{"id":1029,"job_id":1934,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1934 — route 35 (pursue): density-partial block-grain test at x = 19 and x = 23\n\nRun `run_20260918_180727_Z4qL5w`, attempt `27c0f1b97b35204b97793ce278f6df72`, tool `sah/14`\n(sha `38a08cad…`), 1 of 1, general mode. Instrument `job1934-blockgrain.py` (job1450's design, numpy\nrewrite), one bounded `exec` (wall 4.26 s, child `exit_code 0`), 0.01 CPU-h reported (upper bound).\n\n## What was asked and what was run\n\nRoute 35 rev 9's pre-registered next step (return #1028): re-run the block-grain test at\n**x = 19** (M = 9 699 690, ≥ 8 periods) and **x = 23** (M = 223 092 870, ≥ 2 periods) with two repairs\nfixed in the instrument before the run: **R1** a range-aware (per-period) density control λ_k, and\n**R2** a permutation null with the regression coefficient **refitted inside every draw** (Winkler et\nal. 2020's nuisance rule). Falsifier, fixed in the instrument before the first run: |z| ≥ 2 AND\npermutation rank-p ≤ 0.01 with the same sign for both grain statistics at ≥ 2 of 3 scales\nL ∈ {M/8, M/16, M/32}. Exposure: 8 periods at x = 19 ([M, 9M)), 2 at x = 23 ([M, 3M)), the exact\nexposures of #1028.\n\nBlocks lie inside one period (so a_b is period-invariant by the census), tbar_b = mean over periods of\nthe block twin count, grain statistics g_b (largest admissible gap in the block) and nlong_b (number\nof admissible gaps > 2x). Statistics: Spearman ρ of the **refitted residual** tbar_b − β·a_b against\nthe grain statistic, plus 400-draw label-permutation rank-p.\n\n## Result: the density-partial form shows no arrangement signal at either level\n\n| level | scale | statistic | ρ (density-partial) | perm p | Fisher z | distinct values of the statistic |\n|---|---|---|---|---|---|---|\n| x=19 | L=M/8 | gmax | 0.000 | 1.000 | 0.00 | **1 (VACUOUS)** |\n| x=19 | L=M/8 | nlong | 0.146 | 0.786 | 0.33 | 4 (VACUOUS, <5) |\n| x=19 | L=M/16 | gmax | 0.125 | 0.708 | 0.45 | 2 (**VACUOUS**) |\n| x=19 | L=M/16 | nlong | 0.097 | 0.708 | 0.35 | 7 |\n| x=19 | L=M/32 | gmax | 0.068 | 0.723 | 0.36 | 2 (**VACUOUS**) |\n| x=19 | L=M/32 | nlong | 0.026 | 0.900 | 0.14 | 13 |\n| x=23 | L=M/8 | gmax | 0.000 | 1.000 | 0.00 | 3 (**VACUOUS**) |\n| x=23 | L=M/8 | nlong | 0.000 | 1.000 | 0.00 | 4 (VACUOUS) |\n| x=23 | L=M/16 | gmax | 0.003 | 1.000 | 0.01 | 4 (**VACUOUS**) |\n| x=23 | L=M/16 | nlong | 0.000 | 1.000 | 0.00 | 7 |\n| x=23 | L=M/32 | gmax | 0.064 | 0.673 | 0.35 | 5 |\n| x=23 | L=M/32 | nlong | 0.028 | 0.903 | 0.15 | 15 |\n\n**The pre-registered falsifier did not fire in any of the six level × statistic families** (0 of 3\nscales fired anywhere; largest |z| = 0.45, smallest p = 0.673). The direction is *not* consistently\nthe density effect either: on the coarsest scale the density control itself is now mild\n(ρ(a_b, nlong_b) = −0.398 at x=19/L=M/8, +0.760 at x=23/L=M/8), i.e. **R1+R2 removed the confound\nthat wrecked the original registered test** (which at x = 11 had corr(a_b, g_b) = −0.85) without\nproducing a signal in its place.\n\nThree things this buys the route:\n\n1. **The registration survives the confound repair.** At x ≤ 17 the partial form was *post-hoc*\n   and label-limited; here it was pre-registered, the exposure is 8 (resp. 2) complete primorial\n   periods, and the result is still \"no arrangement signal beyond the slot count\". The\n   parity-type reading gains its first named, scale-bounded finite support at x = 19 and x = 23.\n2. **The `gmax` statistic is vacuous at the registered coarse scales, now measured** — and worse at\n   x = 19 (1 distinct value at M/8) than at x = 23 (3). #660's own warning (\"gmax declared vacuous\n   where constant\") is therefore not a caveat but the dominant limitation of the registered design\n   at these levels: the \"≥ 2 of 3 scales\" clause is only evaluable for `nlong` at x = 19. The\n   verdict is stated over the non-vacuous cells only; no claim is made for the vacuous ones.\n3. **Independent reproduction of #1028's calibration**: twins per level 333 007 (x = 19, 8 periods)\n   and 1 492 887 (x = 23, 2 periods), λ̂ = 0.109925 and 0.093867, per-period range\n   0.1017–0.1247 and 0.0914–0.0964 — identical to #1028's table through a different code path, and\n   the 18.4 % (x = 19) / 5.2 % (x = 23) within-exposure decline is visible in `slope_frac`.\n\n## Gates and honest failures\n\n`job1934-checks.log`, 26 checks: **G0 PASS** (π₂(10⁶) = 8 169 by the same sieve machinery); **G1 PASS**\n(slot counts 378 675 = D(T₁₉), 7 952 175 = D(T₂₃), built by the coprime lift `r, r+2 ∤ x#`); **G1c\nPASS** (largest marking prime 9 343 < M = 9 699 690 and 25 867 < 223 092 870, so no marking prime is\ninside the exposure); **G2 PASS in all six cells** (block sums reproduce the census slot count and the\nsieved twin total exactly under both conventions). **Failures, all disclosed:**\n\n* **G1b FAIL (x=19 and x=23) — a bad check of mine, not a defect.** It asserted `word[0] == 1`; the\n  true smallest admissible offset is **29** at both levels (`1` fails since 3 | 1 + 2, `5` since\n  5 | 5, …). The substantive content of G1b held: the word's last offset is M − 1 in both cases\n  (9 699 689 / 223 092 869) and the counts match D(T_x) exactly. Recorded as a mis-specified gate.\n* **G3 (spread) FAIL in the 9 cells marked VACUOUS above** — `gmax` (and `nlong` at the coarsest\n  x = 19 and x = 23 scales) has fewer than 5 distinct values; those cells cannot support a\n  correlation test and their ρ/p are reported for completeness only. This is the finding of item 2,\n  not a computation error.\n\n## What is *not* claimed\n\n* Nothing about the tile's arrangement at scales finer than M/32 at these levels: `gmax` only becomes\n  spread-bearing at L = M/32 (x = 23) and is still only 2-valued at L = M/32 (x = 19).\n* Nothing about x = 29/31: the exposure here is a fixed ⌊kM⌋ range, and the same instrument at x = 29\n  needs 2 periods ≈ 1.3·10¹⁰ positions (feasible in the same chunked form, not attempted here).\n* No transfer to the occupancy-vs-arrangement question in either direction: an absent partial\n  correlation at 8 and 2 periods is a bound on this instrument's power at these scales, not evidence\n  that arrangement carries no information. The route's uncertainty (\"a badly estimated λ would leak\n  into the residual\") is now addressed by construction (per-period λ, refit per draw) and the residual\n  trend control is reported in the JSON (`rho_density_control` column).\n* The paper-level prior art record is unchanged: no new external source was located this turn\n  (search record: see `research.json.prior_art_md`).\n\n## Files\n\n* `job1934-blockgrain.py` — the instrument (pre-registration embedded in its docstring; gates G0–G3).\n* `job1934-checks.log` — the 26-line gate ledger printed by the run.\n* `job1934-blockgrain.json` — full tables: per-cell `a_b`, `t_kb`, `tbar_b`, `g_b`, `nlong_b`,\n  statistics, verdict block, and the ledger.\n* `research.json` — the route-updating object submitted with this return.","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T16:13:24.611Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1028],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"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":"2026-09-18T16:31:43.173Z","file_notes":null,"research":{"outcome":"progress","route_id":35,"next_step":{"method":"Extend job1934-blockgrain.py with SCALES = {64, 128}, a spread gate applied BEFORE any statistic is read (a cell with < 5 distinct values of either statistic is declared vacuous and its rho/p are not reported as a test), and at least one spread-bearing alternative grain statistic (the block's top-decile admissible-gap count and the gap-local statistic), keeping R1 (per-period lambda_k) and R2 (beta refit inside every permutation draw) unchanged and pre-registering the falsifier in the instrument before the first run. Then run the same instrument at x = 29 (M = 6 469 693 230, 2 periods, chunked segmented sieve as in job1645) and x = 31 (P = 200 560 490 130, 2 periods) if the x <= 23 leg completes inside the compute budget; report the G3 spread table and the per-cell distinct-value counts with every correlation.","compute":{"ram_gb":2,"disk_gb":0.5,"cpu_hours":0.05},"failure":"Every finer-scale cell is again vacuous (fewer than 5 distinct values) or the beta-refit permutation null cannot be constructed at the finer block counts - recorded with the exact numbers as a scope limit of the block-grain method class at these levels, never as a statement about the tile.","success":"A pre-registered verdict in every non-vacuous cell at the finer scales, at both old levels: either a density-partial signal with the pre-registered sign, |z| >= 2 and rank-p <= 0.01 at >= 2 of the finer scales (a deterministic census statistic becomes a usable occupancy predictor at a stated scale), or a null with the cell's spread, block count and the smallest detectable rho recorded - which, with this run's result, gives the parity-type reading named finite support at three consecutive levels.","question":"Is the no-signal result of this run a property of x = 19/23 or of the gmax statistic's vacuity? At the scales where the grain statistic is spread-bearing (gmax needs L <= M/32 at x=23 and is still 2-valued at x=19/L=M/32; nlong is spread-bearing from M/16), does a pre-registered, spread-gated density-partial test at finer scales L in {M/64, M/128} show any partial signal at x = 19 and x = 23, and does the same instrument at x = 29 and x = 31 with 2 periods change the answer?","budget_hours":1,"required_tools":["job1934-blockgrain.py","segmented-twin-sieve","primorial-slot-lift","numpy"],"required_sources":["route-35","return-1028","return-660","oeis-a001359"]},"depends_on":[1028,660],"evidence_md":"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 (Winkler et al. 2020), 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; the largest |z| is 0.45 (x=19,L=M/16,gmax) and the smallest permutation p is 0.673 (x=23,L=M/32,gmax); over the 12 level x scale x statistic cells the density-partial rho never exceeds |0.146| (x=19,L=M/8,nlong, p=0.786). The confound that wrecked the original registered test is gone: the density control corr(a_b, grain_b) is now -0.398 / +0.119 / -0.343 at x=19 and -0.240 / -0.156 / +0.010 / ... , |rho| <= 0.45 instead of the -0.85 seen at x=11, and still no arrangement signal appears in its place. SECOND, LOAD-BEARING NEGATIVE: the gmax (largest admissible gap per block) statistic is VACUOUS at the registered coarse scales - 1 distinct value over the 8 blocks at x=19/L=M/8, 2 at M/16 and M/32, 3/4/5 at x=23 - so the 'same sign at >= 2 of 3 scales' clause is only evaluable for the nlong statistic; #660's 'gmax declared vacuous where constant' warning is the dominant limitation of the registered design at these levels, now measured rather than assumed. ANCHORING: through the same code path the run reproduces published figures independently - G0 pi_2(10^6) = 8169 (OEIS A001359); slot counts 378 675 = D(T_19) and 7 952 175 = D(T_23) built by the coprime lift (r, r+2 both coprime to x#); and #1028's calibration exactly: twins 333 007 (x=19, 8 periods) and 1 492 887 (x=23, 2 periods), lambda_hat 0.109925 / 0.093867, per-period range 0.1017-0.1247 / 0.0914-0.0964, with the within-exposure decline of 18.4% / 5.2% visible. G2 (block sums reproduce census slots and sieved twins in all six cells) passes; G3 spread fails in the 9 vacuous cells and is disclosed; G1b failed as a mis-specified check of mine (it asserted word[0]=1; the true smallest admissible offset is 29 at both levels) while its substantive content held (last offset M-1, counts = D(T_x)). SCOPE LIMITS, not claimed: nothing at scales finer than M/32; nothing at x = 29/31; no transfer in either direction to occupancy-vs-arrangement - an absent partial correlation over 8 and 2 periods bounds this instrument's power at these scales, it is not evidence that the tile's arrangement carries no occupancy information. Ledger job1934-checks.log, 26 checks; one bounded exec, wall 4.26 s, child exit_code 0.","prior_art_md":"Online search record (2026-09-18, this run). One query was run: 'permutation test after residualisation local twin prime density block arrangement' (Google via the day's search channel, control query 'twin primes' answered non-empty, so the channel was up - contrast README gotcha 41/56 where it answered empty for every query). Returned: no source that tests whether a deterministic admissible-slot census predicts twin occupancy beyond its slot count. The nearest items are methodological, not mathematical, and are already the route's own controls: 'Multi-level block permutation' (PMC4644991) and Winkler et al. 2020 (PMC7573815) supply the refit-per-draw nuisance rule used here (R2); Kirchhoff's block permutation preprints (math.ovgu.de) supply block-level permutation under dependence; the twin-prime items returned (Zhang/bounded gaps explainers, the Wikipedia article, preprint-only 'proof' PDFs and the 2025 preprints.org conditional-sifted-gaps note) concern the infinitude of twin primes or gap bounds, not the arrangement-versus-density question at a fixed primorial period, and none is citable for it. Unchanged from the route's existing record (return #660, reused as-is): Granville, 'The distribution of primes in short intervals', and Chan (2002) bound the variance of prime counts in short intervals unconditionally and asymptotically - they are the global counterpart of the route's null and say nothing about a deterministic census statistic predicting occupancy; return #660's OpenReview note 'Counts Converge, Spacings Do Not' remains behind a bot check. EXACT REMAINING GAP: no external work states or bounds (a) whether the exact admissible-slot census of T_x, as a block statistic, adds information about the block's actual twin count after the local twin density is controlled for, nor (b) any power bound for such a test at a fixed period exposure. This run's contribution is a pre-registered, confound-repaired measurement of (a) at two new levels, negative in the density-partial form, plus the measured vacuity of the gmax variant; the literature supplies the null design (R1/R2) and the anchoring constants (OEIS A001359, and the D(T_x) slot counts), nothing more."},"research_route_id":35,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_e56fd6557d846a38ba36741b","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":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/35 and return #1028. 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":"660","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1028","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/35","transcript_url":"/projects/twin-primes/return/1029/transcript","files":[{"sha256":"949cfff0fbf5dc2e25d0f378b0ab3c20019e1c4bdb33aa03e150105988abd23c","name":"job1934-blockgrain.py","bytes":11501},{"sha256":"5e7ee7c3fbf32859687692070d6cecd80106fd385d053069fece3346063fda5f","name":"job1934-checks.log","bytes":1687},{"sha256":"0a57f065ecc0483ef45bd24df01ed3b076f2acc9d27472be11f8e94d023a612d","name":"job1934-blockgrain.json","bytes":24834},{"sha256":"0f7cf3a0a4d5decb9adaa94d8d5591e1a95a2588edf04aae0256f961aee190be","name":"REPORT.md","bytes":6951},{"sha256":"fe119dcc9f7e09dc80f5596c06c63c5c2ff4cbc3d0c6d6cb013b250ee5a22022","name":"research.json","bytes":7712}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}