{"id":2548,"job_id":5333,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5333 (route 227, first look / step check) — the held bridge test is still open; no return after its setter touches it\n\nJob #5333, route 227, stage `first_look`, general mode. **Read-only served fetches. cpu_hours 0.**\nNo field built, no moment computed, no published number recomputed.\n\n## Summary\n\nRoute 227 is `proposed`, revision 1, `origin_return_id` **2544**, `last_return_id` **2544**, and its\nonly event is `{id 1280, return 2544, outcome proposed}`. Its served `next_step` is **canonically\nidentical** to `#2544.research.next_step` (the setter's own step; object equality of the full JSON).\nThe step — quoted verbatim as this return's `next_step` — is the **bridge test**: compute exact\nfull-period `M_2k(h)` and the empty-window count `E(h)` at `x = 11, 13, 17` (and 19 if cheap) for\n`kappa = 1` and `kappa = 2`, and test three pre-registered bridges B1/B2/B3 at the dial's window\n`h`, comparing the implied dial exponent with the counted majorant's `2.29 -> 2.94` (`#1457`).\n\nThe returns recorded **after** #2544 are **#2545** (route **226**, `known`), **#2546** (route **31**,\n`progress`) and **#2547** (new route **228**, `proposed`). None is on route 227, and none is on a\nroute the step names — **191** (`known`, last **2339**), **194** (`paused`, last **2368**), **143**\n(`active`, last **2497**). The apparent substring matches are non-substantive: the `\"227\"` in #2545\nand #2546 is the **route count** (\"the whole served register (227 routes)\") and a table cell in\n#2546; the `\"gap-law\"` in #2547 cites **#1456**'s twin-gap law, not route 227's bridge. Neither open\nquestion (`Q-var41`, `Q-hsubpow-K-0829n`) is this bridge.\n\n**The step is therefore still open: outcome `promising`**, with the served step copied exactly as\n`next_step`.\n\n## Why the step is still open (and why it is worth the pursuit)\n\nThe step's decisive content is the **unbought computation** it names: exact side-B objects\n(`M_2k(h)`, `E(h)`) at matched rungs for **both** kappa, and the three bridges evaluated against\nthem. The serving return #2544 supplies only side **A** (`gap_law_eq.py`'s exact kappa=1 census at\n`11#..23#`) and the *proposal* of the bridges. The linked returns settle their own routes without\nexecuting it:\n\n- **Route 191 (#2333, #2339).** #2333 measures the kappa=1 gap law and labels the bridge\n  **conjectural**; #2339 closes the kappa=1 *gate* via CVZ 2003 and states literally that \"Paired\n  gaps and the covering-function bridge remain unresolved\" and \"The proposed gap-to-window bridge is\n  still absent.\" Route 191 is `known`, so it cannot supply it.\n- **Route 194 (#2356, #2368).** #2356 measures side A for kappa=2 only and calls the bridge\n  **CONJECTURAL** (inherited); #2368 is `inconclusive`, and route 194 is `paused`. Side B is not\n  computed there either.\n- **Route 143 (#1457, #2497).** #1457 defines the dial `M_2k(h)` and records the **rising** counted\n  majorant exponent (kappa=2: `2.29` at `x=11` -> `2.94` at `x=19`); #2497 partitions medium\n  denominators of the certificate. Neither uses the gap law, so the A -> B map is still undefined\n  and untested. Route 143 is `active` but its step is its own.\n\nThe proposal's own evidence makes the check non-trivial rather than bookkeeping: #2333 records that\nthe largest gap carries only `6.3e-3` of `M_12` at `23#`, so the gap-law moments are **bulk**-carried\nwhile the dial needs a **tail** statement (empty windows). That is exactly the kind of premise a\ncheap finite test should resolve, and it is still unresolved.\n\n## Scope and disclosures\n\nRecord comparison only; no experiment, no sieve, no gap census, no moment computed; `cpu_hours 0`;\nno asymptotic, exponent or `E_>(x)` claim; nothing about `G2`, `beta_2` or twin-prime infinitude.\nThe step is copied byte-for-byte from the served route record\n(`route_227.json.next_step`). 48 of @Benjaminsen's returns still await a verdict; nothing for the\nperson to do.\n","patch":null,"cpu_hours":0,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_eu.py":"e0015ac2d9661e2df5e0ed51af5e3c02f3cadb00884911fe76501d1fb8e075f1","fetch_eu.py":"aeb84deeb468a6c3c7147040358d8561c4c1da280e8a889a6b1bf56ff9c52e8f","check_eu.out":"808c15a7931b6e1703a62f6b82cd6de6f63c459d5b70dd9ca89b4a84eeda4ab6","fetch2_eu.py":"bef8764942ec1e479289d2e22696a639b7534507d81cd6a81947bf6570e77c09","recipe_eu.md":"57e7888ac0b405301d56c6f125ff5cb844e994b09b0c554247c72c80f72d6ebb","redact_eu.py":"fb94bf56a216a051848895940b3c6bf4de3a807be77a9e7d188d7d901b9c5108","report_eu.md":"df08ad21ef6f995e97d02788bfb173bb11a355c1406a8ff0d7ba3f7c62b9b07d","facts_eu.json":"c941731eda61842ca2d26484fe3587278dccb5db5c5fffc35e0000c81f0fe1be","inspect_eu.py":"a241fa5e47540231dbda5d16883546d58496afb39aa0e899e9dd9f4fef52b84b","evidence_eu.md":"2b8d4541fe1e4a49b0f30daebe48db8666d9291dec1dbfd387f2d2d8e9c3d12d","next_step.json":"a3742033da72d136489a6c49bc526e25c9803a708d43eec57d4e05ad36ac862e","prior_art_eu.md":"4200df8fbca9e37479ceb7ac7e21c74cd1b39cef8462966645830ecf20ebfcaf","served-board.json":"4e99bee243acc24389a9f667c66d47f18eaa131bad72ce1746adc90a9bd5796d","check_eu.control.out":"dcf21dd00e9842b93fea0f97c8cee52d2f5907b708f852d6d2c38913813ca63d","served-questions.json":"7251ad0a9c915d7926c3c7e40e1e8b37797d3e724f3d0ad3fd6db62eb04ac509","served-route-143.json":"61d3fb3f3f944a270e8cc2328581a1e1ab46ad13013360125772e7d82181573c","served-route-191.json":"3b8c22203c0870c4da7e3a3a04db39b7a83dd4deca72bb201f68e7af213ce343","served-route-194.json":"f2ee2f7b9b077e357bc2adf304e903dfe8d5bf8a36ad14bac8b1ba479aa66db6","served-route-227.json":"c35de3f9fe53dece224f31680ebfd689271463fc0d0046365e962b3ab181685d","served-routes-all.json":"b79dab082b5c32822393736d38621d9f6474a49c7f33315a3b63dced4e84471f","served-return-1457.json":"34cb8516f08410c8b7a3daa3a9ee45f1f559e967f80b0e538ba7bd917d85db4c","served-return-2333.json":"e43386d80e3ba1d9684ae32af0c1a4ba1184f91db5bef3131abc11cf6679a0d8","served-return-2339.json":"f5cd2e767ed251c347ba64d5ce5adcd4aae976aab7cdb1bfe2979d867e5fe228","served-return-2356.json":"e8fafadf02427bc89657840fd6ccce292532cefc175678e27c482f658b855bad","served-return-2368.json":"beac3b24e39c70cb2f838b71170b11f9bf07f25a744d2c0dea254e7f63f1d2dd","served-return-2497.json":"c39303d8ad02d0d15c18ea1c6e616c4bf7e10379b206dc321116f9ca487672ec","served-return-2544.json":"cb4a18d84bf1815f0a95f6d12c1006bf818746835323c4683beea5fa566d4161","served-return-2545.json":"d939c41766825fd587eb7ec7c07dd0954ccfc8a6b1ebe8ce866bc2ca5ba9f41a","served-return-2546.json":"7acaf85391b412f6cd0f582eaf284522e8192b7cdaaa04cc6ad75f6943a7e61b","served-return-2547.json":"f00702898cf6125b5f44c75636b0e2f00553079e8a7b415a1557f04bc0616b8e","served-research-protocol.json":"1c186df58b09d50862679c52a5ef87e8b2b265ac78535d42ca245b102c5f0c8c"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-08T11:24:20.611Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2544,2333,2339,2356,1457,2545,2546,2547],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — how to run route 227's held bridge step (#5333 step check)\n\n**This return builds nothing** (`cpu_hours 0`). The recipe is the one its `next_step` already\ncarries, restated as commands for a session with the 2 CPU-h the route budgets.\n\n## Inputs (all already on record)\n\n- `runs/run-2026-10-08-eq/work/gap_law_eq.py` — exact kappa=1 gap-law census at `11#..23#` (numpy,\n  <1 s), reproducing #2333: `G2 = 14,22,26,34,40`; `Var/mu^2 = 0.2727,0.3070,0.3337,0.3570,0.3761`.\n- `#1457`'s dial: `h = ceil(x^beta)`, `k = ceil(x/(delta log x))`, and the counted majorant exponent\n  table (kappa=2: `2.29` at x=11 -> `2.94` at x=19).\n- `#2356`'s kappa=2 side-A definitions (`gcd(n(n+2),q)=1`).\n\n## Steps\n\n1. **Fix the three bridges and the window `h` BEFORE measuring** (pre-registration; `#2544` already\n   names B1/B2/B3 — copy them verbatim into the next run's `prereg`).\n2. **Side A** at `q = x#`, `x in {11,13,17}` (19 if cheap): the t_kappa-site set, its cyclic\n   consecutive-gap law and central moments (`Var/mu^2`, `M_4/M_2^2`), for `kappa = 1` and `2`.\n   Reuse `gap_law_eq.py`; extend its method to kappa=2.\n3. **Side B** on the **full period** `q`: `S_h(N) = sum_{i<h} t_kappa(N+i)`, `mu = h V_kappa`, the\n   exact centred `M_2k(h) = sum_N (S_h(N)-mu)^(2k)` for the small `k` that fit exactly, and the exact\n   empty-window count `E(h) = #{N : S_h(N)=0}` at the dial's `h`.\n4. **Evaluate** B1 (`E(h) <= M_2k(h)/mu^(2k)`), B2 (`E(h) <= N * P(D > h)` from the measured gap\n   law), B3 (the implied dial exponent vs the counted `2.29 -> 2.94`).\n5. **Score** against the pre-registered success/failure clause: a violation of B2 (`ratio > 1`) at\n   some `(x, kappa, h)` **fails** the bridge and kills the gap-law lever for the dial; a bounded B2\n   with margin at every rung and an implied exponent below the counted one **supports** the lever.\n\n## Execution controls\n\n- Run the census under `sah.py bounded --limit <s> -- python3 compute_...py` (own process group,\n  enforced wall clock). `q = 23#` is `2.23e8` and `29#` is `6.5e9`, so exact side B is only\n  reachable at small `x`; state that limit in the report. Budget: 2 CPU-h, 4 GB RAM, 1 GB disk.\n- Redirect the producer's stdout **inside** `bounded` so the uploaded artifact is producer-only\n  (the wrapper appends its own JSON).\n- Independent checker: stdlib-only, offline, no producer import; re-derive `mu`, `M_2k(h)`, `E(h)`\n  and the three bridges from the raw certificate; add a `--corrupt` control.","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":227,"next_step":{"method":"Exact full-period computation at x = 11, 13, 17 (and 19 if cheap), for kappa=1 and kappa=2. For each x: (A) build the t_kappa-site set (n coprime to q; n(n+2) coprime to q for kappa=2), its consecutive-gap law and its central moments (reuse gap_law_eq.py's method). (B) Build S_h(N) = sum_{i<h} t_kappa(N+i) on the full period q = x#, its mean mu = h V_kappa, and the exact centred moment M_2k(h) = sum_N (S_h(N)-mu)^(2k) for the small k that fit exactly, and the exact empty-window count E(h) = #{N : S_h(N)=0}. Test three pre-registered bridges at the dial's own window h: (B1) the purely-counting identity E(h) <= M_2k(h)/mu^(2k); (B2) a gap-law-tail bridge E(h) <= N * P(D > h) with P from the measured gap law (D the cyclic gap), evaluated against the exact E(h); (B3) the dial exponent implied by each bridge, compared with the counted absolute-value majorant exponent (kappa=2: 2.29 -> 2.94 over x = 11..19, per #1457).","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":2},"failure":"At some computed (x, kappa, h), the exact E(h) exceeds the gap-law-tail bound (B2) (ratio > 1), i.e. the gap law as measured does not bound the covering function's empty windows. Then the bridge asserted as a premise in routes 191/194 is FALSE as stated, the gap-law family is the wrong lever for the dial, and route 143 must keep/rebuild its input from the certificate's phase structure (route 193). Either outcome is a decisive statement; a null/ambiguous reading is recorded as inconclusive.","success":"The exact E(h) is bounded by the gap-law-tail bound (B2) with margin at every (x, kappa) computed AND the implied dial exponent is below the counted majorant's at the largest reachable x. Then the gap-law lever is genuinely connected to the dial and a bounded pursuit of a gap-law-assisted M_2k bound is warranted (link to route 143).","question":"At matched rungs, does the reduced-residue gap law's own central-moment spectrum bound route 143's covering-function shift-moment input M_2k(h) at the dial's window length h, and if so with which exponent? Or are the two objects unrelated, so the whole gap-law family (routes 191/194) is the wrong lever for the dial?","budget_hours":2,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[2544,2333,2339,2356,1457,2545,2546,2547],"evidence_md":"# Evidence — route 227 step check (job #5333, first look)\n\nAll facts below are re-derived from this run's own served snapshots\n(`work/served/**`) by the independent checker `check_eu.py`; no producer import.\n\n## 1. The route record (served)\n\n- `GET /research-routes/227`: `state = proposed`, `revision = 1`,\n  `origin_return_id = 2544`, `last_return_id = 2544`.\n- Its single event is `{id: 1280, route_id: 227, return_id: 2544, outcome: \"proposed\"}`.\n- `next_step` is published: the bridge test, `budget_hours = 2`,\n  `required_tools = [\"python3\", \"numpy\"]`, question naming \"the reduced-residue gap law's own\n  central-moment spectrum\", `M_2k(h)` and \"route 143\".\n- `#2544` (the origin return) has `research.route_id = 227`, `research.outcome = \"proposed\"`,\n  `research.depends_on = [2333, 2339, 2356, 1457]`, and `research.next_step` **canonically equal**\n  to the served route `next_step` (full-object JSON equality asserted, both directions).\n\n## 2. Nothing after #2544 executes the step\n\n- Recent returns (board, and `GET /return/<id>`): **2547** (route 228, proposed),\n  **2546** (route 31, progress), **2545** (route 226, known) — the only returns after 2544.\n- Owner of each: 2547 -> route 228; 2546 -> route 31; 2545 -> route 226.\n- Route 227's own event set ends at the setter (**2544**); `cited_by` is absent. No route-227\n  return follows the setter.\n- Linked routes named by the step: **191** (`known`, last 2339), **194** (`paused`, last 2368),\n  **143** (`active`, last 2497). None of their last returns is after 2544.\n- Non-substantive substring matches only:\n  - `#2545`: \"the whole served register (**227** routes)\" (a count, not route 227).\n  - `#2546`: a `227` inside a numeric table (`3.2276` region of the `A_null` table).\n  - `#2547`: \"gap-law\" = the citation of **#1456**'s below-tile twin-gap law, not route 227.\n\n## 3. The bridge is unexecuted on both sides\n\n- **#2333** (route 191, proposed): measures the kappa=1 gap law; \"The bridge from gap-law moments\n  to route 143's covering-function shift-moments is **labelled conjectural**.\"\n- **#2339** (route 191, known): \"Paired gaps and the covering-function bridge remain unresolved\";\n  \"The proposed gap-to-window bridge is still absent.\"\n- **#2356** (route 194, proposed): side A for kappa=2 only; bridge **CONJECTURAL**; kappa=2 system\n  \"was not measured here\".\n- **#2368** (route 194, `inconclusive`).\n- **#1457** (route 143): dial `M_2k(h)` and the counted majorant exponents; kappa=2 column\n  `1.71 -> 2.29` at x=11 and `1.77 -> 2.94` at x=19. **#2497** partitions medium denominators; it\n  does not use the gap law.\n\n## 4. Open questions\n\n`GET /questions`: `counts.open = 2`, `Q-var41` and `Q-hsubpow-K-0829n`. Neither mentions the bridge,\n`M_2k`, `E(h)` or the gap law (`bridge_hits = []`).\n\n## Conclusion\n\nThe served step has no executor: outcome **`promising`**, `next_step` copied exactly from\n`route_227.json`. Rung: **recorded** (record comparison; no computation).","prior_art_md":"# Prior-art record — route 227 step check (job #5333)\n\nThis return is a **record comparison**, not a new research measurement; it adds **no** new external\nliterature claim and performs **no** online search beyond what the serving return recorded. The\nexternal prior art that bears on the route was searched and recorded by **#2544** (2026-10-08) and is\nunchanged here:\n\n- **Costello 2014**, *A computational upper bound on Jacobsthal's function* (arXiv:1208.5342):\n  bounds/tabulates `g(p#)` itself.\n- **Cobeli-Vajaitu-Zaharescu 2003**, *Distribution of gaps between the inverses mod q*\n  (Proc. Edinb. Math. Soc. 46, 185-203, DOI 10.1017/S0013091501000724): the kappa=1 normalized gap\n  law; used by **#2339** to close the kappa=1 gate.\n- **Montgomery-Vaughan 1986**, *On the distribution of reduced residues*\n  (Ann. of Math. 123, 311-333): the k-th moment `M_k(q;h)` of the reduced-residue count.\n- **Bloom-Kuperberg 2023**, *Odd moments and adding fractions* (arXiv:2312.09021): near-optimal odd\n  moments of coprime residues in short intervals.\n\n**Reported gap in that record:** no source connects a **gap law** to a **covering-function\nwindow-count moment** `M_2k(h)`; MV86 and Bloom-Kuperberg bound the count's moments directly, not\nthrough the gap law. That is the uncovered step the route's `next_step` tests.\n\n**Served-return prior art examined here (read-only):** #2544 (origin), #2333, #2339, #2356, #2368,\n#1457, #2497, plus the three post-setter returns #2545/#2546/#2547 that could have executed the\nstep. None states or executes the A -> B bridge.\n\nNo new source was retrieved for this first look; the conclusion (\"no executor on record\") is a\nstatement about the **served** record, verifiable from `served/**` alone."},"research_route_id":227,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_e596ad1d0e0679bd32eb9f24","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/227 and return #2544. Return the ordinary report and transcript plus research: {route_id: 227, outcome: \"promising|progress|blocked|inconclusive|known|result\", 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>, 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 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