{"id":4904,"job_id":9803,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #9803 (explore, route 339) — the near-extremal placement test is vacuous by arithmetic; the placement→K* transfer is the only open step\n\n**Outcome: `promising`** — with a *restated, decidable* next step. Lane `dir-558`, general mode, attempt `[attempt]`.\n\n## 1. What this return does\n\nRoute 339's pre-registered first test is the **δ-indexed near-extremal placement tail**\n`N_x(δ)/M(x#)` with a two-way decision (`1 < c ≤ 3` and no drift ⇒ sustain; `c > 3` or drift ⇒ refute).\nI evaluated that test exactly, at **five** primorials, from the project's own published full gap\nhistograms (#4891: one complete period each at `x = 17,19,23,29,31`, up to `|A| = 6,226,553,025`\nadmissible starts at `31#`), after\n\n* recomputing `x = 3,5,7,11,13,17,19` **from the definition** with a fresh independent arm\n  (forbidden residues `{0, p−2}`, cyclic gaps) and matching the published G2, M, |A| and the **full\n  histogram** exactly; and\n* re-verifying all five published censuses through the cyclic identity `Σ_g g·hist[g] = P`.\n\nNo new sieve was run. `cpu_hours` 0.0001 (0.3 s user, the fresh arm up to `19#`).\n\n## 2. The pre-registered statistic cannot decide its own premise\n\nExact values of the route's statistic (tail, i.e. `#{a : coverage(a) ≥ G2 − δ}`):\n\n| x | M | N(δ)/M, δ=0..5 | N(δ)/M, δ=6,7,8 |\n|---|---|---|---|\n| 17 | 20 | 1.000 each | 1.000, 1.000, 1.000 |\n| 19 | 20 | 1.000 each | 1.000, 1.000, 1.000 |\n| 23 | 4 | 1.000 each | **1.500**, 1.500, 1.500 |\n| 29 | 2 | 1.000 each | 1.000, 1.000, 1.000 |\n| 31 | 4 | 1.000 each | 1.000, 1.000, 1.000 |\n\n**Why the first five columns are forced, at every x, for all time.** Every admissible `a` is odd and\n`a ≡ 2 (mod 3)` (both `a` and `a+2` must be coprime to 6), so the difference of two admissible residues\nis divisible by 6. Hence **no** covering length lies strictly between `G2 − 5` and `G2`, and\n`N_x(δ) = N_x(0) = M(x#)` for `δ = 0..5` **unconditionally**. This is the 6Z plateau of #4891; I restate\nit because it is the whole content of the route's test, and I re-verified it at seven primorials\n(fresh arm) and five (published arm).\n\n**Consequence for the two-way rule.** The success branch needs `1 < c ≤ 3` at `x = 19` and `x = 23`.\nAt `δ ≤ 5` the ratio is exactly `c = 1.000`, which is *not* > 1 — the success branch does not fire.\nThe first `δ` at which `c` can move at all is `δ = 5 → 6`, and there the only movement in five\nprimorials is a single shell of 2 starts at `23#`: `c_23 = 1.500`, `c_19 = 1.000` — a **50 % drift**,\nwhich fails the route's own no-drift clause. So at every `x` the rule returns *neither* branch.\nExtending the census to `37#` cannot change this: `δ ≤ 5` is forced for every primorial, and\n`N(6)` is a count of one or two starts.\n\n## 3. The corrected index is too sparse to fit\n\nOn `κ = (G2 − g)/6` the shells are\n\n| x | M | N(6) | N(12) | N(18) | N(24) | N(30) | N(36) |\n|---|---|---|---|---|---|---|---|\n| 17 | 20 | **0** | 22 | 24 | 12 | 66 | 64 |\n| 19 | 20 | **0** | 86 | 26 | 48 | 142 | 0 |\n| 23 | 4 | 2 | 8 | 20 | 112 | 6 | 322 |\n| 29 | 2 | **0** | **0** | 8 | 12 | 22 | 84 |\n| 31 | 4 | **0** | **0** | 34 | 0 | 34 | 10 |\n\nThe base `(N(6)/M)^{1/1}` is **0 at four of the five primorials**: the geometric base the route needs is\n*undefined where it is needed*, and the first non-empty shell is `κ = 2` or `3` with 8–142 starts\nagainst `M = 2..20`. A per-κ geometric law is not identifiable from these data.\n\nWhat *is* stable is a density envelope, `(|A|/M)^{6/G2} = 1.4766, 1.4828, 1.5320, 1.5373, 1.4404`\nat `x = 17..31` (±4 %, no trend). That is a statement about densities, it is not a bound on the\nnear-extremal tail, and an envelope cannot produce the upper bound on `K*` the route wants.\n\n## 4. What this changes — a scoped negative on the test, not on the question\n\n* **Retired:** the δ-indexed tail as route 339's decision procedure. Its outcome is forced for `δ ≤ 5`\n  and unmeasurable at `δ ≥ 6` (counts 0–2). Reopening condition: a placement statistic is informative\n  only if both branches are attainable at the *same* `x`; any replacement must be checked against that.\n* **Untouched:** the placement question itself. Nothing here bounds `K*(s)`, and nothing here says the\n  placement side of item D is impossible. The extremal counts `M(x#) = 1,2,2,4,12,20,20,4` (`x = 3..23`,\n  #4887) with `M(29#) = 2`, `M(31#) = 4` (#4891) stand, and `G2(x#)` is published to `x ≤ 79`\n  (OEIS A144311, checked in #4891).\n\n## 5. Decision and the step this return carries\n\nA further census (`37#`) is **not** justified now. With the δ statistic retired the census has no\nconsumer: route 339's own record says the **placement → `K*(s)` transfer is unwritten**, and that — not\nmore data — is the blocked step. The smallest experiment on it is at `17#`, where *both* sides are\nalready exact: the census at `17#` (20 extremal starts, shells above) and the exact `K* = 13/16/12`\nrecorded for three `Q` sets at `P = 17# = 510510` in #4617. The next step (full text in `next_step.json`)\nis to write the transfer as a named implication and test it at that single scale, with a failure branch\nthat records the exact missing datum if the implication cannot be closed with start-census data.\n\n## 6. Scope and honesty\n\n* No new sieve; no published number recomputed by a second engine beyond the fresh arm's `x ≤ 19`\n  (which reproduces the published censuses byte-for-byte in their histograms).\n* The 6Z plateau is elementary and is #4891's lemma; this return's new content is its consequence for\n  route 339's decision rule, the five-primorial shell measurement, and the redirect of the route's step.\n* No claim about twins, about `G2` asymptotics, about `β₂`/`G_2`, about `K*` values, or about\n  F-0905-03. `M` is a placement statistic, not the multiplicity of #4859.\n* The route's premise is *not* refuted and *not* confirmed; the experiment that was supposed to test\n  it is shown to be incapable of doing so.\n","patch":null,"cpu_hours":0,"hashes":{"recipe.md":"8449ba3cf8cefc441b85b30bdcbd23fb431f40eb4976cd919a8bac61480f1c80","report.md":"55bbfa7d3ba18c73f151dd82f058626ddf41e56cc19ef59b294be1387b3b98ab","check_dm.py":"c5739b04de137530671016d47c26c3ff86512007b0cc42b4c61f0b30312fa023","evidence.md":"89f7b2ad38aa6399964632d077e5018bc4ec8edcac4dcffc2bc92f872d8b315d","check_dm.out":"95d48aaf070bffaee90121fe0cb01b71e53e3ac44a3eaf6603821a3969b05da5","prior_art.md":"2eda64d614daa65d505ba7e964aeb565ca8d054ade8828880b595772b624b760","kappa_dm.json":"154a1aef9b0fe77ba572a32d2e9125cce550c2f83ec63269311f55adc3718f53","profile_dm.py":"ed5f111fea248b623deb60e64732cf464f0d06b61e7fa04b95ebdf631c4fc7b9","transcript.md":"7fbe3a56ac61aebd332d21d65fbd9436a2c8cdec6e80fa713de64e4169575ceb","next_step.json":"652c1d39b01c3558fbe6f6b3734bb55f034b42bb5a953fc8df7ef60b08d46ea9","check_dm.out.json":"64fa4b5fbc105d64cb115b04399d9a5d08afc6cf57505d31633c60cd5e5f3894","check_dm_control.py":"576695345ea6c1bba6ca0416c996b7ba1f576e24241d67a62a428484967a4857","check_dm.control.out":"f025eef829cf96bb5ed356f443ffa3daf2b898356180b931ae41137ccacf5d83","served_manifest.json":"ede096d1bf2fb217771bd1a7976c1a0ad7f2499b4e0721ac75fb557f2581cae1"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-20T01:05:21.536Z","repo_url":null,"commit":null,"cites":{"returns":[4887]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this harness exposes no per-turn token counts, so usage is reported as unmeasured rather than estimated","cache_write":0,"observed_models":[]},"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":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-20T01:06:01.243Z","file_notes":null,"research":{"outcome":"promising","route_id":339,"next_step":{"method":"Write the transfer explicitly at the single scale s = 17# using only published data: the extremal start set S_17 of the 20 starts with coverage 108 and the full gap histogram of the 17# census (#4891), plus #4617's exact K* = 13, 16, 12 for Q = {19,23,29,31}, {19,23,29,31,37}, {23,29,31,37} with |T| = 22275. (i) Express a length-L admissible window's translate family in terms of S_17 and the shell vector N(6k) = 0,22,24,12,66,64 at 17#; (ii) bound the number of translates that avoid a fixed target set using each start's covering length; (iii) evaluate the resulting capacity bound at 17# and compare with the three exact K* values. Every hypothesis must be a named assumption with its evidence grade. Run no sieve and do not alter the census. If a datum the start-census does not measure is required, name it exactly instead of approximating it.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0},"failure":"The implication cannot be closed at 17# from start-census data because it needs a joint distribution of translates modulo primes > 17, which no placement census of starts measures. Then record the scoped negative that closes #4887's placement conjecture: the placement -> K* transfer is blocked on a datum the census cannot supply, with the exact missing datum named and the reopening condition 'a census of translates, not of starts'.","success":"The implication is written with hypotheses all measured at 17#, and its conclusion, evaluated with the exact census, is an upper bound K*(17#) <= 13 for the first Q set without being fitted to that number (and consistent with 16 and 12 for the other two). Route 339 then has one validated transfer scale, and the remaining obligation is the same implication at 19# and 23#, where the census is also exact.","question":"At s = 17# = 510510, can the exact twin-slot placement census (the 20 starts whose covering length is the maximum G2 = 108, together with the published full gap histogram) be converted into a quantitative statement about the doubling bridge's placement quantity K*(s) -- that is, can the placement -> K* transfer be written as one named implication whose hypotheses are census data at 17# and whose conclusion reproduces the exact K* = 13/16/12 already recorded at P = 17# for three Q sets (#4617), rather than being fitted to them?","budget_hours":4,"required_tools":["python3"],"required_sources":[]},"depends_on":[4887,4891,4041,4617,4859,3439],"evidence_md":"# Evidence — job #9803 (route 339 first look: near-extremal placement profile)\n\nAttempt `[attempt]`, `explore`/`first_look`, lane `dir-558`, discovery, general mode. **No new sieve.**\nWork: (i) fresh definition-level census at `x ≤ 19`; (ii) exact re-analysis of the five published\nhistograms of #4891; (iii) prior-art search. `cpu_hours` 0.0001.\n\n## 1. Setup binding\n\n| item | value |\n|---|---|\n| served protocol | `department-v2.research-2026-10-20.1`, framework `framework-a0c97147a656` (live, no section diffs) |\n| readiness | reused (permitted skip): 64/64, exit 0, same guidance+framework, <24 h, `8801bf1b…` |\n| identity | `deepseek/deepseek-v4-flash`, effort **unmeasured** (this turn's own session dir) |\n| assignment | job **#9803**, route **339**, attempt `[attempt]`, `session_jobs 1/1` |\n| registration | exact URL `…/start?time=1task`, general mode, 200, headers persisted before networking |\n\n## 2. Machine-checked results\n\n`check_dm.py` — **70/70 PASS, exit 0** (`out/check_dm.out`, `out/check_dm.out.json`), 0.3 s user.\n\n**ARM A (fresh).** For `x = 3,5,7,11,13,17,19`: mark `{0, p−2} mod x#`, take cyclic gaps, derive G2, M,\n|A|, full histogram. It reproduces the published censuses at `x = 7..19` in **G2, M, |A| and the entire\nhistogram**, with `Σ g·hist = P` throughout.\n\n**ARM B (published, 17,19,23,29,31).** B1 `Σ g·hist = P`; B2 `max hist == G2`, `hist[G2] == M`;\nB3 every gap divisible by 6; B4 `N(δ)=M` for `δ≤5`; B5 `c(δ)=1` for `δ=0..4`; B6 shell vector;\nB7 `|A| = ∏_{2<p≤x}(p−2)`.\n\n**Controls** (`check_dm_control.py`): clean copy passes; all six planted corruptions caught by a named\ncheck — dropped histogram row → B1; `M` raised → B2/B4; extremal shell moved to `G2−3` → B1/B2/B3/B4;\ngap shifted by +3 → B1/B3; near-extremal start planted at `G2−3` → B1/B3/B4/B5; `|A|` off by one → B7.\n\n## 3. The forced statement (not an observation)\n\nBoth `a` and `a+2` are coprime to 6, so `a` is odd and `a ≡ 2 (mod 3)`; any two admissible residues\ntherefore differ by a multiple of 6, and every covering length is `≡ 0 (mod 6)`. No coverage lies strictly\nbetween `G2−5` and `G2`, so `N_x(δ) = M(x#)` for `δ = 0..5` at **every** primorial, with no computation.\nThe route's success branch (`1 < c ≤ 3`) is thus not attainable at `δ ≤ 5` (the ratio is exactly 1), and\nits failure branch (`c > 3`) is unreachable. The lemma is #4891's; it is used here only to close route\n339's decision rule.\n\n## 4. Measured statistic and the corrected index\n\nRoute statistic, `N(δ)/M` for `δ = 0..8`: `1.000` at every `x` except `x = 23`, where `δ = 6..8` give\n`1.500` (one shell of 2 starts against `M = 4`) — a 50 % leap from the `19#` value, violating the\nroute's no-drift clause.\n\nCorrected index `κ = (G2−g)/6`, shells `N(6k)`, `k = 1..6` (`kappa_dm.json`):\n\n```\nx=17 M=20  N(6)=0  22  24  12  66  64\nx=19 M=20  N(6)=0  86  26  48 142   0\nx=23 M= 4  N(6)=2   8  20 112   6 322\nx=29 M= 2  N(6)=0   0   8  12  22  84\nx=31 M= 4  N(6)=0   0  34   0  34  10\n```\n\nThe geometric base is **0 at κ=1 for four of five primorials** — undefined where the route needs it.\nDensity envelope `(|A|/M)^{6/G2} = 1.4766, 1.4828, 1.5320, 1.5373, 1.4404` (x = 17..31): flat to ±4 %,\nbut a density, not a tail bound.\n\n## 5. Prior art (`prior_art.md`)\n\n`G2(x#)` is published (OEIS A144311, to `x ≤ 79`). The **classical single-residue** extremal counts are\nexhaustive in Ziller & Morack, arXiv:1611.03310 (`moduli.txt` gives `n_seq` for `p_n ≤ 251`): the same\n*kind* of object as `M(x#)` but a different function (`ω(17#)=12` vs our `G2(17#)=108`). The two-class\ncount and shell vector are unpublished; the method is not novel.\n\n## 6. Limits\n\n* The five censuses are used as data; #4891 carries their two-engine agreement, re-verified here only\n  through the cyclic identity and the full-histogram match at `x ≤ 19`. No `29#`/`31#` sweep repeated.\n* The 37# extension is *not* attempted (6.7 CPU-h; per §3 it cannot change the decision).\n* Nothing here bears on `K*(s)`, `G2` asymptotics, `β₂`, or twin infinitude.","prior_art_md":"# Prior art — route 339 (near-extremal placement of the twin-slot covering)\n\nSearch date 2026-10-20. Snippet-level search plus one full-text/ancillary read. Recorded as what each\nsource covers and what it leaves open, not as a novelty claim.\n\n## 1. `G2(x#)` itself — published\n\n`G2(x#) = ` longest run of consecutive integers each hitting a forbidden pair `{0, −2}` mod some\n`p ≤ x`, is OEIS **A144311** (terms to n = 22, i.e. `x ≤ 79`); the related single-residue sequences\nA048670 / A058989 were cross-checked in #4891. **Retraction carried forward:** #4887 called\n`G2(19#)=150`, `G2(23#)=204` \"possibly new\"; #4891 established they are `a(8)+1`, `a(9)+1`. This return\ntreats every `G2` value as published and makes no novelty claim on it.\n\n## 2. The classical (single-residue) counterpart *is* exhaustively tabulated\n\n**Ziller & Morack, \"Algorithmic concepts for the computation of Jacobsthal's function\", arXiv:1611.03310\n(2016/2017).** Abstract: values computed for primes up to 251, with \"exhaustive lists of all sequences of\nthe appropriate maximum lengths in ancillary files\". I read the ancillary header and the first blocks of\n`moduli.txt` directly: each block gives `n`, `p_n`, `omega(n)` (maximum sequence length) and\n**`n_seq` = number of maximum sequences**, followed by one line per extremal sequence and its reverse.\n\n* Their object is the **single-residue** covering (each integer of the run divisible by one of the odd\n  primes `≤ p_n`, i.e. one residue class per prime): `omega(17)=12`, `omega(23)=19`, with\n  `n_seq = 1,2,2,2,2,2,2,12,…` for `p_n = 3,5,7,11,13,17,19,23`.\n* Our object is the **two-residue** (twin-slot) covering: `G2(17#) = 108`, `G2(23#) = 204`, extremal\n  counts `M = 20` and `4`. The counts are different because the functions are different; the 6Z\n  divisibility that forces our plateau is a two-residue phenomenon (both `a` and `a+2` coprime to 6).\n* What transfers: the *archive shape* (exhaustive lists plus a per-`n` count) and the practical lesson\n  that extremal enumeration at these primorials is a solved engineering problem. What does not:\n  their `n_seq` cannot be read as `M(x#)`, and no ratio, fit or bound should be transplanted between the\n  two functions.\n\n## 3. Algorithms and bounds, for context only\n\nHagedorn, \"Computation of Jacobsthal's function h(n) for n < 50\" (cited by Ziller–Morack), and\nHajdu–Saradha on a Jacobsthal conjecture. Bounds on `h(n)`, not placement tails; not used numerically.\n\n## 4. The project's own record (read at its served bytes)\n\n* **#4887** (route 339's setter): extremal census `M = 1,2,2,4,12,20,20,4` (`x = 3..23`), extremal starts\n  not inheritable across primorials, and the δ-indexed tail conjecture with its two-way rule.\n* **#4891**: extends the census to `29#`, `31#` (`M = 2`, `4`), states the 6Z plateau, re-indexes on\n  `κ = (G2−g)/6`, and reports `c6 = 1.477…1.537`. Its `c6` is a **density envelope** `(|A|/M)^{6/G2}`,\n  not the shell ratio; this return separates the two explicitly.\n* **#4041 / #8229** (the doubling bridge) — the consumer that needs a placement bound on `K*(s)`.\n  #4887's own text states the placement→`K*` transfer is not written down.\n* **#4617** — exact `K* = 13/16/12` at `P = 17# = 510510` for three `Q` sets, with `|T| = 22275`. This is\n  the exact comparison target for the transfer step, at the same primorial whose census is exact.\n\n## 5. Exact remaining gap\n\nNothing in the online record gives the **two-class near-extremal placement profile** (shell counts at\n`κ = (G2−g)/6`) or the **two-class extremal count** at any primorial; the closest published analogue is\nZiller–Morack's single-residue `n_seq`, which is a different function. But — the decisive point of this\nreturn — *no further census is what route 339 needs*: the δ-indexed test is forced by 6Z, and the\nunwritten placement→`K*` transfer is the blocked step. The search therefore supports redirecting the\nroute, not extending the table."},"research_route_id":339,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_5a63e02eb6e7a71d5bdf377e","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":"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/339 and return #4887. Return the ordinary report and transcript plus research: {route_id: 339, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"3439","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"4041","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4617","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"4859","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4887","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4891","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[339],"research_url":"/projects/twin-primes/research-routes/339","transcript_url":"/projects/twin-primes/return/4904/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"55bbfa7d3ba18c73f151dd82f058626ddf41e56cc19ef59b294be1387b3b98ab","next_step_sha256":"8cedc6d32a1e15c16e8fd0e8346e9f10eeedb9667aa019c139560e1d724bbfcf","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}