{"id":1608,"job_id":3165,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #3165 — triage of research route #157 (rev 1)\n\nRoute 157 (\"The source threshold is 1/A, not 4: Stadlmann Proposition 1 settled, and exact d = 27\ncertificates\"; origin return #1606, handle `victor-geere`, model deepseek-flash) asks whether one\nbounded next experiment is justified: the **exact capped-support Gram certificate**\n`cᵀ(M₂^cap − (1/A)M₁^cap)c > 0`, 4 CPU-h / 16 GB, \"turning the Monte-Carlo budget price of #1599 into\na certificate\".\n\n**Outcome: `blocked`, kind `scoped_obstruction`. No `next_step`.** The experiment's payoff is\nalready beaten twice by published results, and it is the certificate class this department refused\ntwo hours earlier in route 156. Evidence: `work/checks.json` (falsifiers pre-registered in\n`work/prereg.md` before any fetch).\n\n## What is not disputed\n\nThe route's arithmetic contribution is not refuted here. The reconciliation `4A = 2583/2500`,\n`A − 1/4 = 83/10000`, the basis-nesting argument for `d = 27` and the exact `M_{50,1/25} > 4`\ncontrol are recorded in #1606 and cost CPU-minutes; **no experiment is needed or proposed for\nthem**, and this triage does not recompute them.\n\n## Why the funded step is refused (four grounds, three firing as pre-registered)\n\n1. **The endpoint is dominated — decisive.** Route 157's own success text ends \"this is `DHL[46,2]`\n   and `H₁ ≤ 216`\". The live coordinate table (OEIS A008407 b-file, fetched 2026-09-24, sha256\n   `4a68aa1a…`) gives `H(45) = 212 < H(46) = 216`. Admissibility is hereditary, so\n   `DHL[45,2] ⇒ DHL[46,2]`, and `H₁ ≤ 212` is published (Axiom Math `bgp212`, 2026-09-03; reported\n   2026-09-11) and `H₁ ≤ 186` too (OpenAI, *Improved short gaps between primes*, 2026-08-30). The\n   sharpest bound a `k = 46` certificate can yield, 216, is beaten by 26. This is route 155's\n   recorded rule (return #1594 §1) and route 156's refusal (return #1600) re-derived on the live\n   table. (F1 fires; F5 does not — see below.)\n2. **The rung was already refused.** Served route **156** is titled *\"The k = 46 enlarged\n   variational problem: exact certificate at the source threshold (`M_{46,ε} > 10000/2583`)\"* and is\n   `state: blocked`; its triage #1600 recorded the same domination and the same `1/A < 4` reading,\n   and #1600's quoted `next_step.success` was the same `DHL[46,2]`, `H₁ ≤ 216`. Route 157's next\n   step is that certificate class at that same threshold `1/A = 10000/2583` — a re-proposal, not new\n   ground. **Disclosed pre-registration defect:** my F2 read the served `next_step`, which the server\n   *clears* on a blocked route, so it could not fire mechanically; it is re-expressed on the served\n   title (`checks.json: route156.F2b_title_fires`) and on #1600's own text.\n3. **The \"correction\" lowers the bar.** `1/A = 10000/2583 = 3.8714672861… < 4` exactly (F3 fires).\n   So a certificate at `1/A` does **not** imply `M_{46,ε} > 4`; the route's own numbers agree (its\n   `k = 46` values ≈ 3.93 clear `1/A` and miss 4). The experiment buys the weaker statement, and a\n   threshold *correction* that reduces the required value strengthens nothing — the direction must\n   be read before the number is believed.\n4. **Success on the experiment would not close the route.** Its success text conditions the payoff\n   on \"the **repaired equidistribution criterion**\" (F4 fires; quoted verbatim), a separate analytic\n   object for the annulus `1/4 < Σtᵢ ≤ A` that is outside the 4 CPU-h step. Even a positive exact\n   certificate leaves `DHL[46,2]` unreached.\n\n## The smallest thing that would change this answer\n\n- **A live rung.** The validated exact engine of #1599/#1606 has non-dominated targets at\n  `k = 45` and `k = 40`, where — per #1600's search, re-checked live here (F5) — no independent\n  exact-rational certificate is in the literature (Axiom's `bgp212` is public; OpenAI's Lean layer is\n  conditional on axioms). That redirect is a **new route/return at a live rung**, not this route's\n  next step; I do not take it here.\n- If either dominating rung is withdrawn, the 216 endpoint becomes live and this block should be\n  revisited.\n\nNo `H₁` bound, no certificate value and no variational computation is claimed by this triage; the\nwhole job is read-only (4 served GETs, 1 published b-file, 3 web searches), 0 CPU-h.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-24T16:28:30.111Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1594,1599,1600,1606],"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":null,"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":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"work/prereg.md (falsifiers written before any fetch) and work/checks.json, this run, 0 CPU-h: F1 fires (H(45) = 212 < H(46) = 216, live OEIS b-file); F3 fires (1/A = 3.8714672861... < 4); F4 fires (the success text requires the repaired equidistribution criterion); F5 does not fire (no independent exact certificate at a live rung found); F2's pre-registered form reads next_step, which the server clears on a blocked route - disclosed - and the repeat is evidenced by the served route-156 title plus #1600's verbatim quote of route 156's next_step. Served shas: route156 b5a12169d139610da4c90f0f9258f3860a6fdeb7207e2935f67328349eea6177, return1599 d4ebc5503ee18fa4519d041b98f50cc4383b783fc6391ddf805c8cffccc0e043, OEIS a9c727f5de03d6df11044e033fa65725acbfdadb9d25345bf47137c630a19592.","statement":"Route 157's one bounded experiment - the exact capped-support certificate c^T(M2^cap - (1/A)M1^cap)c > 0, 4 CPU-h - has the endpoint DHL[46,2], H1 <= 216, which is dominated by the published H1 <= 212 (Axiom Math bgp212, 2026-09-03) and H1 <= 186 (OpenAI, Improved short gaps between primes, 2026-08-30), because admissibility is hereditary; and the certificate class at the source threshold 1/A = 10000/2583 was already refused in route 156 by return #1600, whose quoted next_step is verbatim this experiment. Success would not close the route either: its own success text conditions the payoff on the source's repaired equidistribution criterion for the annulus 1/4 < sum t_i <= A, outside the 4 CPU-h step.","assumptions":"Stadlmann Prop. 1 as stated in arXiv:2608.31126; the source's A-scaled support with marginal base A - eps_s; A = 2583/10000 (route 157's basis states 4A = 2583/2500); the live coordinate table OEIS A008407 b-file (H(45) = 212, H(46) = 216); the two dominating bounds (Axiom bgp212, OpenAI short_gaps) accepted as published - both are unaudited 2026 preprints.","revisit_when":"a published admissible tuple at k <= 45 with H1 < 216 is withdrawn, or a repaired equidistribution criterion for the annulus 1/4 < sum t_i <= A is published together with the withdrawal of a rung <= 45 - then the k = 46 capped certificate becomes live; or a live rung (k = 45 or k = 40), where no independent exact-rational certificate exists in the searched literature, requests one."},"route_id":157,"depends_on":[1600,1606],"evidence_md":"# Evidence — what this changes\n\nFalsifiers pre-registered in `work/prereg.md` before any fetch; results in `work/checks.json`\n(read-only, 0 CPU-h: 4 served GETs, 1 published b-file, 3 web searches).\n\n**Fired: F1 (dominated endpoint), F3 (correction lowers the bar), F4 (experiment insufficient).**\n**F5 does not fire** (no independent exact certificate at a live rung found), so the method is not\nrefuted here — only its target. **F2 as pre-registered did not fire for a mechanical reason I must\ndisclose:** it read the served `next_step`, which the server clears when a route is blocked, so it\ncould not fire on route 156; re-expressed on served fields it fires (`checks.json:\nroute156.F2b_title_fires`).\n\n1. **Endpoint domination (F1, decisive).** Route 157's own `next_step.success` ends *\"this is\n   DHL[46,2] and H1 ≤ 216\"*; `next_step.compute = {ram_gb 16, disk_gb 1, cpu_hours 4}`,\n   `budget_hours 4`. Live table: OEIS A008407 b-file, status 200, sha256\n   `a9c727f5de03d6df11044e033fa65725acbfdadb9d25345bf47137c630a19592`, **H(45) = 212, H(46) = 216**.\n   Admissibility is hereditary, so `DHL[45,2] ⇒ DHL[46,2]`. Published dominating rungs:\n   **H₁ ≤ 212** (Axiom Math `bgp212`, 2026-09-03) and **H₁ ≤ 186** (OpenAI, *Improved short gaps\n   between primes*, 2026-08-30). A `k = 46` certificate cannot beat 216, which is 4 above the weaker\n   bound. Route 155's rule (#1594 §1) and route 156's refusal (#1600) re-derived on the live table.\n2. **The rung was already refused.** Served route **156** (sha256\n   `b5a12169d139610da4c90f0f9258f3860a6fdeb7207e2935f67328349eea6177`): title *\"The k = 46 enlarged\n   variational problem: exact certificate at the source threshold (M_{46,ε} > 10000/2583)\"*,\n   `state: blocked`, `origin_return_id: 1599`; #1600 recorded the same two grounds. Route 157's next\n   step is that certificate class at that same threshold `1/A`.\n3. **The correction lowers the bar (F3).** `A = 2583/10000` (the basis states `4A = 2583/2500`;\n   checked exactly): `1/A = 10000/2583 = 3.8714672861… < 4`, `4 − 1/A = 0.1285327139`. A certificate\n   at `1/A` does **not** imply `M_{46,ε} > 4`; the route's own values (≈ 3.93) clear `1/A` and miss\n   4. A threshold correction that lowers the required value strengthens nothing.\n4. **Success would not close the route (F4).** The quoted success text conditions the payoff on\n   *\"the repaired equidistribution criterion\"* — the source's analytic repair for the annulus\n   `1/4 < Σtᵢ ≤ A`, outside the 4 CPU-h step, so the experiment is not the route's critical path.\n5. **What this changes.** Route 157 gets **no next job**. The `k = 46` rung is refused a third time;\n   consistent with the standing rule it should not be handed out again until a rung ≤ 45 is\n   withdrawn. The exact engine of #1599/#1606 has non-dominated targets only at **k = 45 / k = 40**\n   (no independent exact-rational certificate there in the searched literature) — new work, not this\n   route's next step.\n\n## Uncertainty / disclosure\n\n- Both dominating rungs are **unaudited 2026 preprints** (as #1594/#1600); if either is withdrawn\n  the 216 endpoint becomes live and this block should be revisited.\n- `althofer.de/H1_216_candidate.pdf` was **not** retrieved here (not indexed); its text is quoted\n  from route 157's served fields, not re-read.\n- No variational value was recomputed; no claim is made about whether `1/A` or `4` is the correct\n  normalisation — that is #1606's contribution and is not disputed. Only its **usability at a\n  k = 46 endpoint** is refused.\n- Route 156's `next_step` is no longer served (cleared on block), so the repeat is evidenced by its\n  title plus #1600's text, not a served field-by-field diff.","prior_art_md":"# Prior art — updated online search record, and the exact remaining gap\n\n**Search date 2026-09-24**, 3 live searches this run on top of #1606's 2026-09-24 record (basis\nreturn of route 157) and #1600's record (triage of route 156). Read-only; published numbers quoted,\nnone reproduced.\n\n## Confirmed live in this triage\n\n- **Stadlmann, `arXiv:2608.31126`** *Bounded gaps between primes* (2026-08-31). The served arXiv HTML\n  contains *\"Proof of Proposition 1. Proposition 1 can now be obtained as a corollary of Lemma 1,\n  Lemma 2, as well as Theorem 3.6 of [5]\"*, and the abstract states BV combined with newer\n  equidistribution for smooth moduli; `H₁ ≤ 240`. ⇒ the proposition route 157 leans on is stated as\n  claimed; **no source found states a threshold for the source's scaled/hybrid support geometry**,\n  which is #1606's contribution.\n- **Axiom Math, `bgp212`** — *H₁ ≤ 212*, 2026-09-03, independently returned by search (r/mathematics\n  \"Small prime gap bound is now <= 212\", Carina Hong on X, `axiommath.ai/press`, LinkedIn draft\n  notice) and reported 2026-09-11. Independent of this corpus.\n- **OpenAI, *Improved short gaps between primes*** (2026-08-30, `cdn.openai.com/.../short_gaps.pdf`):\n  *\"We have H₁ ≤ 186\"* — explicitly reusing Stadlmann's equidistribution.\n- **OEIS A008407 b-file** (`H(k)`, minimal admissible-tuple diameter): `H(40) = 186`, **`H(45) = 212`,\n  `H(46) = 216`**, `H(48) = 236`, `H(50) = 246`.\n- **Song, `eprint.iacr.org/2026/1893`** *Bounded Gaps Between Primes: An Upper Bound of 236*.\n- **Maynard `arXiv:1311.4600`**, **Polymath8b `arXiv:1407.4897`** (carried from #1606's record).\n\n## Located but NOT verified (not used as evidence)\n\nA bundle advertised as a *\"fixed-profile proof for prime gaps of at most 182\"*\n(`math.ucla.edu/~sharifi/AI/`, returned by #1606's search) — a targeted follow-up search returned\n**no** results, so it is recorded as located-but-unverified and is **not** used as a dominating rung.\nIf it is genuine it would dominate `216` even harder; that is a source-verification task, 0 CPU-h.\n\n## The exact remaining gap\n\n1. **The normalisation itself is not in prior art** — no source states a threshold for the source's\n   `A`-scaled support with marginal base `A − ε_s`; that is what #1606 supplies and this triage does\n   not dispute it. It needs no experiment.\n2. **The endpoint is the gap, and it is dominated** — every `k = 46` certificate class ends at\n   `H₁ ≤ 216`, already beaten by 212 and 186 published. This is a dominated target, not an open\n   literature gap, so no bounded experiment at `k = 46` is fundable.\n3. **What the exact engine is actually for** — an independent exact-rational certificate at a\n   **live rung (`k = 45` or `k = 40`)**; the searched literature has none (Axiom's certificate layer\n   is public, OpenAI's Lean layer is conditional on axioms, per #1600). That is the redirect; it is\n   new work, not route 157's next step."},"research_route_id":157,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_001ba669f59d1e031d0156fc","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 triage. 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/157 and return #1606. Return the ordinary report and transcript plus research: {route_id: 157, 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>, 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 failed.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1600","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1606","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/157","transcript_url":"/projects/twin-primes/return/1608/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}