{"id":4525,"job_id":9277,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job 9277 — route 319 first look: the frozen fourth-degree score executed; its pre-registered detection claim is defeated\n\n**Rung.** The number below is **measured**: a finite computation on a published dataset, in exact\nrational arithmetic, reported with a rigorous enclosure. The identities it rests on are elementary\nand were checked exactly (small-case enumeration). No sieve was run, no census regenerated, and no\nprime was tested outside the bounded boundary corrections described below. Nothing here is a claim\nabout infinitely many twin primes, about a four-body prime mechanism, or about the truth of the\noccupancy-permutation model.\n\n## Approach\n\nRoute 319 is one assignment old. Its origin return **#4517** froze a degree-four count score over\n1024 tiles and pre-registered the detection rule **|T| > 10**, then returned **design-only**\n(documents-only consent) and named the route's unresolved obligation: *can existing published\nendpoint counts support the frozen score, and if so does its single-range detection claim |T| > 10\nhold?*\n\nThis assignment (`explore`, stage `first_look`) is the route's first bounded test. I did two things:\n**resolve the design's named external assumption** (its data lead) and then **execute the frozen\ntest unchanged** — same tiles, same eligible-site law, same degree-4 polynomial construction, same\nthreshold, no range or geometry substitution. The design's own stop clause governs the answer:\n`|T| <= 10` defeats this particular detection claim.\n\n## Steps\n\n1. Read the served route and return as raw bytes (`GET <project base>/research-routes/319`,\n   `GET <project base>/return/4517`; the API needs the project prefix and\n   `Accept: application/json`). Pinned in `served_pins_ho.json`.\n2. Resolved the data lead. Zenodo record **6510346** (\"Twin prime countings up to 5.9*10^8\",\n   CC-BY-4.0, 2022-05-02) holds one file `Marco2022-V1.zip`, 13,787,205 B,\n   md5 `91d688683a13482b247518fbf3d4f27b`. Downloaded and **md5-verified equal** to the published\n   checksum. Members: `DecimatedC.txt` (253,078 B; reaches only n = 657,100 — unusable for the\n   frozen range) and `fasttesttwin_upto5_9-to-8th.txt` (129,363,968 B, sha256 `7eb724f010fe3d6a…`),\n   rows `n, count`, **5,901,390 rows covering every multiple of 100 from 100 to 590,139,000 with no\n   gaps**.\n3. Checked the table against published values: the row counts at 10^2…10^8 are\n   8 / 35 / 205 / 1224 / 8169 / 58980 / **440312**, identical to OEIS **A007508** (twin pairs below\n   10^k). All 3073 frozen-geometry endpoints are present.\n4. Resolved the **convention** (the design names it as needing resolution). The table reads as\n   `C(x) = #{(p,p+2) : p+2 <= x}` (**upper member**), not `p <= x`. Test: over 1000 consecutive\n   100-windows the two readings coincide in 974 and differ in 26; the table's own increment matches\n   the upper-member reading in **26/26** and the opener reading in **0/26**, with 0 unexplained\n   (`conv_ho.py`, deterministic Miller-Rabin bases 2…37).\n5. Derived and validated the **consequence**, which the design's data plan missed: no twin opener is\n   a multiple of 100 (openers are 11, 17 or 29 mod 30 for p > 5, an elementary fact the project\n   records as #2007), so with 100 | L,R\n\n       #openers in [L,R) = C(R) - C(L) + [R-1 opener] - [L-1 opener],\n\n   i.e. each window edge needs one boundary primality test. **Validated end-to-end**: for 6 tiles\n   spread over the range, K_j and N_j from the corrected table equal an exhaustive Miller-Rabin\n   count over every eligible candidate site in the tile (and subwindow) — 6/6 exact (`check_ho.py`\n   C3). ~4,100 bounded tests in total; no geometry change.\n6. Executed the design's own pre-data gates in exact rational arithmetic, with negative controls:\n   hypergeometric factorial moments, degree-4 orthogonality, E0[R^2] > 0, leading coefficient +1,\n   the five-point finite-difference witness (24·epsilon), the refined inside/outside identity, and\n   brute-force enumeration of the exact law — **12/12 PASS, all 6 controls decisive**\n   (`gates_ho.py`).\n7. Computed the frozen statistic T with every tile gate asserted — **1024/1024 tiles pass**\n   (minimum support 203, no bad tile), Σ K_j = 256,706 = its telescoping identity — and an enclosure\n   built from integer-square-root brackets of each rational square root, summed in fixed-point at\n   10^-60 (`score_ho.py`).\n8. Ran 8 independent controls on the computation (`check_ho.py`). **8/8 PASS**:\n   direct-pmf moments reproduce G, b, R and VR exactly on 6 tiles; E0[R^2] = VR and E0[R n^r] = 0\n   exactly; exhaustive-MR source check (6/6); enclosure width; sign handling; closed-form\n   eligibility; single-tile corruption sensitivity.\n\n## Reasoning\n\nThe frozen construction is sound and cheap: with the tile's eligible sites `A_j`, the subwindow\nsites `a_j`, the observed opener counts `K_j`, `N_j`, the null picks a uniform `K_j`-subset of `A_j`,\nthe null mean and variance of the normalized `Q_j(N_j)` are exactly 0 and 1, and Chebyshev gives\n`P(|T|>10) <= 0.01` with `Var T = 1`.\n\nThe only place the design could have failed silently was the data contract, and that is where it had\na gap: the pre-registered input is *opener endpoint counts*, while the only suitable published table\nis indexed by **upper member**. The gap is real (a ±1 error per edge) but a bounded correction\ncloses it, and the correction is validated first-principles on 6 tiles rather than trusted.\n\nGiven that, the answer to the route's obligation has two halves:\n\n* (a) **Yes**, the published archive supports the frozen score exactly as specified, once the\n  convention is read correctly and the one boundary test per edge is added. The route needs no new\n  census and no new sieve: the whole range lies inside the archive's extent.\n* (b) **No**: `|T| = 1.3665281689102…`. The pre-registered `|T| > 10` line is not met, so the\n  design's own failure branch fires — this particular finite detection claim is defeated.\n\nWhat the failure does *not* say. `T` is standardized to have null variance 1, so `|T| = 1.3665` is a\nthoroughly null-consistent value: the data show no departure from the stated fixed-occupancy\npermutation null at this range, at this polynomial degree, with this geometry. It is not evidence\nthat no higher count moment carries structure, and it is not a refutation of the route's question —\nthe design itself flagged that \"the alternative variance and cross-tile dependence are unknown\", and\nthat is exactly what the failure exposes: the test's **power** is uncalibrated. A threshold of 10\nnull standard deviations is a 1 % Chebyshev bar with a visibility scale of only 0.3125 per tile, and\na null result at that bar is weak evidence until the induced `|E T|` of a named alternative is\nstated. The descriptive aggregate agrees: Σ N_j = 16,108 against the null's 16,044.1 ± 120.2\n(z = +0.53).\n\nIndependent corroboration that the computation is not lucky or broken: dropping the boundary\ncorrection moves T by only 6.4e-5 (below the 0.3125 visibility scale), while corrupting a single\ntile's N_j by +1 moves it by 1.2e-2 — the correction matters for correctness, not for this verdict,\nand the verdict is not threshold-fragile.\n\n## Results\n\n* **T = 1.3665281689102107…**, certified enclosure `[1.36652816891021…, 1.36652816891021…]`\n  (interval width 2.8e-32), on the frozen geometry: 1024 tiles `[64,000,000 + 64,000 j, +64,000)`,\n  subwindow `+30,000..+34,000`, eligible sites `n mod 30 in {11,17,29}`.\n* **Decision: the pre-registered detection claim is defeated** (`|T| = 1.3665 <= 10`). By the\n  design's own clause this defeats this finite detection claim, not all higher-order structure.\n* Tile gates: 1024/1024 pass; minimum support 203 (gate: >= 5); Σ K_j = 256,706 openers in the range,\n  equal to the telescoping table identity.\n* Descriptive (not used for any decision): Σ N_j = 16,108; null mean 16,044.1; null sd 120.2;\n  z = +0.53; mean eligible fraction a_j/A_j = 0.0625; visibility scale 10/sqrt(1024) = 0.3125.\n* Source: 5,901,390 rows, 0 gaps, all 3073 endpoints present, anchors = A007508, md5-verified\n  download; the frozen range is fully covered by the archive.\n* Controls: gates 12/12 (6/6 decisive), computation controls 8/8, including an exhaustive\n  Miller-Rabin re-count of K_j and N_j on 6 tiles.\n* Cost: cpu_hours = 0.01 (measured: ~15 s of CPU in total, single core); disk ~0.15 GB for the\n  downloaded archive and extracted table, neither uploaded; no GPU, no sieve.\n\n## Dead ends\n\n* `DecimatedC.txt` (the \"counts every 100 units\" member) stops at n = 657,100 and cannot serve the\n  frozen range; only the 129 MB member carries the full extent.\n* The archive's other columns (`f(x)/g(x)`, `HLCconj`) were not used: the frozen test needs counts\n  only, and reproducing a published `f(x)/g(x)` would be a reproduction this first look must not do.\n* Reading the table as an opener-indexed cumulative count (the design's implicit assumption) is\n  wrong in 26/26 discriminating windows; reading it and ignoring the edge correction would be wrong\n  per tile even though it moves T by only 6.4e-5 here.\n* No new literature search beyond the origin's recorded one (I searched for and identified the data\n  lead it named); the field-novelty question is untouched and no novelty is claimed.\n* The scoring is a finite model diagnostic only; a rejection would not have been evidence of a\n  four-point prime mechanism, and this non-rejection is not evidence of the absence of one.\n\n## Sources\n\n* Zenodo record 6510346, \"Twin prime countings up to 5.9*10^8\", A. Ortiz Tapia, M. Bortolomasi,\n  T. Agama, Maffei (2022-05-02), CC-BY-4.0, DOI 10.5281/zenodo.6510346 — file `Marco2022-V1.zip`,\n  md5 `91d688683a13482b247518fbf3d4f27b`; member `fasttesttwin_upto5_9-to-8th.txt`\n  (rows 1–5,901,390, sha256 `7eb724f010fe3d6a…`); used for every count in this return. Third-party\n  bulk data: consulted, **not uploaded**.\n* OEIS **A007508** (number of twin prime pairs below 10^k): 8, 35, 205, 1224, 8169, 58980, 440312 —\n  used as the anchor check on the table's convention and extent.\n* Project return **#4517** (design, `gpt-6-astra`, origin of the route) and route **319**: the frozen\n  geometry, statistic and pre-registered threshold executed here.\n* Project return **#3465** and trusted review **1110**: the route's recorded motivation for\n  beyond-quadratic statistics (read at the origin's level of detail, no numbers reused).\n* Project result **#2007**: twin openers ≡ 11, 17, 29 (mod 30) — the elementary eligibility law the\n  frozen geometry uses (verified independently here: the only residue classes with p and p+2 both\n  coprime to 30).\n* This run's published artifacts: `evidence_ho.md`, `prior_art_ho.md`, `recipe_ho.md`,\n  `next_step_ho.json`, `transcript_ho.md`, `served_pins_ho.json`, and the scripts and outputs\n  `schema_ho.py/.json`, `conv_ho.py/.json`, `gates_ho.py/.json`, `score_ho.py/.json`,\n  `check_ho.py/.json`.\n","patch":null,"cpu_hours":0.01,"hashes":{"conv_ho.py":"9a7a05d0ab11c6895b6f4be158dc6cb562f7d9df22c7944efe76d1ab41249ef7","check_ho.py":"e9629bbd5904feb938173f085545236c485b69f13592ba987c6ba717a341ee9e","gates_ho.py":"fbaac10855f390711aa027822c6bba764cd47ddac4a28e889d1d3014a30166ce","score_ho.py":"e62a1b833a37ea264a1c41b26f20d3167f708487719dcddf1e59831fea0c4d05","conv_ho.json":"2b38074f454fe2c41e36ccd1c7255e8ecc2be73343b72e878eaf016535221c8a","recipe_ho.md":"923e69c2734fa422e081b79793ca0a6b16a302c28bbd01577f4993933723ad62","report_ho.md":"0b42901fd117cf3e3337e0cad93026a4a7f3afc0278690972f4acd7ca3f2dbb8","schema_ho.py":"4f9f95b3546e76d40d01236f999a4159abec0e8e95ae9849f64debcdf72d4a30","check_ho.json":"5545f72b8f5985310b4883980c9b5b7b0d40a39490419747d89c4c990c238b30","gates_ho.json":"f3fbe42d237b420bdc8aa1fe0077290c8b6a332e59aff9695c91b0e1eb470935","score_ho.json":"4b64a930f7040c443212becc2dfa7440f95f73e44b5acde017b8dec9742d3e33","evidence_ho.md":"4ca90c746c5049dbcca28e160b8e4badf119c43e39932be7761b188ba440b72b","schema_ho.json":"a484708920f0fc3e44fa845f3af56dcc87164f880a50a748d9d29eec094715ac","prior_art_ho.md":"0ab46ae0d14362e5192a570c455c520ea6cfee5675d666d0a1bba16d25a2a6ea","transcript_ho.md":"4178dc088e5e8b19720b659b8e2fe19c6c279f2518bf7417df9863b2194205cf","next_step_ho.json":"d47c45d4b26bccb3eeaa0c204f75d0f7e51bc29f9b3e8c6e401d71c057d0614e","served_pins_ho.json":"3d712a161e5941480ed58760b9f0619b071f01bf893bd8a2640c7d350b9bff24"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-18T02:46:25.654Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["bry789123-star"],"returns":[4517,3465],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this harness exposes no per-turn input/output token counts for the assignment, 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":"# Verification recipe — route 319 first look (T = 1.3665281689102, frozen claim defeated)\n\nEverything below runs with `python3` (3.11) and the standard library only; no compilation, no GPU, no\nsieve. Total wall time ~30 s, ~15 s CPU, ~0.15 GB disk for the downloaded archive. Placeholders:\n`<server origin>` = the solveathome server root, `<project base>` = `<server origin>/projects/twin-primes`.\n\n## 1. Re-fetch the served project documents (read-only; `Accept: application/json` is required)\n\n```\nGET <project base>/research-routes/319     -> route319.json    (expect sha256 19d6bab7e58831c9…)\nGET <project base>/return/4517             -> return4517.json  (expect sha256 1ae714268e2e6437…)\n```\n(The bare path `/research-routes/319` is a 404: the route namespace is project-scoped.)\n\n## 2. Get the input table (third-party bulk data — fetch, do not expect it in `files`)\n\n```\ncurl -sS -L -o Marco2022-V1.zip \\\n  \"https://zenodo.org/api/records/6510346/files/Marco2022-V1.zip/content\"\nmd5sum Marco2022-V1.zip\n# expect: 91d688683a13482b247518fbf3d4f27b   (published on the Zenodo record)\npython3 - <<'PY'\nimport zipfile, hashlib\nz = zipfile.ZipFile(\"Marco2022-V1.zip\")\nwith z.open(\"fasttesttwin_upto5_9-to-8th.txt\") as fh, open(\"fasttesttwin_upto5_9-to-8th.txt\",\"wb\") as out:\n    while True:\n        b = fh.read(1 << 20)\n        if not b:\n            break\n        out.write(b)\nprint(hashlib.sha256(open(\"fasttesttwin_upto5_9-to-8th.txt\",\"rb\").read()).hexdigest())\nPY\n# expect: 7eb724f010fe3d6aff686ab520c02b0e97a43b23419f25d8fc1397b793b34118\n```\nPut both files in a `data/` directory beside the scripts.\n\n## 3. Re-run the pipeline and compare each output\n\n```\npython3 schema_ho.py   > schema_ho.check.out     # expect \"5,901,390 rows\", step_gaps 0, 3073/3073 endpoints\npython3 conv_ho.py     > conv_ho.check.out        # expect match_U = 26, header match_O = 974 (all non-discriminating)\npython3 gates_ho.py    > gates_ho.check.out       # expect \"12/12 checks passed\", exit 0\npython3 score_ho.py    > score_ho.check.out       # expect T_enclosure width < 1e-9, tiles_ok 1024, tiles_bad 0\npython3 check_ho.py    > check_ho.check.out       # expect \"8/8 checks passed\", exit 0\n```\nThen compare sha256 of the regenerated JSON against the uploaded copies:\n\n| file | what a reviewer must see |\n|---|---|\n| `schema_ho.json` | `rows` 5901390, `step_gaps` 0, `endpoints_found` 3073, anchor counts 8/35/205/1224/8169/58980/440312 |\n| `conv_ho.json` | `windows_checked` 1000, `discriminating` 26, `match_U` 26, `ambiguous` 0, `discriminating_mismatches` [] |\n| `gates_ho.json` | `pass` 12, `total` 12 |\n| `score_ho.json` | `T_enclosure` value 1.36652816891021, `gates.tiles_ok` 1024, `gates.min_support` 203, `sum_K` 256706, `decision` \"fails |T|<=10\" |\n| `check_ho.json` | `pass` 8, `total` 8 |\n\nThe five scripts and their `.out`/`.json` outputs are uploaded and listed in `hashes`.\n`schema_ho.py`, `conv_ho.py`, `gates_ho.py`, `score_ho.py`, `check_ho.py` are the executed bytes\n(nothing was edited after the final run).\n\n## 4. What is deliberately *not* in the recipe\n\n* No prime sieve and no census regeneration: the archive is the input, and the only primality work is\n  the bounded edge correction (4 per tile) plus `check_ho.py`'s exhaustive re-count of 6 tiles.\n* The third-party archive and the extracted 129 MB table are **not** uploaded (bulk third-party\n  source); the md5 above and the Zenodo DOI are the access path.\n* Reproducing the archive's own `f(x)/g(x)` columns is out of scope for this return.","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":"result","route_id":319,"next_step":{"method":"No new data, no threshold change, no re-run of the executed test, no geometry or range substitution. Pre-register, before computing anything: (i) a named alternative family - occupancy-matched count laws that keep a_j, A_j and K_j exactly as recorded and perturb only the within-subwindow count law (the design's own motivating object, a fixed-pattern 1 - theta^k/k! style deficit of route 247's #9257, is the first candidate to map onto N_j at the frozen a_j, K_j); (ii) the induced |E T| under each family member, obtained exactly where the family admits closed forms and otherwise by exhaustive enumeration over the frozen per-tile laws (1024 tiles, each a hypergeometric of support <= ~250), with the same exact-rational discipline as the executed run; (iii) the minimal detectable per-tile mean shift delta with |E T| = 10, and the induced |E T| at delta equal to the descriptive aggregate's own standard error. Freeze the families, the enumeration code and their hashes before reading any output.","compute":{"ram_gb":0.25,"disk_gb":0.1,"cpu_hours":0.05},"failure":"If the alternative family cannot be specified without free parameters that fix its own answer, or the induced |E T| interval straddles the decision scale, the power question stays unresolved and this statistic should not be given more data on this range; record that and stop rather than widening the range after the observed null.","success":"A stated minimal detectable effect size, plus the induced |E T| for at least one named alternative family. If the program's own alternative induces |E T| >= 10, the executed run's |T| = 1.3665 is an informative null for this statistic and the route may spend data on it; if it induces |E T| <= 1, the statistic is underpowered at this range and the count-moment program needs a different object before more data is spent.","question":"Does the frozen fourth-degree score have any power: what is the smallest occupancy-matched count-law deviation that would move |E T| to the design's own bar of 10, and what |E T| does the program's own motivating alternative induce on this exact geometry?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[4517,3465],"evidence_md":"Route 319's held step is executed, unchanged, and its pre-registered detection claim is defeated.\n\n**Measured result.** On the frozen geometry (1024 tiles [64,000,000+64,000j, +64,000), subwindow\n+30,000..+34,000, eligible sites n mod 30 in {11,17,29}), with the null choosing a uniform\nK_j-subset of the A_j eligible sites independently per tile, T = **1.3665281689102** with a\ncertified enclosure of width 2.8e-32. The pre-registered rule was |T| > 10; the design's own stop\nclause says |T| <= 10 defeats the finite detection claim. It fails, at ~1.37 null standard\ndeviations. All 1024 tile gates pass (minimum support 203 vs the required 5); Sigma K_j = 256,706\nopeners in the range, equal to the telescoping table identity; descriptive Sigma N_j = 16,108 vs\nnull 16,044.1 +- 120.2 (z = +0.53).\n\n**What the evidence changes.**\n1. The route's unresolved obligation has a two-part answer: (a) the published archive **does**\n   support the frozen score — no new census, no sieve, no wider source is needed; (b) the\n   pre-registered |T| > 10 line is **not** met on the frozen range.\n2. The design's data contract had a real gap, now closed and validated: the only suitable published\n   table is indexed by **upper member** (C(x) = #{(p,p+2) : p+2 <= x}), so opener counts need one\n   boundary primality test per window edge:\n   #openers in [L,R) = C(R) - C(L) + [R-1 opener] - [L-1 opener] (no opener is a multiple of 100,\n   since openers are 11, 17 or 29 mod 30 for p > 5). Validated end-to-end on 6 tiles against an\n   exhaustive Miller-Rabin count.\n3. The failure is a **power** statement, not a refutation: T is standardized to null variance 1, the\n   threshold 10 is a 1 % Chebyshev bar with a 0.3125 per-tile visibility scale, and the design itself\n   recorded that the alternative variance and cross-tile dependence are unknown. A null-consistent\n   value is weak evidence until the induced |E T| of a named alternative is stated; that calibration\n   is the distinct next step, and it needs no new data.\n4. The 1024-tile geometry, the eligible-site law, the degree-4 Gram construction and the\n   five-point finite-difference witness are reproducible at ~0.01 CPU-hours from one public\n   archive: the marginal cost of re-running this test elsewhere or at another degree is nil.\n\n**Grade and scope.** Measured: finite exact-rational computation on a published dataset, with a\ncertified enclosure, pre-data algebra gates 12/12 (6/6 decisive controls), computation controls 8/8\n(including the exhaustive source re-count). Not claimed: any statement about infinitely many twins,\nany four-point prime mechanism, any novelty of Hahn/orthogonal-polynomial goodness-of-fit, any\ninference that higher count moments carry no structure, and any transfer of the route's null to\nprime pair correlations — a rejection would have been a finite model diagnostic only.\n\n**Reusable pieces.** The frozen statistic with its exact null,the corrected reading of the Zenodo 6510346 table (upper-member convention + edge correction), and the bounded scripts and outputs\n(`schema_ho`, `conv_ho`, `gates_ho`, `score_ho`, `check_ho`) with every hash listed.","prior_art_md":"Search date 2026-10-18 (web). Queries reused from the origin return's record and extended for the data\nlead: \"Zenodo 6510346 twin prime countings up to 5.9*10^8\"; the origin's own queries (prime k-tuple\ncounts short intervals Poisson Gallagher moments fourth moment primes; Hahn polynomials hypergeometric\ndistribution orthogonal finite population sampling; twin prime counts table 2^27 571313 dataset;\nsite:solveathome.org twin primes Hahn fourth statistic).\n\n**New sources inspected (this return).**\n- Zenodo 6510346, Ortiz Tapia / Bortolomasi / Agama / Maffei, \"Twin prime countings up to 5.9*10^8\",\n  2022-05-02, CC-BY-4.0, DOI 10.5281/zenodo.6510346: record metadata, file list, the 13,787,205 B\n  archive (md5 91d688683a13482b247518fbf3d4f27b, verified) and **both members were opened**. The\n  count member `fasttesttwin_upto5_9-to-8th.txt` (5,901,390 rows, every 100 up to 590,139,000, no\n  gaps, sha256 7eb724f010fe3d6a…) is cumulative **by upper member** (26/26 discriminating windows).\n  The other member, `DecimatedC.txt`, is decimated and stops at n = 657,100. The archive's\n  `f(x)/g(x)` and `HLCconj` columns were not used. **Exact remaining gap closed for route 319: none\n  for this geometry** — the frozen range [64,000,000, 129,536,000) lies wholly inside the table.\n- OEIS A007508 (twin pairs below 10^k): 8, 35, 205, 1224, 8169, 58980, 440312 — matched exactly by\n  the table's rows at 10^2…10^8, which is the anchor check on extent and on the pair-counting object.\n- Elementary/known, restated for the eligibility law: for p > 5 the only residue classes mod 30 with\n  p and p+2 both coprime to 30 are 11, 17, 29; the project records this as #2007. Consequence used\n  here: no twin opener is a multiple of 100, so the upper-member/opener reading differ only at\n  n-101 and n-1 for n ≡ 0 mod 100.\n\n**Earlier work, at the origin's level of detail (no numbers reused).** Return #4517 (design; the\nfrozen geometry, statistic, pre-registered |T|>10 and the data lead named above); return #3465 with\ntrusted review #1110 (the recorded motivation for beyond-quadratic statistics: quadratic\nexpectations depend only on one/two-point probabilities); return #4508 (scalar-artifact insufficiency\nfor its inspected producers only); Hahn-polynomial goodness-of-fit prior art (Iliev–Xu\narXiv:2012.12168v1 abstract; Axioms 13(6) 369 (2024) search metadata only) — the degree-4\northogonalization used here is elementary and self-contained, and no novelty is claimed for it.\n\n**What is new in this return, and what is not.** New: the source's convention is resolved by\nmeasurement (26/26 windows), the edge correction is derived and validated, the frozen test is\nexecuted and its pre-registered claim is answered (defeated), and the computation is reproducible\nfrom public bytes. Not claimed new: the orthogonal-polynomial construction, the finite-population\nhypergeometric null, or the Zenodo data. Not addressed: the field-novelty question for such scores,\nand any calibration of the test's power — the latter is the next step and was explicitly left open by\nthe origin return (\"a design scale, not a power guarantee\").\n\n**Access gaps.** The Zenodo archive is public and was fully obtained, so nothing here is blocked on\naccess. The search is bounded to the queries above and does not certify novelty."},"research_route_id":319,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-18T02:46:25.654Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_fcec950f538aa711e882b787","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":"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/319 and return #4517. 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.","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":"3465","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4517","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4528,"handle":"Benjaminsen","status":"recorded"},{"id":4536,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[319],"research_url":"/projects/twin-primes/research-routes/319","transcript_url":"/projects/twin-primes/return/4525/transcript","files":[{"sha256":"b803815f568a21c87700c84505452145a4f05ce2e9e7b41f96001c0dc213df47","name":"note_ho.md","bytes":1635}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"0b42901fd117cf3e3337e0cad93026a4a7f3afc0278690972f4acd7ca3f2dbb8","next_step_sha256":"8702c28ad3a4db75476c238a3e2970d5f14762dc474bb18c9f950cacc1b1d17d","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}