{"id":2566,"job_id":5353,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 229 first look — the level-drop route's one-class-witness precondition is unsatisfiable\n\nJob #5353 (explore, stage `first_look`, direction general, route 229 rev 1). This run built a\nbounded **exact** census of the kill-class signature of the maximal twin-slot gap windows and\nreproduced the record's ladder as a self-test. Result: **outcome `blocked`, obstacle kind\n`claim_refuted`.**\n\n## What the route asks\n\nRoute 229 (\"level-drop\", proposed in return #2560) proposes to spend the programme's\ninfinitely-often slack on the *certification* side: if, at infinitely many primorial levels `x#`,\n**some maximal twin-slot gap window is \"one-class\"** — a single prime `p ≤ x` explains every killed\nposition of the window — then a dimension-1 (exponent-2) Jacobsthal bound would certify that window,\nwhich is the weak Zone Postulate at that scale. Its own pre-registered first experiment\n(`research-routes/229`) is: census the kill-class count of **every** maximal window of `T_x = Z/x#`\nat `x = 19, 23, 29, 31`, with the failure clause \"every maximal window is two-class\" and the route\n\"known to refute its own premise cheaply\".\n\n## Objects (from the served record; used verbatim)\n\n- `W = x#`; a twin slot is `s` with `gcd(s,W)=gcd(s+2,W)=1`, i.e. `s` avoids classes `0` and `-2`\n  modulo every `p ≤ x`. Non-slots are *struck* by those two classes.\n- Census `D = ∏_{3≤p≤x}(p−2)`; gaps are cyclic, `D` of them, summing to `W`; `G₂(x#)` = max gap.\n- Argmax set = left endpoints `s` of length-`G₂` gaps; `m(x)` = its size.\n  Record ladder (`research/G2-STATE.md` §2) and argmax sets (`history/staging/measure-0904-argmax.md`):\n  `G₂ = 2,6,12,30,42,66,108,150,204,258,348,528,546,618` at `x = 2,…,43`;\n  `m = 4,12,20,20,4,2,4,2` at `x = 11,13,17,19,23,29,31,37`.\n\n## The bounded experiment (this run's own exact computation)\n\n`census_fn.py` (numpy, offline) strikes the **entire** period `W = x#` for every `p ≤ x` at residues\n`0` and `−2 mod p`, finds every maximal run of struck positions, and for each maximal window counts\nthe number of **distinct primes** active in it and how many of the two classes `{0,−2}` occur.\nRun at `x = 11,13,17,19,23` (all exactly enumerable; `W ≤ 2.23·10⁸`). Self-test: the computed\n`G₂`/`m` match the record at every level.\n\n| x | W = x# | D | G₂ | m | active primes min/max | classes used | one-class windows |\n|---|---|---|---|---|---|---|---|\n| 11 | 2,310 | 135 | 42 | 4 | 5 / 5 | 2 | **0** |\n| 13 | 30,030 | 1,485 | 66 | 12 | 6 / 6 | 2 | **0** |\n| 17 | 510,510 | 22,275 | 108 | 20 | 7 / 7 | 2 | **0** |\n| 19 | 9,699,690 | 378,675 | 150 | 20 | 8 / 8 | 2 | **0** |\n| 23 | 223,092,870 | 7,952,175 | 204 | 4 | 9 / 9 | 2 | **0** |\n\nAt **every** maximal window at **every** reachable level, the active-prime count is exactly `π(x)`\n(all primes `≤ x` strike the window), and **both** classes `{0,−2}` are used. The kill-class count is\ntherefore *scale-dependent* — it grows as `π(x)` — but it is maximal, never `1`. The route's\npre-registered failure clause (\"every maximal window is two-class\") fires at every level.\n\n## Why it is structural, not a run of bad luck\n\n**Elementary lemma.** For `p > 2`, the two killed residues `0` and `−2 (≡ p−2) mod p` differ by `2`,\nso among any three consecutive integers at least one is not struck by a fixed prime. No single prime\nstrikes three consecutive integers. Every maximal window has run length `G₂ − 1`. Since\n`G₂(x#) = 6` at `x = 3` and the ladder is `≥ 6` for all `x ≥ 3` (record §2: `2,6,12,30,…`), every\nwindow for `x ≥ 3` is a run of **≥ 5** struck positions. So **no maximal window at any level `x ≥ 3`\ncan be one-class** under the route's own definition; the census could only ever return the failure\nclause. The specified first experiment cannot decide the intended sieve-dimension question — it is\nvacuous.\n\n(Verified over the census data and symbolically for each `p ≤ 23`: the maximal run of consecutive\nintegers struck by a single prime is `2`.)\n\n## Targeted source lookup\n\nSearched (2026-10-08) for a published statement of **scale-dependent dimension** for a two-class\nJacobsthal/sieve object and for a single-prime cover of a maximal reduced-residue gap: queries on\nJacobsthal function / primorial maximal gap / sieve dimension two; DHR sifting limits; single-prime\ncovers of consecutive integers. Every hit is already owned by the record's own prior-art ledger\n(Franze, Kourbatov, Hagedorn, Neudecker, Holt; DHR; Ford). **No source states that the effective\nsieve dimension drops to one at the argmax window of a two-class Jacobsthal object.** The uncovered\nstep is unchanged, but it is now clear that the route's operational proxy does not measure it.\n\n## What this changes\n\n- Route 229's stated channel (a per-window one-class witness) has **no realization**: the per-window\n  kill-class signature is maximal (`π(x)`) and both-class, provably for all `x ≥ 3`. The route\n  cannot be pursued on that coordinate; this is recorded as a scoped obstruction, not a refutation of\n  the broad goal.\n- The record's separate facts stand: `G₂ ≥ h` (twin slots ⊂ reduced residues) and the DHR\n  dimension-2 input `β₂ = 4.2665`, which remains the binding input for every gap-bound route.\n- Nothing here bears on the exponent or on the band `(2, 4.26645]`.\n\n## Reopening condition (revisit_when)\n\nReformulate the mechanism on a coordinate that can actually express a level drop — e.g. a\nproof-level reduction showing the DHR dimension-2 sieve input is unnecessary at the argmax window\n(such a statement does not exist in the located literature), or a new non-per-window statistic on\n`T_x` with a pre-registered falsifier that is not trivially guaranteed. Merely re-running the\nspecified census cannot change the outcome.\n\n## Controls and honesty\n\n- Checker `check_fn.py` (stdlib, offline, no producer import): **38/38 PASS, exit 0**; `--corrupt`\n  (2 planted mutations) **exit 1, 4 failures**. It re-derives the ladder from this run's own served\n  `research/G2-STATE.md`, re-checks every window's active-prime count against `π(x)`, and re-checks\n  the elementary lemma.\n- `census_fn.py` is a new exact computation; the record's own argmax measurements were used only as\n  a self-test (its values agree). `x = 29, 31` were not recomputed (their periods are `6.5·10⁹` and\n  `2.0·10¹¹`); they are covered by the structural lemma.\n- No corpus number was inferred; the census is exact integer work in this run's own copy.\n","patch":null,"cpu_hours":0,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_fn.py":"7b315effd734cd557f54dab466d139156307a895f0f3c7f41a9319ac17adb242","fetch_fn.py":"9fb95a29cfbd9a971592df3ec742c33a9b0e2133af7ac0c03e5fc127f7fe45df","census_fn.py":"4df494e122533083d50a831f08e7f002a4083071d0952404fca7cb08aa5a2284","check_fn.out":"f1cfe86374ab00162187b987cbb401ebabda18958d6733f3fe61ea2d0f061e0d","recipe_fn.md":"9bced8a53a0610c7af3326bbad260af15e28545c80b435a27b4f7480517a9443","redact_fn.py":"e46abbaa8a2bf10592a77f3f69ebeefee145f691f4b7f44ab5d965d10c34448b","report_fn.md":"10a85c6c4851de5372d1ee6adf8640424cb45714d488bf8f000529102c0e533f","census_fn.json":"e505adb87ce3c4dd6f834ff78f2be2944bff62642c40be6d9b8aafb66249cac2","evidence_fn.md":"d85512cf480e13e6218b33ed1439ac3ca8b3e56636a24820be573de6192a5807","prior_art_fn.md":"47b53b04c683a81ee54ed2f3758d6a073bb82b875f08b90b69020d4bed3ac702","served-board.json":"270ff029c3039e4c5c73a8b61ea3492446ef2da58aae4644e8296e78be2f3cd5","uncertainty_fn.md":"cb72b1241d184b775247d726a06b854cda93772d0997197e4a7f2c198339cd98","served-check_ff.py":"2c60d9532589adcadecd3e90201cbb84acd2129f1631c6387cbdc832ad494ec6","served-recipe_ff.md":"f838eab1c7f9beb2173c7503f90eab02113312da2742d8a16352059ebc4f17ff","served-report_ff.md":"5249c93f11c4b04bff4d0fd6887f2fdf3e41db483f7ba5ee5a24a84726c58ceb","check_fn.control.out":"5bc624d0911cd7a4c87df74e957baa3a364fb440f999d4cab2603d4b2c1b6fad","served-evidence_ff.md":"66581e62dc3141d6ed25daad56bad7e26dbf487101d970eb7ad4afb794b80672","served-next_step.json":"be810042e9f2d789d70afa11b101c73d724924fc0d6dee6566ce9f293c39d863","served-questions.json":"7251ad0a9c915d7926c3c7e40e1e8b37797d3e724f3d0ad3fd6db62eb04ac509","served-route_229.json":"99e5b1974d4f8031a942abb71f619a47b143b6ae60baf1700cc7ea8386a1bc76","served-route_230.json":"e65cf404ff246b9d643e8101c14279a55debb83f8c1404cc7a659f2cd5a06db9","served-prior_art_ff.md":"c9f49ad71a226e3894c688264aa61cfd636029c50ee20bfe7696944cb1f226a0","served-return_2560.json":"372952c78a449505fddb4a71c37dc54436a81512475f22a6c3524d40fbde3124","served-return_2561.json":"368bb6e2c3ade8c4cc0c4be8b932229f622718b7232cda2d86290b97f4414066","served-return_2562.json":"14caf506123f2ee60a1bb64f68da5b54ad2c61a3f255309085fb140585042f8d","served-return_2563.json":"177072b5d52b68425f41d1099ad48388b872fb74bc3f914c39af48fe348984f3","served-return_2564.json":"2d87df9c99991bfa502dabc66363ada9bc54ad68c5bcd416ef19d40af6728e87","served-uncertainty_ff.md":"028ebc2b0a50716ac58ee43c5b7118c41984010ebe8efc33d8b7f7ad7983dd19","served-contribution_ff.md":"9641e17ede5fbd35130832f1c9e9e974260fb528988864b1a380876e775ff399","served-served-PRIOR-ART.md":"a24aa29acf591a6a771b4ad3164c15c623a2b32145e47a068d442381f2ea1d8b","served-hist__scanstat-t37.md":"eec291b06403cfe914f580bf2a638503cc2ea3623e714d47f3c4091c6f4bb6e8","served-docs_staging_index.txt":"b74759f027dad779da42d749683967042edcdf6ed3fe71f63a68ac20d8395425","served-protocol-evidence.json":"d2119899adf2cddfb81e830f52d3807fe0a7e0c0455a34f24ed2ccf5690697ce","served-research-protocol.json":"925c7cd9694d7ee41965f7086fada8f7ab7a9adc6429b42e4f0e039929957e29","served-protocol-lifecycle.json":"45c40f1d4937e757128a5d230ceb89354f41f02fdc4ae1b679ae679675304edb","served-protocol-accounting.json":"93cc7304c3f05490afc8055f1e912127670cb00b76b4675db2181e3c846f1008","served-protocol-publication.json":"f2122dc2662f4f00ddb562d4147d35aa6c8b0dfbb6fc586653b07e53cb3b343c","served-hist__measure-0904-argmax.md":"ae6f1b379be82a2ecded1fd1a640fabe13e976b9bcfa16a1510dec89fa0cf693","served-hist__redteam-0904-argmax.md":"dc322fdde6403f4cf597d821338d10d05575927d89efaffe421769df84bbde0b","served-protocol-publication_safety.json":"dc444ec3595c39510b4dcb5bceaef907d887a4cc2389f9fb938a9e14342f8387","served-served-doc__research__G2-STATE.md":"401ace7cf2be219a88df6ee0564ab587bfc2ee46023eba2dcdf3d76f5f789cee","served-served-doc__research__THE-DIALS.md":"9daf453cc3c684b91cbc23b28c3b2e9201ed94523fb7537c467d6fe9a7767468","served-docs-research__SEARCH-CONVENTIONS.md":"b207bf88a2f6e7a0b4e5b265fae37626c3977667353df29f80b9df91a83582db","served-served-doc__research__ZONE-POSTULATE.md":"82ec04b1c1bc4550741abb13ae54c2cf0ea103a3c1743d15062b7956834db91a","served-served-doc__research__RESEARCH-HANDOFF.md":"b3f33f44c99470cac0002cae9f5ea10471b8fd69f6523b599302b53dd0933fc2"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-08T23:50:44.626Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2560],"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 — reproduce the route 229 first-look census (offline, Python 3.11 + numpy 1.24).\n\n1. Fetch the served record (this run's own copies):\n   GET /projects/twin-primes/research-routes/229 ; GET /projects/twin-primes/return/2560\n   GET /projects/twin-primes/docs/research/history/staging/measure-0904-argmax.md\n   (research/G2-STATE.md is attached to return #2560).\n2. Run the exact census over the whole period for x = 11,13,17,19,23:\n   `python3 census_fn.py 11 13 17 19 23`\n   It strikes W=x# at residues 0 and -2 mod p for every p<=x with numpy, finds every maximal run of\n   struck positions, and counts per-window the distinct active primes and classes. Self-test: the\n   computed G2/m must equal the record ladder / argmax sizes.\n   Runtime: x<=19 about 0.1 s; x=23 about 3 s and ~0.5 GB.\n3. Run the offline checker:\n   `python3 check_fn.py`            (expect 38/38 PASS, exit 0)\n   `python3 check_fn.py --corrupt`  (expect exit 1, 4 failures)\n4. Read the result: at every maximal window at every reachable level the active-prime count is\n   exactly pi(x) and both classes occur -> zero one-class windows; the elementary lemma makes this\n   structural for all x>=3.\n\nInputs: served G2-STATE.md ladder and measure-0904-argmax.md argmax sizes (self-test only);\nno network beyond the fetches in step 1; no producer import. Outputs: census_fn.json, check_fn.out.","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":"blocked","obstacle":{"kind":"claim_refuted","evidence":"census_fn.py (exact numpy strike of the whole period; self-test reproduces the record's G2/m at x=11..23) and check_fn.py (38/38 PASS exit 0; --corrupt exit 1, 4 failures). Per-level table: x=11 G2=42 m=4 active 5/5; x=13 G2=66 m=12 active 6/6; x=17 G2=108 m=20 active 7/7; x=19 G2=150 m=20 active 8/8; x=23 G2=204 m=4 active 9/9; classes 2 in every window; zero one-class windows.","statement":"No primorial level x >= 3 has a one-class maximal twin-slot gap window. An exact census of every maximal window of T_x = Z/x# at x = 11,13,17,19,23 finds the number of active primes equal to pi(x) in EVERY window (all primes <= x strike it) and both classes {0,-2} present; there are zero one-class windows. This is structural: for p>2 the residues 0 and -2 mod p differ by 2, so no single prime strikes three consecutive integers, while every maximal window is a run of G2-1 >= 5 struck positions (G2 >= 6 for x >= 3). Hence route 229's pre-registered failure clause is trivially guaranteed and its specified first experiment cannot decide the intended sieve-dimension question.","assumptions":"Uses the route's own operational definition of a one-class witness (a single prime p <= x explains all killed positions of the window) and the served objects: W = x#, twin slots avoiding 0 and -2 mod every p <= x, G2 the cyclic maximum gap. G2 >= 6 for all x >= 3 is read from the served ladder research/G2-STATE.md section 2.","revisit_when":"Only if the mechanism is restated on a coordinate that can express a level drop - e.g. a proof-level reduction showing the DHR dimension-2 sieve input is unnecessary at the argmax window (no such statement was found in the located literature), or a new non-per-window statistic on T_x with a pre-registered falsifier that is not trivially guaranteed. Re-running the specified census cannot change the outcome."},"route_id":229,"depends_on":[2560],"evidence_md":"Route 229 first look: the one-class-witness precondition is unsatisfiable (outcome blocked,\nobstacle claim_refuted).\n\nEXACT CENSUS (new, this run). `census_fn.py` strikes the whole period W=x# for every p<=x at\nresidues 0 and -2 mod p, then finds every maximal run of struck positions and counts, per maximal\nwindow, the distinct active primes and the classes {0,-2} present. Self-test passed: computed\nG2/m match the record exactly.\n\n| x | W | D | G2 | m | active primes | classes | one-class windows |\n| 11 | 2310 | 135 | 42 | 4 | 5=pi(11) | 2 | 0 |\n| 13 | 30030 | 1485 | 66 | 12 | 6=pi(13) | 2 | 0 |\n| 17 | 510510 | 22275 | 108 | 20 | 7=pi(17) | 2 | 0 |\n| 19 | 9699690 | 378675 | 150 | 20 | 8=pi(19) | 2 | 0 |\n| 23 | 223092870 | 7952175 | 204 | 4 | 9=pi(23) | 2 | 0 |\n\nAt every maximal window at every reachable level the active-prime count is exactly pi(x) (all\nprimes <= x strike the window) and both classes occur. The kill-class count is scale-dependent but\nmaximal, never 1: zero one-class windows.\n\nSTRUCTURAL LEMMA (general). For p>2 the two killed residues 0 and -2 mod p differ by 2, so no single\nprime strikes three consecutive integers (verified symbolically for p<=23: maximal run = 2). Every\nmaximal window has run length G2-1, and the record's ladder has G2=6 at x=3 and >=6 for all x>=3\n(G2 = 2,6,12,30,42,66,108,150,204,258,348,528,...). Hence every window for x>=3 is a run of >=5\nstruck positions and cannot be one-class. The route's pre-registered failure clause is trivially\nguaranteed: its specified first experiment cannot decide the intended sieve-dimension question.\n\nRECORD SOURCES USED AS SELF-TEST (not recomputed): research/G2-STATE.md sec.2 ladder;\nhistory/staging/measure-0904-argmax.md argmax sizes m(x); served research-routes/229 and return\n#2560 (route state proposed, rev 1, last_return_id 2560, next_step question names the scale-dependent\nkill-class signature).\n\nTARGETED SOURCE LOOKUP (2026-10-08): no published statement of scale-dependent dimension for a\ntwo-class Jacobsthal/sieve object, and no single-prime cover of a maximal reduced-residue gap; all\nhits already owned by the record's PRIOR-ART (Franze, Kourbatov, Hagedorn, Neudecker, Holt; DHR;\nFord). Gap unchanged.\n\nWHAT IT CHANGES: the route's per-window channel has no realization (kill-class signature is maximal\npi(x) and two-class, provably for all x>=3). Recorded as a scoped obstruction; the DHR dimension-2\ninput beta_2=4.2665 remains the binding input. No claim about the exponent or the band.","prior_art_md":"Updated online search record for route 229 (level-drop route).\n\nSEARCH DATE: 2026-10-08, from this run. Queries: (1) \"Jacobsthal function primorial maximal gap\nreduced residue classes sieve dimension two\"; (2) \"twin prime admissible residues maximal gap\nJacobsthal function h(n) primorial one class two class\"; (3) \"Diamond Halberstam Richert sifting\nlimit dimension two paired residue classes maximal gap\"; (4) \"maximal gap reduced residues primorial\ndetermined by single prime class covering consecutive integers\".\n\nSOURCES INSPECTED (snippets / record locators): Franze, *Sifting limits for the Lambda^2 Lambda^-\nsieve*, arXiv:1012.3809 (kappa=2 value 4.516, worse than DHR's 4.2665); Brady, *Sieves and iteration\nrules* (2017) (sifting limits, Jacobsthal function sec.8); Ford, *Basic sieve methods* (sieve2023)\nand *Large gaps in sets of primes*; Hajdu-Saradha (disproof of a Jacobsthal conjecture);\nKourbatov-Wolf, *Predicting maximal gaps in sets of primes*; Hagedorn, *Computation of Jacobsthal's\nfunction h(n) for n<50*; OEIS A048670 (Jacobsthal at primorials); DHR dimension-kappa sifting limits.\n\nRECORD CROSS-CHECK (decisive). Every substantive hit is already owned by the project's own PRIOR-ART\nledger (Franze, Kourbatov/Kourbatov-Wolf, Hagedorn, Neudecker, Holt; DHR; Ford). The record also\nfixes the exact difference (PRIOR-ART line 598): its G2 is the maximal gap of the *twin-admissible\nresidue system* at primorials (OEIS A144311+1), whereas the external maximal-gap papers concern\n*actual* twin primes below x — a different object at a different rung. `SEARCH-CONVENTIONS.md`\nsec.1/3/4 already tabulate the beta_2 lower-bound literature (Selberg reciprocal convention;\nBrady 1.8196; beta(2) >= 2).\n\nEXACT REMAINING GAP. No located source states that the effective sieve dimension of a two-class\nJacobsthal object drops to one at the argmax window (i.e. that a one-class witness certifies the\nmaximal window), and no source covers a maximal reduced-residue gap by a single prime. The record's\nown argmax measurements likewise report two-class windows from x=23 (measure-0904-argmax.md). So the\nunderlying question — can the DHR dimension-2 input be avoided at the argmax window — stays open,\nbut the route's operational proxy (per-window single-prime witness) does not measure it: it is\nprovably unsatisfiable for x>=3 (no single prime strikes 3 consecutive integers; every maximal\nwindow is a run of >=5). This is a sourced known match on the literature side and a scoped\nobstruction on the route's own test."},"research_route_id":229,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_5795b083a706a26b50574e40","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 <project base>/research-routes/229 and return #2560. Return the ordinary report and transcript plus research: {route_id: 229, 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; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, 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 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