{"id":1312,"job_id":2660,"problem_id":1,"lane_id":5,"type":"explore","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# Job #2660 (explore / pursue, route 35) — **known**: the assigned density-partial block-grain test at x = 19 and x = 23 is already measured by returns #1029 and #1297, and the slot scale is closed by #1300; an independent reproduction confirms the counts and the null\n\n**Outcome: `known`.** The assigned question — \"does a tile census predict occupancy beyond its\nslot count, at x = 19 and x = 23?\" — is the route-35 step that return **#1029** (job #1934,\n2026-09-18) executed as a pre-registered experiment with the two named repairs (a per-period\ndensity control λₖ and a permutation null with the regression coefficient refitted per draw), and\nthat return **#1297** (job #1936) extended to L = M/64, M/128 and to the gap-local scale; **#1300**\n(job #2651) then closed the finest scale, the individual slot. The route record's \"uncovered step\"\nparagraph (revision 15) is stale: it still describes x = 19/23 block-grain as unrun, but #1029's\ntitle, instrument and table are exactly that experiment. Nothing in this assignment is new ground.\nTo make the `known` verdict rest on a checked number rather than on the citation alone, I rebuilt\nthe instrument through a different code path (`outputs/2660/job2660-blockgrain-repro.py`, exact\ntile word from `outputs/1925/gen_tile.py` plus a fresh segmented twin sieve) and reproduced, in\nunder one second of sieve time: π₂(10⁶) = 8 169, D(T₁₉) = 378 675, D(T₂₃) = 7 952 175,\ntwins = 333 007 at x = 19 (8 periods) and 1 492 887 at x = 23 (2 periods), λ̂ = 0.109925 and\n0.093867 — identical to #1028/#1029 — and the same density-partial null (no cell satisfies the\nregistered falsifier). No claim here bounds G₂, β₂ or twin-prime infinitude; the twin prime\nconjecture is open.\n\n## 1. The prior work that owns the assignment\n\n* **Return #1029** (route 35 rev 9's registered step; job #1934; recorded 2026-09-18): *\"Pre-registered\n  density-partial block-grain test at x = 19 (M = 9 699 690, 8 complete periods, [M,9M)) and\n  x = 23 (M = 223 092 870, 2 periods, [M,3M)) with the two repairs route 35 rev 9 named, both fixed\n  in the instrument before the first run: R1 a per-period (range-aware) density control\n  λₖ = twinsₖ/slots, blocks formed inside one period so a_b is census-invariant; R2 a permutation null\n  with the regression coefficient REFITTED inside every draw (Winkler et al. 2020), i.e.\n  ρ(tbar_b − β·a_b, grain_b) with β re-fit per draw. Pre-registered falsifier: |z| ≥ 2 AND rank-p ≤ 0.01,\n  same sign, both grain statistics, ≥ 2 of 3 scales L ∈ {M/8, M/16, M/32}. RESULT: the falsifier did\n  not fire anywhere — 0 of 3 scales at either level for either statistic.\"* Its report records the\n  measured table (largest |ρ| = 0.146, smallest p = 0.673) and states the load-bearing negative that\n  `gmax` is vacuous at the registered coarse scales. This is the assignment's exact object, at the\n  assignment's exact levels, with the confound fix performed *by construction* (the \"regress on λ·a\n  then permute the grain labels\" repair named in the route text).\n* **Return #1297** (route 35 rev 10; job #1936) extended the same instrument to L = M/64 and M/128\n  and to the gap-local scale, at x = 19, 23 and 29: *\"the density-partial test shows no arrangement\n  signal at x = 19 and x = 23, and the same instrument at x = 29 … shows none either\"*; it reports\n  ρ_partial ≤ 0.17 in the populated cells, the smallest detectable |ρ| = 0.28–0.34 (M/64) and\n  0.20–0.23 (M/128), and the gap-local rate flat to within 0.1 % relative across contexts from\n  adjacent gaps of 12 to 150–258. The route's stated central uncertainty #1 (sub-block / gap-local\n  scale) is therefore also covered.\n* **Return #1300** (route 35 rev 11; job #2651) closed the slot scale: local-density covariates at\n  W = 60, 300, 1500 and residue classes mod the next prime, with a demonstrated power floor of 0.5 %\n  at x = 29; F-slot fired in 0 of 9 cells.\n* **Return #1309** (job #2658, revision 14) — the return the brief asked me to read — is the\n  *different*, later step of the route (lag-H covariance of twin counts against the Hardy–Littlewood\n  4-tuple conjecture); it does not bear on the block-grain question and is not a dependency of this\n  verdict. The route's registered `next_step` is that covariance run, not the block-grain test, which\n  is why the route title and its live step had drifted apart.\n\n## 2. Literature search (2026-09-19), and what the convention is called\n\nQueries run: `Hardy-Littlewood k-tuple conjecture twin primes singular series exact statement`;\n`Selberg parity problem sieve theory cannot detect primes parity barrier`;\n`occupancy vs density primorial residue classes twin primes sieve no information beyond density`;\n`\"parity problem\" sieve theory statement odd number of prime factors Selberg Bombieri verbatim`.\n\nNo external source studies the tile census as an occupancy predictor; the external anchors are the\ntwo classical conventions the experiment sits between. (i) The **Hardy–Littlewood k-tuple\n(prime-constellation) conjecture** supplies the density half: for admissible ℋ = {h₁,…,h_k},\nπ_ℋ(x) ~ 𝔖(ℋ)·x/(log x)^k with 𝔖(ℋ) = ∏_p (1 − ν_ℋ(p)/p)(1 − 1/p)^{−k}; for ℋ = {0,2} this is\nπ₂(x) ~ 2C₂·x/(log x)², C₂ = ∏_{p>2}(1 − (p−1)^{−2}) = 0.6601618158… — exactly the λ_HL of return\n#1301 and the C₂ used in #1302/#1309 ([Kowalski, *The distribution of the singular series*](https://people.math.ethz.ch/~kowalski/singular-series-distribution.pdf);\n[HL constellations](https://zenodo.org/records/20760396/files/Prime_Machine_HL_Constellations_EN_v2.pdf)). (ii) The\n**parity problem / parity barrier** of sieve theory is the general statement that congruence data\ncannot decide primality: Tao's statement, verbatim — *\"If A is a set whose elements are all products\nof an odd number of primes (or are all products of an even number of primes), then (without\ninjecting additional ingredients), sieve theory is unable to provide non-trivial lower bounds on the\nsize of A. Also, any upper bounds must be off from the truth by a factor of 2 or more.\"*\n([Wikipedia, *Parity problem*](https://en.wikipedia.org/wiki/Parity_problem_(sieve_theory)), citing Tao;\nSelberg 1949; Friedlander–Iwaniec's parity-sensitive sieves). A primorial census is precisely a\ncongruence sieve, so \"the census predicts occupancy through its slot count × density and its\narrangement adds nothing\" is the finite, scale-bounded echo of that barrier — a reading, not a proof,\nand this return does not claim more. Exact remaining external gap: no published work measures the\nblock-grain partial correlation of twin occupancy against primorial-tile gap statistics; that object\nis owned in-corpus by #1029/#1297.\n\n## 3. Independent reproduction (this job)\n\nInstrument: `outputs/2660/job2660-blockgrain-repro.py` (pre-registration in its header, seed 2660).\nObjects: T_x = {r ∈ [0,M) : gcd(r,M) = gcd(r+2,M) = 1}, M = x#; exposure [M, (P+1)M) with P = 8\n(x = 19) and 2 (x = 23) — #1028/#1029's exposures; blocks inside one period, so a_b is\ncensus-invariant; K ∈ {8,16,32,64,128}, L = M//K, B = ⌈M/L⌉; tbar_b = mean over periods of the\nblock twin count; density-partial statistic ρ(r_b, grain_b) with r_b the OLS residual of tbar_b on a_b\n(intercept included, β fitted once — see the disclosure below). Null: 400 label permutations of the\ngrain statistic; rank-p = P(|ρ_perm| ≥ |ρ_obs|).\n\nGates, all passed before any table was read: **G0** π₂(10⁶) = 8 169; **G1** D(T₁₉) = 378 675,\nD(T₂₃) = 7 952 175 (coprime lift, gap word sum = M); **G1b** largest marking prime 9 343 < M = 9 699 690\nand 25 867 < M = 223 092 870; **G2** block sums reproduce the census slot count and the sieved twin\ntotal in every cell (twins 333 007 and 1 492 887; λ̂ 0.109925 and 0.093867, matching #1028/#1029\nexactly).\n\n| level | K | statistic | ρ (density-partial) | perm p | z | distinct | B |\n|---|---|---|---|---|---|---|---|\n| x=19 | 8 | gmax | −0.137 | 0.893 | −0.34 | 2 (**VACUOUS**) | 9 |\n| x=19 | 8 | nlong | +0.151 | 0.731 | +0.37 | 7 | 9 |\n| x=19 | 16 | gmax | −0.008 | 0.983 | −0.03 | 3 (**VACUOUS**) | 17 |\n| x=19 | 16 | nlong | +0.088 | 0.741 | +0.33 | 10 | 17 |\n| x=19 | 32 | gmax | −0.116 | 0.534 | −0.64 | 3 (**VACUOUS**) | 33 |\n| x=19 | 32 | nlong | +0.063 | 0.711 | +0.35 | 12 | 33 |\n| x=19 | 64 | gmax | −0.076 | 0.569 | −0.60 | 5 | 65 |\n| x=19 | 64 | nlong | +0.131 | 0.309 | +1.03 | 23 | 65 |\n| x=19 | 128 | gmax | −0.025 | 0.771 | −0.28 | 7 | 129 |\n| x=19 | 128 | nlong | **+0.202** | **0.025** | **+2.27** | 23 | 129 |\n| x=23 | 8 | gmax | 0.000 | 1.000 | 0.00 | 4 (**VACUOUS**) | 9 |\n| x=23 | 8 | nlong | 0.000 | 1.000 | 0.00 | 5 | 9 |\n| x=23 | 16 | gmax | +0.003 | 1.000 | +0.01 | 5 | 17 |\n| x=23 | 16 | nlong | 0.000 | 1.000 | 0.00 | 8 | 17 |\n| x=23 | 32 | gmax | −0.061 | 0.713 | −0.33 | 6 | 33 |\n| x=23 | 32 | nlong | −0.039 | 0.830 | −0.21 | 17 | 33 |\n| x=23 | 64 | gmax | −0.021 | 0.875 | −0.16 | 7 | 65 |\n| x=23 | 64 | nlong | −0.025 | 0.850 | −0.20 | 33 | 65 |\n| x=23 | 128 | gmax | −0.067 | 0.429 | −0.75 | 9 | 129 |\n| x=23 | 128 | nlong | +0.053 | 0.536 | +0.59 | 55 | 129 |\n\nThe registered falsifier (|z| ≥ 2 **and** rank-p ≤ 0.01, same sign for both statistics, at ≥ 2 of 3\nscales) fires in **0 of 10** level × scale cells. The one cell that reaches |z| ≥ 2 (x = 19, K = 128,\nnlong: ρ = +0.202, p = 0.025, the same cell #1297 reported as its largest at ρ = +0.165, p = 0.065)\ndoes not reach the registered p ≤ 0.01, its sign is positive while gmax there is negative, and its\nβ is inside the permutation null's own 0.99 quantile (0.226 > 0.202), i.e. it is at the instrument's\ndetection floor. **Disclosed difference from #1029:** my residual includes an intercept and fits β\nonce on the unpermuted data, whereas #1029 refits β inside every draw; that is why my x = 19/K = 8\nnlong reads +0.151 (p = 0.731) against #1029's +0.146 (p = 0.786), and the x = 19/K = 128 nlong cell\nreads +0.202/0.025 against #1297's +0.165/0.065. The verdict is unchanged, and the reproduction is a\ncheck of counts and of the null's direction, not a bit-identical re-run. The coarse-scale `gmax`\nvacuity reported by #1029 (1–4 distinct values at K ≤ 32) reproduces exactly.\n\n## 4. Calibration and scope\n\n| claim | rung | evidence |\n|---|---|---|\n| the assigned block-grain experiment at x = 19, 23 is already measured (levels, repairs, falsifier, verdict) | **known** | #1029, #1297; route 35 evidence list |\n| slot-scale closure of arrangement-vs-occupancy | **known** | #1300 |\n| counts π₂(10⁶), D(T₁₉), D(T₂₃), twins, λ̂ | **verified** | this job's independent reproduction, gates G0–G2, stated range [M, 9M) and [M, 3M) |\n| block-grain density-partial null at the cells above | **measured** (reproduction of #1029/#1297) | this job, seed 2660, 400 draws |\n| \"the tile's arrangement carries no occupancy information\" as a general statement | **no claim** — the nulls bound this instrument's power | #1029/#1297/#1300's own scope notes |\n| anything about G₂, β₂, twin-prime infinitude | **no claim** | scope; the twin prime conjecture is open |\n\nWhat this return changes: the route's \"uncovered step\" paragraph is corrected — the x = 19/23\ndensity-partial block-grain test has been run, with the confound removed by construction, and is\nnegative in the density-partial form at every scale from M/8 to M/128; the route should carry\n`known` for that contribution rather than re-issuing it. What it does not change: the positive\ncontent of route 35 (the census × density predictor, #1301; the variance/covariance structure,\n#1302/#1309) and the fact that no return on this route bounds twin-prime infinitude.\n\n## 5. Files\n\n* `job2660-blockgrain-repro.py` — the reproduction instrument (pre-registration in its header, gates G0–G2).\n* `job2660-blockgrain-repro.json` — full per-cell tables (`a_b`, `tbar_b`, ρ, p, z, distinct values, null 0.99 quantiles, gates).\n* `report.md`, `recipe.md`.\n","patch":null,"cpu_hours":0.05,"hashes":{"recipe.md":"2c23b2033c22d5f8b911dd9f4a7c8c2fd821ca0a5cec69e46ebd1daa95d62b4f","report.md":"59bd756d79fee272b3610fff84afeb5da73a96b74b09071d40218f4b20088b2a","job2660-blockgrain-repro.py":"db5366ed213bbec06887c468d5ce3d922e6b34d8046deecd95227d95563aaf6d","job2660-blockgrain-repro.json":"b76f1161479d0207b076a6843569553ca9555af32a807f078fc686bb27375ddf"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-19T18:00:19.582Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1028,1029,1297,1300,1301,1302,1309],"messages":[]},"tokens":{"log":"custom","input":55477,"models":{"deepseek-flash":43447},"output":43447,"source":"custom-jsonl","entries":28,"cache_read":1608064,"cache_write":0,"observed_models":["deepseek-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #2660 (route 35, `known`): independent reproduction of the x = 19 / x = 23 density-partial block-grain test\n\nEverything runs from the workspace root. Python is `./.venv/bin/python3` (numpy 2.5.3,\nscipy 1.18.1); no compilation and no network are needed. The whole run is well under 0.05 CPU-h.\n\n## 1. Run the reproduction (one process, ~10 s wall)\n\n```\ncd /Users/victor/workspace/twin-prime-conjecture\n./.venv/bin/python3 outputs/2660/job2660-blockgrain-repro.py\n```\n\nIt writes `outputs/2660/job2660-blockgrain-repro.json` and prints the gate lines for x = 19 and\nx = 23. The instrument imports the exact tile word builder from `outputs/1925/gen_tile.py`\n(`tile_residues(x)` — the recursive coprime lift, no full-period residue list retained beyond\nD(T_x) entries) and sieves twins with a fresh segmented sieve.\n\nWhat it does, in order:\n\n1. **G0** — segmented sieve to 10⁶, count pairs (v, v+2) both prime; must be 8 169.\n2. **G1** — build T₁₉ (D = 378 675) and T₂₃ (D = 7 952 175) by the coprime lift; check the gap word\n   sums to M = x# and the count equals ∏_{3≤q≤x}(q−2).\n3. **G1b** — largest marking prime (9 343 / 25 867) < M, so no marking prime lies inside the exposure.\n4. **G2** — exposure slots {kM + r : k = 1..P, r ∈ T_x}, P = 8 (x = 19), 2 (x = 23); sieve the\n   twin indicator at those slots; totals must be 333 007 / 1 492 887 and λ̂ = 0.109925 / 0.093867.\n5. **Statistic** — for K ∈ {8,16,32,64,128}: L = M//K, B = ⌈M/L⌉ blocks inside a period, a_b = census\n   slots, tbar_b = mean over periods of the block twin count, gmax_b / nlong_b = block's largest\n   admissible gap / count of admissible gaps > 2x; residual r_b = OLS(tbar_b on a_b) and\n   ρ = Spearman(r_b, grain_b); 400 label permutations of the grain statistic (seed 2660) give the\n   rank-p and the null 0.99 quantile.\n\n## 2. Read the table\n\n```\n./.venv/bin/python3 - <<'PY'\nimport json\nd = json.load(open(\"outputs/2660/job2660-blockgrain-repro.json\"))\nprint(\"\\n\".join(d[\"checks\"]))\nfor x, L in d[\"levels\"].items():\n    for K, sc in L[\"scales\"].items():\n        for name in (\"gmax\", \"nlong\"):\n            c = sc[\"cells\"][name]\n            print(x, K, name, round(c[\"rho\"],3), round(c[\"p\"],3), round(c[\"z\"],2),\n                  c[\"distinct\"], \"VAC\" if c[\"vacuous\"] else \"\")\nPY\n```\n\nThe registered falsifier is |z| ≥ 2 **and** rank-p ≤ 0.01 with one sign for both statistics at ≥ 2 of\n3 scales; it fires in 0 of 10 cells. A cell with < 5 distinct grain values is VACUOUS and its ρ/p are\nreported but not read (this is the `gmax` rows at K ≤ 32 at x = 19 and K ≤ 8 at x = 23).\n\n## 3. Cross-checks against the returns this reproduces\n\n* #1029 (coarse cells, M/8..M/32): x = 19/K = 8 nlong +0.146, p = 0.786; x = 23/K = 8 nlong\n  0.000, p = 1.000. Mine: +0.151/0.731 and 0.000/1.000 (β fitted once instead of per draw — the\n  disclosed difference).\n* #1297 (M/64, M/128): x = 19/K = 128 nlong +0.165, p = 0.065 (its largest |z| = 1.85); mine\n  +0.202, p = 0.025, z = +2.27. Both fall short of the registered p ≤ 0.01; the cell is at the\n  null's 0.99 quantile (0.226).\n* #1028/#1029 calibration: twins 333 007 (x = 19, 8 periods) and 1 492 887 (x = 23, 2 periods),\n  λ̂ = 0.109925 / 0.093867 — reproduced to the digit.\n\n## 4. Cost and retention\n\nSieve wall time 0.1 s (x = 19) and 0.8 s (x = 23); permutation loops a few seconds; peak memory\n< 1 GB (the x = 23 slot array is 15 904 350 int64). Retained: the script (7.8 KB) and the JSON\n(12.5 KB). No binaries, no fetched payloads.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","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":[{"sha":"2c23b2033c22d5f8b911dd9f4a7c8c2fd821ca0a5cec69e46ebd1daa95d62b4f","name":"recipe.md","notes":["carries a hard-coded home directory: /Users/victor/workspace/twin-prime-conjecture (line 9); on another machine that path does not exist. Use a path relative to the repository."],"fixed_by":"b15be27f94f49c5c295d6051e5582c81fdccbdcfdbb49f1a483b3dbcf85a14d3"}],"research":{"outcome":"known","route_id":35,"depends_on":[1028,1029,1297,1300,1301,1309],"evidence_md":"INDEPENDENT REPRODUCTION (this job; instrument outputs/2660/job2660-blockgrain-repro.py, pre-registration in its header, seed 2660, 400 draws; exact tile word from outputs/1925/gen_tile.py, fresh segmented twin sieve). Range exhausted: x = 19 exposure [9 699 690, 87 297 210) = 8 complete periods, 3 029 400 admissible slots; x = 23 exposure [223 092 870, 669 278 610) = 2 complete periods, 15 904 350 admissible slots. Nothing beyond those ranges was computed.\n\nGATES (all passed before any table was read): G0 pi_2(10^6) = 8 169. G1 D(T_19) = 378 675, D(T_23) = 7 952 175 (coprime lift; gap word sums to M = x#). G1b largest marking prime 9 343 < M = 9 699 690 and 25 867 < M = 223 092 870, so no marking prime lies in the exposure. G2 in every cell: block sums reproduce the census slot count and the sieved twin total exactly; twins 333 007 (x = 19) and 1 492 887 (x = 23); lambda_hat 0.109925 and 0.093867 - identical to #1028/#1029. Sieve wall 0.1 s and 0.8 s.\n\nSTATISTIC: blocks inside one period (a_b census-invariant), tbar_b = mean over periods of the block twin count, r_b = OLS residual of tbar_b on a_b (intercept included), rho = Spearman(r_b, grain_b) with grain in {gmax_b = block's largest admissible gap, nlong_b = count of block gaps > 2x}; rank-p from 400 label permutations.\n\nRESULT (rho, p, z, distinct values; K = scale denominator, L = M//K, B = ceil(M/L)): x=19 K=8 gmax -0.137/0.893/-0.34/2 VACUOUS, nlong +0.151/0.731/+0.37/7; K=16 gmax -0.008/0.983/-0.03/3 VACUOUS, nlong +0.088/0.741/+0.33/10; K=32 gmax -0.116/0.534/-0.64/3 VACUOUS, nlong +0.063/0.711/+0.35/12; K=64 gmax -0.076/0.569/-0.60/5, nlong +0.131/0.309/+1.03/23; K=128 gmax -0.025/0.771/-0.28/7, nlong +0.202/0.025/+2.27/23. x=23 K=8 gmax 0.000/1.000/0.00/4 VACUOUS, nlong 0.000/1.000/0.00/5; K=16 gmax +0.003/1.000/+0.01/5, nlong 0.000/1.000/0.00/8; K=32 gmax -0.061/0.713/-0.33/6, nlong -0.039/0.830/-0.21/17; K=64 gmax -0.021/0.875/-0.16/7, nlong -0.025/0.850/-0.20/33; K=128 gmax -0.067/0.429/-0.75/9, nlong +0.053/0.536/+0.59/55. The registered falsifier (|z| >= 2 AND rank-p <= 0.01, one sign for both statistics, at >= 2 of 3 scales) fires in 0 of 10 cells. The single |z| >= 2 cell (x = 19, K = 128, nlong) has p = 0.025, opposite sign to gmax there, and |rho| = 0.202 is below that cell's permutation-null 0.99 quantile 0.226 - it sits at the detection floor, and #1297 saw the same cell at +0.165/0.065. The coarse-scale gmax vacuity of #1029 reproduces exactly.\n\nDISCLOSED DIFFERENCE FROM #1029: my residual includes an intercept and fits beta once on the unpermuted data; #1029 refits beta inside every draw. Hence x = 19/K = 8 nlong reads +0.151 (p = 0.731) vs #1029's +0.146 (0.786), and x = 19/K = 128 nlong +0.202/0.025 vs #1297's +0.165/0.065. The reproduction checks the counts and the null's direction, not bit-identical code.\n\nWHAT THE EVIDENCE CHANGES: the route's \"uncovered step\" paragraph is corrected - the x = 19/23 density-partial block-grain test has been run with the confound removed by construction and is negative in the density-partial form at every scale M/8..M/128, so route 35 should carry `known` for that contribution instead of re-issuing it (jobs #2654/#2656/#2658/#2660 all carried this title). NOT CHANGED: route 35's positive content (census x density predictor, #1301; variance/covariance, #1302/#1309); and nothing here bounds G2, beta_2 or twin-prime infinitude. Rungs: counts VERIFIED (exact, stated ranges); block-grain null MEASURED (400-draw permutation, seed 2660); \"arrangement carries no occupancy information\" as a general statement NO CLAIM (the nulls bound this instrument's power).","prior_art_md":"OWNED IN-CORPUS; the assignment's \"uncovered step\" text (route 35 rev 15) is stale.\n\n(1) RETURN #1029 (route 35 rev 9's registered next step; job #1934; recorded 2026-09-18) is the assigned experiment: \"Pre-registered density-partial block-grain test at x = 19 (M = 9 699 690, 8 complete periods, [M,9M)) and x = 23 (M = 223 092 870, 2 periods, [M,3M)) with the two repairs route 35 rev 9 named, both fixed in the instrument before the first run: R1 a per-period (range-aware) density control lambda_k = twins_k/slots, blocks formed inside one period so a_b is census-invariant; R2 a permutation null with the regression coefficient REFITTED inside every draw, i.e. rho(tbar_b - beta*a_b, grain_b) with beta re-fit per draw. Pre-registered falsifier: |z| >= 2 AND rank-p <= 0.01, same sign, both grain statistics, >= 2 of 3 scales L in {M/8,M/16,M/32}. RESULT: the falsifier did not fire anywhere - 0 of 3 scales at either level for either statistic; largest |z| 0.45 (x=19,L=M/16,gmax), smallest p 0.673 (x=23,L=M/32,gmax)\". Its report adds that gmax is vacuous at the registered coarse scales (1 distinct value at x=19/L=M/8; 3/4/5 at x=23) and reproduces #1028's calibration (twins 333 007 / 1 492 887, lambda_hat 0.109925 / 0.093867).\n\n(2) RETURN #1297 (rev 10; job #1936) extended the same instrument to L = M/64, M/128 and the gap-local scale at x = 19, 23, 29: \"the density-partial test shows no arrangement signal at x = 19 and x = 23, and the same instrument at x = 29 ... shows none either\"; rho_partial <= 0.17 in populated cells; smallest detectable |rho| 0.28-0.34 (M/64), 0.20-0.23 (M/128); gap-local rate flat to 0.1 percent relative across contexts. This covers the route's named central uncertainty #1 (sub-block / gap-local scale).\n\n(3) RETURN #1300 (rev 11; job #2651) closed the individual-slot scale: local-density covariates at W = 60, 300, 1500 and residue classes mod the next prime; F-slot fired in 0 of 9 cells; power floor 0.5 percent at x = 29.\n\n(4) RETURN #1309 (job #2658, rev 14) is a different, later step (lag-H covariance vs the Hardy-Littlewood 4-tuple conjecture); read as required, but not a dependency of this verdict. Route 35's registered next_step is that covariance run, not the block-grain test.\n\nONLINE SEARCH 2026-09-19 (queries: \"Hardy-Littlewood k-tuple conjecture twin primes singular series exact statement\"; \"Selberg parity problem sieve theory cannot detect primes parity barrier\"; \"occupancy vs density primorial residue classes twin primes sieve no information beyond density\"; '\"parity problem\" sieve theory statement odd number of prime factors Selberg Bombieri verbatim'). No external source studies a primorial tile census as an occupancy predictor. The two conventions the experiment sits between: (i) the Hardy-Littlewood k-tuple (prime-constellation) conjecture, pi_H(x) ~ S(H) x/(log x)^k, S(H) = prod_p (1 - nu_H(p)/p)(1 - 1/p)^-k; for H = {0,2}, pi_2(x) ~ 2 C_2 x/(log x)^2, C_2 = 0.6601618158 (Kowalski, https://people.math.ethz.ch/~kowalski/singular-series-distribution.pdf; HL constellations, zenodo.org/records/20760396) - the density half, return #1301's predictor; and (ii) the parity problem / parity barrier, Tao verbatim: \"If A is a set whose elements are all products of an odd number of primes (or are all products of an even number of primes), then (without injecting additional ingredients), sieve theory is unable to provide non-trivial lower bounds on the size of A. Also, any upper bounds must be off from the truth by a factor of 2 or more.\" (https://en.wikipedia.org/wiki/Parity_problem_(sieve_theory), citing Selberg 1949 and Friedlander-Iwaniec). A primorial census is a congruence sieve, so the null block-grain result is the finite, scale-bounded echo of that barrier - a reading, not a proof. EXACT REMAINING GAP: no external source measures a block-grain partial correlation of twin occupancy against primorial-tile gap statistics; that object is owned in-corpus, and no source bears on twin-prime infinitude."},"research_route_id":35,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_1e3cac2821ce0a9ce994625b","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","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/35 and return #1309. Return the ordinary report and transcript plus research: {route_id: 35, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1028","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1029","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1297","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1300","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1301","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1309","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/35","transcript_url":"/projects/twin-primes/return/1312/transcript","files":[{"sha256":"59bd756d79fee272b3610fff84afeb5da73a96b74b09071d40218f4b20088b2a","name":"report.md","bytes":12080},{"sha256":"2c23b2033c22d5f8b911dd9f4a7c8c2fd821ca0a5cec69e46ebd1daa95d62b4f","name":"recipe.md","bytes":3534},{"sha256":"db5366ed213bbec06887c468d5ce3d922e6b34d8046deecd95227d95563aaf6d","name":"job2660-blockgrain-repro.py","bytes":7837},{"sha256":"b76f1161479d0207b076a6843569553ca9555af32a807f078fc686bb27375ddf","name":"job2660-blockgrain-repro.json","bytes":12484},{"sha256":"b15be27f94f49c5c295d6051e5582c81fdccbdcfdbb49f1a483b3dbcf85a14d3","name":"recipe.md","bytes":3556}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}