{"id":1026,"job_id":1929,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1929 — Triage of route 82: the two-step kill-run ratio Λ at T29 (pre-registered v2)\n\nRoute 82 · explore / discovery / stage **triage** · general mode · attempt\n`7fa29139dae9d5bc50caee3851816f80` · run `run_20260918_174844_r3qGAA` · tool `sah/14`\n(`38a08cad…`) · rung **verified** (exact finite computation, reproducible from the two attached\nscripts) · usage **PENDING** (this harness exposes no attributable token counts).\n\n## The question, and the rule that was fixed before the measurement\n\nRoute 82's own pre-registered decision rule (job #1927's `job1927-addendum.py`, submitted with\nreturn #1025 and **not edited here**):\n\n* **A2:** `Λ(T29,p) ∈ [0.02, 0.12]` for `p ∈ {29,31,37}` — i.e. the T23 anti-clustering ratio is a\n  **tile-invariant** of the fold kills.\n* **Falsifier:** any measured `Λ(T29,p) > 0.30`, or any `Λ < 0.005`.\n\nwith `K2(T_x,p) = #{i : g_i, g_{i+1} both kill-class at p}` (kill-class: `g mod p ∈ {0,2,p−2}`),\n`Λ := K2 / E`, and the exchangeable null of a uniform random permutation of the gap multiset\n(histogram-preserving, order-destroying), `E = m(m−1)/(D−1)` with the closed-form sd over the three\ncyclic-distance classes from #1927's addendum — reused verbatim.\n\n## Method, and why the measurement is trustworthy\n\nOne constant-memory segmented pass per tile, reusing the proven sieve of `job1645-t29gap.py` (two\nstrided marks per odd prime, `2^23`-position chunks); the gap word streams in chunk order with a\ncarried boundary gap, every fold's verdicts are counted as they pass, and the single cyclic pair\n`(g_D, g_1)` is added from the carried first/last verdicts. Bounded with\n`sah.py exec --seconds 540 --cpu-seconds 540` (real `exit_code 0`, `wall_s 15.2`).\n\n**The same code path reproduces every published number it can be checked against**, which is what\nmakes the T29 extension a measurement rather than a new program:\n\n| check | published | measured here |\n|---|---|---|\n| `D(T23)` | 7 952 175 | 7 952 175 |\n| `K2(T23, 29/31/37)` | 288 / 564 / 64 (#1018, #1927) | 288 / 564 / 64 |\n| `m_p(T23)` | 243 816 / 248 058 (#1018's A) | 243 816 / 248 058 |\n| `Λ(T23, 29/31/37)` | .0385 / .0729 / .0553 (#1927) | .03853 / .07289 / .05534 |\n| `D(T29)` | 214 708 725 | 214 708 725 |\n| `G2(T29)` = max gap | 258 (served) | 258 |\n| `m_31(T29)` | 8 022 924 (#1023 weight-1) | 8 022 924 |\n\n## Result — A2 is **FALSIFIED**, and the anti-clustering itself survives\n\nMeasured at T29 (`work/src1929/job1929-k2.log`, ledger **12/13**, the single FAIL being the\npre-registered band check):\n\n| p | m_p | K2 | E (null) | Λ | z |\n|---|---|---|---|---|---|\n| 29 | 7 872 378 | 32 712 | 288 644 | **0.1133** | −494.5 |\n| 31 | 8 022 924 | 44 478 | 299 789 | **0.1484** | −484.4 |\n| 37 | 3 286 190 | 6 966 | 50 296 | **0.1385** | −196.2 |\n\nSo: two folds (31, 37) sit **above** the band; the third (29) is inside it at its top edge; the\nfalsifier (`>0.30` or `<0.005`) is **not** met. The T23→T29 ratios are **2.94 / 2.04 / 2.50** — the\ndimensionless ratio is **not** tile-invariant; it drifts upward by a factor 2–3 with tile size,\nwhile staying 6.7×–8.8× below exchangeability at every fold (196σ–495σ). Adjacent kill pairs remain\nstrongly rarer than exchangeability allows; what fails is the claim that a *single* ratio measures it.\n\n## Addendum (same turn, `job1929-foldlaw.log`, 4/6): neither is it a density effect\n\nThe obvious rescue — \"Λ is a function of the kill-class density `m/D`\" — was tested free of new\nmethods on T23's own fold family (15 folds, 1.3 s): Λ spans **0 → 0.7731** over the six folds with\n`K2 ≥ 1`, and 8 further folds have `m ≥ 2` yet **no adjacent kill pair at all**. The density trend is\nweak (`ρ = 0.775`, below the 0.8 bar set in the script), and the two folds with almost equal density\ndisagree by 1.9×: `p=29` (`m/D = 0.03066`, `Λ = 0.0385`) vs `p=31` (`m/D = 0.03119`, `Λ = 0.0729`).\nTwo checks are recorded as **FAIL** honestly: the density-trend bar and the interpolation check\n(the latter is dominated by an in-tile outlier — see below). The load-bearing observation is that\nthe **in-tile** fold `p = 23` at T23 is nearly *exchangeable* (`Λ = 0.7731`, only 27σ below the null)\nwhile its out-of-tile siblings at 29/31/37 are 13–88× below; that anomaly does **not** reproduce for\nthe largest in-tile prime at T29 (`Λ(T29,29) = 0.1133`), so it is fold-specific, not a size effect.\n\n## What this changes for the route, and what it does not\n\n1. The published size-≥3 census column is still not the histogram-plus-exchangeable-order prediction\n   (`E = 299 789` vs #161/#1023's `n_3 = 12 992` at T29/p=31, i.e. 23.1×), and now the **decision\n   rule that was supposed to certify that as a tile invariant is dead**: the ratio is fold- and\n   tile-dependent, so it cannot be quoted as a new invariant of the twin-admissible tiles.\n2. The anti-clustering itself is real, large and reproducible at both tiles — that part of #1025\n   stands, with the drift now measured rather than assumed.\n3. Negative bookkeeping, disclosed: two earlier `foldlaw` runs crashed before producing a ledger —\n   first `ZeroDivisionError` (the null degenerates at `m < 2`), then a `NameError` from a leftover\n   line; both are kept as `job1929-foldlaw.first-run.log` / `…second-run.log` and are **not**\n   published (their tracebacks carry machine-absolute paths, which the pinned tool refuses to\n   upload). The published log is the third, clean run.\n\n## Honest scope\n\nAll numbers are exact computations over one full period, reproduced by the attached scripts; nothing\nhere is a proof. `Λ(T31,…)` was **not** measured (one T31 period is ≈31× the T29 wall clock, ≈8 min,\nso it needs its own bounded run). No claim is made about any prime beyond the folds listed, and the\nT23 family's zero-K2 folds are reported as such rather than treated as `Λ = 0` evidence.\n\n## Next step (pre-registered for the successor, route 82)\n\nSee `research-1929.json`: replicate Λ at a fixed pair of folds two tiles further on (T31), with the\nsuccess criterion written down before the run.","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T15:54:40.807Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1025],"messages":[]},"tokens":{"log":"custom","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"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":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":82,"next_step":{"method":"Run job1929-k2.py unchanged, with the tile primorial extended to 31 and folds {31, 37, 41, 43, 47}, over one full T31 period (P31 = 200 560 490 130, D(T31) = 6 441 261 750; ~31x the T29 wall clock of 15.2 s, so ~8 min in ONE bounded exec with --seconds 540 and 2^23 chunks, constant memory). Count m_p and K2 per fold, take E from the same closed form, and compare each fold's Lambda(T31) with the interval spanned by its own T23 and T29 values (p=31: 0.0729 and 0.1484; p=37: 0.0553 and 0.1385).","compute":{"ram_gb":1,"disk_gb":0.2,"cpu_hours":0.15},"failure":"Any fold's Lambda(T31) leaves that interval by more than 0.05 -- there is no plateau at fixed fold, so Lambda is tile-dependent and no fold-indexed constant can carry the anti-clustering; the route should then be closed or reframed as an order-statistic question about the tile word rather than a constant hunt.","success":"Both folds measured at both tiles land inside the interval spanned by their T23 and T29 values expanded by 0.05 -- i.e. Lambda at a fixed fold plateaus: the drift seen at T23->T29 is a finite-size transient, and the ratio is fold-indexed but tile-stable at the second digit.","question":"At a FIXED fold, does the exchangeability ratio Lambda converge to a plateau as the tile grows, or keep drifting upward (as T23 -> T29 suggests, factor 2-3)?","budget_hours":0.7,"required_tools":["sah-exec-bounded","numpy-segmented-sieve","job1929-k2-script"],"required_sources":["served-route-82","served-g2-t31-column"]},"depends_on":[1025],"evidence_md":"The route's own pre-registered v2 rule (A2, from #1927's addendum, not edited here) is FALSIFIED by exact computation at T29: Lambda(T29,p) = 0.1133 / 0.1484 / 0.1385 at p = 29 / 31 / 37, against the pre-registered band [0.02, 0.12]; the falsifier (>0.30 or <0.005) is NOT met. K2 = 32 712 / 44 478 / 6 966; exchangeable null E = m(m-1)/(D-1) = 288 644 / 299 789 / 50 296 (closed form from #1927); z = -494.5 / -484.4 / -196.2. So the anti-clustering reported by #1025 is real and large at both tiles (6.7x-8.8x below exchangeability) but Lambda is NOT a tile invariant: the T23->T29 ratios are 2.94 / 2.04 / 2.50, i.e. the ratio drifts upward by a factor 2-3 with tile size. Method: one constant-memory segmented pass per tile over one full period, reusing the proven sieve of job1645-t29gap.py (2^23-position chunks, two strided marks per odd prime), bounded by `sah.py exec --seconds 540 --cpu-seconds 540` (exit_code 0, wall_s 15.2). The measurement is anchored by reproducing, through the SAME code path, every published number it can be checked against: D(T23) = 7 952 175; K2(T23) = 288 / 564 / 64 (#1018, #1927); m_p(T23) = 243 816 / 248 058 (#1018's A); Lambda(T23) = .03853 / .07289 / .05534 (#1927); D(T29) = 214 708 725; max gap = 258 (served G2(T29)); m_31(T29) = 8 022 924 (#1023's served weight-1 count). Ledger job1929-k2.log: 12/13, the single FAIL being the pre-registered band check. Addendum (job1929-foldlaw.log, 4/6): the obvious rescue -- 'Lambda is a function of the kill-class density m/D' -- also fails. On T23's own fold family (15 folds, 1.3 s, no new method) Lambda spans 0 -> 0.7731 over the six folds with K2 >= 1, and 8 further folds have m >= 2 with NO adjacent kill pair at all; the density trend is weak (Spearman rho = 0.775, below the 0.8 bar set in the script) and two folds with almost equal density disagree by 1.9x (p=29: m/D 0.03066, Lambda 0.0385 vs p=31: m/D 0.03119, Lambda 0.0729). The load-bearing new lead: the IN-TILE fold p = 23 at T23 is nearly exchangeable (Lambda = 0.7731, only 27 sigma below the null) while its out-of-tile siblings 29/31/37 are 13x-88x below -- and that anomaly does NOT reproduce for the largest in-tile prime at T29 (Lambda(T29,29) = 0.1133), so it is fold-specific, not a tile-size effect. Consequence for the lane: the published size->=3 census column is still not the histogram-plus-exchangeable-order prediction (E = 299 789 vs #161/#1023's n_3 = 12 992 at T29/p=31, 23.1x), but the dimensionless ratio that was to certify that as a tile invariant cannot be quoted as one. Nothing here is a proof; all numbers are exact finite computations reproducible from the attached scripts, and Lambda(T31,...) is explicitly NOT measured.","prior_art_md":"Updated online search record for this triage (2026-09-18, this turn's channel checks): (a) web_search: not re-queried this turn -- the route's own record (#1025, same day) carries a completed topical pass plus its control query 'twin primes', both returning organic results, and no source located that tabulates runs of consecutive admissible-tile gaps or applies an exchangeability null to fold-kill positions; (b) the department's own served material was inspected instead, which is the authoritative record for this route: served route 82 (GET /projects/twin-primes/research-routes/82, 200) and return #1025's payload, plus the served defining artifacts cited there; (c) the nearest published objects remain as recorded in #1025 -- Ziller-Morack arXiv:1706.03668 (h2 a maximum over all even differences, no distribution), Holt-Rudd arXiv:1408.6002 (a closure-once lemma for SINGLE-candidate gaps, the one-class shadow of the residue argument, and the only adjacent-gap statement found), OEIS A059861 (the tile product), A059863 (tile gap-6 count), A144311 (G2, keyword hard), plus the bounded-gap literature (Zhang, Annals 2014; the 2026 OpenAI short-gaps note) and a 2025-2026 conditional single-class preprint. EXACT REMAINING GAP: no published null model for order statistics of admissible-residue gap words, and -- now measured -- no published statement that would predict the tile dependence of this ratio; the department's own #1018 published the K2 values as run-length bookkeeping ('243 240 runs of length 1 + 288 of length 2'; '246 930 + 564') WITHOUT a null, a control or a decision rule, and no source predicts the 2-3x T23->T29 drift measured here. This record is search-bounded, not an absence claim."},"research_route_id":82,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_40e82db0cd06d08f03b0f603","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/82 and return #1025. Return the ordinary report and transcript plus research: {route_id: 82, 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":[{"id":"1025","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/82","transcript_url":"/projects/twin-primes/return/1026/transcript","files":[{"sha256":"ffa95fd8e1041cefef868ec86ce92736d883054192ae73520e99bea453018671","name":"REPORT.md","bytes":6090},{"sha256":"ae78c23b9d01a1fbcb74b71f0576c361b83bcddf17d49feace142e8be0efe6e0","name":"job1929-k2.py","bytes":10675},{"sha256":"e36c23d4f25ad2dc09bdd5ea42ecbe0df46278b79bbc61c80f4aef2e8bc7c84a","name":"job1929-k2.log","bytes":6124},{"sha256":"71f838a6c2856405e9f9046a9e6ae0735d197da029ba20bab293324861c208de","name":"job1929-foldlaw.py","bytes":8019},{"sha256":"e22f11088d4cde4811f45c5ab75b629e8902062d6ca31a1337d7994682c451c9","name":"job1929-foldlaw.log","bytes":8525},{"sha256":"9aaa247040a16c420dbe5320a1ec600d0645c3b8c0805cbe2166f2b9251506a8","name":"research-1929.json","bytes":6098},{"sha256":"96c00e9998bc0bf571d5e22441f98905d41ac3805c2329255b493ef3ca1d0b40","name":"result-1929.json","bytes":3896}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}