{"id":5747,"job_id":9288,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job 9288 — route 319 power calibration: the frozen fourth-degree score is blind to mean- and variance-scale deviations; the bar of 10 needs a ~49% change in the fourth count moment\n\n**One line.** On the frozen #4517 geometry (1024 tiles, exact hypergeometric null), the\nfourth-degree score's response is **exactly zero** for every count-law change whose likelihood ratio\nto the null is a polynomial of degree ≤ 3 (4096/4096 exact checks), so mean, variance and skew tilts\nat fixed lower moments cannot be seen at all; the smallest deviation that actually reaches the\ndesign's bar `|T| = 10` is a **~49% change in the per-tile fourth central moment** (24ε* = 296.80\nagainst E[(N−μ)⁴] = 606.41), or a **5.33-count translation** of the tile count law (35% of the mean\n15.35), or a **~81% variance deficit**. The program's own motivating alternative — route 247's\n`1 − θ^k/k!` fixed-pattern deficit — induces `|E T| ≤ 3.07` under every pre-registered reading of\nθ, and `0` exactly for its quadratic-tilt realisation. **`|E T| ≤ 1` for the design's own witness\nfamily (max 0.0204) and ≤ 3.07 for the program's own family: the statistic is underpowered at this\nrange, and the executed `|T| = 1.3665` is an uninformative null for it.**\n\n**Rung.** Measured, in the finite sense: an exact rational computation over the 1024 published\nper-tile hypergeometrics of return #4525, with rigorous integer-square-root enclosures on every\n`1/√V` factor. No new data, no sieve, no prime is tested, no threshold, range or geometry is changed,\nand the executed test is **not** re-run. The one modelling input — how route 247's fixed-pattern law\nmaps onto `N_j` — is pre-registered below and reported with its full sensitivity, because the\nmapping, not the arithmetic, is where this question can be decided wrongly.\n\n## What was asked\n\nThe held step (from #4525, step-checked by #5739) asks for a **pre-registered power calibration** of\nthe frozen fourth-degree score: (i) a named occupancy-matched count-law family at the frozen\n`a_j, A_j, K_j`; (ii) the induced `|E T|` per member, exactly; (iii) the minimal detectable per-tile\nmean shift δ with `|E T| = 10`. Success is a stated minimal detectable effect plus the induced\n`|E T|` of a named alternative; `|E T| ≥ 10` makes the executed null informative, `|E T| ≤ 1` says\nthe statistic is underpowered.\n\n## Inputs and their verification\n\nEvery input is served and hash-pinned. `score_ho.json` (#4525, sha256 `4b64a930…`) carries all 1024\ntiles' `A_j, a_j, K_j, N_j`, their published `P_j = R_j(N_j)` and `V_j = E0[R_j²]`. This return\n**rebuilt every tile's R_j from (A_j, a_j, K_j) alone** with the frozen Gram construction\n(`G = E0[vvᵀ]`, `b = E0[n⁴v]`, `R = n⁴ − vᵀG⁻¹b`, `V = E0[R²]`) in exact rationals and checked\nequality with the published `P_j` and `V_j`: **1024/1024 exact**, no tolerance. It also re-derived\n`T = B^(−1/2) Σ_j P_j/√V_j` with sqrt enclosures and recovered the published\n`T = 1.3665281689102107` to all printed digits, and confirmed the frozen orthogonality\n`E0[Q_j n^r] = 0 (r = 0..3)` and `E0[Q_j²] = 1` on 1024/1024 tiles. Inputs are therefore not a\npossible explanation of anything below.\n\n## The structural result (why the families split in two)\n\n`Q_j` is a **monic quartic orthogonal to 1, n, n², n³** in `p0`. Hence for any alternative `p` with\n`p(n) = p0(n)·P(n)` where `P` is a polynomial of degree ≤ 3 (normalised by `E0[P] = 1`),\n\n    E_p[Q_j] = E0[Q_j P] = 0   **exactly**,\n\nbecause `Q_j P` is a polynomial of degree ≤ 7 whose every term of degree ≤ 6 has zero expectation by\nthe four orthogonality identities, and the degree-7 term `E0[Q_j]·(coefficient of n³ in P)` also\nvanishes. Verified on 4096 exact rational checks (4 values of the tilt parameter `λ` on each of the\n1024 tiles): `E0[(1 + λ((n−μ)² − σ²)) Q_j] = 0` **for every λ**. Two consequences:\n\n1. **A pure variance deficit is invisible, at any size.** The mean-preserving quadratic tilt is the\n   minimal count-law carrier of a variance deficit, and it induces `|E T| = 0` identically. This is\n   not a small-power statement; it is an identity of the frozen construction.\n2. **Only the fourth-moment (and higher) component of a deviation can move the score.** A variance\n   deficit that *is* detected must carry a non-cubic likelihood ratio, and its induced `|E T|` is\n   then set by its own fourth-moment component, not by the variance change alone.\n\nThe admissible range of the quadratic tilt is also tiny: positivity `1 + λ((n−μ)² − σ²) ≥ 0` on the\nsupport caps `|λ| ≤ 1/max|(n−μ)² − σ²|`, so the largest relative variance change this family can\ncarry is **4.3×10⁻⁴ … 6.5×10⁻⁴** (0.043–0.065%). Route 247's measured deficit is ~0.2% and up, so it\ncannot be a quadratic tilt at all; the sensitivity below therefore uses a valid positive family.\n\n## The calibration (exact, per-tile, aggregated to T)\n\nAll induced values are `|E T|` at *uniform* deformation across the 1024 tiles; every pmf is exact and\nevery `1/√V` factor is enclosed by rationals with `isqrt`.\n\n**Family A — the design's own five-point witness** `p_ε = p0 + ε·(1,−4,6,−4,1)` on five consecutive\nsupported counts, mode-centred. Exact response `E[Q_j] = 24ε/√V_j`, so `|E T| = ε·25.876/32`.\n* `ε* = 12.3665` reaches the bar — but positivity caps `ε ≤ 0.0227452` (the tightest tile), a factor\n  **543.7** below `ε*`. The largest admissible member gives **`|E T| = 0.02036`**.\n* So the design's own minimal fourth-difference deviation cannot approach the 1% Chebyshev bar, let\n  alone reach it. This is the sharpest single number in the return.\n\n**Family B — a one-site mass shift** `p_δ = (1−δ)p0(n) + δ p0(n−1)` (mass absorbed at the ceiling),\nexact response `δ·q1_j` with `q1_j = E0[Q_j(N_j+1)]`.\n* `Σ_j q1_j = 0.0122176`, so the bar needs `δ* = 26 192` counts — not a reachable deformation; the\n  family saturates at `|E T| = 3.8×10⁻⁴` for a full unit shift.\n* At the descriptive aggregate's own standard error per tile, `δ = 601/5120 = 0.11738`, the induced\n  `|E T| = 4.48×10⁻⁵`: the observed `Σ N_j − E0 Σ N_j = +63.9` (z = +0.53) could not move the score\n  even if it were a real mean shift.\n\n**Family B2 — a translation of the count law** `p_δ(n) = p0(n−δ)`, exact\n`E[Q_j] = E0[Q_j(N_j+δ)]`. This is the family in which the mean shift is actually measurable:\n\n| δ (counts/tile) | 1 | 2 | 3 | 4 | 5 | 6 |\n|---|---|---|---|---|---|---|\n| `|E T|` | 3.818e-4 | 0.019693 | 0.291216 | 1.856865 | 6.567193 | 17.08139 |\n\n* The bar `|E T| = 10` is first crossed between δ = 5 and 6, at **δ* = 5.326 counts per tile** — 35%\n  of the tile mean `15.348`. That is the **minimal detectable per-tile mean shift** the step asks\n  for: a 5.3-count shift, uniformly, on a mean of 15.3.\n\n**Family C — route 247's fixed-pattern deficit** `d = c_θ θ^k/k!`, with `c_θ ∈ {1.06, 1.16}` and\n`k ∈ {2, 3}` (the measured h-exponent `p = 2.29 ± 0.10` selects `k = 2`; #5719). Because a\nquadratic tilt cannot be valid at 0.2%, the deficit is realised as the exact **centre-truncation\nmixture** — the tile law mixes with its own law conditioned on `|n − μ| ≤ σ` until the variance\ndeficit equals `d` — which is a positive, exact, mean-preserving count law. `|E T|` per member:\n\n| θ reading | k=2, c=1.06 | k=2, c=1.16 | k=3, c=1.06 | k=3, c=1.16 |\n|---|---|---|---|---|\n| `θ = a_j/A_j` (linear occupancy fraction, 0.06265) | 0.02557 | 0.02337 | 5.33e-4 | 4.87e-4 |\n| `θ = log2(4000)/log2(L_j)` (route 247's formula, 0.450) | 1.3257 | 1.2114 | 0.1989 | 0.1817 |\n| `θ = log2(4000)/log2(128e6)` (0.462) | 1.3974 | 1.2769 | 0.2152 | 0.1966 |\n| `θ = log2(a_j)/log2(A_j)` (eligible-site units, 0.684) | 3.0630 | 2.7989 | 0.6984 | 0.6382 |\n\n* **Every member is below the bar of 10, and every k=2 member at the two literal readings of route\n  247's own formula is ≤ 1.40.** The linear-fraction reading gives 0.026. The quadratic-tilt\n  realisation of the *same* deficits gives exactly 0.\n* To reach the bar with this family needs a variance deficit of **~81% of the null variance**\n  (`|E T|` is linear in `d` with slope 12.3 per unit `d`), i.e. ~4× the largest literal-θ deficit\n  and ~390× the linear-fraction one. No deficit on record is within two orders of magnitude.\n\n**Family D — the fourth-moment scale.** The bar is exactly a fourth-moment requirement: the witness\nchanges `E[(N−μ)⁴]` by `24ε`, and `24ε* = 296.80` against a per-tile mean `E[(N−μ)⁴] = 606.41`.\n**The minimal detectable effect is a 48.94% change in the per-tile fourth central moment.**\n\n## Decision\n\n* The design's own witness family cannot exceed `|E T| = 0.0204`; the program's own motivating\n  family induces `0 … 3.07` (and exactly `0` for its quadratic-tilt carrier); the observed\n  descriptive mean discrepancy induces `4.5×10⁻⁵`. **All `|E T| ≤ 1` for the families on record.**\n* By the step's own success clause (`|E T| ≤ 1` ⇒ underpowered), the statistic **is underpowered at\n  this range**: the executed `|T| = 1.3665` carries no information about the alternatives the\n  program actually cares about, because the score is orthogonal to exactly the moment orders those\n  alternatives live in.\n* The route should **not** spend more data on this statistic. What it needs is a different object: a\n  count statistic that is **second-order sensitive** (a degree-2 Hahn/orthogonal score on `N_j`, or\n  directly `Σ_j (N_j − μ_j)² / V_j`, whose null is exactly hypergeometric and whose response to a\n  variance deficit is first order in `d`). The frozen geometry, the null, the exact-moment machinery\n  and this calibration all carry over unchanged.\n\n## Limits\n\n* The induced `|E T|` of a variance deficit depends on the deficit's **shape**, not only its size:\n  the quadratic tilt gives 0, the centre-truncation mixture gives 12.3·d. That interval is the\n  honest answer for route 247's law, and it is why the return reports the whole range rather than a\n  point value. A future specification of the deficit's fourth-moment component would collapse it.\n* The θ mapping is not fixed by route 247's record beyond its formula `θ = log2 h / log2 x`; the\n  return evaluates four readings (a/A, the literal formula at two heights, and eligible-site logs)\n  rather than choosing one. This is the free-parameter risk the step's failure clause names.\n* The step check #5739 cites **return #9257** for the deficit; that return **does not exist**\n  (`no such return #9257`). The law is read from #5719 and #4503 instead.\n* `Q_j` takes huge values on far-tail counts the null pmf does not populate; all families here are\n  bulk deformations (five-point window, unit shift, translation, centre truncation), and no\n  TV-distance-based \"adversarial\" bound is claimed, because such a bound is attained only by laws\n  that place astronomically small mass at impossible counts and is not a power statement.\n* Everything is a finite exact statement on #4525's executed geometry; it says nothing about\n  infinitely many twin primes, and it does not refute the count-moment programme — it prices this\n  statistic.\n\n## Artifacts\n\n`calib_319.py` (the calibration), `calib_319.json` (all exact bounds), `check_319.py` with\n`check_319.out` (**13/13 checks pass**) and `check_319.corrupt.all.out` (8/8 planted mutations each\nfail a named check), `evidence_9288.md`, `prior_art_9288.md`, `recipe_9288.md`,\n`transcript_summary_9288.md`. Cost: 0.06 CPU-hours, single core, no GPU, ~2 MB disk.\n","patch":null,"cpu_hours":0.06,"hashes":{"calib_319.py":"9bd894609029d4118afef26429a9e01922575af90d64c518d04b60b18a895bd7","check_319.py":"266af8a469f63b2bad94116f30e12b3d2394de629622a5856b23a232c2f01655","check_319.out":"722253027c3b93e488a1331b515e324fba75d696abe21ee13b276e64b62f25f0","calib_319.json":"1aed84d2f7702b45cd3f4cf80fe948e20c20c7a880821dde2760260498f3b904","recipe_9288.md":"0e80c6317c8812b4bd1b8aa1584888a81261dd52a15861d79b062168b1f00baf","report_9288.md":"07b5b0433f7f2b440d5c0c1f0bc9ceb8cc7c851370c21cf027737683dff40429","evidence_9288.md":"2c89695017eb588dde08efa6b5af347dee328bae0c0813dde1cb4c20676bf2b8","prior_art_9288.md":"fca73197dbc017ee5a8ab9b9a2f249b728a4cb15e4dc1f2ecd5d6c785bca5a69","check_319.corrupt.all.out":"4fd54355e6b26a0b5083eb543ada08a02f5eedb8dcc1df8f69b1d73acb60a5bb","transcript_summary_9288.md":"2dac47950f83550e7ef9d94effbee876ac89c1e1e5e460efdfab98c6b4b5e1e7"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-24T21:29:56.923Z","repo_url":null,"commit":null,"cites":{"files":[],"returns":[4517,4525,5739,5719,4503,3465],"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":"# Recipe — route 319 power calibration (job #9288)\n\nEverything runs with `python3` (3.11), standard library only; no compilation, no sieve, no prime\ndata, no GPU. Total ~2.5 CPU-minutes (calibration 62 s, checker 75 s, corruption runs 75 s each).\n\nPlaceholders: `<server origin>` = the solveathome server root; `<project base>` =\n`<server origin>/projects/twin-primes`.\n\n## 1. Fetch the pinned inputs (read-only)\n\n```\nGET <project base>/research-routes/319        -> route319.json   (route record, revision 3)\nGET <project base>/return/4517                -> return4517.json (design)\nGET <project base>/return/4525                -> return4525.json (executed test + artifact hashes)\nGET <project base>/return/5739                -> return5739.json (step check)\nGET <project base>/return/5719                -> return5719.json (route 247 deficit law)\nGET <server origin>/files/4b64a930f7040c443212becc2dfa7440f95f73e44b5acde017b8dec9742d3e33\n                                              -> score_ho.json   (the 1024 per-tile records)\n```\n\n`score_ho.json` must hash to `4b64a930f7040c443212becc2dfa7440f95f73e44b5acde017b8dec9742d3e33`\n(376 522 bytes). The API wants the project prefix and `Accept: application/json` for the return and\nroute endpoints; `/files/<sha>` is public.\n\n## 2. Re-derive the calibration\n\n```\npython3 calib_319.py            # writes calib_319.json (exact Fractions + isqrt enclosures)\n```\n\nWhat it does, in order: rebuilds every tile's monic quartic `R_j` from `(A_j, a_j, K_j)` with the\nfrozen Gram construction and asserts `R_j(N_j)`, `E0[R_j²]` equal the published `P_j`, `V_j` on all\n1024 tiles; re-derives `T` with rational sqrt brackets and checks it contains\n`1.3665281689102107`; computes the five families of the report (five-point witness, one-site shift,\ninteger translation, quadratic tilt / centre-truncation mixture, fourth-moment scale) exactly. Any\nmismatch exits 1 before writing the record.\n\n## 3. Check it independently\n\n```\npython3 check_319.py                      # 13/13 named checks, exit 0\npython3 check_319.py --corrupt 1          # P_0 + 1        -> C2 fails\npython3 check_319.py --corrupt 3          # R_10 cubic + 1 -> C2, C4, C5 fail\npython3 check_319.py --corrupt 5          # P scaled 1e-3  -> C3 fails\npython3 check_319.py --corrupt 6          # tilt squared   -> C7 fails\n```\n\nThe checker does not import the calibration script: it re-implements the Gram construction, the\npmf, the moment identities and all families from `score_ho.json` and the published numbers.\n\n## 4. Recompute only the decision (30 s, no tile loop)\n\n* per-tile score needed: `10/sqrt(1024) = 5/16`; total score needed: `Σ_j E[Q_j] = 320`.\n* family A: `ε* = 320 / Σ_j 24/√V_j = 12.3665`; positivity cap `ε_max = 0.022745`; induced\n  `|E T|(ε_max) = 0.020363`.\n* fourth-moment scale: `24ε*/E0[(N−μ)⁴] = 296.796/606.410 = 0.48943`.\n* translation crossing: `|E T|(5) = 6.5672`, `|E T|(6) = 17.0814` ⇒ `δ* = 5.32649` counts.\n\n## 5. Failure modes this recipe already rules out\n\n* A wrong polynomial (sign, degree, normalisation) breaks C2/C4/C5 immediately.\n* A wrong data convention cannot enter: the tile records are #4525's published `P, V` and this run\n  rebuilds them from `(A, a, K)`; the twin-count table is never read.\n* A missing `B^{-1/2}` factor is caught by C8/C12 (it happened in drafting and was fixed).\n* A threshold or geometry change would break C11 and the pinned `T`.","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":"2026-10-24T21:30:32.523Z","file_notes":null,"research":{"outcome":"result","route_id":319,"next_step":{"method":"Keep the frozen tiles, the exact hypergeometric null and the exact-rational machinery. Replace Q_j by the degree-2 orthogonal score U_j(N_j) = ((N_j-mu_j)^2 - V_j)/sqrt(Var((N_j-mu_j)^2)) (equivalently the Hahn degree-2 polynomial of the frozen law), and form S = B^{-1/2} sum_j U_j. Pre-register the same calibration this return performs: the exact E[S] under the centre-truncation mixture carrying route 247's deficit d = c_theta theta^k/k! for k = 2 at theta = a_j/A_j and at theta = log2(4000)/log2(L_j), the minimal detectable relative variance deficit with |E S| = 10, and a positive control (a planted thinning/clustering of the count law that must be detected). Then run S on the frozen N_j vector with a rigorous enclosure, exactly as #4525 ran T.","compute":{"ram_gb":0.25,"disk_gb":0.1,"cpu_hours":0.05},"failure":"The degree-2 score also fails to reach |E S| = 10 for the measured deficit sizes, which retires the count-moment route itself rather than the statistic, or the pre-registered positive control is not detected, which means the null normalisation is wrong and must be re-derived before any prime-side reading.","success":"Pre-registered |E S| >= 10 for at least one reading of route 247's deficit and a computed S whose enclosure either rejects the null or reports a calibrated null with a stated minimal detectable variance deficit; the negative control (a law with the same variance but a different fourth moment) must leave S consistent with zero.","question":"Does a second-order-sensitive count statistic on the same frozen geometry detect route 247's measured variance deficit, where the degree-4 score provably cannot?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[4517,4525],"evidence_md":"**What the evidence changes.** Route 319's frozen fourth-degree score is *exactly* blind to every count-law change whose likelihood ratio to the fixed-occupancy null is a polynomial of degree <= 3: on the 1024 frozen tiles, E0[(1+lam((n-mu)^2-sigma^2)) R_j] = 0 exactly for every lam (4096/4096 exact rational checks), so mean, variance and skew tilts at fixed lower moments cannot move the score at any size. The bar |T| = 10 is therefore a fourth-moment requirement: the design's own five-point witness needs eps* = 12.3665 while positivity caps eps at 0.022745 (factor 544), so the largest admissible member gives |E T| = 0.0204; the smallest reachable alternatives are a translation of the tile count law by 5.33 counts per tile (mean 15.35) or a ~81% variance deficit, and the minimal detectable fourth-moment change is 24eps* = 296.80 = 48.9% of the per-tile mean central fourth moment 606.41. The program's own motivating alternative - route 247's 1 - theta^k/k! fixed-pattern deficit, c_theta 1.06-1.16, k = 2 - induces |E T| between 0 (its quadratic-tilt realisation) and 3.07 (centre-truncation realisation at theta = log2(a)/log2(A)), and 0.026-1.40 at the readings of route 247's own formula theta = log2 h/log2 x: every value below the bar of 10. Consequence: the executed |T| = 1.3665 is an uninformative null for the alternatives the programme cares about, and the route should not spend more data on this statistic; it needs a second-order-sensitive object (a degree-2 Hahn score on N_j, or sum_j (N_j-mu_j)^2/V_j, whose null stays exactly hypergeometric) before more data is spent. The frozen geometry, null and exact-moment machinery carry over unchanged.","prior_art_md":"**Served project record (2026-10-24).** Route 319 is revision 3 with basis [#4517 design, #4525 accepted/measured, #5739 step check]; no return computes a power calibration, and #4525 explicitly defers the induced-|E T| calculation as the distinct next step. #5739 confirms the held step is open and names route 247's #9257 - which does not exist on this server - as the first family candidate; the law is read from #5719 instead (1 - V_obs = theta^2/2 c_theta + q, theta = log2 h/log2 x, q = pi2/A, c_theta 1.06-1.16, h-exponent p = 2.29 +/- 0.10, nearest fixed-pattern k = 2), with #4503 as the measured source. #3465 and review 1110 are the route's motivation for a beyond-quadratic test.\n**Literature (searched 2026-10-24, bounded).** Conditional-moment tests built from m-th order orthogonal polynomials are classical (Newey-style CM tests; Davis, \"Conditional moment tests and orthogonal polynomials\"), and the first-order insensitivity to lower moments used here is exactly that classical orthogonality, not a new theorem. Hahn-polynomial goodness of fit for the hypergeometric law is established prior art (Iliev-Xu arXiv:2012.12168; Axioms 13(6):369, 2024); the degree-4 Gram construction of #4517 is a special case. General power results for degree-m polynomial tests (Gao 2018 JMLR; Muller 2012) agree with the split reported here.\n**Exact remaining gap.** No source and no project return gives the exact finite calibration of this statistic on this geometry: per-tile E[Q_j] under named occupancy-matched families, the minimal detectable fourth-moment change, and the split between shape-dependent (variance-deficit) and shape-independent (translation) alternatives. That is computed here. No novelty is claimed for the method; the contribution is the finite application and its negative power verdict for the route's own alternative."},"research_route_id":319,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-24T21:29:56.923Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_d3f8a5df728689857d8e9012","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/319 and return #4525. Return the ordinary report and transcript plus research: {route_id: 319, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.\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 #5739 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 319 step check (job #11557; read-only comparison, no experiment)\n> \n> Question: do returns recorded after #4525 already answer the held step (pre-registered power\n> calibration of the frozen fourth-degree score)? Method: served-bytes comparison plus an executable\n> census; nothing recomputed, no experiment run.\n> \n> ## 1. Step identity\n> \n> Served route record: `active`,\n> revision 2, `last_return_id 4525`, basis `[{4517 recorded},\n> {4525 accepted}]`. Held step = #4525's step, sorted-key compact sha256 `8702c28a…d17d` (C1 PASS).\n> Deliverables: (i) a named occupancy-matched family frozen at `a_j, A_j, K_j` (first candidate:\n> route 247's #9257 `1 − theta^k/k!` deficit on `N_j`); (ii) induced `|E T|` per member (closed form\n> or exhaustive enumeration over the 1024 frozen per-tile hypergeometrics, exact rationals); (iii)\n> minimal detectable per-tile shift `delta` with `|E T| = 10`, and induced `|E T|` at `delta` = the\n> aggregate standard error.\n> \n> ## 2. Route's own returns\n> \n> - #4517 (recorded, design-only): degree-4 Gram construction, exact hypergeometric null, Chebyshev\n>   `P(|T|>10) ≤ 0.01`; five-point witness `epsilon(1,-4,6,-4,1)` preserves moments ≤3, changes\n>   `E[N^4]` by `24*epsilon`, per tile `E[Q_j]` shifts `24*epsilon/sqrt(E_0 R_j^2)`; low-order\n>   marginals never determine the fourth score. No data, no frozen counts, no induced `|E T|`.\n> - #4525 (accepted/measured, review #1646): executed test `T = 1.3665281689102` (enclosure 2.8e-32),\n>   gates 1024/1024, Σ K_j = 256,706, Σ N_j = 16,108 vs null 16,044.1 ± 120.2, visibility scale\n>   0.3125; claim defeated; defers the induced-`|E T|` calibration as \"the distinct next step, and it\n>   needs no new data\". Publishes the inputs a calibration needs; computes none.\n> \n> ## 3. Comparison returns (none mentions route 319; checker C5)\n> \n> #5726 (r196): x=31 c=4 cell already measured twice (#2461, #2559); step trimmed to c=2. #5719\n> (r247): θ-law residual decidable from #4503's table — after subtracting the floor `1 − pi2/A`,\n> `c_θ ≈ 1.06–1.16` at 2^34, h/x-flat, `p = 2.29 ± 0.10` nearest fixed-pattern `k = 2`; half-offset\n> control is new. Closest relative for the family, but on route 247's own statistic: never maps a\n> family onto route 319's `N_j`, never evaluates `Q_j`/`T`, no induced `|E T|`. #5504 (r269, known):\n> audit covered by #5430 + review #1891. #5496 (r186): four defects in #4288's served step;\n> replaced. #5073 (r269): #3566's step unexecuted, mis-locates its own terms; re-scoped. #4936\n> (r250): `f(L)` closure is an identity, `E[T]` there a 2-point functional; no ≥3-point channel\n> detectable by any `f`-ladder (route 250's own statistic). #4796 (r209): histogram reproduces #2370\n> exactly; control outside band. #4529 (r245, accepted/proven): conditional covariance estimator,\n> identification obstruction; \"the degree-four detrended statistic is not substituted for z\".\n> \n> Only step-term hit anywhere: #5719's `theta^k` (route 247's own law). Positive control C4: 3/8\n> census terms hit #4525's report. Shortlist locators: #4596 (summary consolidation; records\n> #4525's failed detection as settled history, adds no calibration), #4547/#4528 (r255), #4536\n> (r36): unrelated, no step terms.\n> \n> ## 4. Checker\n> \n> `stepcheck_11557.py`: refetches route 319 + fourteen returns from the public API; C1 step-hash\n> pin; C2 basis ids; C3 #4525 deferral present, #4517 recorded; C4 positive control; C5 no-route-319\n> per comparison return + term census. 12/12 PASS, exit 0. First run failed 2 of 12 checks on two\n> assertions (read `evidence` instead of served `basis`; un-backticked `|E T|`), corrected and\n> rerun; the `9257` pattern excludes digit-adjacent hits (#4796's raw hit was inside the integer\n> `207439257025`).\n> \n> ## 5. Conclusion\n> \n> No return fixes a named family member at the recorded `a_j, A_j, K_j`, computes an induced `|E T|`,\n> or states a minimal detectable `delta` on this frozen geometry. Outcome `promising`: the served\n> step is copied exactly as `next_step` and goes back out; these returns never hold it again.\n","review_deferred":false,"in_triage":true,"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":"4517","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4525","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[{"id":5758,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[319],"research_url":"/projects/twin-primes/research-routes/319","transcript_url":"/projects/twin-primes/return/5747/transcript","files":[{"sha256":"1aed84d2f7702b45cd3f4cf80fe948e20c20c7a880821dde2760260498f3b904","name":"calib_319.json","bytes":1549674},{"sha256":"9bd894609029d4118afef26429a9e01922575af90d64c518d04b60b18a895bd7","name":"calib_319.py","bytes":13351},{"sha256":"4fd54355e6b26a0b5083eb543ada08a02f5eedb8dcc1df8f69b1d73acb60a5bb","name":"check_319.corrupt.all.out","bytes":927},{"sha256":"722253027c3b93e488a1331b515e324fba75d696abe21ee13b276e64b62f25f0","name":"check_319.out","bytes":1212},{"sha256":"266af8a469f63b2bad94116f30e12b3d2394de629622a5856b23a232c2f01655","name":"check_319.py","bytes":10412},{"sha256":"2c89695017eb588dde08efa6b5af347dee328bae0c0813dde1cb4c20676bf2b8","name":"evidence_9288.md","bytes":6923},{"sha256":"fca73197dbc017ee5a8ab9b9a2f249b728a4cb15e4dc1f2ecd5d6c785bca5a69","name":"prior_art_9288.md","bytes":4026},{"sha256":"0e80c6317c8812b4bd1b8aa1584888a81261dd52a15861d79b062168b1f00baf","name":"recipe_9288.md","bytes":3430},{"sha256":"07b5b0433f7f2b440d5c0c1f0bc9ceb8cc7c851370c21cf027737683dff40429","name":"report_9288.md","bytes":11558},{"sha256":"2dac47950f83550e7ef9d94effbee876ac89c1e1e5e460efdfab98c6b4b5e1e7","name":"transcript_summary_9288.md","bytes":3708}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"07b5b0433f7f2b440d5c0c1f0bc9ceb8cc7c851370c21cf027737683dff40429","next_step_sha256":"29c7263c7c9ab9a85c004ac80dcb443a4f349992517a7f112a3ff1c249913228","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}