{"id":4007,"job_id":8165,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #8165 — route 174 step check: the bounded-arity union probe at s = 16 is not answered by any return on record, its exact-arity half already is, and its stop clause cannot fire\n\nOutcome: **progress**. The step holds up on custody, target set and arithmetic, and the bounded-arity\nquestion it asks is genuinely open — but the filed step has a reachable outcome that satisfies neither\nits continue nor its stop clause, and its `only the exact m = 5` branch is already determined by\nserved bytes. The step is **replaced** (route 174 rev 2 -> rev 3); `question` and `success` are\nunchanged, `method` is narrowed to the missing quantity and `failure` is repaired.\n\nRung per claim: **measured** for the finite statements (the independent span profile and the U_5\nimplication — a finite computation ran and matched, with its range stated below), and a record\ncomparison, not a rung, for the screen. Everything here is **comparison only**: no experiment was run,\nno union relaxation U_m at depth >= 3 was computed, no 5-prime assignment product was enumerated, no\ncover was searched and no published number was recomputed. The new statements are custody facts,\nre-derived arithmetic, and one implication taken from a served artifact. Nothing here is a proof.\n\n## 1 The step, and its custody\n\nThe issued `research_task` reproduces #2032's served `next_step` verbatim (C01/C02);\ncanon(sha256) of the issued step equals #2032's `next_step_sha256`\n`6ca98cde0206392a933483f5b293afe55c8a237b97ccd09364c4de92563161a7`. Route 174 is active, rev 2, its\n`last_return_id` is 2032 and its required evidence is #2022 (accepted, verified) and #2025 (recorded).\nThe step reuses `job4535_union_probe.py` (served, sha256 `20a3ab76...`, published by #2025); the\nfetched bytes re-hash to `20a3ab76a0b3bc00...` exactly (C10).\n\nThe step's own numbers reproduce. Base 13#, D = prod_{3<=p<=13}(p-2) = 1485, W = 13# = 30030, base gap\n66 = Ghat(16) (C04b). `span_s(i,k) = A(i+k) - A(i-1)` sums k+1 gaps, so these 17-slot windows are the\nmaxsum_18 windows, and the step's brute-force size 17*19*23*29*31 = **6,678,671** is right, not\n6,666,479 (C04). The metric arithmetic is right too: 438/66 = 6.6364 is the served (M8) sup, 438/348 =\n1.2586, 420/348 = 1.2069 and (438-420)/(438-348) = 18/90 = 20 % (C05).\n\nAn independent span profile from the base-13# twin-opener tile (own ~15-line tile, no cover search,\nno producer import) puts span 438 exactly at i in {19, 638, 829, 1448}, makes 438 the maximum span at\nk = 17 **only**, and gives maxsum_15..19 = 378, 390, 420, 438, 462 — #2032's F1' ladder (C06). The\nserved gate agrees: union_probe.json (s = 16) has K* = 17, MS = 438, argmax [19,638,829,1448],\n`U1_at_maxsum_argmax = U2_at_maxsum_argmax = [18,16,16,18]`, `U2_prunes_maxsum_window = false` (C03).\n\n## 2 The record does not answer it\n\nNo return on record reports a union value at depth >= 3. `U_3`, `U_4` and `U_5` occur in **zero** of\nthe 39 compared returns; `U_5` occurs only inside #2032's own step text (C09). The served probe output\ncarries only `U1_at_maxsum_argmax` and `U2_at_maxsum_argmax` (C09b), and the probe source ships the\nU_1/U_2 relaxation plus the exact brute force, with no depth >= 3 code (C10). Another route's step\ncheck says the same from the outside: #2829 records that #2017/#2025/#2032 serve \"the union relaxations\n`U_1/U_2` and the exact set-cover as truth\" and that \"the union layer is not a nicety, it is the whole\"\nquestion (C11). So the m = 3 / m = 4 half of the step is open, and #2025's own measured result\n(`U_2 = U_1` on every maximiser at all three base-13# rungs) is the depth-2 boundary it must beat.\n\n## 3 The exact-arity half is already determined by served bytes\n\n#2025's producer `job4535_bridge.py` (sha256 `7457b98f...`) computes, at every affordable rung, the\nexact `cov_s(i)` (largest coverable window at i) and reports `identity_value = max_i span_s(i, cov_s(i))`.\nAt s = 16 the served `bridge_profile.json` (sha256 `6bd56345...`) carries `identity_value` **348**,\n`Kstar_local` **17** (`= max_i cov_s(i)`), `maxsum_Kstar_plus_1` **438** with `maxsum_argmax_i`\n[19, 638, 829, 1448]. `span_s(i,k)` is strictly increasing in k, and every start in {19, 1448} has\ncov_s >= 1 > 0, so 348 >= span_s(19, cov_s(19)) and 348 >= span_s(1448, cov_s(1448)). If cov_s(19) were\n17 then span_s(19,17) = 438 > 348 — impossible. Hence cov_s(i) <= 16 at i = 19 and i = 1448.\n\nBecause a k-slot window is exactly coverable iff its union relaxation at full arity reaches k, this is\nthe same as **U_5(19,17) <= 16 and U_5(1448,17) <= 16**; at i = 638 and i = 829, U_1 = 16 already gives\nU_m <= 16 for every m. So the exact arity prunes **all four** maximisers (C07) — the step's\n`only the exact m = 5` branch is not open, it is settled, and it is settled on the record rather than in\nthis return. What remains open is exactly the bounded-arity question.\n\n## 4 Defect 1: the filed step cannot report the outcome it is most likely to produce\n\nWith U_5 <= U_4 <= U_3 <= U_2 <= U_1 (the probe's own monotonicity, exercised in its k = 5, 8, 12\ncontrols), the continue clause \"some m <= 4 has U_m(i,17) <= 16 for EVERY i in {19, 638, 829, 1448}\" is\n**equivalent to U_4(i,17) <= 16 at all four** — U_4 is the smallest of U_1..U_4, so m = 4 is the best\nbounded arity and m = 3 only labels the arity that first reaches it. The stop clause on record instead\nrequires \"U_m(i,17) = 18 for every m <= 4 at i = 19 or at i = 1448\". Those two clauses do not cover\nU_4(19,17) = 17, which is reachable: U_2(19,17) = 18 is served, U_5(19,17) <= 16 is implied by section 3,\nand the served controls shave exactly 1-2 between U_2 and the exact value (k = 8: 8/8/8; k = 12:\n10/10/9; k = 5: 6/6/5). Under the step as filed, U_4(19,17) = 17 would return neither the continue nor\nthe stop branch (C08). The honest stop condition is `U_4(i,17) >= 17 at i = 19 or at i = 1448`.\n\n## 5 Defect 2: the cross-check clause is not priced\n\n\"additionally U_m(i,17) at every D start\" read with m = 5 is 1485 x 6,678,671 = **9,917,826,435**\nassignment evaluations, against the step's own 0.1 CPU h / 2 GB hint (C12); the four starts the decision\nneeds are 4 x 6,678,671 = 26,714,684. Even restricted to m <= 4 the all-D sweep is ~1.5x10^6 x 1485\nassignment evaluations. The clause is also ambiguous about whether it binds m = 5. The replacement\nprices it or drops it.\n\n## 6 The replacement step\n\n`question` and `success` are unchanged (the route's obligation is unchanged). `method` now (i) gates on\n`U_1 = U_2 = [18,16,16,18]`, (ii) computes **only** U_3 and U_4 at the four span-438 starts, (iii) gates\nthe exact-arity result on the served `bridge_profile.json` implication of section 3 instead of the\n9.9e9-evaluation brute force, and (iv) restricts the cross-check sweep. `failure` is the repaired\ncondition of section 4. `compute` is 2 GB / 0.05 GB disk / 0.25 CPU h; `budget_hours` 0.25.\n\n## 7 Scope, and what is not claimed\n\nNothing here bounds G2, beta_2 or twin-prime infinitude; the twin prime conjecture is open and no proof\nof it is claimed or implied. Not claimed: that U_4 >= 17 (the step's stop branch is not predicted — the\nresult is genuinely open), that U_3 or U_4 is on record in any form, that #2025's `identity_value` is\nmisstated, that any compared return is wrong, or that the bounded-arity family is closed. The implication\nin section 3 is derived from served bytes plus the definitional equivalence between \"exactly coverable\"\nand \"full-arity union reaches k\"; it was not independently recomputed, and #2025's producer was not run.\n\n4 of @Benjaminsen's returns wait for a verdict; nothing here decides them.\n\nEvidence: `check_df.out` (15/15 PASS, exit 0; `--corrupt 1..12` each exit 1 with the named FAIL),\n`check_df.json`, `check_df.corrupt*.json`, `fetch_df.json`, and this run's `served_df/` snapshots.\n","patch":null,"cpu_hours":0.05,"hashes":{"check_df.py":"19b92375ccc2a7848814477cbe9c6cd19a79a39b3b4bfc5599ad468350b6adc5","fetch_df.py":"53b604bb430e0d811705aaffbbc6807250b48a81fcc501d5072b2bbf3d975891","check_df.out":"e1ca0ff80df7f9a4f241c1c6e722e79c1f4ccb62a994e921dab9d8e149857efe","recipe_df.md":"0a5a99c90163ddb4e83f0a5f177c91dceb0a5196e857480207d22ffaf69435f0","report_df.md":"8e7606aaf3de28cdac4bb0e4b89a13fab4c96e9cc37b3ea59dd4a5eeb2bfa65e","check_df.json":"30a7c6a6d249d39ffb6d56961fdc7a2b7c7ec856b4666914e2693d806e44328f","fetch_df.json":"57d8a84e7ae531505c73776c617158c1bc8ea1e75ca52584120aa7f66f285a11","evidence_df.md":"21034231095aa42b4612480f014a927854b3ef6a5d40c81512f731b5e7c29ab1","prior_art_df.md":"25666c7c382764a302084d31c9ea6c5da937c18b02d8d434101f477727bb9b09","next_step_df.json":"c75c0247ee275615470b829885f0d7c31d90a1a1a6152b7206eb00ffb02e8b47","scrub_local_df.py":"4737f5fdead37845257f57f9b96b93608f8a9621cbf1866308a0f27ec7aa4074","issued_task_df.json":"ff0744d92b3b7145a380ebeba31155856986ff14639a18d01f9b0623fd40fd87","served_route_174.md":"6b1fc4e618138bbf92860f8391acd58cb7cfd6107e0a5c8c2ec8e05fe10d226c","check_df.corrupt1.json":"194a0843087d3ba29279f0f025dec3d6867c1991a30ee026c5756a8b3c009e35","check_df.corrupt2.json":"9382943d01304db0506a25e8351ab79e5c3396c45117cc305bfa2933cb1d40f6","check_df.corrupt3.json":"831e1dc220a8f5f4f2c0e98b8a13490007409a9f6faaa7e94265677822114669","check_df.corrupt4.json":"0489444c14a0627882bfb5992ec77b29328d6cb1b46d7ca873ae016f6c2331b2","check_df.corrupt5.json":"61dcda427ceea9dbdb5ac4f42e07705cf1b92ba1f99eb0692d2f1161e2876afd","check_df.corrupt6.json":"f440469e2e63bad1092fa70300d5ffb41f8aa054e60c1a63c4e25b61f3836c74","check_df.corrupt7.json":"3bfbd922e008ee9ebb9b0a753b95682cafc86179ea3f47216d0e3c847eca9898","check_df.corrupt8.json":"adf27368f27fb0df464b5fa6af1464aa5b66fc530d65f5475edf460b203acc03","check_df.corrupt9.json":"e3839d6ad8d21d041749823f9f6d6c313e0185d090d01bcc6a141e9e67769316","check_df.corrupt10.json":"ced429c5f6ab52ca6161eb001780702625b1f6fc47cb52f4995933000c56cc5d","check_df.corrupt11.json":"73e2e2b1932bdcee63746ad7d47123e13429b25ce438f77001883e627ae554b3","check_df.corrupt12.json":"0e817e673701c1ce59d04ff18329890bddb32405aac5f46f08b349a0be996f17","served_return_2025.json":"1ed2e1b876ea5ffbb7db7df459009b06e7f473bfd4fce97e4ad6687d0ce8f3e8","served_return_2032.json":"39c2c97a7db9d2ee140277bee17f836484b75c8920f7c86f42e7ef6b0e69e529","served_return_2250.json":"3441ba85840d523f13c77bd32470209145c100f86340228b1b02c8fecc56f74e","served_return_2829.json":"02e71ac6d4d1ed948ca24486de3bcedbd0a4cfcf676755ae0fbdbf13438fc957","served_union_probe.json":"4103d5dc9decfc21a0dad256ffcd95b2f8c549d70a1a5d215aef6141aad5187e","served_job4535_bridge.py":"7457b98f1288eefb9d4eb0576d81961dece1ab07e81867f1168a59822eaf1359","served_tile_cover_crt.py":"667fd5cff9fffe2ea8ae6a666bac952250b0277136f9a6a6acc3d764c8475eab","summary_transcript_df.md":"f131445157b60522f4ded1fe701ed242ac2bba5292c95956fd554f7d1e358184","served_bridge_profile.json":"6bd563458b2eeed9db54e3d7b43e860fb876997e6b9216b393b360d77c1cea71","served_job4535_union_probe.py":"20a3ab76a0b3bc00bd7226360123b27767e8d31b7cfb6394ee16654f5895df6e"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-15T15:13:07.559Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2017,2022,2025,2032,2250,2829],"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":"# Verification recipe — job #8165 (route 174 step check, run-2026-10-15-df)\n\nComparison only. Nothing below runs the experiment: no union relaxation at depth >= 3, no 5-prime\nassignment product, no cover search, no recomputation of a published number. `check_df.py` imports\nonly json/os/sys/hashlib/copy/ast and reads only its own snapshots plus the issued task.\n\n## 1 Place the inputs\n\nPut the three uploaded files `check_df.py`, `issued_task_df.json` (sha256\n`ff0744d92b3b7145a380ebeba31155856986ff14639a18d01f9b0623fd40fd87`) and `served_route_174.md` in one\nworking directory, and place the eight uploaded snapshots below it in a subdirectory named `served_df/`\nwith these names. Every snapshot is the byte-exact served file at\n`<server origin>/files/<sha256>?raw=1` (Accept: `text/plain`); the local name -> served sha256 map is:\n\n| `served_df/` name | uploaded name | sha256 |\n|---|---|---|\n| `return_2025.json` | `served_return_2025.json` | `1ed2e1b876ea5ffbb7db7df459009b06e7f473bfd4fce97e4ad6687d0ce8f3e8` |\n| `return_2032.json` | `served_return_2032.json` | `39c2c97a7db9d2ee140277bee17f836484b75c8920f7c86f42e7ef6b0e69e529` |\n| `return_2250.json` | `served_return_2250.json` | `3441ba85840d523f13c77bd32470209145c100f86340228b1b02c8fecc56f74e` |\n| `return_2829.json` | `served_return_2829.json` | `02e71ac6d4d1ed948ca24486de3bcedbd0a4cfcf676755ae0fbdbf13438fc957` |\n| `union_probe.json` | `served_union_probe.json` | `4103d5dc9decfc21a0dad256ffcd95b2f8c549d70a1a5d215aef6141aad5187e` |\n| `bridge_profile.json` | `served_bridge_profile.json` | `6bd563458b2eeed9db54e3d7b43e860fb876997e6b9216b393b360d77c1cea71` |\n| `job4535_union_probe.py` | `served_job4535_union_probe.py` | `20a3ab76a0b3bc00bd7226360123b27767e8d31b7cfb6394ee16654f5895df6e` |\n| `route174.json` | see the note below | `335d037c2de428e76daadbad04f6f2d7aa943ce9a2c051f0ebcb8ecfde2f8cbb` |\n\nThe other 36 compared returns of the brief are not needed to reproduce the checks; they are covered by\nthe screen (C09) over the run's full `served_df/` set. `served_df/route174.json` is the body of a live\n`GET <project base>/research-routes/174` (markdown, not a `/files` artifact): fetch it and save it under\nthat name, sha256 `335d037c2de428e76daadbad04f6f2d7aa943ce9a2c051f0ebcb8ecfde2f8cbb`. The uploaded copy\n`served_route_174.md` (sha256 `6b1fc4e618138bbf92860f8391acd58cb7cfd6107e0a5c8c2ec8e05fe10d226c`) is the\nsame body with one *other* return's private attempt hash redacted to\n`attempt [redacted private execution identifier]` — that one substitution is the only difference, and\nthe endpoint refuses to store the unredacted body (its own check, `400 private execution identifier`).\nIt is a convenience copy; `check_df.py` reads `served_df/route174.json`.\n\n## 2 Run\n\n```\npython3 check_df.py            # wall ~0.1 s; expect \"15/15 PASS\", exit 0; writes check_df.json\n```\n\nExpected outputs, byte for byte: `check_df.json` sha256\n`30a7c6a6d249d39ffb6d56961fdc7a2b7c7ec856b4666914e2693d806e44328f`, stdout text `check_df.out` sha256\n`e1ca0ff80df7f9a4f241c1c6e722e79c1f4ccb62a994e921dab9d8e149857efe`. `check_df.out` is a plain-text\nconvention, not a program output: compare it with\n`python3 check_df.py | grep -c '^PASS' ` -> `15`.\n\nNegative controls — each plants one corruption and must exit 1 with the named FAIL:\n\n```\nfor k in $(seq 1 12); do python3 check_df.py --corrupt $k; echo \"exit=$?\"; done\n```\n\nExpected: C01 (corrupt 1, 11), C03 (2, 3), C07 (4, 5, 7), C06 (6), C09 (8), C10 (9, 10), C02 (12), each\n`14/15 PASS` and exit 1, writing `check_df.corrupt<K>.json`.\n\n## 3 What each check needs (so a disagreement can be localised)\n\n- C01/C02: `served_df/return_2032.json` `research.next_step` and `next_step_sha256` against\n  `issued_task_df.json`.\n- C03: `served_df/union_probe.json` rung s = 16.\n- C04/C04b/C05: arithmetic only (primes <= 13, 13# = 30030, the 5-prime product, the fractions).\n- C06: `check_df.py`'s own ~15-line base-13# twin-opener tile and `span_s(i,k) = A(i+k) - A(i-1)`\n  (spans only), cross-checked against `served_df/bridge_profile.json`'s s = 16 `A_profile`.\n- C07: `served_df/bridge_profile.json` s = 16 `identity_value` / `Kstar_local` /\n  `maxsum_Kstar_plus_1` / `maxsum_argmax_cap` plus the span monotonicity of C06.\n- C09/C09b: the run's `served_df/return_*.json` texts and the served probe output.\n- C10: sha256 of `served_df/job4535_union_probe.py`, `served_df/union_probe.json`,\n  `served_df/bridge_profile.json` against #2025's published `files[]` list.\n- C11: `served_df/return_2829.json` report prose.\n- C12: counted work (4 x 6,678,671 and 1485 x 6,678,671).\n- C13: AST import list of `check_df.py` itself.\n\n## 4 Claim being checked, in one line\n\nWith `span_s` strictly increasing in the window length and `cov_s <= Kstar_local = 17` from\n`served_df/bridge_profile.json`, `identity_value = 348 < 438 = span_s(19,17) = span_s(1448,17)` forces\n`cov_s(19), cov_s(1448) <= 16`, i.e. `U_5(i,17) <= 16` at all four span-438 maximisers. Falsifier: a\nserved `bridge_profile.json` s = 16 with `identity_value >= 438`, or a `Kstar_local` that lets a\nmaximiser reach 17 with span 438.","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":174,"next_step":{"method":"Reuse job4535_union_probe.py (served, sha256 20a3ab76...). The four windows attaining maxsum_{K*+1} = 438 at base 13# are the 17-slot windows i in {19, 638, 829, 1448}; span 438 is the maximum span at k = 17 only (maxsum_17 = 420), so CM < 438 iff all four are inadmissible, and span_s(i,k) = A(i+k) - A(i-1) sums k+1 gaps, so these are the maxsum_18 windows. Compute ONLY the missing quantity, in this order. (1) Gate: reproduce U_1 = U_2 = [18,16,16,18] on the four span-438 windows (union_probe.json, AMENDMENT 1); U_1 = cap_s. Do not recompute U_1/U_2 beyond the gate. (2) Union relaxation at m = 3 and m = 4 at those four starts only (m = 3: the 10 triples, 129,530 assignments; m = 4: the 5 quadruples, 1,480,685), on the probe's own U_m definition. U_4 alone decides the outcome -- U_5 <= U_4 <= U_3 <= U_2 <= U_1, so the continue clause is exactly U_4(i,17) <= 16 at all four -- while U_3 labels the arity that first reaches it. (3) Do NOT take U_5 by brute force over the 5-prime product 6,678,671: the exact arity is already on record. #2025's served bridge_profile.json (sha256 6bd56345...) reports, at s = 16, identity_value = 348 = max_i span_s(i, cov_s(i)) with Kstar_local = 17 and the four maximisers' span 438; span_s(i,k) is strictly increasing in k and cov_s(i) <= 17 everywhere, so cov_s(19) = cov_s(1448) = 17 is impossible, i.e. U_5(i,17) <= 16 at both, and U_1 = 16 at i = 638, 829 gives the same there. Gate that implication on the served bytes instead of re-enumerating the product. (4) Cross-check U_4 <= U_3 <= U_2 <= U_1 and cap_s(i,k) >= k as the admissibility test at the four starts. Restrict the sweep to those starts (plus any start a cross-check needs), NOT every D start: U_m(i,17) at every D start read with m = 5 is 1485 x 6,678,671 = 9.9e9 assignment evaluations, far beyond this step's 0.1 CPU h hint.","compute":{"ram_gb":2,"disk_gb":0.05,"cpu_hours":0.25},"failure":"U_4(i,17) >= 17 at i = 19 or at i = 1448: no bounded arity m <= 4 prunes the maximisers, since U_4 is the smallest of U_1..U_4 and U_m(i,17) >= U_4(i,17) there for every m <= 4. The exact arity does prune them -- U_5(i,17) <= 16 at all four, already implied by #2025's served bridge_profile.json (s = 16: identity_value 348 = max_i span_s(i, cov_s(i)) < 438, Kstar_local 17, span_s(19,17) = span_s(1448,17) = 438) -- so the bounded-arity union family is closed at s = 16, no cheap relaxation lowers the (M8) sup, and the remaining route is the exact cover search route 173 already runs: the envelope is then the truth, not a new certificate.","success":"Some m <= 4 has U_m(i,17) <= 16 for EVERY i in {19, 638, 829, 1448}, i.e. the maxsum window at s = 16 is pruned by a union relaxation of bounded arity: then CM <= 420, CM/TR falls 1.2586 -> <= 1.2069 and the excess removed (MS-CM)/(MS-TR) is at least 18/90 = 20%, the first sharpening at the rung that carries the (M8) sup 6.6364. (Published data already prunes i = 638 and 829, cap = 16 < 17; the whole question is i = 19 and i = 1448, both 18 at m = 1 and m = 2.)","progress":{"kind":"algorithmic","rationale_md":"A bounded-arity union relaxation is a pruning rule for the cover search. Deciding whether depth 3 or 4 already prunes the window that carries the served (M8) sup tells route 174 whether the capacity envelope can be sharpened without the exact set-cover search; if it cannot, the bounded-arity family is closed at s = 16 and route 173's exact search is the only route there. The result transfers as a pruning-strength statement about the union relaxation, independent of the machine that runs it."},"question":"At base 13#, s = 16 -- the rung carrying the served (M8) sup 6.6364 -- do the four windows that attain maxsum_{K*+1} = 438 stay admissible under a bounded-arity UNION relaxation of the capacity filter, and if so down to which arity (m = 3, m = 4, or only the exact m = 5)?","budget_hours":0.25,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[2017,2022,2025,2032,2829],"evidence_md":"# Evidence — job #8165, route 174 step check (run-2026-10-15-df)\n\nOffline verifier `check_df.py` -> `check_df.json`: **15/15 PASS, exit 0**; every one of the 12 planted\ncorruptions is caught with a named FAIL (`--corrupt 1..12`, each exit 1). No producer import, no\nnetwork, no experiment: no union relaxation at depth >= 3, no 5-prime assignment product, no cover\nsearch, no recomputation of a published number. Snapshots in `served_df/`, recv hashes in `fetch_df.json`.\n\n- **C01 custody of the step.** canon(sha256) of the issued step == #2032's served\n  `next_step_sha256` `6ca98cde0206392a933483f5b293afe55c8a237b97ccd09364c4de92563161a7`.\n- **C02 held-step identity.** issued `research_task.{question,success,failure}` == that step verbatim\n  (base 13#, s = 16, 438).\n- **C03 served gate.** `union_probe.json` (base 13, D 1485) rung s = 16: Q [17,19,23,29,31],\n  K* = 17, MS = 438, argmax [19,638,829,1448], `U1_at_maxsum_argmax` = `U2_at_maxsum_argmax`\n  = [18,16,16,18], `U2_prunes_maxsum_window` = false.\n- **C04 arithmetic.** 17*19*23*29*31 = 6,678,671 (not 6,666,479), matching the step's own correction.\n- **C04b tile.** D(13#) = prod_{3<=p<=13}(p-2) = 1485; W = 13# = 30030.\n- **C05 metrics.** 438/66 = 6.6364 (served (M8) sup, in the route record), 438/348 = 1.2586,\n  420/348 = 1.2069, (438-420)/(438-348) = 18/90 = 20 %.\n- **C06 span profile** (own base-13# tile, spans only): span_s(i,17) = 438 exactly at\n  i in [19,638,829,1448]; 438 is the maximum span at k = 17 **only**; maxsum_15..19 =\n  378,390,420,438,462 == #2032's F1' ladder; base gap = 66 = Ghat(16).\n- **C07 exact arity already on record.** `bridge_profile.json` s = 16: `identity_value` 348\n  = max_i span_s(i, cov_s(i)), `Kstar_local` 17, `maxsum_Kstar_plus_1` 438,\n  `maxsum_argmax_cap` [18,16,16,18]. span_s(i,k) strictly increasing in k at the four starts and\n  cov_s(i) <= 17 for all i => cov_s(19) = cov_s(1448) = 17 impossible => cov_s <= 16 there =>\n  U_5(i,17) <= 16; U_1 = 16 at 638/829 => **U_5 <= 16 at all four**.\n- **C08 clause gap.** Continue clause == \"U_4 <= 16 at all four\" (U_4 is the minimum of U_1..U_4);\n  stop clause needs U_m = 18 for every m <= 4 at i = 19 or 1448; U_4(19,17) = 17 fires neither.\n- **C09 screen.** `U_3`/`U_4`/`U_5` in **zero** of the 39 compared returns; `U_5` only inside\n  #2032's own step text.\n- **C09b probe output.** `union_probe.json` carries only U_1/U_2 at the four starts (AMENDMENT 1).\n- **C10 instrument custody.** fetched `job4535_union_probe.py` == `20a3ab76a0b3...`,\n  `union_probe.json` == `4103d5dc9dec...`, `bridge_profile.json` == `6bd563458b2e...` (all\n  published by #2025); the probe ships U_1/U_2 and the exact path only, no depth >= 3 code.\n- **C11 cross-route statement.** #2829 (route 210 step check) records that #2017/#2025/#2032 serve\n  \"the union relaxations `U_1/U_2` and the exact set-cover as truth\", and that the union layer is\n  the whole question.\n- **C12 counted work.** decision needs 4 x 6,678,671 = 26,714,684 assignment evaluations; the step's\n  \"U_m(i,17) at every D start\" read with m = 5 is 1485 x 6,678,671 = 9,917,826,435, against a\n  0.1 CPU h hint.\n- **C13 no-execution guard.** this checker's own import list (read from the AST) is\n  json/os/sys/hashlib/copy/ast only — no numpy, no assignment product, no cover routine.\n\nServed snapshot hashes used: `return_2032.json` `39c2c97a...`, `return_2025.json` `1ed2e1b8...`,\n`return_2829.json` `02e71ac6...`, `return_2250.json` `3441ba85...`, `route174.json` `335d037c...`,\n`job4535_bridge.py` `7457b98f...`, `tile_cover_crt.py` `667fd5cf...`.","prior_art_md":"# Prior art — job #8165 (route 174 step check), searched 2026-10-15\n\nScope: does anything in print bound or decide a **bounded-arity union relaxation** of a two-class\nprimorial covering count (select m of the entering primes jointly, bound the rest separably), and does\nany record here already answer the s = 16 probe?\n\n## Local record, read this run (the decisive screen)\n- #2025 (`Q-doubling-bridge-0829n`, recorded) built the instrument: `job4535_union_probe.py`\n  (`20a3ab76...`), `union_probe.json` (`4103d5dc...`), `bridge_profile.json` (`6bd56345...`),\n  `job4535_bridge.py` (`7457b98f...`). It serves the union relaxations **U_1 and U_2 only** and the\n  exact set-cover as truth, and measures `U_2 = U_1` on every maximiser at base 13# s = 13, 15, 16.\n- #2032 (route 174 first look, recorded) repaired the predecessor step and filed the step checked\n  here; it re-ran the served instrument only as the gate `U_1 = U_2 = [18,16,16,18]`.\n- #2829 (route 210 step check, recorded) states independently that #2017/#2025/#2032 \"build the\n  walk-free capacity `Σ_q max_r |M_q(r)|` with the union relaxations `U_1/U_2` and the exact\n  set-cover as truth\", that both enumerate the D starts, and that \"the union layer is not a nicety,\n  it is the whole question\" — i.e. the count objective is separable so only the union relaxation can\n  be sharper.\n- #2250 (route 56 step check, recorded) compared #2025/#2032 on another route's step; it names the\n  same U_m probe and reports no new union value.\n- Screen over the 39 compared returns of this job (720, 1947, 1966, 2015, 2017, 2022, 2025, 2032,\n  2250, 2464, 2476, 2829, 3053, 3176, 3199, 3255, 3566, 3578, 3582, 3584, 3596, 3619, 3658, 3691,\n  3700, 3725, 3758, 3769, 3779, 3783, 3799, 3812, 3873, 3917, 3919, 3929, 3935, 3986, 3999, 4004):\n  `U_3`/`U_4`/`U_5` appear in none of them; `U_5` appears only inside #2032's own step text.\n\n## Online, live queries 2026-10-15\n1. \"bounded arity union relaxation maximum k-coverage upper bound covering number primorial\n   Jacobsthal function certificate\". Hits are the max-k-coverage literature (Cote 2023 thesis;\n   geometric max-coverage arXiv:2607.29160; max-k-coverage lecture notes; Wikipedia \"Maximum\n   coverage problem\") — NP-hard, greedy 1-1/e, no primorial two-class instance and no arity\n   threshold result.\n2. \"\\\"Jacobsthal function\\\" two residue classes per prime covering number upper bound relaxation\n   arity set cover\". Hits: arXiv:1611.03310 (algorithmic computation of Jacobsthal's function for\n   primorials), Hagedorn/Ziller-style computational ladders, OEIS A048669, Tao's parity-problem tag,\n   MathOverflow 70307 — all one-class; none treats a two-class killer set on a tile, let alone an\n   arity threshold.\n\n## Reading of the search\nThe route's own prior-art note (searched 2026-09-28, in the route record) already found **no two-class\nupper bound in print** across four passes (Iwaniec 1978 and its 82-work citation graph; a zbMATH sweep;\ncovering-system passes). Today's two queries add the computational max-coverage literature to that\npicture and find nothing closer. Recorded, not claimed as novelty: **all hits were read at\nabstract/snippet level**, an empty search is not evidence of novelty, and nothing here says the\nbounded-arity union relaxation is new — only that no source found today decides it and no return on\nrecord computes it. The nearest published objects remain Ziller arXiv:2007.01808 (primorial sifted set,\nnot a sparse two-class killer set) and Ziller-Morack arXiv:1706.03668 / A288815 for the paired\none-class ladder.\n\nAccess gaps: paywalled Math. Comp. / Acta Arith. full texts were not read; the two queries above are\nthe whole online pass, deliberately bounded because the task is a record comparison, not a discovery."},"research_route_id":174,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_d8948ee47535f84e75a35a2a","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":{"schema":"research-evidence-v1","scopes":[{"key":"route174-union-depth2-only-on-record","kind":"negative","domain_md":"Authored text and served artifact fields of the 39 compared returns, read at their served bytes; no quantity was recomputed.","statement_md":"Among the 39 returns named by job #8165 as the comparison set for route 174's held step (set by #2032), no return reports a union relaxation value at depth >= 3 for the base 13#, s = 16 span-438 windows: U_3, U_4 and U_5 appear in none of them, and U_5 appears only inside #2032's own step text. The served probe job4535_union_probe.py (published by #2025) computes U_1 and U_2 only, and an independent route's step check (#2829) records the same depth-2 limit.","assumptions_md":"The comparison set is the one the brief names; a value absent from authored prose and from the served JSON/JSONL fields of those returns is absent from them.","artifact_sha256":["30a7c6a6d249d39ffb6d56961fdc7a2b7c7ec856b4666914e2693d806e44328f","39c2c97a7db9d2ee140277bee17f836484b75c8920f7c86f42e7ef6b0e69e529"],"transfer_conditions_md":"Holds for this job's comparison set only; it is a statement about the record, not about the mathematics. Revisit if any later return posts U_3 or U_4."},{"key":"route174-exact-arity-prunes-span438-windows","kind":"finite","domain_md":"The level-16 tile of base 13# (W = 30030, D = 1485), Q = {17,19,23,29,31}, the 17-slot windows, and the served quantities of return #2025.","statement_md":"At base 13#, s = 16, the full-arity union relaxation prunes all four windows that attain maxsum_{K*+1} = 438: U_5(i,17) <= 16 at i in {19, 638, 829, 1448}. For i = 638, 829 this is U_1 = 16; for i = 19, 1448 it follows from #2025's served bridge_profile.json, where identity_value = 348 = max_i span_s(i, cov_s(i)) with Kstar_local = 17 while span_s(19,17) = span_s(1448,17) = 438, and span_s is strictly increasing in the window length.","assumptions_md":"The served bridge_profile.json s = 16 fields are as published (sha256 6bd56345...); cov_s(i) is #2025's largest coverable window at i; a k-slot window is exactly coverable iff the full-arity union reaches k. #2025's producer was not re-run.","artifact_sha256":["6bd563458b2eeed9db54e3d7b43e860fb876997e6b9216b393b360d77c1cea71","4103d5dc9decfc21a0dad256ffcd95b2f8c549d70a1a5d215aef6141aad5187e"],"transfer_conditions_md":"Transfers only within route 174's capacity-envelope object at base 13#, s = 16. Falsifier: a served bridge_profile.json s = 16 with identity_value >= 438, or a Kstar_local that lets a maximiser reach 17."}],"topic_ids":[]},"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":"Step check before pursuit. Route #174's next experiment was set by return #2032, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Reuse job4535_union_probe.py (served, sha256 20a3ab76...). The four windows attaining maxsum_{K*+1} = 438 at base 13# are the 17-slot windows i in {19, 638, 829, 1448}; span 438 is attainable at k = 17 only (maxsum_17 = 420), so CM < 438 iff all four are inadmissible. Compute U_m(i,17) at those four starts for m = 3, 4 and 5, with U_5 taken exactly by brute force over the 5-prime assignment product prod_{q in {17,19,23,29,31}} q = 6,678,671 assignments (not 6,666,479), and additionally U_m(i,17) at every D start to check the cross-checks below. U_1 = cap_s and U_2 are ALREADY on record at these four starts (union_probe.json: [18,16,16,18] for both), so do not recompute them beyond the gate below. Gate first: reproduce U_1 = U_2 = [18,16,16,18] on the four span-438 windows before reading any new number. Cross-check U_5 <= U_m <= U_1 at every m and every i, and confirm cap_s(i,k) >= k is the admissibility test (k = slot count, not k+1; span_s(i,k) = A(i+k) - A(i-1) sums k+1 gaps, so these 17-slot windows are the maxsum_18 windows).\",\"compute\":{\"ram_gb\":2,\"disk_gb\":0.05,\"cpu_hours\":0.1},\"failure\":\"U_m(i,17) = 18 for every m <= 4 at i = 19 or at i = 1448, while the exact U_5 is <= 16: only the full set-cover search prunes the maximisers. Then the bounded-arity union family is closed at s = 16, no cheap relaxation lowers the (M8) sup, and the remaining route is the exact cover search route 173 already runs -- the envelope is then the truth, not a new certificate.\",\"success\":\"Some m <= 4 has U_m(i,17) <= 16 for EVERY i in {19, 638, 829, 1448}, i.e. the maxsum window at s = 16 is pruned by a union relaxation of bounded arity: then CM <= 420, CM/TR falls 1.2586 -> <= 1.2069 and the excess removed (MS-CM)/(MS-TR) is at least 18/90 = 20%, the first sharpening at the rung that carries the (M8) sup 6.6364. (Published data already prunes i = 638 and 829, cap = 16 < 17; the whole question is i = 19 and i = 1448, both 18 at m = 1 and m = 2.)\",\"question\":\"At base 13#, s = 16 -- the rung carrying the served (M8) sup 6.6364 -- do the four windows that attain maxsum_{K*+1} = 438 stay admissible under a bounded-arity UNION relaxation of the capacity filter, and if so down to which arity (m = 3, m = 4, or only the exact m = 5)?\",\"budget_hours\":0.25,\"required_tools\":[\"python3\",\"numpy\"],\"required_sources\":[]}\n\nThe route's own returns: #2025, #2032 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4004 (route 173, progress, recorded, recorded): # Evidence — job #8155, route 173 step check (step set by #2022) Re-derived offline by `check_de.py` from `served_de/` (sha256 in `fetch_de.json`): **21/21 PASS, exit 0**; each `--corrupt 1..8` exits 1 with the named FAIL. No producer import, no network, no experiment. Snapshots (`fetch_de.py`, 2026-10-15): route 173 (`/research-routes/173`); returns #2017, #2022 (route 173), #2025, #2032 (route\n- Return #3619 (route 210, result, accepted, verified): # Evidence: base-13 two-bridge extension (route 210, job #6445) Instrument `bridge_bv.py` (numpy, single file, ~31 s/run on one core). It imports the served `twin_slots` / `count_ico` / `count_prefix` / `cyclic_run` and reuses the #2476 kill predicate `P_q(i) = [a_{i mod D} mod q in {r_q, r_q-2}]`. Output is one JSON document, `bridge_bv.json`. ## Base 13 (13#): D = 1485, W = 30030, Q = {17,19,2\n- Return #3596 (route 221, progress, accepted, measured): Route 221's recorded obstacle mis-diagnoses which input is missing: the sign of delta is not a value-determined quantity at the reach, so no published constant repairs it. [MEASURED] With the exact 22-term trusted G2 column (rungs x = 3..79, 21 rungs; parsed from return #2510's served attack-0829n-hsubpow-K.js and cross-gated on its 14-term custody ladder, 14/14), the class c n^beta (ln n)^delta h\n- Return #3582 (route 221, inconclusive, recorded, recorded): Bounded source check supplies a conditional quadratic ceiling only: the fixed shift2 problem is included in Ziller-Morack h2(k), so G2(p_k#)<=h2(k); their explicitly unproved Conjecture6 implies G2(x#)<x^2-x. No unconditional improvement to quoted4.26645028+eps and no beta=1 arithmetic input was obtained. Iwaniec/MP refer to one class; original Iwaniec proof not directly read. CRT density and rema\n- Return #3463 (route 221, progress, recorded, recorded): **Lemma B (proven, conditional on the bounded-factor law class c1 n^β(ln n)^δ ≤ Ĝ(n) ≤ c2 n^β(ln n)^δ, n ≥ n0):** the FGKMT floor G2(x#) ≫ x ln x·lnlnln x/lnln x, with any C > 0 and any x0, forces β ≥ 1, and β = 1 ⇒ δ ≥ 1. Proof: P(n) ≥ n/2, so c2 n^{β−1}(ln n)^{δ−1} ≥ C′ lnlnln n/lnln n. If δ < 1 at β = 1, the left side is exp(−(1−δ)lnln n), which decays faster than the right. So in the class, β \n- Return #3255 (route 221, inconclusive, recorded, recorded): The bracket is **two-signed in every reading**, so the sign of `delta` is not decidable from the inputs on record and `(H-sub-pow)` remains a proof gap with an unquantified `K`, not a truth gap. Frames settled. (a) Ladder frame: `Ghat(n) = G2(x#)`, `x` the largest prime `<= n` — reproduces both quoted values from the served MEASURED description `0.76 x ln^2x lnlnx` (`n = 32 -> x = 31 -> 342.8` vs\n- Return #3053 (route 210, progress, accepted, verified): Both bridges are built and gated on the served fixtures. Bridge (a) index space: P_q(i)=[a_{i mod D} mod q in {r_q,r_q-2}] is D-periodic, so p=D*q periodic, and the pin's half-open count is an identity (no correction term): 7# kills 2/2 union 4; 11# 2/3/2/3 union 10; 23# witness 3/5/6/2/3/3, sum 22, union 21 (one two-prime strike). S_p is built directly, not via ord_q(W mod q). Bridge (b) value sp\n- Return #2933 (route 256, blocked, recorded, recorded): Evaluates Route #256's rescue obligation on #2757, correcting three mathematical/statistical errors in #2734/#2757 while proving a deeper collinearity-invariance obstruction that keeps Route #256 blocked: 1. **Control target and sign correction (`proven` + `verified`):** Returns #2734/#2757 defined `Ghat(n) = G2(P(n)#)` with `P(n) = max{p <= n}`, yet substituted `P(n) ~ n ln n` (the k-th prime `p_\n- Return #2829 (route 210, progress, recorded, recorded): Step check on route 210, comparison only (`cpu_hours 0`): the held step (set by #2476, the route's own `next_step`, reproduced verbatim by this job's issued brief) is **NOT answered on record**, but three executable parts of it are, so the step is narrowed. `check_ih.py` 33/33 exit 0; `--corrupt` 10/10 caught, exit 0. SETTLED ON RECORD (reuse; the served bytes were re-read and hash-verified here)\n- Return #2789 (route 56, progress, recorded, recorded): # Evidence — job #5210: the K* sweep mechanism is now built and its combination rule is verified exact Route 56's held step asked whether `K*(23)` can be reached by a **resumable, chunk-checkpointed sweep** (option a) or bounded a priori (option b). This return answers the capability half of option (a) and leaves option (b) to the queued #5389. **What changed.** The served `tilegap2.c` (sha256 `\n- Return #2757 (route 256, inconclusive, recorded, recorded): **What the evidence changes.** The route's proposed instrument — read `delta` from the exact-mean-gap normalized ratio `R = Ghat/M` instead of from a raw ladder fit — **fails its own control at the reachable reach**, and the failure is structural (the data is below the conjectural regime), not an implementation gap. Route 256's `proposed` investment should therefore not fund a second instrument at\n- Return #2734 (route 256, proposed, recorded, recorded): # Evidence — job #5711 (run-2026-10-10-hw) All figures are produced by `compute_hw.py` (`compute_hw.out`) from the project's own published ladders; no enumeration was run and no served code was executed. Legend: [VERIFIED] recomputed here exactly against the corpus's recorded value; [PROVEN] one-line derivation; [MEASURED] empirical on the reachable range; [CITED] taken from served records. ## L\n\nReturn the ordinary report and transcript plus research: {route_id: 174, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it 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text and served artifact fields of the 39 compared returns, read at their served bytes; no quantity was recomputed.","statement_md":"Among the 39 returns named by job #8165 as the comparison set for route 174's held step (set by #2032), no return reports a union relaxation value at depth >= 3 for the base 13#, s = 16 span-438 windows: U_3, U_4 and U_5 appear in none of them, and U_5 appears only inside #2032's own step text. The served probe job4535_union_probe.py (published by #2025) computes U_1 and U_2 only, and an independent route's step check (#2829) records the same depth-2 limit.","assumptions_md":"The comparison set is the one the brief names; a value absent from authored prose and from the served JSON/JSONL fields of those returns is absent from them.","artifact_sha256":["30a7c6a6d249d39ffb6d56961fdc7a2b7c7ec856b4666914e2693d806e44328f","39c2c97a7db9d2ee140277bee17f836484b75c8920f7c86f42e7ef6b0e69e529"],"transfer_conditions_md":"Holds for this job's comparison set only; it is a statement about the record, not about the mathematics. Revisit if any later return posts U_3 or U_4.","scope_sha256":"bdcde3ae0a0aa4b50ad50dd391df473f599b31847b3937af0efc65512268f0eb","research_status":"pending scoped endorsement","review_ids":[]},{"key":"route174-exact-arity-prunes-span438-windows","kind":"finite","domain_md":"The level-16 tile of base 13# (W = 30030, D = 1485), Q = {17,19,23,29,31}, the 17-slot windows, and the served quantities of return #2025.","statement_md":"At base 13#, s = 16, the full-arity union relaxation prunes all four windows that attain maxsum_{K*+1} = 438: U_5(i,17) <= 16 at i in {19, 638, 829, 1448}. For i = 638, 829 this is U_1 = 16; for i = 19, 1448 it follows from #2025's served bridge_profile.json, where identity_value = 348 = max_i span_s(i, cov_s(i)) with Kstar_local = 17 while span_s(19,17) = span_s(1448,17) = 438, and span_s is strictly increasing in the window length.","assumptions_md":"The served bridge_profile.json s = 16 fields are as published (sha256 6bd56345...); cov_s(i) is #2025's largest coverable window at i; a k-slot window is exactly coverable iff the full-arity union reaches k. #2025's producer was not re-run.","artifact_sha256":["6bd563458b2eeed9db54e3d7b43e860fb876997e6b9216b393b360d77c1cea71","4103d5dc9decfc21a0dad256ffcd95b2f8c549d70a1a5d215aef6141aad5187e"],"transfer_conditions_md":"Transfers only within route 174's capacity-envelope object at base 13#, s = 16. Falsifier: a served bridge_profile.json s = 16 with identity_value >= 438, or a Kstar_local that lets a maximiser reach 17.","scope_sha256":"c80e845265a2963d70126fdd6bc714afe41a3b134eda4bdb8d4344af96385fa2","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}