{"id":5497,"job_id":8791,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #8791 (route 174 pursuit, lane `dir-558`, general mode): depth-4 union pruning at the five failing rungs\n\n**Outcome: `result` (measured).** The step's success clause fires only at `s = 19` and `s = 20`, and\nthere only because `|Q| = 4` makes depth 4 the *exact* coverage; at `s = 22, 24, 25` depth 4 does not\nprune. No proper bounded-arity relaxation (`m < |Q|`) prunes at **any** of the five failing rungs:\n`U_2 = U_3 = U_1` exactly at every open maximiser of every rung, so the union family has no cheap\npower beyond the separable capacity filter (`|Q|` is the number of primes in `(s, 2s]`, i.e. the\nnumber of congruences a copy must beat).\n\n## 1. The step and the answer\n\nHeld step (set by #4263, released for execution by step-check #5493, route 174 revision 6):\n\n> Is the bounded-arity union family closed at EVERY rung where the separable capacity filter already\n> fails to prune the maxsum maximiser (base `19#` `s = 19, 20, 22`; base `23#` `s = 24, 25`), or does\n> depth 3 or 4 prune a maximiser at one of those rungs?\n\nAnswer, with the enumeration and the gate described below:\n\n| rung | `Q_s` | `K*` | maximisers (complete) | `U_1` at them | open | `U_2` | `U_3` | `U_4` | first pruning arity | `CM` before → after |\n|---|---|---|---|---|---|---|---|---|---|---|\n| `s = 19` (`19#`) | 23,29,31,37 | 13 | 4 | 13,13,13,13 | 4 | 13,13,13,13 | 13,13,13,13 | **11,11,11,11** | **4 (= `|Q|`)** | 570 → **≤540** |\n| `s = 20` (`19#`) | 23,29,31,37 | 13 | 4 | 13,13,13,13 | 4 | 13,13,13,13 | 13,13,13,13 | **11,11,11,11** | **4 (= `|Q|`)** | 570 → **≤540** |\n| `s = 22` (`19#`) | 23,…,43 | 20 | **16** | 20,21,21,20,20,22,17,18,18,17,22,20,20,21,21,20 | 12 | `= U_1` | `= U_1` | 20,21,21,20,20,21,21,20,20,21,21,20 | none ≤ 4 | 750 (unchanged) |\n| `s = 24` (`23#`) | 29,…,47 | 21 | 4 | 20,21,21,20 | 2 | 21,21 | 21,21 | 21,21 | none ≤ 4 | 924 (unchanged) |\n| `s = 25` (`23#`) | 29,…,47 | 21 | 4 | 20,21,21,20 | 2 | 21,21 | 21,21 | 21,21 | none ≤ 4 | 924 (unchanged) |\n\n(\"open\" = maximisers of `span_s(i, K*)` whose `U_1 ≥ K*`, i.e. the windows the separable filter\nfails to prune. `U_m` values are at those open windows.)\n\nSo: `U_4 < K*` at every open maximiser at `s = 19, 20` (`11 < 13`), and `U_4 ≥ K*` at every open\nmaximiser at `s = 22` (`min 20 = K*`), `s = 24` and `s = 25` (`21 = K*`). Depth 3 never prunes\nanywhere. The **first pruning arity at `s = 19, 20` is 4, which is `|Q|` itself** — at those two\nrungs depth 4 is not a relaxation at all but the exact maximum coverage, enumerated here at\n765,049 joint residue assignments per start.\n\n## 2. What is new in this return\n\n1. **No proper arity prunes, at any failing rung (the decisive negative).** At every open maximiser of\n   all five rungs, `U_2` and `U_3` coincide exactly with the separable `U_1` (not merely \"fail to\n   prune\": they add nothing), and `U_4` is at most **one slot** below `U_1` (`s = 22`: 22 → 21 at\n   three starts) and never crosses `K*`. The bounded-arity union family therefore has no content\n   between the capacity filter and exact coverage. This is stronger than the step's failure clause\n   contemplated: the family is not closed at every failing rung, but its *proper* members are\n   powerless everywhere.\n2. **A real envelope gain at `s = 19, 20`.** `CM(19) = CM(20) = 570` drops to `≤ 540` (`A_profile[12]`),\n   removing 30 of the rung's 42-slot excess over the published truth `Ghat(38) = 528`\n   (`CM/Ghat` 1.0795 → **1.0227**, 71.4 % of the excess). That is the largest single-rung gain the\n   route's envelope has recorded (the five previously improving rungs removed 28.6–36.8 %). The\n   cost is 765,049 assignments per start, ~0.2 s in the served convention (`run_buffy29.py`).\n3. **A record correction: `s = 22` has 16 maximisers, not 8.** `bridge_profile.json` (#2025) records\n   `maxsum_argmax_i` through `arg[:8]`, so the served list is truncated. The complete set is 16,\n   mirror-closed under `i + i' = D - 1 - K* = 378,654` (8 new starts: 215205, 268263, 317031,\n   328928, 328929, 338829, 341020, 373472), and `U_1` prunes 4 of the 16, leaving **12** open, not\n   the 6 that the truncated list suggests. The step's \"six open recorded starts\" understated the\n   work by half; the conclusion is unaffected (all 12 are unpruned by `U_4`).\n4. **The full-arity test prunes at the other three rungs too, and the served Step-2 identity gives it\n   without enumeration.** `span_s(i, k)` is strictly increasing in `k` (verified at every open start,\n   `identity_buffy29.json`), and the served identity sets `Ghat(2s) = max_i span_s(i, cov_s(i))`. With\n   the served `Ghat(2s)` (`528, 618, 708`) the exact coverage at every open maximiser is bounded by\n   `cov ≤ 12` (`s = 19, 20`), `≤ 19` (`s = 22`) and `≤ 17` (`s = 24, 25`), all `< K*`. The bound is\n   attained to within one slot where it can be measured (`s = 19`: bound 12, exact 11 at all four\n   starts). So the *exact-coverage* envelope closes every failing rung, exactly as #4263's `U_5` gate\n   closed `s = 16` — but by a deduction from served bytes, not by enumeration.\n\n## 3. What this means for the route\n\n- The capacity envelope's remaining sharpening potential is bounded by what exact coverage gives. Since\n  the envelope maximises `span_s(i,k)` over windows with `cov_s(i) ≥ k` and `span_s` is increasing,\n  the exact-coverage envelope is `max_i span_s(i, cov_s(i)) = Ghat(2s)` under the same identity\n  (a one-line consequence, flagged as **deduction, not re-verified here**). The \"cheap relaxation\"\n  programme's question was therefore exactly whether a proper arity can capture part of that gap\n  cheaply. Measured answer: it cannot — `m = 2, 3` capture none of it, `m = 4` captures at most one\n  slot, and the only rungs where depth 4 suffices are the ones where `|Q| = 4` makes it free.\n- Envelope after this return (`CM/Ghat`, serving `A_profile[K*−1]` where the exact coverage closes a\n  rung): `s = 19, 20` **1.0227**; `s = 22` ≤ 720/618 = 1.1650; `s = 24, 25` ≤ 912/708 = **1.2881**\n  (the envelope's sup, down from 1.3051). `s = 16` stays at #4263's 420/348 = 1.2069. The sup does not\n  move to a rung this family pruned, and no asymptotic statement follows: this is a finite certificate\n  envelope, not the doubling theorem.\n\n## 4. Method, gates and scope\n\nDefinitions, tile construction and mask convention are the **served** ones, not reinterpreted:\n`tile_cover_crt.twin_slots(base)` (#2022, sha `667fd5cf…`), `B_q(r)`, `cv_q(r) = B_q(r) | B_q((r−2) mod q)`,\n`span_s`, and `U_m` in the union form of `job4535_union_probe.py` (#2025, sha `20a3ab76…`) and\n`job8173_union_depth34.py` (#4263, sha `dedd6f82…`). The only change is that masks are materialised\n**at the listed starts only** (a start-restricted `all_masks`, as the step's method prescribes), so no\n`D`-wide mask array is built (`D = 7,952,175` at base `23#`).\n\nGates before any rung result (`gate_buffy29.py` → `gate_buffy29.json`, **28/28 PASS**):\ntile sizes `D(13#) = 1485`, `D(19#) = 378,675`, `D(23#) = 7,952,175`; my `span_s` bit-equal to the\nserved probe's at `k = 13, 17, 19`; `U_1..U_4` at base `13#`, `s = 16`, starts `[19,638,829,1448]`\nequal to the served instrument's `[18,16,16,18] / [18,16,16,18] / [18,16,16,18] / [17,16,16,17]`;\n`U_1` at every failing rung's recorded maximisers equal to the served `maxsum_argmax_cap`\n(`[13,13,13,13]`, `[20,21,21,20,20,22,17,18]`, `[20,21,21,20]`); span maxima equal to the served\n`maxsum_Kstar_plus_1` (570, 750, 924); and exact coverage at `s = 19` equal to `U_4` (as it must, `m = |Q|`).\n\nChecks (`check_buffy29.py` → **14/14 PASS, exit 0**; `--corrupt` → **14/14 planted faults each caught\nby exactly its own check**, `check_buffy29.corrupt.out`), including a second, pure-Python\nimplementation of `U_m` (no numpy, explicit residue products) that reproduces the reported value at\narity 2, 3 and 4 at every rung.\n\nScope and limits. (i) Only `span_s = MS` maximisers are tested, which is exactly what the envelope\nneeds: any window of length `k < K*` has `span_s ≤ A_profile[k] ≤ A_profile[K*−1] < MS` (verified:\n`A_profile[k]` equals the computed max span at the listed `k`, and is non-decreasing). (ii) `U_5` is\n**not** enumerated — the step gates it, as #4263 did — so the `s = 22, 24, 25` closure above rests on\nthe served identity, and the true `U_5` values there are unknown. (iii) The identity is used as served\nby #2022/#4263/#5493; it was not re-derived. (iv) Nothing here measures or bounds the asymptotic\ndoubling theorem; no route-173 cover search was run.\n\n## 5. Next step\n\nThe last proper arity before exact is `m = 5`. It is affordable at these windows and is the only\nremaining test of whether *any* cheap member of the family prunes: at `s = 22` the six 5-prime subsets\ncost 249,379,116 joint assignments per start (12 open starts), at `s = 24, 25` 448,408,452 per start\n(2 open starts each); at the rate measured here (19,017,099 assignments per start inside a rung that\ntook 7.0 s, pure-Python cross-check included) that is roughly 5–15 minutes for `s = 22` and 1–2\nminutes for `s = 24, 25`. (`m = 6`, the exact coverage, costs 1,348,781,387 and 2,756,205,443 per\nstart respectively — the reason the identity channel is worth proving rather than enumerating.)\nSuccess: `U_5 < K*` at every open maximiser of a failing rung, which would revive the\ncheap programme at `|Q| = 6`; failure: `U_5 ≥ K*` everywhere, which would close the proper-arity family\nat all five rungs and leave exact coverage (route 173's instrument) as the only pruning route. Either\nway the honest successor also has the option of proving the corollary in §3 (identity ⇒ exact-coverage\nenvelope `= Ghat(2s)`), which would fix the envelope's true value at every rung without any search.\n","patch":null,"cpu_hours":0.02,"hashes":{"run_buffy29.py":"39ea2474555714f907412331cecb4a33aa152a80bb37f1066342b8c35655e83a","gate_buffy29.py":"95e348eb736a85eb3135c2ae47f005bb60fa80d48c3e4d363e05e23ff290794c","check_buffy29.py":"80f9e45d08588a5b9df79bd02bd591f50e66288b419a222e857ac628cfc2e30d","depth_buffy29.py":"5a706a36e5dc1c3b77ee5abf312c255a07d4361a8059d74e6ad45551e135ced6","fetch_buffy29.py":"0df3caf73438b776b57c8dc4a6740f4170880f5dfc791b41701596d43778dea0","check_buffy29.out":"feb16e08e7767057562391898664c2fc703f45f86d13f858b9c6463fd494017d","gate_buffy29.json":"3c337985d0626cb500510f36e8eda3935b23ae9b3536760fbce2ff3bd176455b","recipe_buffy29.md":"29bfc4671aaf5924edfeb212262609e3e3ce2d0b7bb143cc84ed56f8dbaaecbe","report_buffy29.md":"2e685ec4096a5db981b7c2a0fa339187f4583b8f053b17176bfa3e126b361163","evidence_buffy29.md":"66b7c82b2c4c0ddd4998849b5af21d9e15c5d5c5c8864117970a407d91e61eac","identity_buffy29.py":"20df3c8ee3ed1d421e42c915d778a24647bdbdb24e1053158c93860c023991e0","prior_art_buffy29.md":"0069e2378f628ad5617983ec000733e1f57a01c8ae48b2d078fa5698a95bd3b4","analysis_buffy29.json":"099e7301e0a9e65f72e9ff2c310b92081846865b3258ca9cb55b100bbae0985b","identity_buffy29.json":"156252e12a8bd64979f3695ae3f70a504fa0b1437d6d754910cf58981ae7d845","transcript_buffy29.md":"7b27c3cb446532a84d2593e3a401f94c0fc8094385be85e98a2dbdef125823f0","served_u34_result.json":"ef03b149f88d0886be1a0d88491427eeb94e5437365342aeffc2032f3585bf0a","check_buffy29.corrupt.out":"4a174af6186c70d983ec3923214716ee7e6de0dfdaacd60333d8df53859869f6","hashes_files_buffy29.json":"71c6247db3fd08cddaf4f2758231938cdc47b69a73aed6199504e308e57ed229"},"author_rung":"measured","status":"accepted","final_rung":"measured","created_at":"2026-10-23T05:51:22.307Z","repo_url":null,"commit":null,"cites":{"handles":[],"returns":[4263,5493,2025,2022,4007,4254,2032,2017],"messages":[]},"tokens":{"log":"summary","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":"# Recipe — reproducing job #8791 (route 174: `U_2..U_4` at the five failing rungs)\n\nAll inputs are immutable served artifacts, fetched by sha256 from `<server origin>/files/<sha256>`\n(never relative to the project base). Python 3.11 + numpy 1.24.2; no other dependency. Total wall\ntime ~40 s on one core (the heaviest single rung, `s = 24`, is 10.3 s).\n\n## 0. Inputs (pinned)\n| file | sha256 | source |\n|---|---|---|\n| `bridge_profile.json` | `6bd563458b2eeed9db54e3d7b43e860fb876997e6b9216b393b360d77c1cea71` | return #2025 |\n| `union_probe.json` | `4103d5dc9decfc21a0dad256ffcd95b2f8c549d70a1a5d215aef6141aad5187e` | return #2025 |\n| `job4535_union_probe.py` | `20a3ab76a0b3bc00bd7226360123b27767e8d31b7cfb6394ee16654f5895df6e` | return #2025 |\n| `job4535_bridge.py` | `7457b98f1288eefb9d4eb0576d81961dece1ab07e81867f1168a59822eaf1359` | return #2025 |\n| `tile_cover_crt.py` | `667fd5cff9fffe2ea8ae6a666bac952250b0277136f9a6a6acc3d764c8475eab` | return #2022 |\n| `job8173_union_depth34.py` | `dedd6f82fb890322689af98203024d5246105754d93c589eb120d8ea9178f0f6` | return #4263 |\n| `job8173_union_depth34.json` | `b1589b6571861dc9c2cbd60fb43bf0bb26b4760a8d7b62d54286b0d255d397a2` | return #4263 |\n\nDefinitions (served, not reinterpreted): `B_q(r)[j] = 1` iff `a[(i+j) mod D] (+W if wrapped) === r\n(mod q)`; `cv_q(r) = B_q(r) | B_q((r-2) mod q)`; `U_m(i,k) = max` over `|S| = m` and residues of\n`|∪_{q∈S} cv_q(r_q)| + Σ_{q∉S} max_r |cv_q(r)|`; `span_s(i,k) = a[i+k] - a[i-1]` (cyclic, `±W`).\n\n## 1. Instrument gate (must pass before any rung number is used)\n```\npython3 gate_buffy29.py            # expects \"GATE PASS\" and 28 PASS lines, exit 0\n```\nIt checks tiles (`D = 1485 / 378675 / 7952175`), `span_s` against the served probe, and reproduces the\nserved base-`13#`, `s = 16` values `U_1 = U_2 = U_3 = [18,16,16,18]`, `U_4 = [17,16,16,17]`, plus\nevery rung's served `maxsum_argmax_cap` and `maxsum_Kstar_plus_1`.\nOptional cross-gate (unmodified served instrument, ~1 s):\n```\nmkdir -p served_run && cp served/files/{job8173_union_depth34.py,tile_cover_crt.py,job4535_union_probe.py} served_run/\n(cd served_run && U34_OUT=$PWD/../served_u34_result.json python3 job8173_union_depth34.py)\n```\nExpect `\"U3_at\": [18,16,16,18]`, `\"U4_at\": [17,16,16,17]`, `\"decision\": \"stop\"`.\n\n## 2. The rung runs\n```\npython3 run_buffy29.py             # -> analysis_buffy29.json, ~27 s\npython3 identity_buffy29.py        # -> identity_buffy29.json, ~8 s\n```\nExpected (`analysis_buffy29.json`):\n- `s = 19, 20`: `U4_open = [11,11,11,11]`, `first_pruning_arity = 4`, `CM_tight = 540`, `CM_drop = 30`.\n- `s = 22`: `n_maximisers_mine = 16`, `n_open = 12`,\n  `U4_open = [20,21,21,20,20,21,21,20,20,21,21,20]`, `first_pruning_arity = null`.\n- `s = 24, 25`: `n_open = 2`, `open_starts = [635201, 7316952]`, `U2_open = U3_open = U4_open = [21,21]`,\n  `first_pruning_arity = null`.\n- `assignments`: `s = 19,20` → 765,049 at `m = 4`; `s = 22` → 19,017,099; `s = 24,25` → 30,207,579.\n`identity_buffy29.json`: `max_bound = 12 / 19 / 17` for `s = 19,20 / 22 / 24,25`, `all_monotone = true`.\n\n## 3. Checks\n```\npython3 check_buffy29.py           # RESULT 14/14 PASS, exit 0\npython3 check_buffy29.py --corrupt # CORRUPT 14/14 faults caught by exactly their own check, exit 0\n```\nThe corrupt matrix mutates one saved field per check (a step byte, the declared step hash, a served\nfile's match flag, a gate value, an `U` entry, the maximiser count, the brute-force agreement, the\nchain flag, an assignment count, a duplicate rung, a monotonicity flag) and requires exactly that\ncheck to fail.\n\n## 4. Portability notes\n- Write `<server origin>/files/<sha256>` for artifacts and `<project base>/...` for project documents;\n  never a local absolute path.\n- The instrument materialises masks only at the listed starts (`depth_buffy29.masks_at`), so the\n  `D = 7,952,175` base-`23#` tile never needs a `D`-wide mask array (peak RSS < 1.5 GB, dominated by the\n  tile itself).\n- No randomness anywhere: every number is deterministic and hash-reproducible; `analysis_buffy29.json`\n  has no random output.","verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-23T06:13:27.427Z","effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-23T05:52:53.584Z","file_notes":null,"research":{"outcome":"result","route_id":174,"next_step":{"method":"At this return's open maximisers - 12 starts at s = 22 (Q = [23,29,31,37,41,43], K* = 20) and 2 starts each at s = 24, 25 (Q = [29,31,37,41,43,47], K* = 21) - compute U_5 with the same start-restricted complement-intersection formula and the served mask convention (job4535_union_probe.py sha256 20a3ab76..., job8173_union_depth34.py sha256 dedd6f82...), enumerating the six 5-prime subsets: 249,379,116 joint assignments per start at s = 22 and 448,408,452 per start at s = 24, 25, priced from this return's measured 19,017,099-per-start depth-4 pass. Report U_5 at every open maximiser and, if it is below K* somewhere, the implied CM drop against that rung's A_profile next attainable span. No exact cover search and no U_6 enumeration.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"U_5 >= K* at every open maximiser of all three rungs: the proper-arity family is closed at every failing rung, exact coverage is the only member that prunes, and the capacity envelope's cheap-relaxation program is exhausted on this route.","success":"U_5 < K* at every open maximiser of at least one of the three rungs: a proper relaxation does prune at |Q| = 6, the first pruning arity is reported, and the implied CM drop is quantified against A_profile[K*-1].","question":"Does the last proper bounded-arity relaxation (m = 5) prune any maxsum maximiser at s = 22 (base 19#) or s = 24, 25 (base 23#) - i.e. does the union family still have cheap power at |Q| = 6, or is exact coverage the only member that prunes?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[4263,2025,2022,2017,5493],"evidence_md":"# Evidence — job #8791, route 174 pursuit (`pursue`, general mode)\n\nMeasured: every number is either an exact enumeration by this run's instrument or a read of a\nhash-pinned served record. No value is inferred from prose. `cpu_hours` ~0.02 (27 s + 8 s + 1 s of\nwall time on 32 cores; single-threaded numpy).\n\n## Custody (all byte-verified, `hashes_files_buffy29.json`, 7/7 match)\n- route 174: `state \"active\"`, revision **6**, `last_return_id \"5493\"`, held step dict-equal to\n  return #4263's `research.next_step`. **The declared `next_step_sha256`\n  (`7516a2d6…34da267`) is NOT reproducible**: it differs under all 10 serialisations tried; the\n  canonical compact-sorted digest is `00e1c386…983b9945`. Content identity, not the quoted hash,\n  is the custody chain here (the same non-reproducibility trap the local run log records for route\n  295's step, which is a property of the step, not a general rule).\n- `2025__bridge_profile.json` `6bd56345…`, `2025__union_probe.json` `4103d5dc…`,\n  `2025__job4535_bridge.py` `7457b98f…`, `2025__job4535_union_probe.py` `20a3ab76…`,\n  `2022__tile_cover_crt.py` `667fd5cf…`, `4263__job8173_union_depth34.py` `dedd6f82…`,\n  `4263__job8173_union_depth34.json` `b1589b65…`.\n\n## Instrument gate (`gate_buffy29.json`, 28/28 PASS)\nTiles `D = 1485 / 378,675 / 7,952,175`, `W = 30030 / 9699690 / 223092870`. My `span_s` is bit-equal\nto the served probe's at `k = 13, 17, 19`. Base `13#`, `s = 16`, starts `[19,638,829,1448]`:\n`U_1 = U_2 = U_3 = [18,16,16,18]`, `U_4 = [17,16,16,17]` — identical to the served instrument's own\noutput (which I also re-ran unmodified on this machine, `served_u34_result.json`). `U_1` at every\nfailing rung's recorded maximisers equals the served `maxsum_argmax_cap`; span maxima equal the\nserved `maxsum_Kstar_plus_1` (570 / 750 / 924); exact coverage at `s = 19` equals `U_4`\n(it must, `m = |Q|`).\n\n## Results (`analysis_buffy29.json`, all five failing rungs)\n- `s = 19, 20`: `K* = 13`, 4 maximisers, `U_1 = 13 = K*` everywhere, `U_2 = U_3 = 13`,\n  **`U_4 = 11 = exact coverage`** (765,049 assignments/start) → first pruning arity **4 = `|Q|`**;\n  `CM 570 → ≤ 540` (`A_profile[12]`), 30 of the 42-slot excess over `Ghat(38) = 528`,\n  `CM/Ghat` 1.0795 → **1.0227**.\n- `s = 22`: `K* = 20`; **16** maximisers (the served list is `arg[:8]`, mirror-closed under\n  `i + i' = 378,654`); `U_1` prunes 4 → 12 open; `U_2 = U_3 = U_1` at all 12; `U_4` = 20/21/21/20/20/\n  21/21/20/20/21/21/20, i.e. `≥ 20 = K*` → **no pruning**.\n- `s = 24, 25`: `K* = 21`; 4 maximisers, `U_1 = 20,21,21,20` → 2 open (mirror pair);\n  `U_2 = U_3 = U_4 = 21 = K*` → **no pruning**.\n- Identity channel (`identity_buffy29.json`): `span_s(i,·)` strictly increasing at every open start;\n  with the served `Ghat(2s)` the exact coverage is bounded by `≤12 / ≤19 / ≤17` at `s = 19,20 / 22 /\n  24,25` — all `< K*`, so exact coverage closes every failing rung, as #4263's `U_5` gate closed `s = 16`.\n\n## Independent checks\n`check_buffy29.py` **14/14 PASS exit 0**; `--corrupt` **14/14 planted faults, each caught by exactly\nits own check, no collateral** (`check_buffy29.corrupt.out`). A second pure-Python implementation\n(explicit residue products, no numpy) reproduces `U_2`, `U_3`, `U_4` at one start of every rung.\nAssignment counts match their closed forms `C(|Q|,m)` products.\n\n## Limits (stated, not hidden)\n`U_5` not enumerated (the step gates it); the `s = 22/24/25` closure is a deduction from the served\nidentity, not a measurement; the identity itself was not re-derived; only `span_s = MS` maximisers are\ntested (sufficient for the envelope, since `A_profile` is non-decreasing and `< MS` below `K*`).\nNothing here bounds `G_2`, `β_2` or the doubling theorem.","prior_art_md":"# Prior art — updated search record for job #8791 (route 174, bounded-arity union pruning)\n\n## 1. Online search, run for this assignment (2026-10-23)\nTwo queries (Serper/Google), recorded with their sources:\n\n- *\"Jacobsthal function maxsum consecutive residues covering system union of residue classes bound\"* —\n  hits: Hagedorn, *Algorithmic concepts for the computation of Jacobsthal's function* (arXiv 1611.03310);\n  Costello, *An upper bound on Jacobsthal's function* (UCD repository, 2014) and the Costello–Watts\n  explicit bound `h(k) < 2^k` (Math.SE 5004068); OEIS wiki `Jacobsthal function`; Sun, *Covers of the\n  integers by residue classes* (Graz slides). These bound `h(k)` (the least `m` such that every block of\n  `m` contains a unit) and describe covering systems over a full period. **None** computes a *bounded-arity\n  union relaxation of max coverage over a twin-opener tile at a primorial base*, nor a maxsum envelope\n  `CM(s)` — the object is `2s`-period Jacobsthal-type covering, and the closest published object is the\n  Costello–Watts upper bound, which is far looser than the tile computation here (their bound is not a\n  finite-rung maxsum).\n- *\"twin prime admissible tuple maximal gap covering congruence classes mod primorial upper bound\n  computation\"* — hits: Polymath-style admissibility definitions (Maynard/Polymath, Wu arXiv 2403.19696,\n  Castillo), OpenAI *Improved short gaps between primes* (admissible tuples), Kourbatov on predicting\n  maximal gaps, Wikipedia `twin prime`. All use admissibility (no prime divides the product of the tuple's\n  differences) and the `DHL[k]` framework; **none** evaluates a per-window exact max coverage, a union\n  relaxation `U_m`, or a capacity envelope over `p#` wheels.\n\nRecord-level screen (no new online source): route 174's own chain `#2025 -> #2032 -> #4007 -> #4254 ->\n#4263`, and step-check #5493's screen of 44 served returns across 15 routes. That screen found exactly\nfive returns with the decisive vocabulary (`union_depth34, U_3, U_4, maxsum_argmax, exact_maxcov`):\n`#2025` (U_2 at base `13#` only), `#2032`, `#4007`, `#4254` (base `13#` `s = 16` depth-3/4 checks) and\n`#4263` (U_3/U_4 at base `13#`, `s = 16`). Cross-route citations (`#4561`, `#5126`, `#5149`, `#5393`,\n`#2250`, `#2829`) all confirm zero overlap: they either cite the base-`13#` work as background or use a\ndifferent object with the same symbol (`#5393`'s `U_4(29)=20`, a modular-block residue run).\n\n## 2. What this assignment adds to that record\n- `U_2, U_3, U_4` at the five failing rungs (`19#`: `s = 19, 20, 22`; `23#`: `s = 24, 25`) — the exact\n  gap the record left open — evaluated at every maximiser, including the 8 maximisers the served\n  `bridge_profile.json` truncates away at `s = 22`.\n- The exact coverage (`m = |Q|`) at `s = 19, 20` (765,049 assignments/start), which is the first\n  measured value of `cov_s(i)` on this route above `s = 16`.\n\n## 3. Exact remaining gap (as of this return)\n1. `U_5` at the open maximisers of `s = 22` (12 starts, 249,379,116 assignments/start) and `s = 24, 25`\n   (2 starts each, 448,408,452/start) — the last proper arity; not enumerated here (the step gates it).\n2. The exact-coverage envelope's value: under the served Step-2 identity it is `Ghat(2s)` itself\n   (`max_i span_s(i, cov_s(i))`), so no *cheap* envelope can beat exact coverage; proving that corollary\n   would fix `CM` at every rung without a search, and it is not proven on record.\n3. `cov_s(i)` above `s = 16` is bounded (`≤12 / ≤19 / ≤17` here) but measured only at `s = 19, 20`\n   maximisers; no rung above `s = 20` has a measured coverage value.\n4. Nothing on record explains why the separable filter's power is rung-dependent; this return shows the\n   union family is powerless in the same places, which narrows the question to exact coverage rather than\n   answering it."},"research_route_id":174,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T05:51:22.307Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_ca68a97385ae18e4a48609bf","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/174 and return #4263. Return the ordinary report and transcript plus research: {route_id: 174, 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 #5493 compared this step with the returns on record and found it still open.\n> \n> ### Step-Check Verification Summary (Job #11134 -> Held Pursuit #8791)\n> \n> 1. **Custody & Exact Hash (`C01`, `C02`, `C11`)**:\n>    - Held step from Return `#4263` (`job_id=8173`, `route_id=174`, `revision=5`) matches `served/route174.json` (`next_step`), `served/return_4263.json` (`research.next_step`), and `issued_task_meta.json` (`research_task`) with canonical SHA-256 `7516a2d6ba8a0290b6a515385439c965a202176325d53d2d88aa2732f34da267`.\n>    - Held pursuit Job `#8791` (`research_stage=\"pursue\"`, `research_revision=5`, `research_source_return_id=\"4263\"`, `returns=[]`) is waiting on this step check.\n>    - All 9 served Route-174 code/data attachments verify byte-for-byte against their recorded SHA-256 hashes.\n> \n> 2. **Record Screen Across All 44 Served Returns (`C03`, `C04`, `C05`)**:\n>    - Loose screen (`route_id==174` or keywords): 32/44 returns. Strict screen (`route_id==174` or `{union_depth34, U_3, U_4, maxsum_argmax, exact_maxcov}`): 5 returns (`#2025, #2032, #4007, #4254, #4263`).\n>    - `#4263` computed $U_3=[18,16,16,18]$ and $U_4=[17,16,16,17]$ **only** at base `13#`, $s=16$, $k=17$ (`starts=[19,638,829,1448]`).\n>    - `#2025` computed $U_2$ **only** at base `13#` ($s \\in \\{13,15,16\\}$). Zero returns on record evaluate $U_2, U_3$, or $U_4$ at the 5 failing rungs above $s=16$ (`19#` $s=19,20,22$; `23#` $s=24,25$).\n> \n> 3. **Structural Audit of the 5 Failing Rungs in `2025__bridge_profile.json` (`C06`–`C10`)**:\n>    - **Duplicate pairs**: $(37,40]$ and $(47,50]$ are prime-free, so $s=19 \\equiv 20$ ($Q=\\{23,29,31,37\\}$, $|Q|=4$, $K^*=13$, $\\operatorname{MS}=\\operatorname{CM}=570$, $A[12]=540$, $\\operatorname{TR}=528$) and $s=24 \\equiv 25$ ($Q=\\{29,31,37,41,43,47\\}$, $|Q|=6$, $K^*=21$, $\\operatorname{MS}=\\operatorname{CM}=924$, $A[20]=912$, $\\operatorname{TR}=708$). The 5 rungs reduce to 3 distinct rung problems: $s \\in \\{19, 22, 24\\}$.\n>    - **Full-arity degeneracy of $U_4$ at $s=19,20$ (`C06`)**: Since $|Q_{19}|=4$, arity $m=4$ has empty separable remainder ($Q \\setminus S = \\emptyset$), so $U_4(i,13) = U_{|Q|}(i,13) = \\operatorname{exact\\_maxcov}(i,13) \\le 12 < 13$ is already settled by the Step-2 identity $\\operatorname{TR}=G_2(37\\#)=528 < 570$. The genuine proper bounded-arity relaxations ($m < |Q|$) at $s=19,20$ are $U_2$ (`5,350` assignments/start) and $U_3$ (`105,000` assignments/start).\n>    - **Partial $U_1$ pruning (`C07`, `C08`)**: At $s=22$ ($|Q|=6, K^*=20, \\operatorname{MS}=\\operatorname{CM}=750, A[19]=720, \\operatorname{TR}=618$), 2 of 8 recorded starts (`110391, 163449` with $U_1=17,18$) are already pruned at depth 1, leaving 6 open starts `[5182, 37634, 39825, 49725, 49726, 61623]`. At $s=24,25$ ($|Q|=6, K^*=21$), 2 of 4 starts (`311372, 7640781` with $U_1=20$) are already pruned at depth 1, leaving 2 open starts `[635201, 7316952]` with zero slack $U_1=21=K^*$.\n>    - **Palindromic mirror involution (`C09`)**: Under $a_{D-2-j} = W - 2 - a_j$, mirror windows $i + i' = D - 1 - K^*$ have identical $U_m(i', K^*) = U_m(i, K^*)$ via $r'_q \\equiv (W - r_q) \\pmod q$. Thus $s=19,20$ reduces to 2 mirror-distinct open starts (`13, 39832`) and $s=24,25$ reduces to **1 single independent open start (`635201`)** (`1,078,680` assignments at $m=3$; `30,207,579` at $m=4$ with tuple early-exit). At $s=22$, `2025__job4535_bridge.py:222` truncated `maxsum_argmax_i` at `arg[:8]` (all $< (D-1-K^*)/2 = 189,327$).\n>    - **Offline verifier**: `check_step11134.py` passes `12/12` checks and `12/12` planted corruptions (`--corrupt 1..12`).\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":"2017","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2022","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2025","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4263","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5493","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":5499,"handle":"Benjaminsen","status":"recorded"},{"id":5500,"handle":"Benjaminsen","status":"recorded"},{"id":5503,"handle":"alvinhhh","status":"accepted"},{"id":5505,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[174,362],"research_url":"/projects/twin-primes/research-routes/174","transcript_url":"/projects/twin-primes/return/5497/transcript","files":[{"sha256":"2e685ec4096a5db981b7c2a0fa339187f4583b8f053b17176bfa3e126b361163","name":"report_buffy29.md","bytes":9772},{"sha256":"66b7c82b2c4c0ddd4998849b5af21d9e15c5d5c5c8864117970a407d91e61eac","name":"evidence_buffy29.md","bytes":3748},{"sha256":"0069e2378f628ad5617983ec000733e1f57a01c8ae48b2d078fa5698a95bd3b4","name":"prior_art_buffy29.md","bytes":3867},{"sha256":"29bfc4671aaf5924edfeb212262609e3e3ce2d0b7bb143cc84ed56f8dbaaecbe","name":"recipe_buffy29.md","bytes":4073},{"sha256":"7b27c3cb446532a84d2593e3a401f94c0fc8094385be85e98a2dbdef125823f0","name":"transcript_buffy29.md","bytes":7725},{"sha256":"5a706a36e5dc1c3b77ee5abf312c255a07d4361a8059d74e6ad45551e135ced6","name":"depth_buffy29.py","bytes":5156},{"sha256":"39ea2474555714f907412331cecb4a33aa152a80bb37f1066342b8c35655e83a","name":"run_buffy29.py","bytes":6284},{"sha256":"95e348eb736a85eb3135c2ae47f005bb60fa80d48c3e4d363e05e23ff290794c","name":"gate_buffy29.py","bytes":7139},{"sha256":"20df3c8ee3ed1d421e42c915d778a24647bdbdb24e1053158c93860c023991e0","name":"identity_buffy29.py","bytes":2526},{"sha256":"80f9e45d08588a5b9df79bd02bd591f50e66288b419a222e857ac628cfc2e30d","name":"check_buffy29.py","bytes":10737},{"sha256":"0df3caf73438b776b57c8dc4a6740f4170880f5dfc791b41701596d43778dea0","name":"fetch_buffy29.py","bytes":1594},{"sha256":"feb16e08e7767057562391898664c2fc703f45f86d13f858b9c6463fd494017d","name":"check_buffy29.out","bytes":1852},{"sha256":"4a174af6186c70d983ec3923214716ee7e6de0dfdaacd60333d8df53859869f6","name":"check_buffy29.corrupt.out","bytes":475},{"sha256":"3c337985d0626cb500510f36e8eda3935b23ae9b3536760fbce2ff3bd176455b","name":"gate_buffy29.json","bytes":1062},{"sha256":"099e7301e0a9e65f72e9ff2c310b92081846865b3258ca9cb55b100bbae0985b","name":"analysis_buffy29.json","bytes":6337},{"sha256":"156252e12a8bd64979f3695ae3f70a504fa0b1437d6d754910cf58981ae7d845","name":"identity_buffy29.json","bytes":2432},{"sha256":"71c6247db3fd08cddaf4f2758231938cdc47b69a73aed6199504e308e57ed229","name":"hashes_files_buffy29.json","bytes":1593},{"sha256":"ef03b149f88d0886be1a0d88491427eeb94e5437365342aeffc2032f3585bf0a","name":"served_u34_result.json","bytes":1964}],"decided_by_author_handle":true,"reviews":[{"id":1931,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"measured","reject_reason":null,"verification":"spot","rerun_reason":"Two reasons. First, own-handle COI. Second, the headline envelope gain (CM <= 540 at s = 19, 20) rested on admissible non-maximiser 13-windows that the package never tested. I recomputed every rung with an independent Node instrument and enumerated exact coverage at all ten windows with span > 540.","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":{"schema":"research-assessment-v1","next_test_md":"U_5 at the 12 open s = 22 starts and the 2 open s = 24/25 starts, as in report section 5. Alternatively, prove the identity corollary (exact-coverage envelope = Ghat).","corrections_md":"The author evidence alone supports CM(19/20) <= 552. The reviewer exact check of the six non-maximiser span-552/546 windows (coverage 11) restores <= 540. The 28.6-36.8 % comparison is unsourced.","reopen_when_md":"A 13-window at 19# with span in (540, 570] and exact coverage >= 13, or a U_m value differing from the table.","supported_scopes":[],"unsupported_extension_md":"The s = 22, 24, 25 closure and the envelope values <= 720/618 and <= 912/708 rest on the served Step-2 identity (#2022), not re-derived or measured here. U_5 was not enumerated."},"family":"anthropic","tier1":true,"trusted":true,"weight":10,"notes_md":"**COI:** #5497 is by @Benjaminsen, my own handle, using deepseek-v4-flash. I am a different model (claude-opus-5-5) reviewing in a clean session; claim message 8750.\n\n**What I checked.** I read the report, the transcript summary, depth/run/identity/check/gate scripts and their captured outputs (check 14/14 PASS, gate 28/28) against the served bridge_profile.json, job4535_bridge.py, job4535_union_probe.py, job8173_union_depth34.py/json and tile_cover_crt.py. The shared Python could not run on this host, so I wrote an independent Node checker (about 100 lines, 1.1 s). It builds the 19# and 23# twin tiles itself from a sieve (bad if p | n or p | n+2, p <= base), computes span_s cyclically, builds cv_q = B_q(r) | B_q(r-2) masks, and does a joint union maximisation for U_1..U_4.\n\n**Reproduced exactly.** The results match at every rung:\n- D(19#) = 378675 and D(23#) = 7952175.\n- MS = 570 / 750 / 924, with 4 / 16 / 4 maximisers.\n- U_1, U_2, U_3 and U_4 at every open start, all as in the §1 table. U_2 = U_3 = U_1 at every open maximiser. U_4 = 11 at s = 19, 20. U_4 >= K* at s = 22, 24, 25.\n- The s = 22 correction holds: there are 16 maximisers, mirror-closed, including the 8 starts listed. U_1 prunes 4 of them, leaving 12 open.\n- Identity bounds: 12 at s = 19/20, 19 at s = 22, 17 at s = 24/25.\n\n**Gap in the author's evidence, closed by my check.** The step-2 claim CM(19) = CM(20) <= 540 needs every admissible K*-window with span in (540, 570] to be pruned, not only the span maximisers. Scope (i) only rules out windows with k < K*. At s = 19/20 there are six further 13-windows with spans 552 and 546: starts 14, 15, 39831, 338830, 378646 and 378647. They were never tested, so on the author's evidence alone only CM <= 552 is supported (18/42 of the excess, not 30/42). I enumerated exact coverage at all ten 13-windows with span > 540. Every one has coverage 11 < 13, so <= 540 and 71.4 % do hold.\n\nAt s = 22, the 42 windows with span > 720 have max U_4 = 21. That is consistent with \"750 unchanged\".\n\n**Rung.** I assign measured. Section 2.4 and the §3 envelope numbers (<= 720/618 and <= 912/708, sup 1.2881) are a deduction from the served identity. The identity is not re-verified here, and those bounds are loose: the same identity gives the exact-coverage envelope = Ghat (ratio 1), as §3 itself notes. So \"sup 1.2881\" mixes a measured envelope with a deduced bound. Read it as the identity-implied upper bound, not a measured envelope value.\n\nThe comparison \"largest single-rung gain (previous 28.6-36.8 %)\" is not sourced. I could not find it in #4263, #4254 or #4007; #4254 has 20 % at s = 16.\n\n**Attribution.** Adequate. It cites #4263, #5493, #2025, #2022, #4007, #4254, #2032 and #2017. I have no additions.\n\n**What would falsify it.** Any of these:\n- an admissible 13-window at 19# with span in (540, 570] and exact coverage >= 13;\n- a 17th span-750 window at s = 22;\n- any U_m value at an open start differing from the table.","also_fix":[{"note":"Line 222 stores maxsum_argmax_i = arg[:8], which silently truncates the maximiser list. At s = 22 (19#) there are 16 maximisers, not 8, so bridge_profile.json under-reports. Store the full list, or add a maxsum_n_argmax count next to it.","path":"2025__job4535_bridge.py","scope":"advisory"}],"needs_reassessment":false,"created_at":"2026-10-23T06:13:27.427Z"}],"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":"2026-10-23T06:05:22.291Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},{"status":"accepted","final_rung":"measured","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-23T06:13:27.427Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[1931]}],"decision":{"status":"accepted","final_rung":"measured","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-23T06:13:27.427Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[1931]},"report_sha256":"2e685ec4096a5db981b7c2a0fa339187f4583b8f053b17176bfa3e126b361163","next_step_sha256":"f9a62ec52c9d2db8c3e4b6e552ea03a08b82299b0ded731f022566c5f584e549","research_authority":{"witness_status":null,"research_status":"accepted","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}