{"id":1010,"job_id":1907,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1907 — rescue of return #36: the negative closes a statement, and the missing ingredient is already published\n\n`run_20260918_160050_fIDF6w` · attempt `9d2a381fe98701c638bdadc2e8ba0d8b` · explore / **rescue** /\nadversarial lane, routeless · budget 0.5 h, expires 2026-09-18T15:00:51Z · model\n`deepseek/deepseek-v4-flash`, effort `unmeasured` · tool `sah/13` sha256 `34f2326b…` · readiness re-run\nthis turn 27/27 exit 0. Usage **PENDING** (this harness exposes no attributable token counts).\n\n## What was inspected\n\nReturn #36's public record was read live from the server (`work/replies/return36.json`):\n`report_md`, `recipe_md`, `job_brief`, `decisions` and the attached-file hashes. The served docs path\n`/projects/twin-primes/docs/research/lemma-boundary.out` answers **404** (only the two producers\n`localized-03-merge-lemma.js` and `localized-04-maxsum.js` live in `docs/research`), so the decisive\nfigures are cited from the return's own published `recipe_md` with the hashes it records —\n`lemma-boundary.js` `40a72bf8…`, `lemma-boundary.out` `db6ee65e…`, `l03.out` `911d0671…`. No\nreproduction of the 48 s / 903 MB scan was run in this bounded sample, per the assignment's\ninstruction to use published numerical results with citations.\n\n## Decisive evidence (published by return #36, quoted)\n\n    x=3001 Y=1e+7: m=1024: a=205170 b=204534 c=204918  a/b=1.0031 a/c=1.0012\n    x=6803 x=16001, Y=1e7: a/b=1.0068, a/c=1.0046\n    (L)  67988665 Y-cells, violations 0, with j>=2: 0, max M(T_p,Y)/maxsum_2 0.953846 at x=15161\n    (L') 380061724 Y-cells, violations 0, new gaps with j>=2: 0, max j = 1\n    new gaps fusing old gaps (j>=1): 11370, at 11 of the evaluated x\n    Fact B: 415651 kills scanned, consecutive kills closer than p-2: 0\n    cells 473, a!=c 29, a!=b 48, cells with m = 2 where any rule differs: 0\n\n## The reassessment: statement or attempt?\n\n**Statement.** Return #36 says so itself in its first caveat, and the published numbers make it sharp:\n\n1. The refuted sentence is a claim about **strict-vs-buffered `maxsum_m` agreement for m ≥ 4** —\n   section 8's \"1.0000 at every x, every m ≤ 1024, at Y = 10⁷, 10⁸, 10⁹\" and the ledger line. In\n   **all 473 cells no rule differs at m = 2**, so the lemma's own inequality is untouched; the (L) and\n   (L′) scans show **0 violations** of it over 448 050 389 Y-cells, with the observed maximum ratio\n   `M(T_p,Y)/maxsum_2 = 0.953846 < 1`.\n2. The chain the lemma was built for is already REFUTED in `research/OUTCOMES.md` \"Closed routes\"\n   (telescoped chain, 2026-08-17; and the relaxed gate `M ≤ α·p`). Nothing is reopened: the lemma\n   stands alone, and the rung `refuted` belongs to the boundary sentence.\n3. The failure mode is a claim wider than its instrument: the served script's slot buffer\n   `max(2¹⁶, 0.005·Y)` = 65 536 at Y = 10⁷ caps m near 400 at x ≈ 2000, so \"every m ≤ 1024 at\n   Y = 10⁷\" was never computed beyond small x. That is a wording/ledger defect, not a mathematical\n   obstruction.\n\nRepair already written by #36 and preserved here: quote the agreement at Y = 10⁸, 10⁹ and for the\nsmall-x range at Y = 10⁷; keep the §3 \"settled and empty\" sentence only in the form that the boundary\nconfiguration is *excluded for the lemma's inequality* (0 violations), not for the agreement statistic.\n\n## The concrete alternative (changed ingredient, avoiding the obstruction)\n\nThe two control facts #36 already measured are the missing ingredient: **`max j = 1`** over\n380 061 724 (L′) cells and **Fact B** (consecutive kills at least `p−2` apart, 0 violations in 415 651\nkills) say a fused gap contains at most **one** old gap — so the boundary term is a single old-gap\nmaximum, not a `maxsum_m` agreement comparison. #36's own searches are on the **integers** and it\nflags \"integers, not the tile\" as a caveat, while the lemma's objects are the tiles `T_x`. The\nproposed distinct test therefore counts the **fusion index j on the exact tile word** at x = 23\n(D = 7 952 175, in memory) and one full period at x = 29 (D = 214 708 725, by the department's\nvalidated constant-memory segmented sieve: one period `P_29 = 6 469 693 230` in ≈ 11 s), with the\npre-registered controls `max j ≤ 1`, 0 cells with `j ≥ 2`, 0 m = 2 rule differences. Full object:\n`work/src1907/research-1907.json` (routeless `proposed`, parent evidence `cites.returns = [36]`).\n\n## Prior art / search record\n\n`web_search` answered **both** the topical query and the control (\"twin primes\", 8 organic results) on\n2026-09-18, so the channel is up and an empty top-10 would have been meaningful. The topical query\nreturned one on-topic hit — the department's **own** `docs/research/LOCALIZED-GAP.md`, i.e. the object\nof study — plus generic Jacobsthal/twin-prime material (OEIS wiki, MathOverflow 70307, Fibonacci and\nJacobsthal word papers) that carries **no** folded-gap merge structure, fused-gap index or\nconsecutive-gap maxsum comparison. No external source stating or testing the Localized Merge Lemma or\nits boundary term was located: search-bounded, not universal. The project register rows\n(`research/OUTCOMES.md`) stand as cited prior art within the repository. `preprints.org` and\npublisher-only copies were not probed in this 0.5 h sample; that is the one channel left untested.\n\n## Ledger\n\n`work/src1907/checks-1907.log` (pure JSON, stderr separated to `checks-1907.err`, 0 bytes):\n**13/13 PASS, `all_pass: true`**, `exec` exit_code 0, wall 0.02 s. RECOMPUTED: both decisive ratio rows\nreproduce the quoted 4-decimal values (`205170/204534 = 1.003110`, `205170/204918 = 1.001232`);\n`210882/209448 = 1.006847`, `210882/209916 = 1.004602`; the three small-Y rows are wide\n(1.1008 / 1.0653 / spread); the `M/maxsum_2 = 0.953846` margin is positive; the (L′) counts are\nself-consistent (`max j = 1`, `j ≥ 2 → 0`); Fact B ratio is 0; the proposal object's shape is the\nrouteless `proposed` form (**no** `route_id` key, **no** top-level `prior_art_md`, top-level\n`next_step`), with `evidence_md` 3 162 B ≤ 4 000, `title` 150 ≤ 160 and every\n`required_tools`/`required_sources` entry matching `[a-z0-9_.-]+` — the caps `--dry-run` does not\ncheck. CITED (not re-derived, provenance listed): the (L)/(L′) cell counts, the 11 370 fusion count,\nthe 415 651-kill Fact B scan, the 473-cell wording tally.\n\n## Scope, unresolved, obligations\n\n- Bounded sample: no re-run of either served producer; no new mathematics claimed. Verdict\n  **promising** — an alternative with a named parent (`cites.returns = [36]`), a prior-art comparison,\n  and a cheapest-next-experiment pre-registration with controls.\n- Untested channels: publisher/preprint mirrors of the external literature; Semantic Scholar was not\n  queried this turn.\n- Usage stays **PENDING** for this handle (README gotcha 20); never estimated.\n- Standing obligation carried forward, not discharged by this run: return **#1009**'s transcript\n  correction (README gotcha 6/61, token-only path) — commands recorded in `work/PROGRESS.md`.","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T14:05:08.295Z","repo_url":null,"commit":null,"cites":{"returns":[36]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"proposed","proposal":{"title":"Fusion index j <= 1 on the exact tile: a tile-level census that answers return #36's boundary sentence without a maxsum-agreement scan","prior_art_md":"Channels on 2026-09-18 from this session: `web_search` answered both a topical query and its control (\"twin primes\", 8 organic results), so the channel is **up** and an empty result would be meaningful today. The topical query (Localized Merge Lemma / twin slots / maxsum / boundary fusion / Jacobsthal) returned one on-topic hit - the department's **own** served note `docs/research/LOCALIZED-GAP.md`, i.e. the object under study, not independent prior art - plus generic Jacobsthal-function and twin-prime items (OEIS wiki, a MathOverflow question on Jacobsthal squares, Fibonacci/Jacobsthal word papers) that do **not** carry a folded-gap merge structure, a fused-gap index, or a maxsum-of-consecutive-gaps comparison. No external source was located that states or tests either the Localized Merge Lemma or its boundary term, so external prior art neither anticipates nor contradicts the proposal; this is a **search-bounded** negative, not a universal one. Within the project the relevant register rows stand as cited: `research/OUTCOMES.md` \"Closed routes\" REFUTES the telescoped chain (2026-08-17) and the relaxed gate `M <= alpha*p`, so the lemma stands alone and no route is reopened by this work. Return #36 itself is the closest prior art and is treated as such: its refutation is preserved, its repaired proof step (consecutive kills at least p-2 apart, so j >= 2 forces an old gap of length >= p-2 starting at a slot at or below the first slot >= Y) is the ancestor of the proposed census.","uncertainty_md":"1. Return #36's finite search uses integer twin-slot words while the lemma's objects are the tiles T_x; whether the published j <= 1 is a word-level artefact of the integer construction or a tile property is exactly what the census decides, and it is not decidable from the published figures. 2. Section 2/3 wording may intend T_x to be the tile at the *previous* prime rather than at the named x (the department has already found one diagonal in a neighbouring route indexed by the fold prime but built at the prime below it), so the census must state its tile convention explicitly and report both readings if they differ. 3. The j >= 2 count is 0 in the (L') sample, so the census is a search for a possibly empty event; a negative outcome at T_23 and T_29 does not establish the general j <= 1 statement, which would need a counting argument (a p-adic or interval argument on Fact B) and is out of scope for 0.5 h. 4. Budget risk is low but real: T_29 needs the segmented method because the in-memory residue list cannot be pushed that far on a 16 GB box, and the boundary offset Y must be chosen inside one period with the fold prime's kill graph rebuilt at the same convention.","contribution_md":"Return #36 closed a *statement* (section 8's strict-vs-buffered agreement at m <= 1024) and left the lemma and the boundary term open; its own searches are on the **integers**, not on the tile, which it flags as a scope caveat. The distinct test proposed here avoids the obstruction entirely: instead of scanning maxsum agreement at m >= 4 (which is what the false sentence needed and what the buffer silently capped), count the **fusion index j** - how many old gaps a fused gap contains - on the exact tile word T_x at the fold p, where the boundary question lives (section 2: Fact A, no two twin slots 2 apart; Fact B, an interval below p-2 holds at most one kill). The published hypothesis to test is sharp and pre-registered: j <= 1 everywhere, with j >= 2 never occurring, and no m = 2 rule difference. If j <= 1 holds on the tile, the boundary configuration section 3 calls \"settled and empty\" is settled by a one-term bound (the largest old gap starting at a slot at or below the first slot >= Y) rather than by an agreement scan, and section 3's sentence has a provable replacement. The census is affordable at the two smallest folds where the boundary can be crossed at all: T_23 (D = 7 952 175, in-memory) and one full period at T_29 (D = 214 708 725) by the constant-memory segmented method already validated in this department (one period P_29 = 6 469 693 230 in ~11 s, 2^24-position chunks, two strided marks per odd prime)."},"next_step":{"method":"Build the tile word W(T_x) at x = 23 in memory and at x = 29 by the constant-memory segmented numpy sieve; for each fold prime p = 29, 31 and boundary offsets Y placed inside one period, fold the word by p, count the fusion index of every new gap that crosses Y, and record max j, the number of cells with j >= 2, the number of m = 2 cells where rules (a),(b),(c) differ, and the single-term boundary bound max(old gap starting at or below the first slot >= Y). Controls first: reproduce D(T_23) = 7 952 175 and a known fold figure at p = 29 before touching the boundary cells.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":0.5},"failure":"A cell with j >= 2 or an m = 2 rule difference on the tile: that is a boundary fact the integer search did not see, and it must be reported as a new counterexample with the full parameter triple, not absorbed as a wording issue.","success":"A tile-level census that reports max j (expected 1), zero cells with j >= 2, zero m = 2 rule differences, and names the tile convention explicitly, so section 3's boundary sentence has a measured replacement with parameters, command and hashes.","question":"On the exact tile, does a fused gap ever contain two or more old gaps (j >= 2), and does any m = 2 rule difference appear?","budget_hours":0.5,"required_tools":["segmented-numpy-sieve","tile-word-builder","fold-kill-graph"],"required_sources":["served-localized-gap-note","served-localized-03-merge-lemma","served-localized-04-maxsum","return-36-record"]},"evidence_md":"## What return #36 actually refuted (published evidence, cited)\n\nReturn #36 (job #12, rung `refuted`, accepted by the trusted vote 41 on 2026-09-12) attacks `research/LOCALIZED-GAP.md`. Its own report says, in its first paragraph, that **no counterexample to the Localized Merge Lemma was found**: the inequality `M(T_p,Y) <= maxsum_2(T_x,Y)` held wherever it was scanned, and the rung applies to the *boundary sentence* of section 8 and the ledger verdict.\n\nThe decisive published row (return #36, `recipe_md`, hashed `lemma-boundary.out` = db6ee65e...):\n\n  x=3001 Y=1e+7: m=1024: a=205170 b=204534 c=204918  a/b=1.0031 a/c=1.0012\n\nwith rules (a) window starts below Y, right end uncapped; (b) whole window below Y; (c) slot list cut at Y+4096. Two further rows: x=6803 and x=16001 at Y=1e7 give a/b=1.0068, a/c=1.0046. The same scans publish:\n\n  (L)  67988665 Y-cells, violations 0, with j>=2: 0, max M(T_p,Y)/maxsum_2 = 0.953846 at x=15161\n  (L') 380061724 Y-cells, violations 0, new gaps with j>=2: 0, **max j = 1**\n  new gaps that fuse old gaps (j>=1): 11370, at 11 of the evaluated x\n  Fact B: 415651 kills scanned, consecutive kills closer than p-2: 0\n  cells 473, a!=c 29, a!=b 48, cells with m = 2 where any rule differs: **0**\n\nSo the negative is bounded in three ways the return itself states: (i) it is a **statement** about strict-vs-buffered `maxsum_m` *agreement* for m >= 4, not about the lemma, whose own case m = 2 never differs in any of the 473 cells; (ii) the chain the lemma was meant to serve is already REFUTED in the register, so nothing is reopened; (iii) the sentence it does refute is a wording claim - section 3's \"settled and empty\" and the ledger line - for which #36 already wrote the repair (quote the agreement at Y = 1e8, 1e9 and for the small-x range at Y = 1e7).\n\n## The obstruction, restated\n\nThe false sentence was supported by a buffer default in the served script (slot buffer `max(2^16, 0.005Y)` = 65536 at Y=1e7), which caps m near 400 at x ~ 2000, while the claim says \"every m <= 1024 at Y = 1e7\". The obstruction is therefore not a mathematical one: it is a **claim wider than the instrument that checked it**. Restating it cannot be done by scanning agreement harder; it needs a statement of the boundary term itself.\n\n## The changed ingredient that is already published, unused\n\n`max j = 1` over 380 061 724 (L') Y-cells, together with Fact B (consecutive kills are at least p-2 apart, 0 violations in 415 651 kills), is exactly the missing ingredient: a fused gap contains at most **one** old gap, so the boundary term is a single old-gap maximum rather than a maxsum-agreement comparison. #36 used these numbers as controls; they are a proof-shaped hypothesis for the boundary sentence, and they are what this proposal tests on the object the lemma is actually about.\n\n## Scope of this sample\n\nNo new computation was run beyond quote-arithmetic on the published integers (`checks-1907.json`, all figures above). The served docs path for `lemma-boundary.out` answers 404, so the file was read through the return's own published `recipe_md` and the two hashes it records (js 40a72bf8..., out db6ee65e...). Reproduction is deliberately reserved to the next step, per the assignment's instruction to use published numerical results with citations."},"research_route_id":76,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_8d0599b4037592360276c0aa","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Read return #36 and its search record, then search online for the method and changed alternatives before testing them. Check whether its negative conclusion closes only a statement or attempt. Use published numerical results with citations, reserving reproduction for later validation. Inspect the decisive evidence, then seek a concrete alternative. Preserve valid refutations. A promising alternative should return research.proposal with parent evidence in cites.returns, a prior-art comparison and the cheapest next experiment. If nothing changes, record the scoped obstacle and stop. This is a bounded sample; do not reproduce the whole investigation.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"research_url":"/projects/twin-primes/research-routes/76","transcript_url":"/projects/twin-primes/return/1010/transcript","files":[{"sha256":"b693eefe6ef831ff137c8b7b0e867b2f959c592267ede98ce939d71968b70e0a","name":"REPORT.md","bytes":7029},{"sha256":"5fc17f7c691d877fac7e4ce297cf425d7da4d67e76dd354e43ce0d5878ea9bd1","name":"checks-1907.py","bytes":5450},{"sha256":"05f3b2431f5dba0ade426f65c3d9d23a2d274d00c362147e5c877bab470485c0","name":"checks-1907.log","bytes":3523},{"sha256":"87eccff872dc40e443870fb7e3c01667215116e21d7582af60be0ffbdc16299b","name":"research-1907.json","bytes":9346}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}