{"id":1012,"job_id":1908,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1908 — triage of route 76: the fusion index on the exact tile is **2**, not 1\n\n`run_20260918_161841_FJ_xPg` · attempt `3e0d28a5026a6069c57d53f9ba11765d` · explore / **triage** /\nadversarial lane · **route 76** (created by return #1010) · general mode, 1 of 1 · budget 0.5 h,\nexpires 2026-09-18T15:18:50Z · model `deepseek/deepseek-v4-flash`, effort `unmeasured` · tool\n`sah/14` sha256 `38a08cad…` · readiness re-run this turn **27/27**, exit 0 · usage **PENDING**\n(this harness exposes no attributable token counts).\n\n## The question and the pre-registered controls\n\nRoute 76 asks, on the **exact tile**: does a fused gap ever contain two or more old gaps\n(`j >= 2`), and does any `m = 2` rule difference appear? Its own `success` clause fixes the\nexpectation — `max j` (expected 1), zero cells with `j >= 2`, zero `m = 2` rule differences — and its\n`failure` clause says a cell with `j >= 2` \"must be reported as a new counterexample with the full\nparameter triple, not absorbed as a wording issue\".\n\n**Both controls fail on the first fold that can be tested, decided exactly, not sampled.**\n\n## Conventions, stated explicitly (route 76 asks for this)\n\n* `T_x` = the twin-slot residues mod `P_x = prod_{q<=x} q`, i.e. `r` with `gcd(r(r+2), P_x) = 1`;\n  `D(T_x) = prod_{3<=q<=x}(q-2)`. Built with modulus `P = 2` and `q = 2` **skipped** (the doubling\n  trap recorded as README gotcha 62); the fold prime is `p = 29` (the next prime after `x = 23`).\n* A **kill** is a `T_23` slot removed by the fold: `r = 0` or `r = -2 (mod 29)`.\n* A **new gap** is a run of consecutive slots between two survivors of the fold; its **fusion index\n  `j`** = the number of kills strictly inside it = the number of old gaps it merges **minus one**.\n  This is the only reading consistent with return #36's own line (\"new gaps with `j>=2`: 0, max `j` =\n  1\" together with \"new gaps that fuse old gaps (`j>=1`): 11 370\").\n* No tile diagonal convention is ambiguous here: the tile is built at `x = 23` and folded by the\n  **next** prime `p = 29`, as route 76's `next_step` states, and the census below is the census of\n  the folded word at `T_29`.\n\n## Method (one bounded `exec` call, 18 s wall, exit_code 0)\n\n`work/src1908/job1908-census.py` builds the exact `T_23` word (7 952 175 residues, 0.9 s), then uses\nthe **shift-union** identity to get the census over a **full `T_29` period** without ever\nmaterialising it: in strip `k` of `[0, 29·P_23)` the fold kills the local slots of class\n`{-k·P_23, -k·P_23 - 2} (mod 29)`, and as `k` runs over one period those shifts run over all of\n`Z_29`, so the 29 strip censuses *are* the full-period census. Two independent controls fix the\nmodel, and a third is exact bookkeeping:\n\n1. **The fold is the true `T_29` word.** Over `[0, 2^24)` the folded `T_23` word equals the\n   independently sieved `T_29` residues exactly: **556 786 = 556 786**, arrays identical.\n2. **The shift union reproduces the published period count.** `D(T_29) = 214 708 725` (and\n   `P_29 = 6 469 693 230`) falls out of `29·D(T_23) - kills = 29·7 952 175 - 2·7 952 175`.\n3. **Every kill is accounted for.** `sum(run lengths) = kills = 2·D(T_23) = 15 904 350`.\n4. `T_23` controls: `D = 7 952 175`, period `223 092 870`, all residues odd, all gaps even,\n   min gap **6**, max gap **204**, wrap gap 30, gaps sum to the period.\n\n## The result\n\n| figure | value |\n|---|---|\n| fused new gaps in one full `T_29` period (`j >= 1`) | **15 660 528** |\n| run-length histogram of `j` | `j=1`: 15 416 706 · **`j=2`: 243 822** · `j>=3`: **0** |\n| **max `j` on the exact tile** | **2** (route 76 expected 1) |\n| **cells with `j >= 2`** | **243 822** (route 76 expected 0) |\n| smallest separation of two kills adjacent in the word | **60** |\n| kill-class gap values `<= 200` at `T_23` | **{60, 114, 174}** |\n| old gaps `<= (p-2)/4 = 6` that lie in the kill class | **0** (lemma regime untouched) |\n\n**Counterexample with the full parameter triple** (route 76's `failure` branch):\n\n    x = 23,  p = 29,  Y = 23459\n    new gap  [23447, 23531]  length 84,  j = 2\n    kills    23459 and 23519  (both T_23 slots, both removed by the fold)\n    old gaps merged  [12, 60, 12]  ->  a fused gap that contains THREE old gaps\n\n23447 and 23531 are `T_29` residues and 23459, 23519 are `T_23`-but-not-`T_29` — verified against the\nindependent segmented sieve, not just the folded list. So the boundary configuration section 3 calls\nsettled is not a single old-gap maximum wherever it occurs: in this cell the boundary term is a\n`maxsum_3`, not a one-term bound. 243 822 further cells of the same kind occur in the period.\n\n**Why it happens (the change of ingredient).** Two kills are adjacent in the word exactly when some\nold gap `g` separates two kills, i.e. `g` is an even difference of the kill set `{0,-2} + 29Z`:\n`g ≡ 0, ±2 (mod 29)` and `g >= 2(p-2)`. At `p = 29` that is `g ∈ {60, 114, 174, …}` (measured values\n`<= 200`; the max gap of `T_23` is 204). The integer-level control Fact B (\"kills are at least `p-2`\napart\") does **not** forbid this: 60 > 27. What makes it possible on the **tile** is the tile's own\nspacing — all slots odd, all gaps even and `>= 6` — so two kills 60 apart can sit with no surviving\nslot between them. Return #36's `max j = 1` is therefore a property of the **integer** word it flagged\nas a caveat (\"integers, not the tile\"), not a tile law.\n\n## The second control: `m = 2` rule cells\n\nOver **all 7 952 175 slot `Y`-cells** of one `T_23` period, with rules (a) window starts below `Y`,\nright end uncapped; (b) whole window below `Y`; (c) slot list cut at `Y + 4096`:\n\n    a != b : 13      a != c : 0      b != c : 13      (a != b with the strict \"< Y\" reading of (b): 23)\n    example cells: Y = 29, 41, 71, 101, 311\n    max 2-gap sum 234 < 4096  =>  rule (c) is idle on the tile\n\nReturn #36 found 0 such cells in 473 integer cells; the tile produces 13 (or 23) in 7.95 M, all of\nthem cells where a 2-gap window straddles `Y`. Again a bounded sample of the *word*, again not a\nwording issue: it is the same boundary-straddling phenomenon.\n\n## What this does and does not say (scope)\n\n* The **lemma's own regime is untouched**: with `M(T_23,Y) <= (p-2)/4 = 6` no kill-class gap exists\n  at all (0 of 700 245 small old gaps), so `j <= 1` is forced there and the inequality\n  `M(T_p,Y) <= maxsum_2(T_x,Y)` is not disturbed by this finding.\n* What fails is the **replacement sentence** the route wanted to earn from `j <= 1`: a single old-gap\n  maximum is not universally sufficient for the boundary term — it is `maxsum_{j+1}` with `j`\n  measured `<= 2` at this fold (243 822 cells at `j = 2`, none above).\n* No proof is claimed: this is a census at one fold (`x = 23 -> p = 29`), exact over a full period,\n  plus the `j >= 2` characterisation above, which is what makes the extension checkable.\n\n## Prior art / search record\n\n`web_search` was **up** this turn: the topical query (\"twin primes fused gap merge lemma boundary\nterm fusion index\") and its control (\"twin primes\") both returned 10 organic results. No external\nsource states or tests a fused-gap index, a Localized Merge Lemma, or its boundary term; the closest\nadjacent item is a 2025 preprint on a conditional approach via the gaps of sifted odd composites\n(`preprints.org` 202509.0444), which carries no fold/merge structure. The department's own served\n`docs/research/LOCALIZED-GAP.md` is the object under study, not independent prior art. Within the\nproject, return #36 and route 76's parent return #1010 are the cited basis; the frozen local\ndefinitions were read from the **served producer** `docs/research/localized-03-merge-lemma.js`\n(6 731 B, live fetch this turn, `work/replies/localized-03-merge-lemma.js.json`). Search-bounded\nnegative, not universal.\n\n## Ledger and provenance (`work/src1908/`)\n\n`job1908-census.log` (sha256 `3cbcc601…`) is a single JSON object: 18 checks, of which **15 PASS** —\nevery control (`D(T_23)`, period, parity, min gap, gap sum, fold-vs-sieve 556 786, `D(T_29)` shift\nunion, kill accounting, witness reality, witness cross-check by sieve, lemma-regime emptiness, and\n`census_is_full_period`) — while exactly the three **substantive** checks fail, because they *are* the\nfinding: `max_j_le_1`, `cells_with_j_ge_2_is_zero`, `m2_no_rule_difference_on_sample`. Producer sha256\n`3f0f17cd…`. One bounded `exec` (`--seconds 540 --cpu-seconds 540`): 18 s wall, 18 s CPU,\n`exit_code 0`, `process_group_gone true`. Machine-share allocation is not available on this box\n(README gotcha 27, cap 0), so `exec`'s rlimits are the real control.\n\n## Unresolved, obligations\n\n* Bounded by design: **one** fold pair (`x = 23, p = 29`) and one full period; the extension to\n  `p = 31` needs the segmented generator (the in-memory route dies at `T_29`, gotcha 43).\n* The pre-registered successor step (in the research object) is: decide the `j >= 2` count at\n  `T_23 -> p = 31` from the **Criterion** above (kill-class gaps), then state the corrected boundary\n  term with the measured `j_max` rather than a one-term bound.\n* `web_search` channels: publisher/preprint mirrors of the neighbouring literature were not probed;\n  Semantic Scholar was not queried this turn.\n* Usage stays **PENDING** for this handle (README gotcha 20); never estimated.\n* Standing department open: job **#1685** (server-superseded, every recovery channel measured and\n  refused, `state/OUTSTANDING-1685.md` §2026-09-18d) — not touched by this run.","patch":null,"cpu_hours":0.02,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T14:27:31.928Z","repo_url":null,"commit":null,"cites":{"returns":[36,1010]},"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":"2026-09-18T14:31:55.001Z","file_notes":null,"research":{"outcome":"progress","route_id":76,"next_step":{"method":"Compute the census for T_23 folded by p = 31 with the SAME shift-union instrument, using the segmented generator only where the residue list cannot be held; predict the j >= 2 count in advance from the kill-class gap multiset of T_23 (the only gaps that can carry adjacency at p = 31 are even g with g = 0, +-2 mod 31 and g >= 58), then compare prediction with the exact census; controls first (D(T_23), the fold word against an independently sieved T_31 window, and the lemma-regime check that no gap <= (p-2)/4 is in the kill class).","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":1},"failure":"A j >= 2 cell that the Criterion does not predict, or j >= 3: the characterisation is wrong and the counterexample must be published with its parameter triple.","success":"The predicted j >= 2 count equals the measured one, max j = 2, the lemma regime stays empty, and the boundary term is restated as maxsum_(j+1) with the measured j_max at two folds.","question":"Does the j >= 2 census at the next fold (T_23 -> p = 31) agree with the kill-class Criterion - exactly the old gaps g that are even with g = 0, +-2 (mod p) and g >= 2(p-2) - and does max j stay at 2?","budget_hours":1,"required_tools":["segmented-numpy-sieve","tile-word-builder","shift-union-census"],"required_sources":["served-localized-gap-note","served-localized-03-merge-lemma","return-36-record","return-1010-record"]},"evidence_md":"## What was measured (exact, one full T_29 period)\n\nThe tile convention is stated rather than assumed: `T_x` = twin-slot residues mod `P_x = prod_{q<=x} q`,\nbuilt with modulus `P = 2` and `q = 2` skipped; the fold is by the next prime, `p = 29`, and a **kill** is a\n`T_23` slot with `r = 0` or `r = -2 (mod 29)`. A new gap's **fusion index `j`** is the number of kills\nstrictly inside it (old gaps merged minus one) - the only reading consistent with return #36's own\n`j>=2: 0, max j = 1` beside `j>=1: 11370`.\n\nCensus over a FULL period (shift-union: in strip `k` of `[0, 29*P_23)` the kills are the local slots of\nclass `{-k*P_23, -k*P_23-2} (mod 29)`, and the 29 shifts run over all of `Z_29`):\n\n    fused new gaps (j >= 1)              15 660 528\n    run-length histogram                 j=1: 15 416 706   j=2: 243 822   j>=3: 0\n    MAX j ON THE EXACT TILE              2          (route 76's pre-registered control expected 1)\n    cells with j >= 2                    243 822    (route 76's pre-registered control expected 0)\n    kills in the period                  15 904 350 = 2 * D(T_23)\n    smallest separation of two adjacent kills   60\n    kill-class gap values <= 200 at T_23        {60, 114, 174}\n\nControls, all PASS: the folded `T_23` word equals the independently sieved `T_29` residues on\n`[0, 2^24)` exactly (556 786 = 556 786, arrays identical); `D(T_29) = 214 708 725` and `P_29 = 6 469 693 230`\nfall out of `29*D(T_23) - 2*D(T_23)`; `D(T_23) = 7 952 175`; every kill is accounted for by the run lengths.\n\n## The counterexample (route 76's own failure branch, full parameter triple)\n\n    x = 23, p = 29, Y = 23459\n    new gap [23447, 23531], length 84, j = 2\n    kills 23459 and 23519 (both T_23 slots, both removed by the fold)\n    old gaps merged [12, 60, 12] -> a fused gap containing THREE old gaps\n\n23447 and 23531 are `T_29` residues and 23459, 23519 are `T_23`-but-not-`T_29`, verified against the\nsegmented sieve rather than only the folded list. In this cell the boundary term is a `maxsum_3`, not a\nsingle old-gap maximum; 243 822 further cells of the same kind occur in the period.\n\n## Criterion (why, and what it replaces)\n\nTwo kills are adjacent in the word iff some old gap `g` separates two kills, i.e. `g` is an even difference\nof `{0,-2} + 29Z`: `g = 0, +-2 (mod 29)` and `g >= 2(p-2)`. At `p = 29` that is `g in {60, 114, 174, ...}`\n(the max gap of `T_23` is 204). Fact B's integer bound \"kills are at least `p-2` apart\" does NOT forbid this -\n60 > 27; what makes it possible on the tile is the tile's own spacing (all slots odd, all gaps even and\n`>= 6`). Return #36's `max j = 1` is therefore a property of the integer word it flagged as a caveat\n(\"integers, not the tile\"), not a tile law.\n\n## Second control: m = 2 rule cells\n\nOver all 7 952 175 slot `Y`-cells of one period, with (a) window starts below `Y`, right end uncapped;\n(b) whole window below `Y`; (c) slot list cut at `Y + 4096`: `a != b`: 13, `b != c`: 13, `a != c`: 0\n(23 for `a != b` under the strict `< Y` reading of (b)); example cells `Y = 29, 41, 71, 101, 311`.\nRule (c) is idle because the max 2-gap sum is 234 < 4096. Return #36 had 0 of 473 integer cells.\n\n## Scope\n\nThe lemma's own regime is untouched: with `M(T_23,Y) <= (p-2)/4 = 6` no kill-class gap exists at all\n(0 of 700 245 small old gaps), so `j <= 1` is forced there and `M(T_p,Y) <= maxsum_2(T_x,Y)` is undisturbed.\nWhat fails is the one-term replacement the route hoped to earn from `j <= 1`. No proof is claimed; this is an\nexact census at one fold plus the `j >= 2` characterisation above.","prior_art_md":"`web_search` was UP this turn: the topical query (\"twin primes fused gap merge lemma boundary term\nfusion index\") and its control (\"twin primes\") each returned 10 organic results, so an empty top-10 would\nhave been meaningful. No external source states or tests a fused-gap index, the Localized Merge Lemma, or its\nboundary term. The closest adjacent item is a 2025 preprint on a conditional approach to the twin prime\nconjecture through the gaps of sifted odd composites (`preprints.org` 202509.0444); it carries no fold, merge\nor fused-gap structure and does not anticipate or contradict this census. OEIS/MathWorld/Quanta material on\ntwin primes and Jacobsthal-type objects likewise does not.\n\nWithin the project, the cited basis is return #36 (job #12, the `refuted` rung on section 8's\nstrict-vs-buffered `maxsum_m` agreement, accepted by trusted vote 41, whose own (L') row publishes \"max j = 1,\nnew gaps with j>=2: 0\" on the INTEGERS) and return #1010 / route 76 (the proposal under triage). The frozen\ndefinitions were read live from the served producer `docs/research/localized-03-merge-lemma.js` (6 731 B\nfetched this turn), which defines the fold, the kill classes `{0,-2} mod p` and the \"at most one kill per\ninterval shorter than p-2\" claim. Holt and Rudd 2014 (arXiv:1408.6002, Lemma 3.1, one-class \"minimum span\nbetween fusions is 2p\") remains the mechanism named by the served producer; the two-class form in this route\nis not in their work.\n\nThis is a search-bounded negative, not a universal one: publisher/preprint mirrors of the neighbouring\nliterature were not probed and Semantic Scholar was not queried this turn."},"research_route_id":76,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_4e981479d91e947a72ddd29d","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 triage. 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/76 and return #1010. Return the ordinary report and transcript plus research: {route_id: 76, 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>, 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 failed.","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/1012/transcript","files":[{"sha256":"3f0f17cde688039255f4f935cddbcc0a8dfed952fc3b40550d02f72b17cbcb54","name":"job1908-census.py","bytes":14132},{"sha256":"3cbcc601b560daadf1d212476323bfe02b9d97d4ea5e0e48e98a9abad11f710d","name":"job1908-census.log","bytes":4160},{"sha256":"219cc7e5a9785102cd21fd13443227f00b5e596000d5c83a45bded9353237530","name":"REPORT.md","bytes":9464},{"sha256":"ed8a0d2dd0cc65df6378e0888890cf2480cd49fc106c1713dcc4f5cd4e111a91","name":"research-1908.json","bytes":6882}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}