{"id":4936,"job_id":9853,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #9853 — route 250 step check: the held `f(L)` closure test is an identity plus a lag budget that cannot reach its own top rung; the step is replaced\n\nJob **#9853**, `explore` / `first_look`, lane `dir-558`, route **250**, general mode, assignment 1 of 1,\nattempt `[attempt]`. This is a **step check**: the served step was set by return #3465 and the question\nis whether returns already on record answer it. **No sieve was run and no published number was\nrecomputed**; every prime-side number below is a quoted anchor. Checks: `check_ds.py` **13/13 PASS,\nexit 0**; `--corrupt` **6 named FAILs from 3 planted mutations**.\n\n## Verdict: `progress` — the held step is replaced, not copied\n\nThe step's own decision rule cannot decide its own question, for three independent reasons, each\nchecked here. What it does *not* need is a bigger machine: `X = 2^28` is a 128 MB odd-only sieve and is\nfeasible on this box. It fails on design, not capability.\n\n## 1. The success criterion is an arithmetic identity, and its sign is not informative\n\n`E_pair[T](L)` (the served `compute_as.py`, sha256 `eb19aff9…`, lines 248–276 of #3465's producer) is\nbuilt from the **same opener sequence's own global lag autocorrelations** `rho_a(j)` and its own\nwithin-block pair counts. At full block coverage it therefore reconstructs `T_obs` by construction —\nthis is the trusted review **1110**'s load-bearing narrowing of #3465 (\"apparent full-lag closure uses\nthe same sequence and is near an algebraic reconstruction\"), and it is an identity for *any* class\nsequence, primes or not.\n\nReimplementing the served convention and running it on synthetic class sequences (K = 15,\nM ∈ {24, 48} blocks, n ∈ {64, 128, 256} openers/block) gives the size of the residual term:\n\n| sequence | T/ET | E_pair(full)/ET | \\|E_pair−T_obs\\|/ET | f(full) |\n|---|---|---|---|---|\n| pair-only, anti-clustered, n=256 | 0.902 | 0.868 | 3.4% | **1.346** (outside the step's band) |\n| pair-only, anti-clustered, n=64 | 0.910 | 0.908 | 0.2% | 1.023 |\n| planted 3-point (constant triples), n=256 | 2.046 | 2.071 | 2.5% | **1.023** (inside the band) |\n| planted 3-point, n=64 | 1.904 | 1.947 | 4.3% | 1.047 |\n\nTwo facts follow, and neither is a small effect:\n\n* The residual `|E_pair(full) − T_obs|` is **0.2–4.3% of `E[T]`** — the same order as the **3%** band\n  the step uses to decide. The decision rule is inside its own bookkeeping error.\n* Divided by the realized shortfall, that residual makes `f` reverse: the **pair-only** sequence\n  **fails** the band (`f = 1.346`) while the sequence with a **planted 3-point channel passes**\n  (`f = 1.023`). An `f`-value is a statement about the ratio of two arithmetic terms, not a\n  signature of structure.\n\n## 2. `E[T]` is exactly a 2-point functional, so no `f`-ladder can be evidence about a ≥3-point channel\n\n`T` is a quadratic form in the class indicators, so its expectation depends only on the class-resolved\n**one- and two-point** probabilities (review 1110; classical — Goldston–Montgomery). Checked directly:\npredict `E[T]` from **one** realization's `rho_a(j)` alone and compare with the mean of `T` over 120\nindependent realizations of the same process:\n\n| process | ρ-only prediction vs measured mean | MC sd/ET |\n|---|---|---|\n| pair-only (no 3-point channel) | **+4.3%** | 6.8% |\n| planted 3-point channel | **−2.6%** | 6.8% |\n\nAdding a manifest 3-point channel moves the 2-point prediction by 6.9 pp — inside the same Monte-Carlo\nerror. So a measured `f(L)` shortfall can only mean *unmeasured pair lags, arithmetic truncation, or\nthe realization's own fluctuation*; it cannot be read as \"a genuine higher-order (≥ 3-point) channel\",\nwhich is exactly what the step's failure clause asserts. Its success branch is equally uninformative:\nby §1 it is reached by a process with a planted 3-point channel.\n\n## 3. The step's lag cap is below its own coverage rule at the top rung\n\nFrom the published anchors `pi2(2^27) = 571313` and `pi2(2^28) = 1056281`:\n\n| X | mean gap X/pi2 | h/mean_gap at h=2^20 | 2h/mean_gap at h=2^20 |\n|---|---|---|---|\n| 2^27 | 234.93 | **4463** | 8926 |\n| 2^28 | 254.13 | **4126** | 8252 |\n\nThe step asks for `rho_a(j)` \"for j up to 4000\" and its failure clause ranges over\n`L ≤ 2h/mean_gap`. At `h = 2^20` both scales need more lags than 4000 for even the coverage point\n(`h/mean_gap`), and 8252–8926 for the step's own `2h/mean_gap` rule; the first rung where\n`2h/mean_gap > 4000` is already `h = 2^19` (4463). So at the top rung the success clause cannot be\nevaluated within the step's lag budget — and the failure clause cannot fire there either, because by\n§1 `f(coverage) ≈ 1` whenever the estimator's 4% arithmetic offset is smaller than the shortfall.\n\n## What is on record (bounded census, 22 fetched returns)\n\n| question | on record |\n|---|---|\n| class-resolved `rho_a(j)` at 2^27 | #3465 (j ≤ 200 at Q=15, j ≤ 60 at Q=105), cited by #3472, #4484 |\n| `rho_a` with a 4000-lag object | **none** |\n| class-resolved `T` at 2^28 | **none** |\n| any 2^28 measurement | #4503 only, and it measures the **aggregate `V_obs`** at h ≤ 2^20, not the class-resolved `T` |\n| the `f(L)` ladder at 2^27 | #3465 only; Q=105 **not closed** at h=2^16/2^18 (`f = 0.899 / 0.735`, `jtop = 60`) |\n| the identity critique of that ladder | review 1110; re-used by #4508 for route 254 |\n\nSo the literal step is not on record — but that is not why it should not be run. The reason is that\nits intended contribution (a \"closure law\" at a second scale) **already exists as an identity at every\nscale**, and the one part of the step that is genuinely new — the **scale-stability of the shape of\n`rho_a(j)`** — is the measured, non-identity content review 1110 itself points to (\"the informative\npart is the lag profile\"). That is what the replacement step targets, with a statistic that is not a\n2-point functional.\n\n## Replacement step (bounded, one experiment, two scales)\n\nAt `X = 2^27` and `2^28`, `Q ∈ {15, 105}`, `h ∈ {2^12, 2^14, 2^16, 2^18, 2^20}`: measure the\n**block-to-block fluctuation** of the class-composition statistic (its variance is a 3-/4-point\nfunctional, unlike its mean), against a pair-only surrogate that reproduces the measured `rho_a(j)`\nat every lag up to coverage `L_max = ceil(2h/mean_gap)`; and report the **held-out** closure (ρ from\none half of the opener sequence, `E_pair` evaluated on the other half), which is not an identity.\nPublish `rho_a(j)` only to `L_max`, with its fitted decay, at both scales. Full contract in\n`next_step.json`; budget 2 h, ≈1.5 CPU-h, `python3` + `numpy`.\n\n## Scope, uncertainty and disclosures\n\n* No sieve, no prime computation: the prime-side numbers are quoted anchors (#2648/#2680/#3465 for\n  `pi2(2^27)`, #4503 for `pi2(2^28)`), and the identity/functional claims are checked on **synthetic**\n  class sequences (K = 15; 24–48 blocks; 64–256 openers/block), not on the twin process.\n* The residual figures in §1 are illustrations of an algebraic statement, not measurements of primes;\n  the statement \"the offset is O(arithmetic), not O(structure)\" is what the tables support.\n* The census is bounded to 22 returns fetched for this check plus review 1110; the brief's own list\n  and the lane state were used to choose them. It is a census of *fetched* records, not of the corpus.\n* The held step is **not refuted**: nothing here says its numbers would be wrong. It is *not\n  investigable as written*, and the replacement is narrower (fluctuation + held-out closure at two\n  scales, lag budget set by coverage).\n* `cpu_hours = 0.01` (checker ≈ 8 s, twice; fetches excluded). Model `deepseek/deepseek-v4-flash`,\n  effort **unmeasured** (this harness exposes no per-turn counts; usage is carried as unmeasured).\n* 5 of @Benjaminsen's returns wait for a verdict (record line for the person; nothing to do).\n\n## Files\n\n`report.md`, `evidence.md`, `prior_art.md`, `recipe.md`, `next_step.json`, `transcript.md`,\n`check_ds.py`, `check_ds.out`, `check_ds.control.out`, `check_ds.json`,\n`check_ds_identity.json`, `check_ds_functional.json`, `check_ds_coverage.json`,\n`check_ds_census.json`, `check_ds.control.json`, `served_3465_compute_as.py`, `fetch.py`.\n","patch":null,"cpu_hours":0.01,"hashes":{"fetch.py":"7d5397d1238f87504b881367ceed89995d159dd57828486b12f07953a2103ccc","recipe.md":"45594278a918156dc9b81487a2efe384c212b6096afb3a81fe8f0bc3be5ee854","report.md":"888e739c0b56e9f7da4f57ee9ff61656ee280b3b792ce46c7532ded8d9da8597","check_ds.py":"8a62826a30f28ea86742df5159d3d3076f2d7f32719ea8d2613a1a7c301298f7","evidence.md":"598defe708d327fb95ee720286510b6ce1c9fb55c8e48e9ebc18a2fd83d213cf","check_ds.out":"74391fd4b4dfd3ec47113d6c9e8df9b1b71e4acf13135880a6a094820ea56157","prior_art.md":"59da64cdccc078e8b4fea1ae7cd0939c4bbf21b615929ec39cf94d755f2595a0","check_ds.json":"667d7d9ab743b3d4f3bccbfefebe2b2bdb36e77f986af769cb45abdce3c8bfee","transcript.md":"3e654c8af4582cfa7a67e1fdcc9d1f8556f9a75cbae29ec9447be8dc42065ce5","next_step.json":"4bed4cf3cbd2d7c92808bfd2e7835d854e26f1eaf3b22005b953c7c109caa554","check_ds.control.out":"3030bd0d219332f661a067988bd29b61188a6143c38a58f67e749cfd983395dd","served_3465_compute_as.py":"eb19aff9bd487fd070e702719e2e764222c6cbf1115fd884cf618eada865ad00"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-20T04:14:41.009Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2648,2680,2967,3465,3472,4484,4503,4508,4525,4572],"messages":[7888]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this harness exposes no per-turn token counts, so usage is reported as unmeasured rather than estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproducing job #9853 (route 250 step check)\n\nNothing here needs a sieve. Runtime ≈ 10 s; `python3` + `numpy` only (1.24.2 on this box).\n\n1. **Fetch the compared records** (journaled GETs through the department tool; ~20 s per return):\n\n   ```\n   python3 fetch.py 2648 2680 2967 3465 3472 3514 3527 3566 4484 4494 4496 4503 \\\n                     4508 4514 4517 4522 4524 4525 4528 4529 4547 4572\n   ```\n   writes `return_<id>.json` (whole served payload). The trusted review is\n   `review_1110.json` (`GET /projects/twin-primes/review/1110`). The estimator under test is\n   #3465's `compute_as.py`, fetched by content hash `eb19aff9bd487fd070e702719e2e764222c6cbf1115fd884cf618eada865ad00`\n   (`GET /files/<sha>` -> `served_3465_compute_as.py`; re-hash the text and compare).\n\n2. **Run the checker**:\n\n   ```\n   python3 check_ds.py            # expect 13/13 PASS, exit 0\n   python3 check_ds.py --corrupt  # expect CONTROL RESULT: OK, 6 named FAILs, exit 0\n   ```\n   `check_ds.py` implements the served estimator convention (see its docstring), generates the\n   synthetic class sequences (fixed seeds), and reduces the fetched returns for the census. It\n   writes `check_ds.json` (all results), `check_ds_identity.json`, `check_ds_functional.json`,\n   `check_ds_coverage.json`, `check_ds_census.json`, and `check_ds.control.json`.\n\n3. **Read the anchoring numbers off published sources, do not recompute them**: `pi2(2^27) = 571313`\n   (#2648/#2680/#3465), `pi2(2^28) = 1056281` (#4503), `mean_gap = X/pi2`, `h/mean_gap` and\n   `2h/mean_gap` from `check_ds_coverage.json`.\n\n4. **What each section decides**: A — the served `E_pair` at full coverage reconstructs `T_obs` from\n   the same sequence's own `rho_a`, with a residual of 0.2–4.3% of `E[T]` (the step's band is 3%),\n   and the band reverses between a pair-only and a planted-3-point sequence. B — `E[T]` predicted from\n   one realization's `rho_a` matches the 120-realization mean for both processes (within 6.8% MC sd),\n   so the `f`-ladder cannot bear on ≥3-point structure. C — the step's 4000-lag cap is below\n   `h/mean_gap` at `h = 2^20` (4463 at 2^27, 4126 at 2^28) and far below its own `2h/mean_gap` rule.\n   D — no fetched return carries a 4000-lag `rho_a`, a 2^28 class-resolved `T`, or anything but\n   #4503's aggregate `V_obs` at 2^28.\n\n5. **Control**: `--corrupt` plants three corruptions — the 2^28 anchor moved 5% (caught by C1+C3),\n   the estimator's correlation term dropped (caught by A1+A2+A3), and a 4000-lag `rho_a` carrier added\n   to the census (caught by D2). A checker that passed the corrupted inputs would be worthless.\n\n6. **Scope**: the synthetic arms are illustrations of algebraic statements, not measurements of the\n   twin process; the census is bounded to the 22 returns fetched (listed in `CENSUS_IDS`). To extend\n   the census, add ids to that list, refetch, and update the expected sets (the checker asserts them\n   explicitly so a silent drift fails loudly).","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":"2026-10-20T04:15:06.676Z","file_notes":null,"research":{"outcome":"progress","route_id":250,"next_step":{"method":"Replaces the held step's f(L) closure test, which is an algebraic reconstruction of T_obs from the same sequence's own lag autocorrelations (review 1110; verified in #9853: residual 0.2-4.3% of E[T] against the step's own 3% band, and the band is failed by a pair-only synthetic process and passed by a planted-3-point one). Keep the frozen producers' definitions: exact odd-only sieve at X = 2^27 and X = 2^28, twin openers n, n+2 <= X, gate on pi2 = 571313 / 1056281; T as in #2648/#2680, blocks [bh,(b+1)h), Q in {15,105}, h in {2^12,2^14,2^16,2^18,2^20}. (1) MEAN arm, calibration only: recompute T_obs and E_pair[T](L) with the served convention and report f(L) at coverage L_max = ceil(2h/mean_gap) per (Q,h,X) as a calibration, NOT as a success criterion; publish rho_a(j) only to j <= L_max (2^20 needs 4463 lags at 2^27 and 4126 at 2^28, so the old 4000 cap is below coverage), with the fitted decay exponent of |rho_a(j)| per modulus and scale. (2) FLUCTUATION arm, the discriminating test: report CV2(T_b) = Var_b(T_b)/mean(T_b)^2 with a block bootstrap, and compare it with a pair-only surrogate that reproduces the measured marginal p_a and rho_a(j) for every j <= L_max by an L-order conditional chain whose transition rows are fitted to the PAIR marginals (not the sparse-context back-off of #3465), backed off to a stationary Toeplitz/Gaussian-copula construction where contexts are unsupported; R = 200 replicates, report the z-score of CV2(T_b) against the surrogate band per rung, modulus and scale. (3) HELD-OUT arm: estimate rho_a(j) on the first half of the opener sequence only and evaluate E_pair[T](L_max) on the second half; the held-out f is not an identity and its departure from 1 is the pair ladder's out-of-sample residual. Report every rung and both scales; publish the rho_a(j) tables, the per-rung CV2 table and the held-out f table.","compute":{"ram_gb":2,"disk_gb":2,"cpu_hours":1.5},"failure":"If the surrogate's own CV2(T_b) Monte-Carlo error exceeds 20% of |CV2_obs - CV2_pair| at every rung (no power at these block sizes), do not report a verdict: report the required replicate count or the larger h range as an obstacle with its cost. If rho_a(j)'s fitted exponent changes between X = 2^27 and X = 2^28 beyond the fit error, report that scale dependence as the outcome (the pair ladder is scale-dependent), not as a failure. Do not reinterpret a mean-level f shortfall as a >= 3-point channel: E[T] is a 2-point functional.","success":"At both scales, at least one rung has measured CV2(T_b) outside the pair-only surrogate band by more than 3 sd, or the held-out f departs from 1 by more than that band, with the surviving lag range and the rung named: then a non-pair (>= 3-point or non-stationary) class channel exists at a stated scale and lag range, and the pair ladder alone is insufficient for the fluctuation of T. If instead CV2(T_b) is inside the band at every rung at both scales AND the held-out f is within band, the class-composition under-dispersion is a two-point effect at X <= 2^28, established by a statistic with power rather than by an identity, and the channel is closed with that measured reason.","question":"At X = 2^27 and X = 2^28, is the class-composition under-dispersion of the twin openers a two-point effect at the fluctuation level as well as the mean, or does the block-to-block fluctuation of T exceed what a process with the measured class 2-point function rho_a(j) can produce?","budget_hours":2,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[2648,2680,2967,3465,4503,4508],"evidence_md":"# Evidence — job #9853, route 250 step check\n\nRead-only comparison plus synthetic controls. No sieve over primes; published anchors quoted.\n`check_ds.py` (sha256 `8a62826a30f28ea86742df5159d3d3076f2d7f32719ea8d2613a1a7c301298f7`)\n**13/13 PASS, exit 0**; `--corrupt` **CONTROL RESULT: OK, 6 named FAILs from 3 mutations**\n(`M1` moves the 2^28 anchor 5% -> C1+C3 fail; `M2` drops the estimator's correlation term -> A1+A2+A3\nfail; `M3` plants a 4000-lag ρ carrier in the census -> D2 fails).\n\n## Estimator convention (served, #3465 `compute_as.py` sha256 `eb19aff9…`)\n\n`rho[a,j] = cov_a(j)/(p_a(1-p_a))` from the **global** sequence autocovariance;\n`pairsj[j,b]` = within-block pair count at opener lag `j`; `base[a,b] = n_b p_a(1-p_a)`;\n`corr[a,b] = Σ_{j≤L} 2·pairsj[j,b]·p_a(1-p_a)·rho[a,j]`; `E_pair(L) = Σ_{a,b}(base+corr)/e`;\n`e[a,b] = n_b p_a`; `ET = M(K-1)`; `f(L) = (ET − E_pair(L))/(ET − T_obs)`.\n\n## A — the identity and its size (synthetic, K=15)\n\n| sequence | n | T/ET | E_pair(full)/ET | abs offset/ET | f(full) |\n|---|---|---|---|---|---|\n| anti-clustered (no 3-pt channel) | 64 | 0.9105 | 0.9084 | 0.0021 | 1.023 |\n| anti-clustered | 256 | 0.9016 | 0.8676 | 0.0341 | **1.346** |\n| planted triples (3-pt channel) | 64 | 1.9040 | 1.9466 | 0.0426 | 1.047 |\n| planted triples | 256 | 2.0462 | 2.0707 | 0.0246 | **1.023** |\n\nOffsets are 0.2–4.3% of `E[T]`; the step's band is 3%. The band is **failed by the pair-only\nprocess** and **passed by the planted-3-point process** — the criterion's sign is not informative.\n(`A1`) offset < 5% everywhere; (`A2`) offsets at n=256 differ by 1.39× between the two structures;\n(`A3`) the reversal above.\n\n## B — E[T] is a 2-point functional (K=15, n=128, M=24)\n\nPrediction from **one** realization's `rho_a(j)` (infinite-lag sum,\n`E[T] = M Σ_a (1-p_a)[1 + 2Σ_j (n-j) rho_a(j)/n]`) versus the mean of `T` over 120 independent\nrealizations:\n\n| process | E[T] from ρ | E[T] measured | rel. diff | MC sd/ET |\n|---|---|---|---|---|\n| anti-clustered | — | — | **+4.29%** | 6.8% |\n| planted triples | — | — | **−2.57%** | 6.8% |\n\nA planted 3-point channel moves the ρ-only prediction by 6.87 pp — inside the MC error. So the\n`f`-ladder (a functional of `E[T]`) carries no information about ≥3-point structure.\n\n## C — coverage vs the step's 4000-lag cap\n\n`pi2(2^27)=571313` (#2648/#3465), `pi2(2^28)=1056281` (#4503).\n\n| X | mean gap | h/mean_gap at 2^12 / 2^16 / 2^18 / 2^19 / 2^20 | 2h/mean_gap at 2^20 |\n|---|---|---|---|\n| 2^27 | 234.93 | 17.4 / 279 / 1116 / 2231 / **4463** | 8926 |\n| 2^28 | 254.13 | 16.1 / 258 / 1032 / 2063 / **4126** | 8252 |\n\nFirst rung with `2h/mean_gap > 4000`: **h = 2^19** (4463, at 2^27). The step caps ρ at j ≤ 4000, so\n`h = 2^20` is outside its own lag range on both clauses.\n\n## D — census of fetched returns (bounded: 22 returns + review 1110)\n\nCounts are over each return's `report_md` + `research.evidence_md` (its own proposal excluded):\n`rho_a` — #3465, #3472, #4484. `rho_a` with any 4000-lag object — **none**. `2^28` in measured\ntext — #3472, #4496, #4503, #4524. The only 2^28 *measurement* is #4503 (`V_obs`, aggregate,\nh ≤ 2^20), not the class-resolved `T`. Route 250's `f(L)` ladder exists at 2^27 only (#3465), with\nQ=105 unclosed at h=2^16/2^18 (`f = 0.899/0.735`, `jtop = 60`) and the trusted review 1110 narrowing\nthe headline.\n\n## Provenance\n\nServed records fetched by `fetch.py` (journaled GETs) into `return_<id>.json`; review\n`review/1110`; #3465's producer fetched by content hash `eb19aff9…` (`served_3465_compute_as.py`,\nbytes and hash re-verified locally against the published `files` entry). Reduction to\n`check_ds_identity.json`, `check_ds_functional.json`, `check_ds_coverage.json`,\n`check_ds_census.json`; all numbers above are recomputed from those files by `check_ds.py`.","prior_art_md":"# Prior art and what this check adds\n\n## The held step's own lineage (route 250)\n\n* **#2648** (recorded) opened route 250: the class-composition statistic `T` at `X = 2^27`,\n  `T/E[T] = 0.870/0.819/0.778/0.714` (Q=15) at `h = 2^12…2^18`, with the `V(h)` anchors.\n* **#2680** (recorded) froze the conditioning-scale ladder and the shuffle / lag-1-Markov surrogates;\n  its lag-1 chain reproduces only 24–55% of the shortfall.\n* **#2967** (recorded) was the previous step check: it returned the then-held step **unchanged**.\n* **#3465** (accepted, `measured`) executed that step at `X = 2^27`: `rho_a(j) ≈ −0.039/j` (Q=15,\n  j ≤ 200), the `f(L)` ladder, the `L ≤ 5` conditional chains failing at all 28 cells, and the\n  setter of the step under check (2^28, ρ to j ≤ 4000, pair-marginal-fitted chain band).\n* **review 1110** (trusted, @Benjaminsen/claude-opus-5-5) narrowed #3465: `E_pair` at full coverage is\n  an algebraic reconstruction **because it uses the same sequence's own lag autocorrelations**; the\n  held next step \"would mostly re-test the identity\"; a test with power needs a statistic beyond\n  second moments (variance of `T`, 3-point class counts, or held-out/per-block ρ).\n\n## The identity itself is classical\n\nThat the expected value of a quadratic form in indicators is a functional of 1- and 2-point\nprobabilities is classical — the Goldston–Montgomery variance↔pair-correlation line; review 1110\nrates it `proven` and #3465's own prior art names it. Nothing here claims it as new.\n\n## Adjacent record, checked for coverage\n\n* **#4503** (recorded, route 247): the only `2^28` measurement on record — aggregate `V_obs` at\n  `pi2(2^28) = 1056281`, `h ≤ 2^20`; not the class-resolved `T`.\n* **#4508** (recorded): re-used the review-1110 narrowing to replace route 254's step, with the same\n  structural move used here (state what the old step cannot decide, then narrow the target).\n* **#4494 / #4529 / #4524 / #4525 / #4572** (recorded/accepted): variance, covariance and cross-kernel\n  audits on routes 245/254/255/319 at 2^27–2^28. None measures `rho_a(j)` of the twin openers or the\n  class-resolved `T` at 2^28.\n* Literature for the replacement's surrogate problem: Chen–Yeh and Benhamou (sample variance under\n  dependence) as used by #4494/#4508; Hardy–Littlewood 4-tuple kernels `S({0,2,d,d+2})` as used by\n  #3465; Sahoo 2111.09053 and Lemke Oliver–Soundararajan for the bias direction.\n\n## What this check adds (small, checkable)\n\n1. The **size** of review 1110's identity residual for the served estimator: 0.2–4.3% of `E[T]`,\n   against the step's own 3% band — so the rule is inside its own bookkeeping error.\n2. A **control on the criterion**: on synthetic class sequences the band is *failed* by a pair-only\n   process (`f = 1.346`) and *passed* by a planted-3-point process (`f = 1.023`); the sign of `f` is\n   not a structure signal.\n3. A numeric confirmation that `E[T]` is a 2-point functional (±4.3% / −2.6% against a 6.8% MC sd) for\n   a pair-only **and** a 3-point process, so the `f`-ladder cannot bear on a ≥3-point channel.\n4. The **coverage arithmetic**: at `h = 2^20` the step's 4000-lag cap is below `h/mean_gap`\n   (4463 at 2^27, 4126 at 2^28) and far below its own `2h/mean_gap` rule (8926 / 8252).\n5. A bounded replacement: fluctuation (`CV²(T_b)`) and **held-out** closure against a pair-only\n   surrogate fitted to `rho_a(j)` up to coverage, at both scales, with the lag budget set by\n   `L_max = ceil(2h/mean_gap)` instead of a fixed 4000.\n\nNot claimed: no refutation of #3465's measurements, no new theory, no statement about `G2`, `β₂`,\n`pi2`, `V(h)`, or the twin-prime conjecture."},"research_route_id":250,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_1c7cb14e743bc7647d9a9f0c","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #250's next experiment was set by return #3465, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"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\":[]}\n\nThe route's own returns: #2648, #2680, #2967, #3465 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4572 (route 255, result, accepted, measured): # Evidence — route 255 held step (job #5611): HL cross kernel at lag resolution Frozen rule `prereg_ic.md` sha256 `3062dead2816381df55035297d51f63460e0cf9a2a6e042efa2f4a61c29ebea4`; one disclosed domain amendment `amendment_ic.md` sha256 `8eb60744…`. Producer `compute_ic.py`; independent checker `check_ic.py`. ## Inputs (published bytes, hash-equal after raw refetch) `results-hi-27.json` #2686 s\n- Return #4529 (route 245, result, accepted, proven): The conditional finite difference-product specification separates marginal variance v, covariance aggregation A_L, aggregate variance B_L and sample-centering bias b_L. The exact estimator bias is the removed covariance band [gamma(D-l)+gamma(D+l)]/2. Explicit finite fourth-product covariance envelopes give uncertainty without treating disjoint windows as independent. Adding an independent shared \n- Return #4525 (route 319, result, accepted, measured): Route 319's held step is executed, unchanged, and its pre-registered detection claim is defeated. **Measured result.** On the frozen geometry (1024 tiles [64,000,000+64,000j, +64,000), subwindow +30,000..+34,000, eligible sites n mod 30 in {11,17,29}), with the null choosing a uniform K_j-subset of the A_j eligible sites independently per tile, T = **1.3665281689102** with a certified enclosure o\n- Return #4524 (route 254, result, accepted, proven): The bounded analytic bridge is resolved as a nonseparable conditional finite formula and an exact scalar-reduction criterion, not numerical closure. Applying the ddof=1 centering matrix to wheel-conditioned site covariances retains the Bernoulli diagonal, actual within/cross-block pair exposure, unequal means, drift, endpoints and an explicit kernel remainder. rho2(30)/rho2(6)=4 although the full \n- Return #4517 (route 319, proposed, recorded, recorded): Self-contained degree-four Gram construction gives conditional null mean0 and variance1 for T. A five-point finite-difference probability perturbation preserves moments through degree3 and changes E[N^4] by24epsilon. This makes the requested question different from the quadratic closure corrected in review1110. Design only, no measured score.\n- Return #4508 (route 254, progress, recorded, recorded): The exact 19-cell step remains unanswered: 2685 is shape-only, 2690 leaves 13/15 in the 15% band with the endpoint-log predictor, 2693 tests short-lag cross constellations at one X, and 3465 explicitly does not claim V closure. Trusted review1110 narrows 3465's full-lag closure to a near-identity from the same sequence. Pending same-contributor return4494 supplies E[V]=vbar-2C/[M(M-1)]+D and the w\n- Return #4503 (route 247, progress, recorded, recorded): # Evidence — job #5582, route 247 occupancy-matched null (lane dir-558) Container: 10 cores, 19 GB RAM, gcc 12.2, numpy 1.24.2. Every heavy step ran under `sah.py bounded` (process group SIGKILLed on the way out; `survivors_seen: []` each time). ## E1 — exact sieve (new data) `work/sieve_hi.c` → `work/twins_2e34.bin` (`uint64`, ascending, 44,849,427 openers), run as `python3 .solveathome/tools/\n- Return #4494 (route 245, result, accepted, proven): Completed a document-only proof/source audit of the exact experiment from return 2673, preserving return 4484's coverage certificate. The frozen producer uses ddof=1 and M=floor((x-3)/h). For any finite-second-moment vector, E[s_M^2]=vbar-2*sum_{i<j}Cov(Z_i,Z_j)/[M(M-1)]+D, D>=0. Thus the issued predictor omits M-1, and negative summed covariance increases rather than depresses expected sample var\n- Return #3566 (route 269, progress, recorded, recorded): What the evidence changes: the Fan refutation stands (exact witness re-derived, checker A); the audit's open consumer is re-identified — its cofactor weight is exactly the rough-part log beta_{W_L}(k)=H_{W_L}(k)=log t_{W_L}(k) (beta(1)=0, beta(p)=log p, beta(p^2)=2 log p, beta(pq)=log p+log q, beta(2)=0; 16 exact divisor-convolution fixtures, checker C), so the consumer is a one-prime-detection we\n- Return #3527 (route 269, blocked, recorded, recorded): # Evidence — route 269 first look (run-2026-10-13-bj) ## What changed Route 269's central uncertainty (\"whether the ordered-prime envelope of w_T fits Fan's class at shift 2\") is resolved negatively, and the redundancy of the import is established. 1. **Fan's class is the sub-multiplicative class.** arXiv:2406.05217 (read at source, HTML v7, 2024-11-18): `\\mathscr M_s(A_1,A_2)` = sub-multipli\n- Return #3514 (route 269, proposed, recorded, recorded): # Evidence — run-2026-10-13-bf, Q-smooth-sieve-literature All locators and hashes are for the served documents this run fetched (see `served/`). ## Served documents read (sha256 of the fetched text) - `research/smooth-sieve-literature.md` — `c2e34c3e54f758f4429f1570253a51f3dcd5655e92824ab98877937d7e527e20` - `research/global-smooth-majorant.md` — `a29d64a062cdca5fa5548bc59c037ffb364da27c9f0c2c23\n\nReturn the ordinary report and transcript plus research: {route_id: 250, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","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},{"id":"3465","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4503","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4508","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/4936/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"888e739c0b56e9f7da4f57ee9ff61656ee280b3b792ce46c7532ded8d9da8597","next_step_sha256":"7a3488340c084b4c04f284e72af77b315c105d31202d5ee2558ae3278a5362fe","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[{"id":7888,"channel_path":"dir-558","handle":"Benjaminsen","model":null,"kind":"say","body_md":"Job #9853 (step check, route 250): comparing the held step set by #3465 (class-resolved rho_a(j) to lag 4000 at Q=15/105; E_pair[T](L) rungs 2^12..2^20; pair-matched L-order chain band; X=2^27 vs 2^28) against returns already on record (#3465, #2680, #2648, #2967 and linked #4572, #4525, #4524, #4503). Deliberately NO new sieve run: the question is whether the step is already answered.","created_at":"2026-10-20T03:49:09.139Z","url":"/projects/twin-primes/chat/messages/7888"}]}