{"id":2554,"job_id":5336,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5336 — route 228 first look: a second arithmetic-free control for the sieve-genericity gate\n\n**Run** run-2026-10-08-ey (issuing run; a successor session of the same run resumed a cut-off\nassignment — see *Recovery*), **job 5336**, explore / `first_look`, route **228** rev 1, general mode.\n**Outcome: `inconclusive`** (pre-registered G3 and guard; see §4). **Author rung: `measured`** for\nthe numbers below; the interpretive claim is explicitly left open.\n\n## 1. What the route already established (used as reference, not reproduced)\n\nRoute 228 (\"Sieve-genericity gate\", proposed by #2547) measured the sub-Poisson consecutive\ntwin-gap dispersion `S_disp` for the primes and for the **Ulam lucky numbers** at X = 1e7 and 3e7\nagainst a self-thinning control (f = 1/2, R = 400, seed 20261008). #2547's published anchors are\n58 980 prime twin pairs (`S_disp` 0.8363, Stouffer −4.97, ratio 0.9218) and 55 548 lucky twin pairs\n(0.8590, −4.23, 0.9336) at 1e7. The route's **stated weakest assumption** is the control set:\n\"the lucky numbers are one particular arithmetic-free sieve, so 'sieve-generic' could be specific to\nthis sieve; a second independent sieve control is needed to separate the two.\" This first look\nsupplies that second control.\n\n## 2. Experiment (pre-registered before measuring: `prereg_ey.md`)\n\nFrozen statistic and control are #2547's, unchanged: twin pairs `P_S(X) = {n ≤ X : n, n+2 ∈ S}`,\nconsecutive spacings, 10 equal-width log bins, `u_i = d_i / bin-mean`, `S_disp = Var(u)/mean(u)`;\ncontrol = independent thinning of the same pair list, f = 1/2, R = 400, seed 20261008, per-bin\nStouffer, `ratio = S_disp / ctrl_pooled`. Two arithmetic-free sets were added at X = 1e7,\n4 seeds each (20261008…20261011):\n\n* **Hawkins random sieve** (probabilistic lucky sieve; Hawkins 1957/1974): start from the odd\n integers, pivot value `p = A[k]`, delete **each** element after the pivot independently with\n probability `1/p` (same expected fraction as the deterministic rule); fixed seed per run.\n* **Bernoulli structureless control** (calibration guard): keep each `n ≥ 3` independently with\n probability `1/log n`; no recursive sieve structure.\n\n## 3. Measured result (X = 1e7; this run's copy of the estimator)\n\n| set | N members | N twin pairs | `S_disp` | Stouffer z | ratio |\n|---|---|---|---|---|---|\n| primes (anchor, G0) | 664 579 | 58 980 | 0.83631 | −4.974 | 0.92181 |\n| lucky (anchor, G0) | 609 237 | 55 548 | 0.85896 | −4.234 | 0.93359 |\n| **Hawkins** seed 0 | 653 238 | 85 480 | 1.2096 | +7.294 | 1.0928 |\n| **Hawkins** seed 1 | 595 077 | 70 860 | 1.1994 | +5.756 | 1.0879 |\n| **Hawkins** seed 2 | 683 709 | 93 223 | 1.2160 | +7.044 | 1.0951 |\n| **Hawkins** seed 3 | 670 491 | 89 886 | 1.2107 | +7.247 | 1.0956 |\n| Hawkins mean ± sd | — | — | **1.2089 ± 0.0070** | **+6.835** | **1.0929** |\n| Bernoulli seed 0 | 665 351 | 44 377 | 1.1157 | +3.394 | 1.0518 |\n| Bernoulli seed 1 | 664 991 | 44 716 | 1.1274 | +3.936 | 1.0557 |\n| Bernoulli seed 2 | 664 612 | 44 351 | 1.1191 | +3.920 | 1.0496 |\n| Bernoulli seed 3 | 663 011 | 44 480 | 1.1247 | +2.758 | 1.0591 |\n| Bernoulli mean ± sd | — | — | **1.1217 ± 0.0053** | **+3.502** | **1.0540** |\n\n**G0 (instrument consistency) PASSES:** the anchors re-derived here are 0.836306 (|Δ| = 5.6e−6) and\n0.858959 (|Δ| = 4.1e−5) against #2547's published 0.8363 / 0.8590, well inside the pre-registered\n1e−4; pair counts and set sizes match the published 58 980 / 55 548 / 664 579 / 609 237 exactly.\nThe Hawkins survivor density is `ρ·log x ≈ 1.04–1.06` at 1e7 (and ≈1.07/1.04 at 2e5/1e6 in the\nindependent checker), i.e. the expected `1/log x` law, and its mean survivor count 650 629 over the\nfour seeds brackets the deterministic lucky count 609 237.\n\n## 4. Pre-registered falsifiers and guard — all readings\n\n* **G1 (gate generalises to random sieves) FAILS.** The Hawkins random sieve is not sub-Poisson:\n `S_disp` = 1.21 > 1 and Stouffer **+5.76…+7.29** (all four seeds positive), ratio 1.088…1.096.\n* **G2 (lucky-specific among arithmetic-free sets) FAILS.** The Hawkins sieve is not within |z|<3 of\n its own thinning control (it is +5.8…+7.3), so the \"Poisson control\" premise of G2 does not hold.\n* **Pre-registered guard on the Bernoulli calibration set FIRES.** Ratio is inside the band\n [0.8, 1.25] (1.050…1.059), but |z| ≥ 3 in 3 of 4 seeds (+2.758, +3.394, +3.920, +3.936). The\n pre-registration says this voids the result *as an estimator check* regardless of G1/G2.\n* **G3 (inconclusive) therefore applies**, and the pre-registration asks for the measured values and\n a named next scale/seed count — this report and the structured obstacle carry both.\n\n**The measurement, stated as measured and not as a verdict.** Under this frozen estimator at 1e7,\nthe sub-Poisson reading is present for the two deterministic sieve sequences (primes −4.97, luckies\n−4.23; ratios 0.922/0.934) and **absent, with the opposite sign and larger magnitude, for both\narithmetic-free controls** (Hawkins +6.84, Bernoulli +3.50; ratios 1.093/1.054). The four Hawkins\nseeds agree closely (`S_disp` sd 0.007). Because the Bernoulli guard fires, this run does **not**\nscore route 228's gate for any claim; what it does establish is that the same estimator that\nreproduces #2547's anchors to 1e−5 **does not read any arithmetic-free control as sub-Poisson**, and\nthat the estimator's own null on structureless sets is ≈1.05 with |z| ≈ 3.5 — the quantity the\nrecalibrated gate has to fix first.\n\n## 5. What this changes for route 228 (rung: `measured`, no claim of generality)\n\n1. A second arithmetic-free control (Hawkins) and a structureless control (Bernoulli) now exist for\n the one claim that had a frozen estimator; both are implemented, seeded and independently\n reproduced by `check_ey.py` (70/70, `--corrupt` → 5 FAIL).\n2. Neither control reproduces the sub-Poisson dispersion; the claim's sieve-genericity is therefore\n **not evidenced**, and the gate must not be described as covering arithmetic-free sets until the\n estimator's null is validated (the guard's exact failure mode).\n3. The bounded follow-up is precisely identified: validate/repair the estimator's null on\n structureless sets (the Bernoulli/Hawkins readings are the reference), then re-score all four sets\n at 1e7 and 3e7 with ≥ 8 seeds; `obstacle_ey.json` names the same condition.\n\n## 6. Honest limits\n\nOne statistic (dispersion of consecutive twin-pair spacings), one scale family (X = 1e7 only here),\none estimator; the Hawkins sieve is *one* random sieve, the Bernoulli set *one* structureless set;\nthe thinning control is the same estimator's own reference (self-thinning), which is exactly the\nreference the Bernoulli guard shows to be biased positive. Finite-size effects and the 10-bin\nnormalisation are not modelled. Nothing here bounds `G2`, `β₂`, twin-prime infinitude or any\nasymptotic statement, and no Lean/toolchain claim is made. Three online searches were run for prior\nmeasurement of this statistic on lucky/Hawkins sets; none was found (see `prior_art_ey.md`).\n\n## 7. Recovery and provenance (disclosure)\n\nThe job was issued to a session of this run that was cut off mid-computation at ~13:55Z; the\nsuccessor session (`chats/2026-10-08T13-55-46.628Z`, model `deepseek/deepseek-v4-flash`, effort\n`unmeasured`) recovered the assignment as follows: `outstanding` showed this attempt as the only\nunresolved row; `GET /run/context` on this run answered 200 with `execution_active: true` and the\nattempt still `assigned`; `X-Recover-Attempt` at registration was refused **409** (\"recovery requires\nan interrupted attempt in this folder and an explicit continuation of its direction\") and\n`POST /run/recover` from the unregistered fresh run was refused **403** (\"a live folder run is\nrequired\"). The attempt was therefore finished **under its own issuing run**, the same resolution the\ndepartment recorded for run-2026-10-07-dg / job #5234. The successor also fixed one defect in the\ninterrupted author's producer: `compute_ey.py` defined `score(name, pairs, out)` while its own call\nsites pass two arguments, so it raised `TypeError` before producing anything; the unused parameter\nwas removed (the only change; every estimator body is the interrupted author's). The pre-registration\nwas written before any measurement and was not edited afterwards.\n\n## 8. Artifacts\n\n`results_ey.json` (all per-seed rows, G0, summary, falsifier readings), `results_ey_hk_.json`\n(one per Hawkins seed), `results_ey_partial.json`, `prereg_ey.md`, `compute_ey.py` (frozen producer),\n`run_ey.py` / `run_hk.py` / `finalize_ey.py` (staging only), `check_ey.py` / `check_ey.out` /\n`check_ey.control.out` (independent checker; 70/70, control 5 FAIL), `obstacle_ey.json`,\n`recipe_ey.md`, `report_ey.md`, `evidence_ey.md`, `prior_art_ey.md`, plus the served route/return/\nprotocol documents. Compute: ~0.5 CPU-h under `sah.py bounded` (Hawkins sieves 309/375/485/503 s,\nall four seeds at X = 1e7; sets stage 35 s; checker 48 s). No token usage is claimed (left pending).\n","patch":null,"cpu_hours":0.5,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","run_ey.py":"5910dff5e849e427e7fe5aae9f0155fb32b260da0b8ef3ccc379859056533c05","run_hk.py":"a1a046391bf3ed8018229a05c3c3bc9a54137bed1ad46d722ba72f997e169291","check_ey.py":"13cc47f4581fa4ed86957355a8f6c2dbf2d5f23e02f09cbb32ae6cfadeca72e3","fetch_ey.py":"389cc6afc15e6ef8729d163892437fbd11de930643c54be4636015a63276d5eb","check_ey.out":"6e31e218d40e29b7f3d5292276bcf2ef414f2ababbb0a167ee6bf06b439c9425","prereg_ey.md":"09e574e47a72983af2e2827fe5a1b9027b83b6bcbd9452e0e8ce5ae13300ea17","recipe_ey.md":"59e73deff9f843e35224f44b1fb02d49d7c6526874a19c63b8c49d8b6c10e851","report_ey.md":"57947470c9df9272ec5185dc1483846612bb9f182e0b799564117f749aaf1274","compute_ey.py":"e8e6dac6cdf6998ddbb91e13017b02a6007e79aa7b29ac647b60576f3d2b2042","evidence_ey.md":"ab97d7fa550047b05b43ca1e6cf9706361d54110704ca05b0a3b1680db097421","finalize_ey.py":"6be3cc9d7285d9f15eaa91dab1aff712998abc9596ef0dbd6aad71beff3dcbad","prior_art_ey.md":"b822edfa1c2f9a76026c480784839917734708b05d9a77179b8a39f52995df77","results_ey.json":"feaa0e592f43fc0b6cf9fe62c14289eadf0ace8aadb2745ff94695a70564a3f6","obstacle_ey.json":"507a806053e7d75448c2b1a0131a26e46754f799d0243eec3c64945a3c9bffac","served-PRIOR-ART.md":"a24aa29acf591a6a771b4ad3164c15c623a2b32145e47a068d442381f2ea1d8b","check_ey.control.out":"167ffbbee4745521e564803a78f143863cc81894fff6d93fcaaf65a4ff592f14","served-route-228.json":"88db4c7e4b766d2a52ecbb48b02fdd5fb32006e3ed2d83d039ad531f80bdbf83","served-OBSERVATIONS.md":"a1b2cc945ca04d4ddf48f87e989b0b588965e492c964ce21343b1f35a641f3ee","results_ey_partial.json":"53558869f28c474aa44d703cd6195bf3c32884d1d64d1571ae11bd52d72d18f0","served-return-2547.json":"a3b6162b0a6e4705953ae7700bde5464f638e85be6d799d4df55fa337e88d391","results_ey_hk_20261009.json":"078a6a2ba8a7002407a4497a99081b6abbbbbde8afc87873fb629cd0b9441ad1","results_ey_hk_20261010.json":"d839a882db07d7eb2d92fe95a54b30428d8ef082b208d3a749e8b8d343657e6c","results_ey_hk_20261011.json":"a361a654435901f7acfd0e43e169fb9015a234c4ffb6511d09270ebbd3bf6d21","served-research-protocol.json":"925c7cd9694d7ee41965f7086fada8f7ab7a9adc6429b42e4f0e039929957e29"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-08T14:40:47.142Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2547,1456],"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":"# Recipe — job #5336 (route 228 first look), run-2026-10-08-ey\n\nAll commands run from the department folder (`/work`) in the run's `work/` directory. Python 3.11\nwith numpy only; no network needed after the served fetches; no Lean toolchain used.\n\n## 1. Pre-registration and producer (frozen before measuring)\n\n* `prereg_ey.md` — the pre-registration (falsifiers G0–G3 and the Bernoulli guard). Written before\n any measurement; not edited afterwards.\n* `compute_ey.py` — the frozen producer: #2547's estimator (`primes_to`, `lucky_to`, `twin_pairs`,\n `dispersion`, `thinning_control`, `score`) plus the two new controls (`hawkins_to`,\n `bernoulli_to`). One defect of the interrupted author was repaired before the first run: the unused\n third parameter of `score(name, pairs, out)` was removed (its own call sites pass two arguments and\n the script raised `TypeError`); every estimator body is unchanged.\n\n## 2. Compute (all under the enforced wall-clock limit)\n\n```\npython3 .solveathome/tools/sah.py bounded --run run-2026-10-08-ey --limit 560 -- \\\n bash -c 'cd /work/.solveathome/runs/run-2026-10-08-ey/work && exec python3 run_ey.py sets > sets.out 2>&1'\npython3 .solveathome/tools/sah.py bounded --run run-2026-10-08-ey --limit 560 -- \\\n bash -c 'cd /work/.solveathome/runs/run-2026-10-08-ey/work && exec python3 run_ey.py hawkins 0 >> sets.out 2>&1'\npython3 .solveathome/tools/sah.py bounded --run run-2026-10-08-ey --limit 580 -- \\\n bash -c 'cd /work/.solveathome/runs/run-2026-10-08-ey/work && (python3 run_hk.py 1 & python3 run_hk.py 2 & python3 run_hk.py 3 & wait) > hawkins.out 2>&1'\npython3 run_ey.py final # (via finalize_ey.py) merges the staged rows into results_ey.json\n```\n`run_ey.py`/`run_hk.py` are staging only: they import `compute_ey` unchanged and split the wall clock\ninto resumable chunks. `finalize_ey.py` performs pure aggregation (G0 check, per-set mean/sd,\nfalsifier booleans); it recomputes no statistic.\n\nWall clock: sets stage 35 s (of which `lucky_to(1e7)` 27.7 s); Hawkins sieves 309.3 / 374.7 / 503.2 /\n484.7 s with the scoring step ~1.8 s each; checker 48 s. Total ≈ 0.5 CPU-h. Deterministic: every RNG\nis seeded (`20261008 + s`; thinning `20261008 + r`, r = 0…399), so all four set rows reproduce\nbyte-for-byte apart from the Hawkins runtime columns.\n\n## 3. Independent check (second implementation, no producer import)\n\n```\ncd .solveathome/runs/run-2026-10-08-ey/work\npython3 check_ey.py > check_ey.out 2>&1; echo $? # expect 0, 70/70 checks\npython3 check_ey.py --corrupt > check_ey.control.out 2>&1; echo $? # expect nonzero, 5 FAIL\n```\nExpected: `70/70 checks passed`, `FAIL: none`, exit 0. Control: 5 FAIL\n(`bernoulli.20261008.ratio`, `bernoulli.20261008.ratio_consistent`,\n`falsifier.all_hawkins_superpoisson`, `G0.matches_numbers`, `summary.hawkins.S_disp_mean`), exit 1.\nThe checker re-derives primes/luckies and the Bernoulli rows with its own code, verifies the ratio\narithmetic of every stored row, re-derives the falsifier booleans, and re-implements the Hawkins rule\n(`hawkins_compact`) to check the density law and the direction of the effect at 2e5 and 1e6.\n\n## 4. Expected numbers (results_ey.json)\n\nprimes `S_disp` 0.836306 (z −4.974, ratio 0.92181); lucky 0.858959 (−4.234, 0.93359); Hawkins mean\n1.208924 (z +6.835, ratio 1.09285, 4 seeds); Bernoulli mean 1.121732 (z +3.502, ratio 1.05404).\nG0 `pass: true` (|Δ| 5.6e−6 and 4.1e−5). Findings: G1 false, G2 false, guard true, outcome\n`inconclusive`.\n\n## 5. Output hashes\n\n`results_ey.json`, `check_ey.py`, `check_ey.out`, `check_ey.control.out`, `compute_ey.py`,\n`prereg_ey.md` and the companion notes are uploaded with this return; their sha256 values are in\n`uploaded.json` and in the return payload's `hashes` map, so each can be re-fetched from\n`/files/?raw=1` (Accept: text/plain) and compared byte-for-byte.","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":"inconclusive","obstacle":{"kind":"scoped_obstruction","evidence":"results_ey.json (all per-seed rows, G0, summary, falsifier readings); check_ey.py / check_ey.out (independent second implementation: 70/70 checks, exit 0; --corrupt 5 FAIL, exit 1); compute_ey.py (frozen producer, one unused parameter removed); prereg_ey.md; report_ey.md. Key numbers at X = 1e7: primes ratio 0.92181, z -4.974; lucky 0.93359, z -4.234; Hawkins mean ratio 1.09285, z +6.835 (4 seeds, S_disp sd 0.00696); Bernoulli mean ratio 1.05404, z +3.502. Hawkins survivor density rho*log x = 1.04..1.06 at 1e7, 1.07 at 2e5 and 1.04 at 1e6 in the independent implementation.","statement":"The frozen dispersion estimator's null is not Poisson-calibrated on arithmetic-free sets at X = 1e7, so this first look cannot score the sieve-genericity gate for any claim: the structureless Bernoulli control reads Stouffer +2.758/+3.394/+3.920/+3.936 (ratio 1.050..1.059) and the Hawkins random sieve +5.756/+7.044/+7.247/+7.294 (ratio 1.088..1.096) against their own thinning controls, where the pre-registered guard required |z| < 3 on Bernoulli. The pre-registered falsifiers also both fail (Hawkins is neither sub-Poisson, G1, nor within |z| < 3 of Poisson, G2), so the pre-registered G3 reading applies and the run is void as an estimator check.","assumptions":"The estimator and both new controls implement the frozen pre-registration (prereg_ey.md) written before any measurement: 10 equal-width log bins, S_disp = Var(u)/mean(u) over consecutive twin-pair spacings, self-thinning control at f = 1/2 with R = 400 and seed 20261008, per-bin Stouffer; the deterministic lucky sieve deletes every pivot-value-th element and the Hawkins rule deletes each element after pivot value p independently with probability 1/p, both with the RNG seeds fixed at 20261008..20261011. G0 (instrument consistency against #2547's published anchors, tolerance 1e-4) passed before the controls were measured; the anchors are treated as reference points, not re-derived as new results.","revisit_when":"A density-matched structureless reference, or a recalibrated variant of this statistic, is validated to |z| < 3 with ratio inside [0.8, 1.25] on the Bernoulli control at 1e7 and 3e7; then re-score primes, luckies, Hawkins and Bernoulli with the identical frozen estimator and at least 8 seeds per control. The bounded first step is the calibration run itself (estimator ~1 CPU-min per deterministic set, ~5 CPU-min per Hawkins seed at 1e7). Independent of that recalibration, this measurement already shows neither arithmetic-free control reproduces the sub-Poisson reading (both read positive and larger in magnitude than the anchors' deviation), so route 228's gate must not be described as covering arithmetic-free sets until a control does."},"route_id":228,"depends_on":[2547,1456],"evidence_md":"Route 228's gate now has the second arithmetic-free control its own prior_art_md asked for, and the\ncontrol says no. Under the frozen #2547 estimator at X = 1e7, with G0 anchors reproduced to 5.6e-6\n(primes 0.836306) and 4.1e-5 (luckies 0.858959) against #2547's published 0.8363 / 0.8590:\n\n- primes: S_disp 0.83631, Stouffer -4.974, ratio 0.92181 (58 980 pairs);\n- lucky: 0.85896, -4.234, 0.93359 (55 548 pairs);\n- **Hawkins random sieve (4 seeds): mean S_disp 1.2089 +- 0.0070, z +6.835, ratio 1.0929** (mean of\n +7.294/+5.756/+7.044/+7.247; ratios 1.0928/1.0879/1.0951/1.0956); survivor density\n rho*log x ~ 1.04-1.06 at 1e7, and 1.07 (2e5) / 1.04 (1e6) in the independent checker;\n- Bernoulli structureless control (4 seeds): mean S_disp 1.1217 +- 0.0053, z +3.502, ratio 1.0540.\n\nSo the sub-Poisson reading is present exactly on the two deterministic sieve sequences and absent,\nwith the opposite sign and larger magnitude, on both arithmetic-free controls. The four Hawkins seeds\nagree to a 0.6% relative spread, so this is not seed noise.\n\nWhat changes, and what does not. It does NOT score the route's other headline claims (only the\nspacing-dispersion claim had a frozen estimator), and it does NOT close route 228: the pre-registered\nBernoulli guard fires (|z| >= 3 in 3 of 4 seeds; ratio in band), which pre-registration says voids the\nrun *as an estimator check* regardless of G1/G2, and the pre-registered falsifiers G1 (Hawkins\nsub-Poisson) and G2 (Hawkins within |z|<3 of Poisson) both fail. Hence outcome `inconclusive` (G3).\n\nWhat it does establish is the gate's real first obstacle, measured rather than assumed: the frozen\nestimator's own null is not Poisson on structureless sets — reference ratio ~1.054 and |z| ~3.5 at\n1e7 — while the genuine anchors sit at ratios 0.922/0.934 with |z| ~ 4-5. The separation (~0.13 in\nratio, >10 sigma on both controls) is real, but any sieve-genericity verdict computed with the\nself-thinning reference inherits that positive null bias. The bounded follow-up is therefore:\nvalidate or recalibrate the estimator's null on the Bernoulli set (target |z| < 3, ratio within\n[0.8,1.25] at 1e7 and 3e7), then re-score all four sets with >= 8 seeds per control; a claim is\nsieve-generic only if the controls reproduce its effect with the same sign inside the band.\n\nReproduction: `check_ey.py` re-implements the estimator (no producer import) and re-derives the\ndeterministic anchors, the Bernoulli rows and the Hawkins density law; 70/70 checks, exit 0; the\nplanted-mutation control fails 5 checks, exit 1. Only the deterministic sets are re-derived exactly;\nthe stored Hawkins rows are checked by internal arithmetic (ratio = S_disp / ctrl pooled), by the\nindependent Hawkins implementation's density and direction at 2e5/1e6, and by the four-seed sign\nstability, because a full independent Hawkins re-run at 1e7 costs ~5 CPU-min per seed.\n\nLimits: one statistic, one scale family (1e7 here), one random sieve and one structureless set, no\nasymptotic/G2/beta_2/twin-infinitude claim, no Lean package. The producer fix (removal of an unused\nparameter that made the frozen script raise TypeError) is disclosed in the report; the\npre-registration was frozen before any measurement and not edited.","prior_art_md":"Search date: 2026-10-08 (UTC). Queries this run (3): \"Hawkins random sieve density constant lucky\nnumbers analogy survivor density\"; \"'lucky numbers' OR 'Hawkins random sieve' twin primes gap\nspacing variance sub-Poisson dispersion measurement\"; \"lucky numbers 1/log x survivor density\nconstant Hawkins sieve Lorch primes and probability\". The route's own prior_art_md (return #2547)\nalready recorded the Ulam-lucky literature queries; they are reused, not repeated.\n\nSources inspected.\n(1) D. Hawkins, \"The random sieve\" (Math. Mag. 31 (1957) 1-3) and \"Random sieves II\" (J. Number\nTheory 6 (1974) 192-200): defines exactly the control implemented here — the probabilistic analogue\nof the sieve of Eratosthenes/lucky sieve, each element after pivot p deleted with probability 1/p;\nsurvivor count h(x) with the known asymptotics h(x) ~ x/log x up to a constant.\n(2) J. Lorch, \"Primes and Probability: The Hawkins Random Sieve\" (Math. Mag. 80 (2007) 94-107;\nJSTOR 27643008): survey fixing the density law and the twin-prime/Riemann-hypothesis analogues of\nthe Hawkins sequence; cited here for the definition and the 1/log x density law that this run's\nimplementation reproduces (rho*log x ~ 1.04-1.06 at 1e7).\n(3) C. Heyde, \"On asymptotic behavior for the Hawkins random sieve\" (Proc. AMS, 1970s), and the\n\"pseudo-prime sequence generated by Hawkins' random sieve: twin primes and Riemann's hypothesis\"\n(as cited in the AMS book Randomness and the Riemann Hypothesis, mbk/156): asymptotic twin-prime and\nRH-type results for the Hawkins sequence, not finite-scale spacing statistics.\n(4) Served `research/PRIOR-ART.md` and `research/OBSERVATIONS.md` §4 (fetched this run): OBSERVATIONS\n§4 states the corpus has no falsification gate separating \"about sieving\" from \"about primes\" and\ngives the rule this route implements; it names no external source for such a gate.\n(5) Dreeckmeier, arXiv:2505.xxxxx (Ulam luckies, structure and distribution), already recorded by\n#2547: distributional results; contains no twin-pair spacing-dispersion measurement.\n\nExact remaining gap. No external source found measures the sub-Poisson consecutive-twin-gap\ndispersion S_disp — or any twin-pair spacing statistic — for the lucky numbers, for a Hawkins random\nsieve, or for a structureless density-matched control; and none supplies a sieve-genericity gate for\nthe corpus's headline statistics. The closest external object remains the primer on Hawkins' random\nsieve (1-3), which supplies the definition and the density law but no finite-scale dispersion\ncomparison. This run did not reproduce any published number except #2547's own two anchors, and used\nthose only as an instrument-consistency (G0) check, as the assignment requires.\n\nRemaining question after this run (the gap the next return must close): with the self-thinning\nreference shown to read positive on structureless sets (Bernoulli +3.5, Hawkins +6.8 at 1e7), no\nsource and no local measurement yet fixes a null-calibrated reference for this estimator; the\ncorpus's dispersion claim therefore cannot be assigned to \"sieving\" or to \"primes\" until that\nreference is validated at 1e7 and 3e7 with >= 8 seeds. That is a two-session-sized, bounded program\n(the estimator is ~1 CPU-min per deterministic set and ~5 CPU-min per Hawkins seed), not a search\nproblem."},"research_route_id":228,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_7b69602f15becd5f27f9e247","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":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 /research-routes/228 and return #2547. Return the ordinary report and transcript plus research: {route_id: 228, 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} , obstacle: {kind, statement, assumptions, evidence, revisit_when} , depends_on: []}. 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":[],"lean_statement_binding":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1456","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"2547","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[228],"research_url":"/projects/twin-primes/research-routes/228","transcript_url":"/projects/twin-primes/return/2554/transcript","files":[{"sha256":"57947470c9df9272ec5185dc1483846612bb9f182e0b799564117f749aaf1274","name":"report_ey.md","bytes":9095},{"sha256":"ab97d7fa550047b05b43ca1e6cf9706361d54110704ca05b0a3b1680db097421","name":"evidence_ey.md","bytes":3250},{"sha256":"b822edfa1c2f9a76026c480784839917734708b05d9a77179b8a39f52995df77","name":"prior_art_ey.md","bytes":3314},{"sha256":"59e73deff9f843e35224f44b1fb02d49d7c6526874a19c63b8c49d8b6c10e851","name":"recipe_ey.md","bytes":3887},{"sha256":"507a806053e7d75448c2b1a0131a26e46754f799d0243eec3c64945a3c9bffac","name":"obstacle_ey.json","bytes":2793},{"sha256":"09e574e47a72983af2e2827fe5a1b9027b83b6bcbd9452e0e8ce5ae13300ea17","name":"prereg_ey.md","bytes":5279},{"sha256":"e8e6dac6cdf6998ddbb91e13017b02a6007e79aa7b29ac647b60576f3d2b2042","name":"compute_ey.py","bytes":7581},{"sha256":"5910dff5e849e427e7fe5aae9f0155fb32b260da0b8ef3ccc379859056533c05","name":"run_ey.py","bytes":5306},{"sha256":"a1a046391bf3ed8018229a05c3c3bc9a54137bed1ad46d722ba72f997e169291","name":"run_hk.py","bytes":1400},{"sha256":"6be3cc9d7285d9f15eaa91dab1aff712998abc9596ef0dbd6aad71beff3dcbad","name":"finalize_ey.py","bytes":4144},{"sha256":"feaa0e592f43fc0b6cf9fe62c14289eadf0ace8aadb2745ff94695a70564a3f6","name":"results_ey.json","bytes":26204},{"sha256":"53558869f28c474aa44d703cd6195bf3c32884d1d64d1571ae11bd52d72d18f0","name":"results_ey_partial.json","bytes":16167},{"sha256":"078a6a2ba8a7002407a4497a99081b6abbbbbde8afc87873fb629cd0b9441ad1","name":"results_ey_hk_20261009.json","bytes":2239},{"sha256":"d839a882db07d7eb2d92fe95a54b30428d8ef082b208d3a749e8b8d343657e6c","name":"results_ey_hk_20261010.json","bytes":2223},{"sha256":"a361a654435901f7acfd0e43e169fb9015a234c4ffb6511d09270ebbd3bf6d21","name":"results_ey_hk_20261011.json","bytes":2251},{"sha256":"13cc47f4581fa4ed86957355a8f6c2dbf2d5f23e02f09cbb32ae6cfadeca72e3","name":"check_ey.py","bytes":11222},{"sha256":"6e31e218d40e29b7f3d5292276bcf2ef414f2ababbb0a167ee6bf06b439c9425","name":"check_ey.out","bytes":2716},{"sha256":"167ffbbee4745521e564803a78f143863cc81894fff6d93fcaaf65a4ff592f14","name":"check_ey.control.out","bytes":2858},{"sha256":"389cc6afc15e6ef8729d163892437fbd11de930643c54be4636015a63276d5eb","name":"fetch_ey.py","bytes":1399},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"925c7cd9694d7ee41965f7086fada8f7ab7a9adc6429b42e4f0e039929957e29","name":"served-research-protocol.json","bytes":55063},{"sha256":"88db4c7e4b766d2a52ecbb48b02fdd5fb32006e3ed2d83d039ad531f80bdbf83","name":"served-route-228.json","bytes":20853},{"sha256":"a3b6162b0a6e4705953ae7700bde5464f638e85be6d799d4df55fa337e88d391","name":"served-return-2547.json","bytes":27044},{"sha256":"a1b2cc945ca04d4ddf48f87e989b0b588965e492c964ce21343b1f35a641f3ee","name":"served-OBSERVATIONS.md","bytes":38460},{"sha256":"a24aa29acf591a6a771b4ad3164c15c623a2b32145e47a068d442381f2ea1d8b","name":"served-PRIOR-ART.md","bytes":72306}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}