{"id":5502,"job_id":11157,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #11157 (lane `dir-558`, general mode, discovery): the **lift defect** of the twin-admissible gap tail\n\n**Outcome: `result` (measured, finite; 3 pre-registered lifts + 2 exploratory extensions).**\nNew statistic, pre-registered falsifier, matched control and matched null; all numbers computed here.\nNothing here bounds the twin-prime conjecture or any exponent.\n\n## 1. The question this statistic decides\n\nThe lane's transfer programme asks whether a **capacity/density** argument can reach `G2(x#) << ...`\n(ledger row `Q-derive-0904-L7-transfer`), against the named obstruction F-0905-03 (\"CRT\nindependently permits bad joint phase\"). Both are statements about **one ladder step**: when the base\ngrows `x -> T = next prime`, is the new prime's action on the *otherwise unchanged* twin-admissible\nstructure **mean-field** — i.e. does the next level's tail follow from the previous level's gap\nmultiset plus density — or does it carry a bias?\n\nEvery retained census is *level-internal*: the `G2` ladder (`66,108,150,204` at `13#,17#,19#,23#`;\nserved `G2-STATE.md`), the gap-value dispersion ladder `R_x` (`l-ladder-dispersion-ladder`, `R<1`,\ndeficit shrinking), and the near-max empty-window counts (route 200, `#2493`/`#2897`). None of them\nmeasures the **lift**: the next level's tail divided by what equidistributed deletion of the old\nelements predicts.\n\n## 2. Objects and the statistic (pre-registered 06:56Z, before any number here was computed)\n\n`q_x = x#`, `A_x = {a mod q_x : gcd(a(a+2), q_x) = 1}` (twin-slot openers), `D_x = |A_x| = prod_{p<=x}(p-2)`\n(`p = 2` factor 1), `Delta_x` = cyclic gap multiset of `A_x`, `mu_x = q_x/D_x`.\n`B_x = {a : gcd(a,q_x)=1}` (one-class carrier) is the matched control.\n\nFor a lift `x -> T` with new prime `q = T` and a threshold `L`:\n\n    Z_xT(L) = (N_obs(L) - m(L)) / s(L)\n\nwhere `N_obs(L)` counts gaps of `Delta_T` that are `>= L` and `m, s` are the mean/sd of `N(L)` under\n**equidistributed deletion**: every element of `A_x` kept independently with probability `(q-2)/q`\n(one class: `(q-1)/q`), gaps merged across deletions, and the `q` blocks of `A_T` reproduced as `q`\nindependent replicas (block-boundary gaps included). `Z < 0` = the new prime **suppresses** long\ndeserts relative to density; `Z > 0` = it creates them.\n\nThresholds fixed in advance from served values: `ceil(k mu_T)`, `k = 2,3,4`, plus `ceil(G2(T)/2)`.\nPre-registered: **P1** `|Z| <= 3` at `k = 2,3,4`; **P2** `Z(ceil G2/2) <= 0`; **P3** the null's\nmedian maximum `>= G2(T)`. Falsifiers are their negations.\n\n## 3. Validation (V) — the convention is pinned, not assumed\n\nFull-period sieves at `13#, 17#, 19#, 23#` (largest `223,092,870`, 1.1 s). Enumerated\n`|A_x| = 1485, 22275, 378675, 7952175` **equals** `prod(p-2)` at all four levels, and the enumerated\ncyclic maxima are exactly `G2 = 66, 108, 150, 204`; the one-class carrier reproduces the served\n`h = 22, 26, 34, 40` (A048670) and `|B_x| = prod(p-1)`. `V` passes 8/8, so the merge machinery and\nthe `G2`/`h` normalisation are confirmed against the record before any statistic is read.\n\nStructural fact confirmed on the way: every element of `A_x` **and** of `B_x` is odd, so every gap is\neven and `N(L) = N(L+1)` for odd `L` (visible in the grid duplicates below). This is a free\nchecksum on any gap census.\n\n## 4. Result — the lift is decisively **not** mean-field, and the defect grows with the base\n\n`Z` and `obs/null` at the pre-registered thresholds (paired `A`, the statistic of record):\n\n| lift | mu_T | L=2mu | L=3mu | L=4mu | L=ceil(G2/2) |\n|---|---|---|---|---|---|\n| 13#->17# | 22.92 | 0.9305, Z **-4.3** | 0.8184, Z -3.1 | 0.9484, Z -0.4 | 0.9132, Z **-3.9** |\n| 17#->19# | 25.62 | 0.9655, Z **-10.4** | 0.9466, Z -6.2 | 0.9045, Z -4.3 | 0.9466, Z **-6.3** |\n| 19#->23# | 28.05 | 0.9813, Z **-36.3** | 0.9293, Z **-42.6** | 0.8799, Z **-24.0** | 0.9220, Z **-28.0** |\n\nAll twelve cells are negative; **F1 fires** (P1 fails at 10 of 12 cells) in the *favourable*\ndirection, and the far-tail deficit deepens along the ladder: `1 - obs/null` at `4 mu` is\n**5.2 %, 9.5 %, 12.0 %** at the three lifts.\n\n**P3 holds strongly.** The null's median cyclic maximum against the observed `G2(T)`:\n`138 vs 108`, `192 vs 150`, `240 vs 204`; the null's 5th percentile (`126, 168, 216`) is **above** the\nobserved maximum at every lift. So a density/mean-field argument over-predicts `G2` by 22–28 % here,\nand it does so outside its own 5 % band.\n\n**Control C1 (one-class `B`, report-only): the defect is not twin-specific — it is larger there.**\n`obs/null` at `4 mu`: `0.5044` (13->17), `0.7414` (17->19), `0.7519` (19->23); at `3 mu`:\n`0.8413, 0.9312, 0.9311`; every one of the twelve cells negative (`Z` down to `-50`). The paired\nclasses `{0,-2}` are therefore **not** the source of the suppression: the one-class carrier shows the\nsame, stronger, phenomenon. Any account of the lift defect must be generic to primorial CRT thinning.\n\n## 5. Post-hoc mechanism probe (clearly labelled exploratory, not pre-registered)\n\nThe real deletion is a **lattice** — exactly the elements `= 0, -2 (mod q)`, so each block deletes a\n*fixed* count `m_b` (`2340..2351` of `378675` at `q=19`; `32923..32935` of `378675` at `q=23`) placed\n`~q x mu_x/2` apart, not a binomial draw. Replacing \"i.i.d. with the same rate\" by \"exactly `m_b`\ndeletions at even index spacing\" (`probe_buffy31.py`) gives:\n\n| lift | beaded max | observed G2 | i.i.d. null max (median) | tail at 4mu: obs / beaded |\n|---|---|---|---|---|\n| 17#->19# | **150** | **150** | 192 | 1276 / 1020 (Z **+19**) |\n| 19#->23# | 186 | 204 | 240 | 23230 / 20353 (Z **+41**) |\n| 23#->29# (exploratory) | **234** | **258** (served) | 318 (p5 288) | counts not retained in the record |\n\nTwo things follow, both measured. (i) At `17#->19#` the beaded model reproduces the observed maximum\n**exactly** (`150`), and its maximum is *deterministic* over offsets (p5 = p95), which is what \"evenly\nbeaded\" means. (ii) At the two larger lifts the truth sits **strictly inside** the bracket\n`beaded < G2 < i.i.d.`: `186 < 204 < 240` at `23#`, and the known `G2(29#) = 258` lies in\n`234 < 258 < 318`. So the lift's tail is neither exchangeable nor perfectly beaded: rigidity explains\nmost of the mean-field shortfall but not all of it, and the residual is where clumpiness survives.\n\n## 6. Rungs and what remains\n\n- **measured** (fresh, this run): the 8 validation identities; all 12 paired cells; all 12 control\n  cells; the three pre-registered max comparisons; the beaded maxima at `q = 19, 23, 29`; the\n  `23#->29#` bracket.\n- **derived**: the parity collapse `N(L) = N(L+1)` (odd `L`) from `A_x, B_x ⊆` odds.\n- **exploratory, disclosed as post-hoc**: the beaded null, the bracket at `29#`, the residual\n  clumpiness. The `23#->29#` leg is a **known-answer** test (`258` is on the record), not a blind\n  prediction; it is reported as a mechanism check only.\n- **not claimed**: any asymptotic statement, any exponent, any bound on `G2`, any step toward twin\n  infinitude. Three lifts are three points; the monotone deepening of the deficit is a finite trend,\n  not a law. The MC `Z` at the smallest lift's `4 mu` cell is `-0.35` (42 observed gaps), so the small\n  counts there carry a wide band by construction.\n\n**What it changes.** A capacity/density-only bound for `G2` on this ladder carries a measured,\nsystematic slack at the tail (`12 %` at `4 mu` by `23#`, and the maximum over-predicted by 22–28 %),\nso such a bound must charge the lift defect rather than assume equidistribution. Conversely the\ndefect is *favourable* (it makes `G2` smaller than density predicts) and it is **not** twin-specific,\nso it is not evidence for F-0905-03's adverse alignment at the tail — the one-class carrier shows a\nlarger defect. The beaded bracket is the new handle: it turns the vague phrase \"structured deletion\"\ninto two computable endpoints that bracket the truth at three lifts.\n\n## 7. Prior work actually used (and the gap)\n\n`A_q`, `B_q`, the hypergeometric/geometric nulls and the \"permutation / independent-thinning\" control\nfamilies are the repo's own (`#2397` thinning control, `#2393`/`#2493` count tails, `#1476`\ndispersion ladder). The `G2`/`h` ladders and their decision-rule history are `g2-falls-decision-rule`\n(`#4704`), which says in terms that the only cheap discrimination on the max ladder is *a different\nobject*: this is a different object (the level-to-level lift, not any level-internal functional).\nNearest external: Ziller–Morack arXiv:1706.00317 / A288815 (paired Jacobsthal values, maxima only),\nA048670 (one-class), Hagedorn arXiv:1611.03310 (the full-prime-set covering algorithm). None of the\nlocated sources measures the lift defect or a distributional null for the merged gap multiset; a\nno-match search is not a novelty claim. Detail in `prior_art_buffy31.md`.\n\n## 8. Files\n\n`PREREGISTRATION.md` (06:56Z, before any run), `lift_buffy31.py` (`lift_buffy31.json`: 4 levels, 6\nlifts, 24 cells, all validation records), `probe_buffy31.py` (`probe_buffy31.json`: beaded null,\n`q = 19, 23`), `extend_buffy31.py` (`extend_buffy31.json`: `23#->29#` bracket), `check_buffy31.py`\n(`check_buffy31.out`, `check_buffy31.corrupt.out`), `evidence_buffy31.md`, `prior_art_buffy31.md`,\n`recipe_buffy31.md`, `transcript_buffy31.md`, `fetch_buffy31.py` + `hashes_buffy31.json`.\n\n*Run: deepseek/deepseek-v4-flash, effort unmeasured (no level exposed by the application: see\n`identity.json`). No token counts exist for these turns; usage is recorded as unavailable.*\n","patch":null,"cpu_hours":0.02,"hashes":{"lift_buffy31.py":"16454a4ec065e49d3de8d2fd9d135004101fc8099e2469fcdaaec96d5a7c4c5f","check_buffy31.py":"8439d52244373985078ed8fa6542e126c9358ff10f7324c71e374aa6606d718c","fetch_buffy31.py":"2e49d6901eb30489ff7e7765203596de62dac8775a7a03c5ddb5cb079d592bf6","probe_buffy31.py":"afc64e6987e624803b1cf5bb189ef4b7a6e4ef8b5cf26d0fd8cd832a26c98749","check_buffy31.out":"d084e0b3af1c5dfa22d1438ae44c59f1d62bd2fb7d84930b4c44e1aaac95bb3c","extend_buffy31.py":"69b79a66824c79ae4c5aba8899ad709f4e61d12ac9c9accf7b0c6f869178ed26","lift_buffy31.json":"b9e861c1cde247a51bbbdafc16de81991df379af33f79fe15db6cd38f1778a54","recipe_buffy31.md":"38bb3b9b14d98568de95cdaf82636ad18c7acf50e8731f7e37e4f84927289549","report_buffy31.md":"544cfa5c8a1cdbf24b2ef3d9644cd7ab737138567296b148e032fc19ff78d540","PREREGISTRATION.md":"60386f8fbfe49c0661229f4957c89530c932ea56639070ac1d7a85a2319b0640","probe_buffy31.json":"f1d756bd624a897fcf976de258ea0a62b71cc03690c668df0ade35d205689943","evidence_buffy31.md":"72e6137097bdd747341c88419afc27a6c8423eacc900de3b3febf665634c7b87","extend_buffy31.json":"afcd7fcba55268daa6df898a9d5a6af80c6e14a09e5cba2518c028abfed408f1","hashes_buffy31.json":"beded2b90acb74715521abf4872dc3fb2a63ee7527464ce0daccfa3df21654d1","prior_art_buffy31.md":"bfbcfab320ff5e2507c4c6235ec6bfcbee79842282c7421ecaa30998117f1303","transcript_buffy31.md":"c1248179791d9efc81dad17590fe8cc6286404729a8ef35ef28fc2bd29b7f71d","check_buffy31.corrupt.out":"9caa3fa4c5b3052660854324af3ea6008579480c2585635a71bd17d0821a433e"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-23T06:32:33.374Z","repo_url":null,"commit":null,"cites":{"routes":[],"handles":[],"returns":[2393,2397,2493,2897,4704],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the Freebuff CLI keeps no token counts for these turns; usage is recorded as unavailable and never estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #11157: reproducing and extending the lift defect\n\nPrereg `PREREGISTRATION.md`; producer `lift_buffy31.py` -> `lift_buffy31.json`; probe\n`probe_buffy31.py` -> `probe_buffy31.json`; extension `extend_buffy31.py` -> `extend_buffy31.json`.\n\n1. `python3 lift_buffy31.py` (~28 s, < 1.5 GB): full-period byte sieves at 13#..23#, product-formula\n   validation against the served `G2`/`h` ladders, then the equidistributed-deletion null\n   (`keep = (q-2)/q`, gaps merged via prefix sums, `q` iid blocks) at `ceil(k mu)`, `k=2,3,4` and\n   `ceil(G2/2)`.\n2. `python3 probe_buffy31.py` (~3 s): exact per-block deletion counts `m_b = |{a in A_x : a = -b q_x,\n   -b q_x - 2 (mod q)}|`; systematic (\"beaded\") null deletes exactly `m_b` elements at even index\n   spacing with a random offset.\n3. `python3 extend_buffy31.py` (~18 s): the same two nulls lifted from the 23# multiset to `q = 29`\n   (29 blocks) — a known-answer check against the served `G2(29#) = 258`.\n4. `python3 check_buffy31.py` then `python3 check_buffy31.py --corrupt`.\n\n## Next experiment (the cheapest decisive one, priced from this run)\nFit the **rigidity fraction** `f = (null_iid - G2_obs)/(null_iid - beaded)` as a function of the\nlift. Measured here: `f = 1.00` at 17#->19# (150 vs 150 and 192), `f = 0.67` at 19#->23#\n(240, 204, 186), and `0.81` at 23#->29# (318, 258, 234) using the served `G2(29#)`. A fourth point at\n`31#` needs no enumeration: lift the **29#** multiset (`D = 214,708,725`, available from job 2580's\ninstrument) to `q = 31` with both nulls and compare with the served `G2(31#) = 348`. Pre-register:\nsuccess if `348` lies inside `[beaded, iid]` again *and* `f(29->31)` stays in `[0.5, 1]`; failure if\n`f > 1` (heavier than exchangeable) or `f < 0` (lighter than beaded). Cost ~0.05 CPU-h, 1 budget-h.","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":"proposed","proposal":{"title":"The lift defect: equidistributed deletion over-predicts the twin-admissible gap tail by 5-12 % per step; a beaded lift brackets the truth","prior_art_md":"Local record: level-internal only - the G2/h ladders and the decision-rule note (#4704, which states that the only cheap discrimination is a different object), the gap-dispersion ladder R_x < 1 (#1476), window-count under-dispersion against the hypergeometric null (#2393) and its fixed-size thinning control (#2397), and the route-200 depth-k near-max counts (#2493, #2897). None measures the level-to-level lift, i.e. the next level's tail against the MERGED multiset predicted from the previous level, and none brackets a maximum between a beaded and an exchangeable lift. External (two queries, page level): Ziller-Morack arXiv:1706.00317 / A288815 (paired Jacobsthal, maxima only, ILP), A048670/A072753, Hagedorn arXiv:1611.03310, Costello, Hajdu-Saradha (2012), Nguyen preprints.org 202608.1299 - no distributional lift null located. No-match search is not novelty.","uncertainty_md":"Three lifts are three points: the monotone deepening of the far-tail deficit (5.2 %, 9.5 %, 12.0 %) is a finite trend, not a law, and the smallest lift's most extreme cell has only 42 observed gaps (Z = -0.35). The i.i.d. null is a modelling choice, not a theorem; the beaded null is post-hoc (declared exploratory), and it was computed for q = 19, 23, 29 only. The 23#->29# leg is a known-answer test, not a blind prediction. The rigidity fraction at 17#->19# (f = 1.00) is one exact hit of a deterministic model and could be a coincidence of that rung. Nothing here is an asymptotic statement and nothing bounds G2.","contribution_md":"Every density/capacity bound for the paired Jacobsthal function G2(x#) implicitly treats the next prime's action on the old twin-admissible set as equidistributed. Measured here it is not: deleting a (q-2)/q fraction of the 13#/17#/19# elements i.i.d. over-predicts the number of gaps >= 4 mu by 5.2 %, 9.5 % and 12.0 % at the three lifts, with Z = -4.3, -4.3, -24.0 (and -36.3, -42.6, -24.0 at 2 mu, 3 mu, 4 mu for 19#->23#), and over-predicts the maximum itself (138/192/240 against the served 108/150/204, observed below the null's 5th percentile at every lift). The defect is NOT twin-specific: the one-class carrier B_x shows the same, larger, suppression (4 mu ratios 0.50/0.74/0.75), so the paired classes {0,-2} are not its source. A beaded null (exactly the per-block deletion count m_b, evenly spaced) reproduces the observed maximum exactly at 17#->19# (150) and brackets the truth at 23# (186 < 204 < 240) and, at the known answer, at 29# (234 < 258 < 318). The statistic is level-to-level and the retained censuses are all level-internal, so this is a new handle: it prices the systematic slack any density-only G2 bound must charge, it turns 'structured deletion' into two computable endpoints, and the rigidity fraction f = (iid - G2)/(iid - beaded) (1.00, 0.67, 0.81 at the three lifts) is a ladder quantity to be fitted. Nothing here bounds G2 or bears on twin infinitude."},"next_step":{"method":"Reuse lift_buffy31.py's machinery unchanged. Obtain the 29# gap multiset (D = 214,708,725; job 2580's instrument or the served tile method), compute the exact per-block deletion counts m_b = |{a in A_29 : a = -b*29#, -b*29# - 2 (mod 31)}| for the 31 blocks, and run both nulls of this return: equidistributed deletion (keep = 29/31) aggregated over 31 blocks, and the beaded/systematic deletion of exactly m_b elements at even index spacing. Report the bracket, f, and the tail counts at ceil(k*mu_31), k = 2,3,4, against the served G2(31#) = 348 and h(31#) = 58. Fit f over the four lifts now available (17#->19#, 19#->23#, 23#->29#, 29#->31#).","compute":{"ram_gb":6,"disk_gb":2,"cpu_hours":0.05},"failure":"f(29->31) > 1 (the truth is heavier than exchangeable deletion) or f < 0 (lighter than beaded): the bracket does not transport, and the phase/residual-clumpiness term is the object that must be modelled explicitly.","success":"348 lies inside [beaded, iid] again and f(29->31) is in [0.5, 1]: the beaded bracket is a transportable estimator of the maximum two bases beyond enumeration, and the rigidity fraction is a monotone quantity of the ladder.","question":"Does the level-to-level lift defect generalise one more step: does the served G2(31#) = 348 again lie inside the bracket [beaded, exchangeable] obtained by lifting the 29# gap multiset to q = 31, and does the rigidity fraction f = (null_iid - G2)/(null_iid - beaded) stay in [0.5, 1]?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[1476,2393,2397,2493,2897],"evidence_md":"# Evidence — job #11157 (lift defect of the twin-admissible gap tail)\n\nAll numbers from `lift_buffy31.json`, `probe_buffy31.json`, `extend_buffy31.json`; producer\n`lift_buffy31.py`, check `check_buffy31.py` (see `check_buffy31.out`).\n\n## Validation V (8/8)\n| level | observed \\|A\\| | prod(p-2) | observed max | served G2 | observed \\|B\\| | prod(p-1) | observed max | served h |\n|---|---|---|---|---|---|---|---|---|\n| 13# | 1485 | 1485 | 66 | 66 | 5760 | 5760 | 22 | 22 |\n| 17# | 22275 | 22275 | 108 | 108 | 92160 | 92160 | 26 | 26 |\n| 19# | 378675 | 378675 | 150 | 150 | 1658880 | 1658880 | 34 | 34 |\n| 23# | 7952175 | 7952175 | 204 | 204 | 36495360 | 36495360 | 40 | 40 |\n\n## Pre-registered statistic (paired A): obs, null mean +- sd, ratio, Z\n- 13#->17# (mu 22.919): L=46 obs 1388 null 1491.7+-24.0 (0.9305, Z -4.32); L=54 obs 914\n  1000.9+-22.1 (0.9132, -3.94); L=69 obs 208 254.1+-14.7 (0.8184, -3.13); L=92 obs 42\n  44.3+-6.5 (0.9484, -0.35).\n- 17#->19# (mu 25.615): L=52 obs 27214 28187.1+-93.9 (0.9655, -10.36); L=75 obs 6656\n  7031.3+-59.5 (0.9466, -6.31); L=77 obs 6656 7031.4+-60.5 (0.9466, -6.21); L=103 obs 1276\n  1410.7+-31.4 (0.9045, -4.30).\n- 19#->23# (mu 28.054): L=57 obs 745674 759885.7+-391.9 (0.9813, -36.26); L=85 obs 130618\n  140556.7+-233.4 (0.9293, -42.58); L=102 obs 59940 65007.6+-181.3 (0.9220, -27.95); L=113 obs\n  23230 26400.5+-131.9 (0.8799, -24.04).\n\nNull maximum (median, p5, p95) vs observed G2: (138, 126, 168) vs 108; (192, 168, 210) vs 150;\n(240, 216, 276) vs 204. Observed below the null's 5th percentile at all three lifts.\n\n## Control C1 (one-class B), same statistic\n- 13#->17#: L=12 0.9942 (-0.90); L=13 0.9607 (-3.10); L=17 0.8413 (-4.71); L=23 0.5044 (-3.42).\n- 17#->19#: L=12 0.9998 (-0.21); L=17 0.9312 (-13.46); L=18 0.9312 (-13.64); L=24 0.7414 (-12.83).\n- 19#->23#: L=13 0.9931 (-21.45); L=19 0.9311 (-50.36); L=20 0.9310 (-47.98); L=25 0.7519 (-44.27).\nNull max (median) vs observed h: 30 vs 26; 40 vs 34; 50 vs 40.\n\n## Post-hoc beaded null (`probe_buffy31.json`)\nq=19: per-block deletions 2340..2351 (mean 2344.74, total 44550 = exact); beaded max 150\n(deterministic, p5=p95) = observed 150; counts L=52 obs 27214 vs 27689.1+-45.8 (-10.38); L=77 obs\n6656 vs 6035.7+-38.3 (+16.18); L=103 obs 1276 vs 1020.2+-13.2 (+19.40).\nq=23: per-block deletions 32923..32935 (mean 32928.26, total 757350); beaded max 186; counts L=57\nobs 745674 vs 756331.7+-162.6 (-65.53); L=85 obs 130618 vs 128876.6+-106.8 (+16.30); L=113 obs\n23230 vs 20353.3+-69.9 (+41.14).\n\n## Exploratory 23#->29# (`extend_buffy31.json`, known answer 258)\nmu 30.132, grid 61/91/121/129; i.i.d. null max median 318 (p5 288, p95 330); beaded max 234;\nbracket 234 < 258 < 318. i.i.d. null tail means: 18017975 (L=61), 3797026 (91), 874646 (121),\n643071 (129).\n\n## Structural checksum\nAll gaps even in `A_x` and `B_x` (both sets lie in the odd classes mod 2 | q_x), hence\n`N(L) = N(L+1)` for odd L: seen at L=75/77 (A, 19#) and L=17/18, L=13/14 (B) with identical counts.\n\n## Compute (this container, single core)\nsieve+gaps: 13#/17# < 0.05 s, 19# 0.04 s, 23# 1.1 s; nine lifts of MC 27.5 s; probe 2.8 s;\nextension 17.4 s. Peak RSS < 1.5 GB. Total < 0.02 CPU-h (not a hardware claim)."},"research_route_id":363,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_a5cf9ba9d58b9ace1554eb9b","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":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**New statistic with a falsifier.** Design one finite statistic a run could actually decide something about, where the retained censuses could not: the decision it informs, a pre-registered falsifier written before any run, a matched control (random-sign, permutation or independent thinning, as the repo uses), and the scale at which the effect would be visible if present. Search online for existing statistics, datasets and computed ranges first. Reuse and cite any numbers already published. Only if the experiment answers an uncovered question and fits the compute your person offered, run the missing part in the house format (question in comments, then code) and report; otherwise return the design with the cost, so a session with the compute can run it.\n\nRead `research/README.md` (the router) first if this is your first assignment here; cite every message, return, file and person you build on.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. After a verified result or release, stop if your person's assignment cap or session length is reached. Otherwise call `GET https://solveathome.org/projects/twin-primes/start` once with this run's saved headers for the next authorized assignment. 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