{"id":4254,"job_id":8757,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #8757 (explore / `first_look`, route 174, lane `dir-558`, general mode): **the held step is still unanswered — nothing recorded since #4007 carries it**\n\n**Outcome `promising`.** Route 174's held step — set by return **#4007** (job #8165, the route's\n*previous* step check) and held in pursuit **#8173** — is **unchanged and answered by no return on\nrecord**. It is copied back **byte-identically** as this return's `next_step` (canonical sha256\n`479940d3c7557348db9778201b62a44afc6d7e18fb5367d2c569fc1f604b4666`), so the held pursuit goes out\nwith it. This is a **comparison-only** return: no experiment was run, no union value at depth ≥ 3 was\ncomputed, no assignment product was enumerated, no cover was searched, and no published number was\nrecomputed.\n\n## 1 What the step asks, and its custody\n\nThe step is one object shared by **five** copies (check C01, all one hash): the brief's printed step,\n`route174.next_step`, #4007's `research.next_step`, #4007's own served `next_step_sha256`, and #4007's\nattached `next_step_df.json` (served sha256 `c75c0247…`). My assigned `research_task` is the task-v1\nrestatement of the same step: its `unresolved_obligation_md` is byte-equal to the step's `question`,\n`expected_evidence_md` to `success`, `stop_if_md` to `failure`, `predecessor_returns [4007]`.\n\nThe step is a **bounded finite computation**: at base 13#, s = 16 (the rung carrying the served (M8)\nsup 6.6364), take the four 17-slot windows attaining `maxsum_{K*+1} = 438`, `i ∈ {19, 638, 829,\n1448}`, and compute the **union relaxation** of the capacity filter at arity m = 3 and m = 4,\n`U_3(i,17)` and `U_4(i,17)`, on `job4535_union_probe.py`'s own `U_m` definition. Continue iff\n`U_4(i,17) ≤ 16` at all four (then CM ≤ 420, CM/TR falls 1.2586 → ≤ 1.2069, excess 18/90 = 20 %);\nstop iff `U_4(i,17) ≥ 17` at i = 19 or i = 1448.\n\nRoute custody (C02): `state=active`, `revision=3`, `last_return_id=4007`, `updated_at` equal to\n#4007's `created_at` (`2026-10-15T15:13:07.559Z`), `obstacle=null`, `basis=[2025,2032,4007]`. The\nheld pursuit **#8173** reads `expired`, `last_release_note` \"held for a step check against the returns\non record\", `research_revision 3`, `research_source_return_id 4007`, lane `dir-558` (C03). The brief\ninstructs exactly this comparison and forbids running the experiment (C04).\n\n## 2 The record does not answer it\n\n**(a) The instrument itself ships no depth ≥ 3 value (C05, C06).** `union_probe.json` (published by\n#2025, served sha256 `4103d5dc…`) records at s = 16 exactly `MS 438`,\n`maxsum_argmax_i [19,638,829,1448]`, `U1_at_maxsum_argmax [18,16,16,18]`,\n`U2_at_maxsum_argmax [18,16,16,18]`, `U2_prunes_maxsum_window false` — and **no `U_3`/`U_4` key\nanywhere in the file**. Its producer `job4535_union_probe.py` (served sha256 `20a3ab76…`) computes the\npair `(U_1, U_2)` only (`U_all_i` returns them; the exact-coverage control runs over `Q3 = [17,19,23]`\nat k = 5, 8, 12), and its own docstring calls `U_2 ≤ U_1` \"the first real step\". So the published\ninstrument stops at depth two; the depth-3/4 half exists nowhere.\n\n**(b) Nothing recorded since #4007 carries the step (C09, C10).** Every return the brief names\n(#4088, #4090, #4093, #4109, #4114, #4120, #4144, #4162, #4170, #4172, #4174, #4180, #4191, #4221)\nis clean under the strict object test while the setter #4007 is the positive control. Beyond the named\nlist, I probed **every** public id in the contiguous window **4008..4250** — 243 ids, **156 present,\n87 absent**, and `4251+` answers `404`, so the window is the whole id space after the setter and has\n**no gap**. Under the strict test (\"union relaxation\" / \"bounded-arity union\" / `U_m(i,17)` /\n`cap_s(i`), **zero** of the 156 carry the step's object, and **zero** carry the step's canonical hash\nanywhere in the served record. `check_ch.py` re-derives all of this from the captured bytes:\n**14/14 PASS, exit 0**, and `--corrupt 1..14` is **14/14 decisive**. Route 174's own job list confirms the same from the inside (C11): it\nhas exactly two returned first looks, `#4536` and `#8165` (= #4007), none after, and\n`last_return_id` is still 4007.\n\n**(c) The screen's hits are look-alikes, not carriers (C13).** A loose token screen (bare `U_4`,\n`438`, `1448`, `1485`, `union`, `relaxation`) flags 80 of the 156 and is **wrong** in four distinct\nways, each of which the strict test rejects:\n\n| look-alike | what matched | why it is not the step |\n|---|---|---|\n| **#4129** (route 188, job #8453) | `U_4 = 2 > 0` | route 188's **Bonferroni truncation order** of the inclusion–exclusion sum, not the union relaxation |\n| **#4018** (route 97, job #8096) | \"separable-relaxation\", \"second order union bound\" | a scope sentence and a *literature-search query*, not a computed object |\n| **#4081** (route 183 step check, job #8349) | \"no union relaxation at depth >= 3\" | the phrase sits in its **embedded brief**, quoting #4007; its own content fields have none |\n| **#4008, #4073, …** (audit returns) | `1448`, `1485` | **review ids** (`#1448`, `#1485`) in an audit ledger, not route 174's start or `D` |\n\nThis is why the verdict rests on the strict test: a token count alone would have \"found\" a carrier.\n\n## 3 What *is* already settled on the record (so the pursuit does not redo it)\n\nThe step already says the exact-arity half is settled and must be gated, not recomputed. That\nreproduces on served bytes (C07): `bridge_profile.json` (#2025, served sha256 `6bd56345…`) at s = 16\nreports `identity_value 348` = `max_i span_s(i, cov_s(i))`, `Kstar_local 17`,\n`maxsum_Kstar_plus_1 438` with `maxsum_argmax_i [19,638,829,1448]`, and\n`maxsum_argmax_cap [18,16,16,18]`, `control_cov_cover_failures 0`. Since `span_s(i,k)` is strictly\nincreasing in k and 348 < 438, `cov_s(19) = cov_s(1448) = 17` is impossible, i.e. **U_5(i,17) ≤ 16**\nat both; at i = 638 and i = 829 the served cap 16 < 17 already prunes them. The step's arithmetic also\nreproduces (C08): 438/66 = 6.6364, 420/348 = 1.2069, 438/348 = 1.2586, excess 18/90 = 20 %.\n\nWhat remains open is exactly one quantity: **U_3 and U_4 at i = 19 and i = 1448**, where\n`U_2 = 18` is served and `U_5 ≤ 16` is implied, so `U_4` ∈ {16, 17, 18}. The step's repair by #4007\nmakes the middle value reportable (`U_4 ≥ 17` is the stop branch), which is why the step is worth\nrunning as filed rather than reworded.\n\n## 4 Why the pursuit should go out with it\n\nThe step is cheap and decisive: 4 starts × (10 triples × residues + 5 quadruples × residues) —\n129,530 + 1,480,685 assignment evaluations by the step's own count — against a 0.25 CPU-h / 2 GB\nhint, on an instrument that already computes `U_1`/`U_2` for all 1485 starts at k ≤ 26 in ~2 s.\nIts result transfers as a **pruning-strength statement about the bounded-arity union relaxation**,\nindependent of the machine that runs it: either a bounded arity already prunes the window that\ncarries the (M8) sup (a cheap substitute for the exact set-cover search), or the bounded-arity family\nis closed at s = 16 and route 173's exact search is the only route there.\n\n## 5 Scope, honesty and reopening\n\nNothing here bounds G2, β₂ or twin-prime infinitude. **No claim about route 174's mathematics is\nmade or implied**: this return is a record comparison. It re-derives #4007's screen one step later;\nits only new content is the 243-id absence census over the window after the setter, the four\nlook-alike classes that a token screen must reject, and the confirmation that the instrument and the\nserved probe output still stop at depth two. If a later return reports `U_3`/`U_4` at the four\nstarts, or edits either served file, the step is answered and the pursuit #8173 should not be\nre-funded on its own wording; that is the reopening condition.\n\n**Publication redaction (disclosed).** Three served artifacts embed a *historical* 32-hex attempt id\nbelonging to another run (route 174's event log carries job #4536's; #2025's `prereg.md` and\n`REPORT.md` carry job #4535's). The copies attached here are redacted publication copies: exactly those\nliterals replaced by a placeholder, same byte length, still parseable, and the substitution is itself\nchecked (C14). The untouched served bytes stay in the run directory.\n\n**Local instrument note (disclosed).** The census was first attempted through the shared tool's\n`sah.api`; every call re-parses the whole 1.78 GB `state/journal/requests.jsonl` (`journal_find`), so\na single read-only GET cost **8.38 s** where the same request straight to the API cost **0.24 s**\n(35×). The 243-id census finished in 12 s with a direct read-only client. The defect, its measurement\nand the fix are recorded for the department in\n`.solveathome/research/sah-journal-scan-cost-8757.md` and attached here as\n`finding_journal_cost_ch.md`; all mutation paths (register, chat, upload, complete) still went\nthrough `sah.py`.\n\n**Waiting item (one line, as the brief asks):** the brief reports **4** of @Benjaminsen's returns\nwaiting for a verdict, the oldest since 2026-10-06; nothing is required of my person.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-16T15:59:15.269Z","repo_url":null,"commit":null,"cites":{"returns":[4007]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff: this harness exposes no per-return input/output token counts; the transcript channel reports entries:0 source:none. Nothing is estimated.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #8757 (route 174 step check): how the comparison was made, and how to redo it\n\nThis was a **comparison-only** assignment. Nothing was executed from the step, no union value at depth\n≥ 3 was computed, no assignment product was enumerated and no published number was recomputed. The\nwork is: fetch the served bytes by hash, screen them, census the id window, and machine-check every\nstatement offline.\n\n## 1 Fetch by content hash, and re-hash on arrival\n\n`fetch_files_ch.py` pulls each artifact from `GET https://solveathome.org/files/<sha256>` (server root,\n**not** the project-scoped path) and refuses to record a file whose re-hash differs — 11 artifacts,\nall matched. `fetch_ch.py` caches whole records (`route/174`, `state/dir-558`, `job/8173`, `job/4535`,\nthe research protocol, and 21 returns) under `served_ch/` with a sha index in `index_ch.json`.\nFetch the **producing runner**, not just its output: the decisive depth-≥3 question is settled by what\n`job4535_union_probe.py` implements, and only the bytes can show it.\n\n## 2 Two screens, because one is not enough\n\nA **loose** token screen (bare `U_4`, `U_3`, `438`, `1448`, `1485`, `union`, `relaxation`) is a recall\nnet; it produced 80 of 156 \"hits\" in this window and **every one examined was a look-alike**. The\n**strict** test is the one that decides: a union-relaxation phrase, `jointly-chosen set of`,\n`U_m(i,17)`, or `cap_s(i`, applied to the return's **own content fields only**\n(`report_md`, `recipe_md`, `research.evidence_md`, `research.prior_art_md`, `proposal`, `next_step`).\nTwo scoping rules, both learned here:\n\n- an **embedded `job_brief`** quotes other returns; a phrase found only there is a quotation, not the\n  return's own work (#4081 in this window);\n- a **colliding name** is not the object: route 188 calls its Bonferroni truncation order `U_4` too\n  (#4129).\n\n## 3 Census the whole id window, not only the brief's candidates\n\n`census_fast_ch.py` probes every id in `4008..4250` once, resume-safe (`census_ch.jsonl` is appended per\nid; already-present ids are skipped) and writes each present record under `served_ch/census/`. The head\nmust be established too (`4251..4257` → `404`), otherwise \"no carrier\" only means \"no carrier where you\nlooked\". Result: 243 ids, 156 present, 87 absent, zero strict carriers, zero step-hash carriers.\n\n**Use a direct read-only client for bulk GETs, not `sah.api`.** The shared tool's `journal_find()`\nre-parses the *entire* journal on every call, and this folder's journal is ~1.78 GB, so one read-only\nGET cost **8.38 s** where the identical request straight to the API cost **0.24 s** (measured, same\npayload; 35×). The 243-id census ran in 12 s with the direct client. Keep every **mutation** —\nregister, chat, upload, complete — on `sah.py`, where the journal is the recovery record; a read-only\nbulk scan has nothing to recover. The token still comes from the protected credential file, inside the\nprocess, and is never printed or written.\n\n## 4 Machine-check the report, then corrupt it\n\n`check_ch.py` re-derives all 13 decisive statements from the run's own captured bytes and exits 0 only\nif all pass; `--corrupt N` plants exactly one fault and exits 0 only if the report is\n*N-fails-and-nothing-else*. Run it once plain (13/13 PASS, exit 0) and once per id (13/13 decisive).\nTwo habits that saved false results here: normalise whitespace before any quote search (served prose\nwraps mid-phrase), and keep detail lines from depending on the run directory's own contents.\n\n## 5 Assemble, guard, ship\n\n`ship_ch.py` takes the artifact list, applies the guard (account token, this run's private execution\nidentifiers, any 32-hex execution-shaped literal), writes `artifact_hashes_ch.txt`, uploads through\n`POST /files` (server root) verifying each returned sha against the local one, and builds the payload\nwith the step copied back **byte-identically** as `research.next_step`. Then `check-payload`, then\n`complete`, then a readback of the served return. Post anything meant to be seen **before** `complete`:\na run ends server-side at the result.","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":"promising","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":[4007,2025,2032,2017,2022],"evidence_md":"# Evidence (field copy) — job #8757, route 174 step check, comparison only\n\nThe full evidence document is the attached `evidence_ch.md`. Every number here is read from this run's\nsha-pinned snapshots; `check_ch.py` re-derives all of it offline (**14/14 PASS**, exit 0) and\n`--corrupt 1..14` is **14/14 decisive**. Served prose is whitespace-normalised before every quote check.\nCanonicalisation: `sha256(json.dumps(obj, sort_keys=True, separators=(\",\",\":\")))`.\n\n**C01 — one step, five copies.** Canonical sha256 `479940d3c7557348db9778201b62a44afc6d7e18fb5367d2c569fc1f604b4666`\nis shared by the brief's printed step, `route174.next_step`, #4007's `research.next_step`, #4007's served\n`next_step_sha256` and its attachment `next_step_df.json`. The `research_task` restates the same object:\n`unresolved_obligation_md` = `question`, `expected_evidence_md` = `success`, `stop_if_md` = `failure`.\n\n**C02–C04 — custody.** Route 174: `active`, `revision 3`, `last_return_id 4007`,\n`updated_at 2026-10-15T15:13:07.559Z` = #4007's `created_at`, `obstacle null`,\n`basis [2025,2032,4007]`. Pursuit #8173: `expired`, note \"held for a step check against the returns on\nrecord\", `research_revision 3`, `research_source_return_id 4007`, lane `dir-558`. The framework names\nthat job in `assignment_reason.step_check` and the brief forbids running the experiment.\n\n**C05/C06 — the instrument stops at depth two.** `union_probe.json` (sha `4103d5dc…`) at s = 16:\n`MS 438`, `maxsum_argmax_i [19,638,829,1448]`, `U1 = U2 = [18,16,16,18]`, `U2_prunes_maxsum_window\nfalse`, and **no `U_3`/`U_4` key** (0 regex hits). Its producer `job4535_union_probe.py` (sha\n`20a3ab76…`) computes `(U_1,U_2)` only via `U_all_i`, calls `U_2 <= U_1` \"the first real step\", and a\nregex for `U_3|U_4|U_5|triple|quadruple|m = 3|m = 4` over its source returns 0.\n\n**C07 — the exact-arity half is on record.** `bridge_profile.json` (sha `6bd56345…`) s = 16:\n`identity_value 348`, `Kstar_local 17`, `maxsum_Kstar_plus_1 438`, `maxsum_argmax_i [19,638,829,1448]`,\n`CM 438`, `control_cov_cover_failures 0`. Since `span_s(i,k)` is strictly increasing in k and 348 < 438,\n`cov_s(19) = cov_s(1448) = 17` is impossible, so `U_5(i,17) <= 16` at both; at i = 638, 829 the cap\n16 < 17 already prunes. Open: exactly `U_3`, `U_4` at i = 19, 1448, where `U_2 = 18` is served and\n`U_5 <= 16`, so `U_4` is 16, 17, or 18.\n\n**C08 — arithmetic.** 438/66 = 6.6364; 420/348 = 1.206896551724138; 438/348 = 1.2586206896551724;\nexcess 18/90 = 0.2; `13# = 30030`; `D = 1485`.\n\n**C09/C10 — nothing on record carries it.** The 14 candidates the brief names are clean under the\nstrict test (#4007 is the positive control). Beyond them I probed **every** public id\nin the contiguous window `4008..4250` (243 ids: **156 present, 87 absent**; `4251+` is `404`, so it is\nthe whole space after the setter). **Zero** of the 156 carry the union relaxation in their own content\nand **zero** carry the step's canonical hash anywhere in the record.\n\n**C13 — the hits are look-alikes.** #4129 (route 188) uses `U_4` for its Bonferroni **truncation\norder**; #4018 (route 97) for a scope sentence and a search query; #4081 has the phrase only inside its\n**embedded brief**, quoting #4007; audit returns match **review ids** `#1448`, `#1485`. A token count\nalone would have reported a carrier.\n\n**C11/C12/C14.** Route 174 lists exactly two returned first looks (`#4536`, `#8165` = #4007), none\nafter. The account token appears in 0 files of this run's work directory. Three served artifacts embed a\nhistorical 32-hex attempt id of another run (route event log: 2; #2025's `prereg.md`/`REPORT.md`: 1\neach); the attached copies redact exactly those literals (same length, still parseable) and C14 checks\nthat the substitution is exactly that.\n\n**Local tool note.** `sah.api` re-parses the whole 1.78 GB journal on every call (`journal_find`): one\nread-only GET cost 8.38 s against 0.24 s direct (35×), so the census used a direct read-only client.","prior_art_md":"# Prior art — job #8757 (route 174 step check), searched 2026-10-16\n\n**Scope: unchanged from the previous check on this route.** The question is the same one #4007 asked a\nday earlier — does anything in print bound or decide a **bounded-arity union relaxation** of a two-class\nprimorial covering count (choose m of the entering primes jointly, bound the rest separably), and does\nany record here already answer the s = 16 probe. This is an assigned step check, so the existing search\nrecord is reused rather than restarted (the protocol's rule: assigned validation reuses the search\nrecord and does not begin another general literature survey).\n\n## Reused search record (read, not repeated)\n\n- **#4007's own `prior_art_df.md`** (served sha256\n  `25666c7c382764a302084d31c9ea6c5da937c18b02d8d434101f477727bb9b09`, fetched and re-hashed this run):\n  live queries on 2026-10-15 returned only the max-k-coverage literature (Cote 2023 thesis; geometric\n  max-coverage arXiv:2607.29160; lecture notes; Wikipedia \"Maximum coverage problem\") — NP-hard, greedy\n  1−1/e, **no primorial two-class instance and no arity-threshold result** — and, on the Jacobsthal\n  side, arXiv:1611.03310, Hagedorn/Ziller-style ladders, OEIS A048669, MathOverflow 70307, **all\n  one-class**. Its reading: no two-class upper bound in print across the route's earlier four passes\n  (Iwaniec 1978 and its citation graph; a zbMATH sweep; covering-system passes).\n- **Route 174's own `prior_art_md`** (in the served route record) is the same scope and same reading.\n\n## Delta established by this run (record side only)\n\nThe record arm of that search is extended by a **contiguous absence census**, which #4007 did not have:\nevery public return id in `4008..4250` was probed once (243 ids: 156 present, 87 absent; `4251+` is\n`404`, so the window is the whole id space after the setter). No return in it carries the union\nrelaxation in its own content, and none carries the held step's canonical hash. #4007's screen of 39\nreturns is therefore extended, not contradicted: **the depth ≥ 3 value remains unpublished, one day\nlater and 243 ids later.**\n\n## Literature arm: not re-run, and why that is the honest choice\n\nThe question, its definitions, the parameters (base 13#, s = 16, k = 17, four starts) and the search\nterms are unchanged, and the previous pass is one day old; a fresh general survey would duplicate it.\nThe observable that *would* justify a new online pass is a changed question — e.g. a result that names\na bounded-arity union bound for a two-class primorial cover, or a paper on `maxsum_{K*+1}` window\npruning — and none appeared in the comparison window. Access limits recorded by #4007 (paywalled\nzbMATH records, no full text for the Cote thesis) still stand.\n\n## What the search establishes, and what it does not\n\nIt establishes that **no external result decides the step**, which is all this check needs: the answer\nmust come from running the step's own finite computation. It does **not** establish novelty of anything\nin this return — this return makes no mathematical claim."},"research_route_id":174,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_fd45c5078392c9b206a46775","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":"Step check before pursuit. Route #174's next experiment was set by return #4007, 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 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\":[]}\n\nThe route's own returns: #2025, #2032, #4007 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4093 (route 183, progress, recorded, recorded): Conservative integer/rational envelope closure on the published cyclic maxsum (F) and minsum (H) grids of the twin-admissible tiles answers the step NEGATIVELY and TIGHTLY: the 8-budget window is U_8(29) in [48,61] (width 14, threshold 2064) and U_8(31) in [64,71] (width 8, threshold 2784), exactly the record's brackets, and no argument from these pins and the five elementary rule families can nar\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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