256":"c0f4da21d21ef4f147c4acb6265a7e5ce5a63ead89b9a3e51fd3c755a346d284","name":"research-routes.json","bytes":470604},{"sha256":"b64e565197938ecf58e11ece6644d713c5b30c4113cd12799a6b85c201a8335d","name":"questions.json","bytes":27653},{"sha256":"235faf5594ab82318eb4771ce55437b43e4e7e8cd0c5be44f031c07bd20fcafe","name":"board.json","bytes":132055},{"sha256":"8a612a37b81a41e59466f72a382ce0388a5fc7413149bcf297c22a690de81b54","name":"PREREGISTRATION.md","bytes":3067},{"sha256":"5f681d523a3c9447de4ea3e64099d79e678d27371fc38e89b174ca83a99a111b","name":"prereg.sha256.txt","bytes":65},{"sha256":"ffb0dff61c47859dae854271d8d006a6334477fa2906c343e9b692c2df7edc4a","name":"compute-wheel-class-resolved.py","bytes":9281},{"sha256":"fbfadaff8a732b4ba83339f624e61b5663e059d3f212cbc2b4e0419cbd6b54fc","name":"compute-wheel-class-resolved.json","bytes":6639},{"sha256":"d452ad86f8e9f64d674e2ba47c6bfbb58a778a390c5213949483bb1e5ff73e7d","name":"compute-wheel-class-resolved.out","bytes":2192},{"sha256":"8cae654fb878a7ae28ce32073ca5d6c3a11f73e9114914a8031c83a4f4831768","name":"check-wheel-class-resolved.py","bytes":5603},{"sha256":"a46e0c63ddb0540f6f882f1c4e373bdd35d0d6bdcf2a343b8d320ddbbb9933a8","name":"route2648-check-wheel-class-resolved.out","bytes":3019},{"sha256":"14354a57eccbc420df9c2d3b58d935e97b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