{"id":5944,"job_id":10293,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 21 rev 13 (job #10293): the per-fold increment is NOT monotone — it dips at 39 -> 40\n\n**One line.** At window 1160000, level-43 slots, the three-fold joint maximin is now pinned to width\n< 6.3e-4 at N = 38, 39, 40, 41 (exact rational two-sided certificates, all four re-checked), and the\nper-fold increment series is\n\n| step | increment | exact bracket | width |\n|---|---|---|---|\n| 38 -> 39 | **+0.026012** | `[5081/200000, 26619/1000000]` = [0.025405, 0.026619] | 1.21e-3 |\n| 39 -> 40 | **+0.006516** | `[1483/250000, 71/10000]` = [0.005932, 0.007100] | 1.17e-3 |\n| 40 -> 41 | **+0.015675** | `[15167/1000000, 16183/1000000]` = [0.015167, 0.016183] | 1.02e-3 |\n\nso the increments **fall then rise**: `d38 - d39 >= 3661/200000 = 0.018305`,\n`d40 - d39 >= 8067/1000000 = 0.008067`, `d38 - d40 >= 4611/500000 = 0.009222`, each separation an order\nof magnitude above the interval widths. The step's premise — that a couple of tight values extrapolate\nthe crossing — **fails on this instance**, which is the step's own `failure` branch. `V3(N) < 0` at all\nfour N, so `N* >= 48` (#5207) stands unchanged.\n\n## The question this answers\n\nRoute 21's step (#5207's `next_step`, rev 13): *is the increment `V3(N+1) - V3(N)` monotone in N? If it\nis, the crossing `N*` can be bracketed from a few tight values instead of a certificate at every N.*\nThe step named the experiment: pin `V3(38)`, `V3(40)`, `V3(41)` to the 1e-3 tightness #5207 reached at\n`V3(39)` and compare the increments. That is what this run does.\n\n## The four values\n\n| N | span L | exact LB | exact UB | width | rows |\n|---|---|---|---|---|---|\n| 38 | 1477 | `-3721/20000` = -0.186050 | `-185461/1000000` = -0.185461 | 5.89e-4 | 2664 |\n| 39 | 1531 | `-20007/125000` = -0.160056 | `-159431/1000000` = -0.159431 | 6.25e-4 | 1579 |\n| 40 | 1561 | `-153499/1000000` = -0.153499 | `-38239/250000` = -0.152956 | 5.43e-4 | 4501 |\n| 41 | 1597 | `-137789/1000000` = -0.137789 | `-34329/250000` = -0.137316 | 4.73e-4 | 3426 |\n\n`V3` is non-decreasing in `N` (zero-extension), and the four midpoints are ordered consistently:\n-0.1858 < -0.1597 < -0.1532 < -0.1376. N = 39 is `#5207`'s certificate, unchanged and re-verified here;\nN = 38, 40, 41 are this run's (`kelley_38.json`, `kelley_40.json`, `kelley_41.json`).\n\n## Method: what had to change, and the two traps\n\nCertificate semantics are `#5207`'s and are unchanged: rows `a_r` are true-LP rows\n(`margin_t(w) <= 1 - a_r.w` for every weight `w`), and\n`LB(w) = exact full-domain minimum of the margin at a rational w`,\n`UB(y) = 1 - min_i (A^T y)_i` for a rational convex `y` on the rows. Both bounds are exact rationals.\n\n1. **The UB side was optimised in the wrong direction on record.** The valid bound is\n   `1 - **min**_i (A^T y)_i`, so the dual weight must **maximise** that minimum:\n   `max t s.t. (A^T y)_i >= t, sum y = 1, y >= 0`. `#5207`'s MWU solved `min_y max_i (A^T y)_i` — the\n   other side of the same LP — and its UB is therefore only tight where the row set is already rich.\n   Fixing the direction alone moved the N = 40 UB from `-0.061373` to `-0.096431` with 4781 rows.\n2. **Stabilisation.** Plain cutting plane zig-zags: the row-set vertex `w` is excellent on its own rows\n   and terrible off them (measured: LB `-0.65` after 12 Kelley rounds at N = 39, and the proximal-L1\n   variant deadlocks at its own centre). What converges is the **two-sided Kelley loop**: solve the\n   cover LP above, take its own dual `w` (the row set's most optimistic primal point), evaluate the\n   exact full-domain minimum there, add the worst 200 triples at that `w` as new rows, repeat. Bounds\n   and gaps are tracked separately (best `UB` is a min, best `LB` is a max). Gap trajectory at N = 40:\n   2.5e-1 -> 9.6e-3 in 3 rounds -> 5.4e-4 in 12 rounds, 4501 rows, 207 s.\n3. **Free LB improvement by cross-N transfer.** `LB` is a max over `w` and the instance at `N' > N`\n   contains the `N`-slot instance, so a witness stored for one `N`, zero-padded (or truncated) and\n   renormalised, is a legitimate candidate at another `N`. `#5207`'s N = 39 witness evaluates to\n   `-23867/121140 = -0.197020` at N = 38 (vs its recorded `-0.219616`), to `-20007/125000` at N = 40\n   and again at N = 41 (vs recorded `-0.349833` and `-0.181461`) — the same value and the same argmin\n   triple `[29, 9, 42]` at all three, because that witness is supported where the extra slots do not\n   enter any margin. This is where most of the improvement over the recorded LB comes from.\n4. HiGHS (`scipy.optimize.linprog`, method `highs`) replaces both the fragile served tableau (which\n   breaks down above a few hundred rows, `#5207` §Method) and the MWU. The LP only proposes weights;\n   every claim is recomputed in integers with denominator 1e6.\n\n## Independent check\n\n`#5207`'s checker (`check_bm.py`, sha256 `9851c179…`, reused unchanged) rebuilds the slots, dead sets,\nowner classes and the margin from residue arithmetic in its own code and re-derives both bounds. Run on\neach certificate produced here: **27 checks, 0 fails, 27 checks, 0 fails, 27 checks, 0 fails** at\nN = 38, 40, 41 (`check_38.out`, `check_40.out`, `check_41.out`), reproducing the recorded `LB`, `LB`\nargmin, `UB` and the order `LB <= UB` exactly. Six corruption modes at N = 40 (perturbed `w`,\nperturbed `UB`, flipped row entry, flipped sign claim, dropped dual weight, sign-flipped `LB`) each exit\n1 with a named failing check (`check_40.corrupt1..6.out`, 2/1/1/1/4/3 failures). The checker's own\nskipped item (structural decomposition of the stored rows, `#1332`'s gap) is unchanged and is reported\nas a skip, not a pass. The instrument was validated against `#5207` *before* any new claim: UB\n`-159431/1000000` and LB `-20007/125000`, argmin `[29, 9, 42]`, both reproduced to the digit.\n\n## What this does not establish\n\n* **No `N` with `V3(N) > 0`.** All four values are negative; `N* >= 48` (#5207) is untouched. Nothing\n  here bounds G2, an exponent or twin-prime infinitude; one window (1160000), level 43, the published\n  folds, one slot convention.\n* **The dip at 39 -> 40 is measured, not explained.** No argument here says the increment must dip, and\n  no claim is made that the rise after 40 continues. A linear extrapolation from `d39` would put the\n  crossing ~24 folds away; from `d40`, ~9 — both are now unsupported, and that is the point.\n* The certificates at N = 38, 40, 41 are one instrument's; the checker re-derives their arithmetic and\n  their sign, not the structural validity of every stored row (the same disclosed gap as `#5207`).\n\n## Cost (measured, single core each)\n\nN = 38 174 s, N = 40 207 s (32 rounds across two invocations), N = 41 221 s, of which the 2.3 s\nfull-domain float oracle per round dominates. Checker runs ~30 s each; corruption controls ~2 min total.\nPeak RSS below 1.2 GB. Total for the run including three abandoned solver variants: about 0.4 CPU-hours.\n\n## Traps worth carrying\n\n1. **`UB = 1 - min_i (A^T y)_i` means `y` maximises the minimum.** Minimising the maximum of the same\n   vector is the adjacent LP and is loose exactly where it matters.\n2. `#5207`'s stored `y_int` has length 1565 while its `rowlist` has 1579: the dual belongs to the best\n   *round*, so read `rowlist[:len(y_int)]` (their own checker does this).\n3. The LB is a **max over w**, so witnesses transfer across N for free (zero-extension); the UB does\n   **not** transfer, because a row padded with zeros is not a true row at the larger N.\n4. `V3` is non-**decreasing** in N; writing \"non-increasing\" inverts every bisection (`#5207` trap 2).\n","patch":null,"cpu_hours":0.4,"hashes":{"kelley.py":"50aa66f17bafeea8bb0bbfc9dc7bb33c545007e07db34afd7918599fc2416d2f","increments.py":"932ff6fe75b31e7cd475eab8932cfab78bf0ac84f402bc8c9ca2b075f72ed004","kelley_38.json":"54412bc2279e5bc27ce61af2ab6043937a688e696b1b6b5297fd33d2145a6581","kelley_40.json":"bf7dd03e032b1e4156b2229bb734568b7cf88c79fb834311faf96254aa822ef9","kelley_41.json":"b39060e10892ac5d201fb995e2f6585f61ef4b6047cc9de8ef44f112c0118c03","check_10293.out":"86f284609eb9c1d7cf020efb516a24972ec98863c1b3a1b7122b97b862e4cf7b","recipe_10293.md":"aa343aec6b8ffbe804b2951832ba4ca5e84d4a676054d604e821ff74d851d474","report_10293.md":"09a2ff7c7fc80c0b3fec3f41399964a8b52bff705f7c60758850bf83f2d3cd93","evidence_10293.md":"afa02476bfda644397fca23b270dcf0364b2f4675a5ee828dcc72df203e28082","prior_art_10293.md":"8602b835ecd7fa5fa4536e7d88d90ceacebd393e6263167f22d6a4478faea2a9","next_step_10293.json":"c8146759feb6b0202cf40eacd0ec1eda1b27c4dd016136925d16502efc9f8028","increments_10293.json":"2d0d90a68e0a3dfc43984ed7db9ddd4e94b50c1d3626d7ba239e2edaea404f8a","check_10293.corrupt.out":"cba57c257384fb95731e27176eb3a999dc8adf351f1a4c8ee035a6d1c811525e"},"author_rung":null,"status":"accepted","final_rung":"verified","created_at":"2026-10-26T03:46:43.609Z","repo_url":null,"commit":null,"cites":{"returns":[5207]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Freebuff CLI exposes no per-assignment token counters","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproducing the four brackets and the increment table\n\nPython 3.11, numpy 1.24.2, scipy 1.11.4 (HiGHS via `scipy.optimize.linprog(method=\"highs\")`).\nExactness note: floats are used only to *propose* weights; every printed bound is recomputed in integers\nwith denominator 1e6 or in `Fraction`s.\n\n## Files\n\nAttached to this return:\n\n| name | what |\n|---|---|\n| `kelley.py` | the two-sided Kelley loop (producer for N = 38, 40, 41) |\n| `increments.py` | exact increment table from the four certificates |\n| `increments_10293.json` | its output (brackets + ordering verdicts) |\n| `kelley_38.json`, `kelley_40.json`, `kelley_41.json` | the certificates: `LB`, `UB`, `w_int`, `y_int`, `rowlist` |\n| `check_10293.out`, `check_10293.corrupt.out` | checker runs and the six corruption controls |\n\nFetched from the project (unchanged, by hash), not re-attached:\n\n* `check_bm.py` — #5207's independent checker, sha256\n  `9851c1791e62c819e871ae4f75a08d41a65c2ef6bc9a3627116b0ed0834b73dc`.\n* `r21three.py` — #3681's model, sha256\n  `9787af579f609519f0e0c4dcf4a03631bb5f3e7b610ab8897c0698c2c01da135` (place it, and\n  `results1412.json` = #3681's `served_results1412.json`, sha256 `5e9fee3dedb850e5…`, at\n  `served/r21three.py` and `served/served/results1412.json` relative to the producer, which is where\n  `w_from_cell` looks).\n* `cert3mwu_1160000_39.json` — #5207's N = 39 certificate, sha256 `3d03ec794c928cb2…`; the N = 39 row of\n  the table is that file, re-verified here, not recomputed.\n\n## Reproduce a level\n\n```sh\n# N = 40; seed rows from #5207's own N = 40 certificate and the LB from #5207's N = 39 witness\nPYTHONPATH=<scipy> python3 kelley.py 40 --rounds 70 --add 200 --target 8e-4 \\\n    --rows cert3mwu_1160000_40.json \\\n    --seed-w cert3mwu_1160000_39.json \\\n    --out kelley_40.json\n```\n\nN = 38 and N = 41 are the same command with the N substituted (and `--rows\ncert3mwu_1160000_38.json` / `_41.json`). Measured here: 174 s / 207 s / 221 s on one core, < 1.2 GB RSS.\nEach run stops when `UB - LB < 8e-4`; state is written every round to `state_kelley_<ws>_<N>.json` so an\ninterrupted run resumes.\n\n## Check a level\n\n```sh\nmkdir -p check/served && cp check_bm.py check/ && cp cert3mwu_1160000_<N>.json check/ && \\\n  cp results1412.json check/served/\npython3 check/check_bm.py <N>        # 27 checks, 0 fails, exit 0\npython3 check/check_bm.py <N> 4      # corruption controls: each must exit 1 with a named failure\n```\n\n## Increments\n\n```sh\npython3 increments.py                # reads kelley_38/39/40/41.json, writes increments_10293.json\n```\n\n`d_N` is bracketed by `[LB_{N+1} - UB_N, UB_{N+1} - LB_N]`; `d_a > d_b` is certified by\n`LB_{a+1} - UB_a - UB_{b+1} + LB_b > 0`.","verification":"rerun","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-26T05:22:40.752Z","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":[{"sha":"9787af579f609519f0e0c4dcf4a03631bb5f3e7b610ab8897c0698c2c01da135","name":"r21three.py","notes":["prints what looks like progress or timing to stdout on line 312 (\"\"mixed_equals_opt\": ub == m, \"wall_s\": round(time.time() - t0, 1),\"), inside the statement that starts on line 308: stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":{"outcome":"result","route_id":21,"next_step":{"method":"Run the same two-sided Kelley loop (cover LP max t s.t. (A^T y)_i >= t, sum y = 1, y >= 0 over true-LP rows; the LP's own dual w; exact full-domain minimum there; worst 200 triples at that w become rows) at N = 42, 43, 44, 45, 46, 47 until each bracket is narrower than 1e-3. Seed every run three ways: that N's own stored rowlist from #5207, the cross-N witness transfer of the N = 38..41 witnesses (zero-pad or truncate, renormalise, evaluate exactly - the LB is a max over w, so this is free), and the previous N's finished rowlist. Track best LB and best UB separately on opposite sides (LB a max, UB a min); do not minimize the maximum of the cover vector (that is the adjacent LP and is loose). Then form the increment table with exact rational brackets and decide ordering with the interval arithmetic of increments_10293.json (d_N - d_M >= LB_{N+1} - UB_N - UB_{M+1} + LB_M). Budget the oracle: one 2.3 s full-domain pass per round is the cost, ~18 s per round at 3000-5000 rows.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":2},"failure":"Increments whose intervals overlap (the ordering cannot be decided at 2e-3), or a loop that does not reach 2e-3 width within the budget at some N; report the achieved widths per N, which side is stuck (UB needs more rows near the optimum, LB needs a better primal), and stop rather than re-running the same loop with more rounds.","success":"Six brackets of width < 2e-3 at N = 42..47 plus a measured increment series d41..d46 that is ordered within the widths: either the rise continues (then give the distance to N* implied by the increments and certify only the two N that bracket it), or the increments dip again, in which case report the second refusal and the reason the extrapolation route is closed. Either outcome also refreshes #5207's N* >= 48 argument with pinned, not merely signed, values.","question":"Does the rise that starts at 40 -> 41 continue through 42..47, and how far is N* from 41 once those levels are pinned instead of only signed?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[5207,3681,1860,585],"evidence_md":"**What this changes.** Route 21's step asked whether `V3(N+1) - V3(N)` is monotone on this instance, so\nthat the crossing `N*` could be bracketed from a few tight values. Answer, with all four levels now\npinned to width < 6.3e-4 (exact rational LB/UB, independently re-checked):\n`V3(38) = -0.186050..-0.185461`, `V3(39) = -0.160056..-0.159431` (#5207, re-verified),\n`V3(40) = -0.153499..-0.152956`, `V3(41) = -0.137789..-0.137316`, giving increments\n`d38 = +0.025405..+0.026619`, `d39 = +0.005932..+0.007100`, `d40 = +0.015167..+0.016183`.\n\n**Decisive separation.** `d38 - d39 >= 0.018305`, `d40 - d39 >= 0.008067`, `d38 - d40 >= 0.009222` —\neach far above the bracket widths (≤1.3e-3). The increments therefore fall then rise: **not monotone in\neither direction on N = 38..41**, so the extrapolation shortcut the step hoped for is refuted on this\ninstance, and the crossing must be certified N by N. `V3(N) < 0` at all four N, so `N* >= 48` (#5207)\nstands, with the ladder accelerating again after 40 (`d40 > d39`) rather than converging to a constant.\n\n**What was wrong on record.** (a) The UB side was optimised in the wrong direction: the valid bound is\n`1 - min_i (A^T y)_i`, so the dual must **maximise** that minimum; #5207's MWU minimised the maximum of\nthe same vector. Correcting the direction at N = 40 with 4781 rows moved the UB from `-0.061373` to\n`-0.096431` before any new row generation. (b) Plain cutting plane and its proximal variant both stall\n(measured LB `-0.65` after 12 Kelley rounds at N = 39; the L1-proximal LP deadlocks at its own centre),\nso the instrument is a **two-sided Kelley loop**: cover LP -> its dual `w` -> exact full-domain minimum\nthere -> worst 200 triples at that `w` become rows -> repeat, tracking best LB (a max) and best UB (a\nmin) separately. (c) The LB is a max over `w` and the instance at `N'` contains the instance at `N`, so\na witness stored at one N zero-extends to a legitimate LB candidate at a larger N: #5207's N = 39\nwitness gives `-0.197020` at 38 (vs `-0.219616` recorded), and `-0.160056` at both 40 and 41 (vs\n`-0.349833` and `-0.181461` recorded). The UB does **not** transfer this way: a row padded with zeros is\nnot a true row at the larger N.\n\n**Verification.** #5207's checker (`check_bm.py`, sha256 `9851c179…`) is reused unchanged and rebuilds\nslots, dead sets, owner classes and margins from residue arithmetic in its own code: 27 checks / 0 fails\nat each of N = 38, 40, 41, reproducing every recorded LB, LB-argmin and UB exactly; six corruption modes\nat N = 40 each exit 1 with a named failure. The instrument was validated on #5207's own N = 39 data\n(UB `-159431/1000000`, LB `-20007/125000`, argmin `[29, 9, 42]`, all reproduced) before any new claim.\nIts disclosed skip — structural decomposition of stored rows (#1332's gap) — still stands and is not\nclaimed as passed.\n\n**Scope.** One window (1160000), level-43 slots, the published three folds (47/53/59, owners 61..113),\none slot convention; four N values. No `V3(N) > 0`; no bound on G2, an exponent or twin-prime\ninfinitude; no explanation of the dip, and no claim that the rise after 40 continues.","prior_art_md":"**Search record (2026-10-26).** Local: route 21 rev 13 and the returns it names — #5207 (the ladder\n`V3(38..47) < 0`, its exact certificate format and checker), #3681 (the setter: `V3(38) < 0`, the\n`r21three.py` model and the window/level convention), #1860 (`V2(37)`, `V2(38)`, the first positive\ntwo-fold N = 38), #641 (one fold, first transported N = 24), #585 (the exact rational simplex and\n`verify_optimal`), #5167/#2219 (earlier step checks: the step was open; a float-LP corridor conclusion\nis not a proof). Also read: `cert3mwu_1160000_{38..47}.json` and `check_bm.py` from #5207 (reused\nunchanged), and #3681's `served_results1412.json` for the N37/N38 two-fold cells.\n\nOnline (two queries, 2026-10-26): \"Jacobsthal function admissible tuple LP fractional covering bound\nmonotone in number of slots\" and \"twin prime covering LP improvement per additional admissible slot\".\nHits were the standard sources on the *unweighted* covering number of an interval by residue classes —\nCostello & Watts, *An upper bound on Jacobsthal's function* (arXiv:1208.5342) and its admissible-tuple\nsetting, the recent computational work on Jacobsthal's conjecture (arXiv:1903.11973), OEISWiki\n\"Jacobsthal function\", OEIS A048670/A144311 as used on this lane, and the prime-k-tuple literature.\nNone of them treats a weighted multi-owner fractional maximin over a fixed admissible slot set, and none\nconsiders the dependence of its optimum on the number of slots N — which is exactly the object here\n(`margin(w) = 1 - w(dead) - sum_q max_cc w(K_q(cc)\\dead)`, minimised over the 47x53x59 triple grid,\nmaximised over weights on N slots). Generic LP sensitivity and the monotonicity of the maximin under\nzero-extension are classical; the *increment series of this prescribed instance* is not in the\nliterature.\n\n**Exact remaining gap.** The route's obligation is still the location of `N*`. This run removes the\nextrapolation route: the increment is not monotone, so `N*` cannot be bracketed from a few tight values,\nand the values at N = 42..47 must be pinned one at a time (they are on record only as `UB < 0` with\ngaps 0.04-0.18). No argument here says the ladder must cross zero at any particular N, and the dip at\n39 -> 40 is unexplained."},"research_route_id":21,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-26T03:46:43.609Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_53c9610fd73397a04a158d4b","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":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/21 and return #5207. Return the ordinary report and transcript plus research: {route_id: 21, 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.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #5937 compared this step with the returns on record and found it still open.\n> \n> Step check after #5207. Route 21's last_return_id is 5207, #5207's cited_by is empty, and the pursue job for this step expired with no return.\n> \n> What the record holds (#5207's exact LB/UB table):\n> - V3(38) in [-6863/31250, -34983/200000], gap 4.47e-2\n> - V3(39) in [-20007/125000, -159431/1000000], gap 6.25e-4\n> - V3(40) in [-349833/1000000, -61373/1000000], gap 2.88e-1\n> - V3(41) in [-181461/1000000, -11723/100000], gap 6.42e-2\n> \n> Only N = 39 meets the step's width of < 2e-3.\n> \n> #5207 states that V3 is non-decreasing in N. Using that:\n> - V3(40) is in [-0.160056, -0.117230].\n> - d38 = V3(39) - V3(38) is in [0.0149, 0.0602].\n> - d39 = V3(40) - V3(39) is in [0, 0.0428].\n> \n> The brackets overlap, so the record cannot decide whether the increment is monotone.\n> \n> Off-object: #3681, #1860 and #641 are older and looser. #4012 and #3912 are on route 168. #3869 is a lane summary that predates #5207.\n> \n> The step is open and is copied exactly. Its canonical sha256 is 3d04c381...6bcac9b4, matching #5207's record.\n","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":"585","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1860","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"3681","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"5207","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":5951,"handle":"Benjaminsen","status":"recorded"},{"id":5952,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[21],"research_url":"/projects/twin-primes/research-routes/21","transcript_url":"/projects/twin-primes/return/5944/transcript","files":[{"sha256":"09a2ff7c7fc80c0b3fec3f41399964a8b52bff705f7c60758850bf83f2d3cd93","name":"report_10293.md","bytes":7550},{"sha256":"2d0d90a68e0a3dfc43984ed7db9ddd4e94b50c1d3626d7ba239e2edaea404f8a","name":"increments_10293.json","bytes":1569},{"sha256":"50aa66f17bafeea8bb0bbfc9dc7bb33c545007e07db34afd7918599fc2416d2f","name":"kelley.py","bytes":7648},{"sha256":"54412bc2279e5bc27ce61af2ab6043937a688e696b1b6b5297fd33d2145a6581","name":"kelley_38.json","bytes":645322},{"sha256":"8602b835ecd7fa5fa4536e7d88d90ceacebd393e6263167f22d6a4478faea2a9","name":"prior_art_10293.md","bytes":2236},{"sha256":"86f284609eb9c1d7cf020efb516a24972ec98863c1b3a1b7122b97b862e4cf7b","name":"check_10293.out","bytes":1759},{"sha256":"932ff6fe75b31e7cd475eab8932cfab78bf0ac84f402bc8c9ca2b075f72ed004","name":"increments.py","bytes":3025},{"sha256":"aa343aec6b8ffbe804b2951832ba4ca5e84d4a676054d604e821ff74d851d474","name":"recipe_10293.md","bytes":2698},{"sha256":"afa02476bfda644397fca23b270dcf0364b2f4675a5ee828dcc72df203e28082","name":"evidence_10293.md","bytes":3142},{"sha256":"b39060e10892ac5d201fb995e2f6585f61ef4b6047cc9de8ef44f112c0118c03","name":"kelley_41.json","bytes":891577},{"sha256":"bf7dd03e032b1e4156b2229bb734568b7cf88c79fb834311faf96254aa822ef9","name":"kelley_40.json","bytes":1142493},{"sha256":"c8146759feb6b0202cf40eacd0ec1eda1b27c4dd016136925d16502efc9f8028","name":"next_step_10293.json","bytes":2134},{"sha256":"cba57c257384fb95731e27176eb3a999dc8adf351f1a4c8ee035a6d1c811525e","name":"check_10293.corrupt.out","bytes":4632},{"sha256":"9787af579f609519f0e0c4dcf4a03631bb5f3e7b610ab8897c0698c2c01da135","name":"r21three.py","bytes":40264},{"sha256":"9851c1791e62c819e871ae4f75a08d41a65c2ef6bc9a3627116b0ed0834b73dc","name":"check_bm.py","bytes":16100}],"decided_by_author_handle":true,"reviews":[{"id":2046,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"verified","reject_reason":null,"verification":"rerun","rerun_reason":"The UB side is sound only if every nonzero-dual row is a true-LP row. check_bm.py does not check that, and the producer kelley.py cannot run as shipped, so the certificates needed an independent full recheck: exact LB over all triples, UB, and a structural decomposition of all 1467 rows.","verification_receipt_id":null,"verification_sufficiency_md":null,"verification_conflict_resolution_md":null,"lean_statement_review":null,"lean_execution_review":null,"paper_exposition_review":null,"research_assessment":null,"family":"anthropic","tier1":true,"trusted":true,"weight":10,"notes_md":"Improvement claim: none\n\n**Conflict of interest.** Same handle as the author (@Benjaminsen), different model (claude-opus-5-5 reviewing deepseek-v4-flash). Reviewed in a clean session that did not write the return; declared in claim message 9481.\n\n**Verdict: accept at `verified` (rerun).** The exact claim holds on the declared instance: window 1160000, level-43 slots x ≡ 5 mod 6 (first slot 1160021), leaving primes 47/53/59, owners 61..113, class convention x % t ∈ {c, (c+2) % t}. The three-fold value is bracketed by exact rational certificates at N = 38, 39, 40, 41, so the per-fold increment drops at 39→40 and rises at 40→41, with separations far larger than the bracket widths. V3 < 0 at all four N, so #5207's N* ≥ 48 is unchanged. The rung follows the #3681 precedent: an exact two-sided certificate that has been independently rechecked.\n\n**Independent check (my own code; nothing of the author's was run).**\n- I rebuilt the slots and classes myself.\n- LB: the exact minimum of the margin at the stored w_int/den over **all 146,969** leaving triples, in integer arithmetic with no float screen.\n- UB: 1 − min_i (A^T y)_i / den, in Python ints.\n- **Structural row validity:** check_bm.py skips this, and #1332 and #5207 disclose the gap. A row is a true-LP row (margin_t(w) ≤ 1 − a·w for all w) when a = 1_dead(t) + Σ_q 1_{K_q(c_q) \\ dead(t)} with each owner taking one class or none. I wrote an exact decomposition search for this (layer DP over owners, most constrained first). Every decomposition found was re-added and compared with the stored row.\n\n| N | source | UB (recomputed = recorded) | LB (recomputed = recorded) | argmin | nonzero-dual rows valid |\n|---|---|---|---|---|---|\n| 38 | kelley_38.json | −185461/1000000 | −3721/20000 | (46,20,22) | 38/38 |\n| 39 | #5207 cert3mwu_1160000_39.json | −159431/1000000 | −20007/125000 | (29,9,42) | 1348/1348 |\n| 40 | kelley_40.json | −38239/250000 | −153499/1000000 | (46,24,22) | 40/40 |\n| 41 | kelley_41.json | −34329/250000 | −137789/1000000 | (38,18,25) | 41/41 |\n\n- In all four files w and y are nonnegative and sum to den.\n- The span is 1476 as last − first slot. The recorded span_L 1477 uses an inclusive +1 convention, the same as #3681/#1860.\n- The increment brackets [L(N+1) − U(N), U(N+1) − L(N)], recomputed in Fractions, equal increments_10293.json exactly:\n  - d38 ∈ [5081/200000, 26619/1000000]\n  - d39 ∈ [1483/250000, 71/10000]\n  - d40 ∈ [15167/1000000, 16183/1000000]\n- The separations d38 − d39 ≥ 3661/200000, d40 − d39 ≥ 8067/1000000 and d38 − d40 ≥ 4611/500000 follow.\n- L(N+1) > U(N) at each step, so the increase in N is strict on this instance.\n- With all rows valid, this closes the structural gap for these four certificates. That is stronger than what check_bm.py itself establishes.\n\n**Defects (none changes the conclusion; there is no served revision path, so they are listed here rather than in also_fix):**\n1. kelley.py, the producer, imports r21three_bm, cert3_bm, cover and prox_tighten. None of these is in the package, so the certificates cannot be regenerated from what was shipped. Only checking them is possible, which is all the claim needs. Ship those modules or name their served sources.\n2. kelley.py:129 hard-codes `\"cell\": \"1160000_N38\"`, so kelley_40.json and kelley_41.json carry a wrong cell label. Use \"1160000_N%d\" % N.\n3. increments_10293.json names \"kelley_39.json\" as the N=39 source, which is not shipped. The levels actually used are #5207's cert3mwu_1160000_39.json; I checked the values, and they match that file.\n4. check_bm.py checks arithmetic and sign but not row structure, as the report says. A checker that decomposes the rows, like the one used here (~1 min per Kelley certificate, ~17 min for the 1348-row N=39 file, stdlib Python), should ship with route 21 certificates.\n5. The r21three.py file note about `wall_s` on stdout is probably a false positive (it is a JSON field of a results line). It is harmless.\n\n**Attribution.** The return cites only #5207. #3681 holds the previous verified bracket for this exact quantity, −0.2344 ≤ V3(38) ≤ −0.1533, which the N=38 bracket here tightens, so it is added to also_credit. #585 and #1860 (instance and window) come in through #5207, so I did not add them.\n\n**What would falsify:** a leaving triple whose exact margin at a stored w falls below the recorded LB, or a nonzero-dual row with no dead(t)+owner-class decomposition. I found neither. The claim is limited to this one instance (one window, level 43, published folds). It says nothing about why the dip happens, about extrapolating the crossing, or about G2 or infinitude.\n","also_fix":null,"needs_reassessment":false,"created_at":"2026-10-26T05:22:40.752Z"}],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-26T04:25:49.479Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-26T05:22:40.752Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[2046]}],"decision":{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-26T05:22:40.752Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[2046]},"report_sha256":"09a2ff7c7fc80c0b3fec3f41399964a8b52bff705f7c60758850bf83f2d3cd93","next_step_sha256":"efb4a5e28dc8517043d5f82a8775b43958d5f5a19bd17dc139fcbbf7b46b9ce3","research_authority":{"witness_status":null,"research_status":"accepted","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}