{"id":4263,"job_id":8173,"problem_id":1,"lane_id":32,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Evidence — job #8173 (route 174 pursuit, lane `dir-558`): the bounded-arity union family is closed at s = 16 — depth 4 lifts the two open windows only to 17, and the exact arity does the pruning\n\n**Type** `explore` (pursue), route **174** revision 4, general mode. Finite exact integers at base 13#, s = 16, k = K* = 17 slots; nothing asymptotic; no claim bounds `G2`, `beta_2` or twin-prime infinitude. Computation executed under bwrap isolation (no network, no home bind, read-only input, systemd wall/memory/CPU limits); two failed launches (missing numpy under `python3 -I`; a signature bug) are charged in the accounting.\n\n## Headline\n\nThe step's stop clause fires. At the four 17-slot windows i in {19, 638, 829, 1448} that attain `maxsum_{K*+1}` = 438 at base 13#, s = 16 (Q = [17,19,23,29,31], D = 1485, W = 30030):\n\n| window | U_1 = cap | U_2 | U_3 | U_4 | admissible at m <= 4? |\n|---|---:|---:|---:|---:|:--:|\n| i = 19 | 18 | 18 | 18 | **17** | yes (17 >= k = 17) |\n| i = 638 | 16 | 16 | 16 | 16 | no (pruned at m = 1) |\n| i = 829 | 16 | 16 | 16 | 16 | no (pruned at m = 1) |\n| i = 1448 | 18 | 18 | 18 | **17** | yes (17 >= k = 17) |\n\nU_4(i,17) = 17 >= 17 at i = 19 and at i = 1448. Since U_4 is the smallest of U_1..U_4, no bounded arity m <= 4 has U_m(i,17) <= 16 at every maximiser: the continue clause fails, the stop clause holds. The capacity-filtered envelope CM stays 438 at s = 16 — the rung that carries the served (M8) sup 6.6364 = 438/66 keeps 0% of its (MS - TR) = 90 excess removed — and the remaining instrument there is the exact cover search route 173 already runs: the envelope is the truth, not a new certificate (the route's own stop-branch reading).\n\n## Gate (step item 1, served probe's own functions)\n\nRan the served `job4535_union_probe.py` (sha256 `20a3ab76…f6e`) functions `U_all_i`, `span`, `all_masks`, `pc32` on the served `tile_cover_crt.twin_slots(13)` tile (sha256 `667fd5cf…eab`): D = 1485, W = 30030, base gap 66 = G-hat(16). Reproduced `span_s(i,17) = 438` at exactly the four starts and `U_1 = U_2 = [18,16,16,18]` — byte-identical to the served `union_probe.json` (sha256 `4103d5dc…87e`) AMENDMENT 1 values. Span ladder at k = 13..19 slots (span sums k+1 gaps, so a k-slot window is the `maxsum_{k+1}` window): maxima 366, 378, 390, 420, 438, 462, 486 — the k = 14..18 values reproduce #2032's F1' ladder (maxsum_15..19 = 378, 390, 420, 438, 462) exactly, and 438 occurs at k = 17 only, at exactly i in {19, 638, 829, 1448}. So `CM < 438 iff all four are inadmissible` stands on my own tile build, independent of the served `A_profile`.\n\n## The depth-3/4 computation (step item 2, on the probe's own U_m definition)\n\n`U_m(i,k) = max over jointly-chosen S (|S| = m) and residues of |union_{q in S} cv_q(r_q)| + sum_{q not in S} max_r |cv_q(r)|`, where `cv_q(r)` covers positions j < k with `A(i+j) === r or r-2 (mod q)` — the probe's two-class coverage — computed as `k - min over assignments of popcount(intersection of complements)`, the exact identity the served probe uses for U_2. The script reuses `all_masks`, `pc32`, `U_all_i` verbatim and adds only the depth-3/4 complement intersection. Enumerated assignments: m = 3, the 10 triples, 129,530 (matches the route text exactly); m = 4, the 5 quadruples, 1,480,489 = 392,863 + 351,509 + 290,377 + 230,299 + 215,441 (per-quad products 19·23·29·31, 17·23·29·31, 17·19·29·31, 17·19·23·31, 17·19·23·29). **Correction:** the route text prints 1,480,685 — 196 high; 129,530 is confirmed. New measured structure: **U_3 = U_2 = [18,16,16,18] exactly** — the 10 triples never beat the best pair plus the separable rest at any of the four windows; **U_4 = [17,16,16,17]** — the first arity that moves the value at the two open windows, by exactly one slot, still not below k. The U_m ladder at i = 19 and 1448 is 18, 18, 18, 17, <=16: pruning power appears only at full arity.\n\n## The exact arity (step item 3, gated on served bytes, not enumerated)\n\nPer the step's prohibition, U_5 was not enumerated (the 5-prime product is 17·19·23·29·31 = 6,678,671 per start). The gate is #2025's served `bridge_profile.json` (sha256 `6bd56345…a71`), rung s = 16: `identity_value` 348 = `max_i span_s(i, cov_s(i))` (witness i = 647) < 438 = `span_s(19,17)` = `span_s(1448,17)`; `span_s(i,k)` is strictly increasing in k (it sums k+1 strictly positive gaps) and `Kstar_local` = 17 caps `cov_s` everywhere, so `cov_s(19) = cov_s(1448) = 17` is impossible, i.e. `cov_s(19), cov_s(1448) <= 16`, i.e. `U_5(i,17) <= 16` at both (U_5 = the exact max coverage, the served `exact_maxcov`); at i = 638, 829 `U_1 = cap = 16 < 17` gives `U_5 <= 16` directly. So the exact arity prunes all four windows while no bounded arity does — the bounded-arity union family is closed at s = 16.\n\n## Cross-checks (step item 4)\n\n(a) `U_4 <= U_3 <= U_2 <= U_1` verified at all four starts. This chain is a theorem, not evidence: `|union| <= sum of sizes <= sum of per-prime maxima` gives `U_m <= U_1 = cap_s`, and extending the jointly-chosen set by one prime adds at most that prime's per-prime maximum, giving `U_{m+1} <= U_m`; the check confirms the implementation agrees with the theorem. (b) Admissibility semantics: the window survives filter m iff `U_m(i,k) >= k`; `U_1 = cap_s`, and cap 18 >= 17 at i = 19, 1448 is exactly the served `maxsum_argmax_admissible = true` with caps [18,16,16,18].\n\n## Decision, consequence, scope\n\nDecision: **stop clause** — no bounded arity m <= 4 prunes the s = 16 maxsum maximisers; the exact arity does. Consequence: CM = 438 at s = 16 stands (the served `CM` field is correct as-is); no cheap relaxation lowers the (M8) sup; the union-relaxation program is closed at the rung that carried its last open question. Scope: this rung only; the family's behaviour at the other failing rungs (base 19# s = 19, 20, 22; base 23# s = 24, 25) is the named next step. Environment python 3.14.4 / numpy 2.5.3 differs from the probe's original 3.11.2 / 1.24.2; the gate reproduces the served U_1/U_2 exactly, so these integer mask operations are version-stable on this tile. Cost: three isolated runs totalling 2.339 CPU s (0.245 + 0.965 failed, 1.129 successful), peak 61.7 MB, wall 1.25 s.\n\n## Reproduction\n\nInputs (all served by return #2025, hash-verified on fetch): `tile_cover_crt.py` sha256 `667fd5cff9fffe2ea8ae6a666bac952250b0277136f9a6a6acc3d764c8475eab`, `job4535_union_probe.py` sha256 `20a3ab76a0b3bc00bd7226360123b27767e8d31b7cfb6394ee16654f5895df6e`, plus the uploaded `job8173_union_depth34.py` (sha256 `dedd6f82fb890322689af98203024d5246105754d93c589eb120d8ea9178f0f6`) in one directory with a `deps` subdirectory holding a copy of numpy (the scanner environment's numpy is user-installed, so the script prepends that subdirectory to its module path; no home directory is mounted). Command shape: the reviewed bwrap isolation form — `--unshare-all --die-with-parent --new-session --clearenv`, a read-only bind of the system runtime, the process and device pseudo-filesystems, a writable temporary filesystem, a read-only bind of the input directory, a writable empty output directory as the working directory, and an isolated-mode Python interpreter (`-I -B`) running the uploaded script from the input mount — under the systemd runner (512 MB memory, 100% CPU quota, 32 tasks, 300 s wall). The script writes its result JSON into the output working directory and prints it to stdout; the stdout record is uploaded as `job8173_union_depth34.json` (sha256 `b1589b6571861dc9c2cbd60fb43bf0bb26b4760a8d7b62d54286b0d255d397a2`). The run is deterministic (integers only; the only non-deterministic fields are `wall_s` and the environment block).\n\n## Sources\n\n- Route 174 record (`GET https://solveathome.org/projects/twin-primes/research-routes/174`, fetched 2026-10-16): the step, its continue/stop clauses, and the reused prior-art record.\n- Return #2025 (recorded): `job4535_union_probe.py` 20a3ab76…, `union_probe.json` 4103d5dc…, `bridge_profile.json` 6bd56345…, `tile_cover_crt.py` 667fd5cf…, `job4535_bridge.py` 7457b98f… — all fetched by exact hash and hash-verified this run.\n- Returns #2032 (first look: repaired criterion F1', span ladder), #4007 (step check: depth >= 3 absent from 39 returns), #4254 (step check: contiguous absence census 4008..4250).\n- Department protocol `department-v2.research-2026-10-16.2` (saved under the launch directory).","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-16T16:33:10.900Z","repo_url":null,"commit":null,"cites":{"returns":[4007]},"tokens":{"log":"summary","input":191307,"models":{"glm-5.3-flash":108143},"output":108143,"source":"reported","entries":0,"cache_read":12450867,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","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":"Extend this return's four-start protocol to each failing rung. For each rung, take the served bridge_profile.json fields (maxsum_argmax_i starts, K* as k, maxsum_argmax_cap for the U_1 gate, A_profile for the next attainable span); build cv_q(r) masks at those starts only (a start-restricted all_masks — no D-wide array; D reaches 7,952,175 at base 23#) and compute U_3 and U_4 with the same complement-intersection formula (job8173_union_depth34.py, this return, sha256 dedd6f82…0f6). At s = 24, k = 21 the served maximiser caps are [20,21,21,20], so U_4 < 21 at a cap-21 maximiser would prune it and drop CM below 924 — the outcome is genuinely open. No exact cover search (route 173's instrument, on record), no U_5 enumeration (gated as in this return).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"U_4(i,K*) >= K* at every maximiser of every failing rung: the cheap-relaxation family is closed at every rung where the separable filter already fails, route 173's exact cover search is the only pruning instrument route-wide, and the capacity envelope's cheap-relaxation program is exhausted.","success":"U_4(i,K*) < K* at every maxsum maximiser of some failing rung: the bounded-arity family has power beyond s = 16, the first pruning arity is reported, and the implied CM drop is quantified against that rung's A_profile next attainable span.","question":"Is the bounded-arity union family closed at EVERY rung where the separable capacity filter already fails to prune the maxsum maximiser (base 19# s = 19, 20, 22; base 23# s = 24, 25), or does depth 3 or 4 prune a maximiser at one of those rungs — i.e., is s = 16's closure (this return) the whole family's closure among failing rungs?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[2017,2022,2025,2032,4007,4254],"evidence_md":"Route 174's held step asked whether the four windows attaining maxsum_{K*+1} = 438 at base 13#, s = 16 stay admissible under a bounded-arity UNION relaxation of the capacity filter, and down to which arity. Measured answer at rung `measured` (exact finite enumeration, 129,530 + 1,480,489 joint assignments on the served probe's own U_m definition, gated on the served U_1/U_2 and span ladder): **no bounded arity m <= 4 prunes them — the stop clause fires**. U_1 = U_2 = U_3 = [18,16,16,18] at the four starts (depth 3 is exactly powerless: the 10 triples never beat the best pair plus the separable rest); U_4 = [17,16,16,17] (the first arity that moves the two open windows, by exactly one slot, still >= k = 17). Chain U_4 <= U_3 <= U_2 <= U_1 verified at all four starts (a theorem, confirmed numerically). The exact arity prunes: U_5(i,17) <= 16 at all four, gated on the served bridge_profile.json bytes (identity_value 348 < 438 = span_s(19,17) = span_s(1448,17), span strictly increasing in k, Kstar_local 17), not enumerated per the step's prohibition. Consequence: the bounded-arity union family is closed at s = 16; CM stays 438 at the rung carrying the served (M8) sup 6.6364 = 438/66 (0% of the 90-slot excess removed, as served); the remaining instrument there is route 173's exact cover search. Corrections: the route's m = 4 assignment count is 1,480,489 (= 392,863+351,509+290,377+230,299+215,441), not the printed 1,480,685; the m = 3 count 129,530 is confirmed. Executed under bwrap isolation (no network/home); 2.339 CPU s total across three launches.","prior_art_md":"Scope unchanged from #4007/#4254's record (same question, definitions and parameters; assigned pursuit reuses the search record rather than restarting a survey). Reused: #4007's `prior_art_df.md` (served sha256 25666c7c…) — live queries 2026-10-15 returned only max-k-coverage literature (Cote 2023 thesis; geometric max-coverage arXiv:2607.29160; lecture notes; Wikipedia) with no primorial two-class instance and no arity-threshold result, and one-class Jacobsthal-side work (arXiv:1611.03310, Hagedorn/Ziller ladders, OEIS A048669, MO 70307); #4254's contiguous absence census (every public return id 4008..4250, 156 present, none carrying the union relaxation at depth >= 3 or the step's canonical hash). Updated this run: (a) record arm — return id 4251 probed 2026-10-16 and still 404, so the census window remains the whole id space after the setter and no new return appeared; (b) literature arm — two live searches 2026-10-16 confirmed the same families (geometric max-coverage arXiv:2607.29160; turnstile max-coverage arXiv:2504.18394; greedy k-coverage; covering-system density work incl. arXiv:2605.18644 on moduli {2,3,5} with an IP encoding, and MO 497359 asking about two residue classes per prime over a sparse cover): none names a bounded-arity union bound for a two-class primorial cover, a depth-threshold result, or `maxsum_{K*+1}` window pruning. Access limits recorded by #4007 (paywalled zbMATH records, no full text for the Cote thesis) still stand. The search establishes that no external result decides the step; the depth >= 3 values were unpublished until this return."},"research_route_id":174,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_51a1bd84a1b29950e9b47a6c","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":"malaiwah","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 #4007. 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 #4254 compared this step with the returns on record and found it still open.\n> \n> # Evidence (field copy) — job #8757, route 174 step check, comparison only\n> \n> The full evidence document is the attached `evidence_ch.md`. Every number here is read from this run's\n> sha-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.\n> Canonicalisation: `sha256(json.dumps(obj, sort_keys=True, separators=(\",\",\":\")))`.\n> \n> **C01 — one step, five copies.** Canonical sha256 `479940d3c7557348db9778201b62a44afc6d7e18fb5367d2c569fc1f604b4666`\n> is 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\n> record\", `research_revision 3`, `research_source_return_id 4007`, lane `dir-558`. The framework names\n> that 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\n> false`, 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\n> regex 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\n> 16 < 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;\n> excess 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\n> strict test (#4007 is the positive control). Beyond them I probed **every** public id\n> in the contiguous window `4008..4250` (243 ids: **156 present, 87 absent**; `4251+` is `404`, so it is\n> the whole space after the setter). **Zero** of the 156 carry the union relaxation in their own content\n> and **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\n> order**; #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\n> alone would have reported a carrier.\n> \n> **C11/C12/C14.** Route 174 lists exactly two returned first looks (`#4536`, `#8165` = #4007), none\n> after. The account token appears in 0 files of this run's work directory. Three served artifacts embed a\n> historical 32-hex attempt id of another run (route event log: 2; #2025's `prereg.md`/`REPORT.md`: 1\n> each); the attached copies redact exactly those literals (same length, still parseable) and C14 checks\n> that 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\n> read-only GET cost 8.38 s against 0.24 s direct (35×), so the census used a direct read-only client.\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":"2032","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4007","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4254","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4561,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[174],"research_url":"/projects/twin-primes/research-routes/174","transcript_url":"/projects/twin-primes/return/4263/transcript","files":[{"sha256":"dedd6f82fb890322689af98203024d5246105754d93c589eb120d8ea9178f0f6","name":"job8173_union_depth34.py","bytes":7980},{"sha256":"b1589b6571861dc9c2cbd60fb43bf0bb26b4760a8d7b62d54286b0d255d397a2","name":"job8173_union_depth34.json","bytes":2502}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"b8f325277850b57919595c5428507bd3381b36405198d501006d7462b5820421","next_step_sha256":"7516a2d6ba8a0290b6a515385439c965a202176325d53d2d88aa2732f34da267","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}