{"id":1600,"job_id":3137,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Triage report — route 156 (job 3137, run-2026-09-24-v)\n\n**Outcome: `blocked` — scoped obstruction of the payoff, not a failed proof attempt.**\n\n## The assignment's question\nRoute 156 rev 1 (origin return #1599, handle `victor-geere`) funds an exact Lemma-7.2 engine at\n`k = 46` and asks for the *capped*-support certificate `c^T(M2 − (1/A)M1)c > 0`,\n`1/A = 10000/2583`. Its stated payoff: \"with the verified diameter-216 46-tuple and the repaired\nequidistribution criterion gives `DHL[46,2]` and `H₁ ≤ 216`\". The proposed experiment is 8 CPU-h\n(route budget 4 h): converge the engine to `d = 23/25/27` at 1024-bit precision and price the capped\nsupport by inclusion–exclusion instead of Monte-Carlo.\n\n## Decision\n**Not fundable.** The route's only record-level payoff, `H₁ ≤ 216`, is dominated by two rungs below\nit, live in the coordinate table fetched this run (OEIS A008407 b-file, sha256\n`a9c727f5…9592`): `H(45) = 212` (Axiom Math's public `DHL[45,2]`, 2026-09-03) and `H(40) = 186`\n(OpenAI's `DHL[40,2]`). Admissibility is hereditary, so `DHL[45,2] ⇒ DHL[46,2]` and the sharpest bound\na `k = 46` certificate can yield is `H(46) = 216 > 212`. This is the department's own recorded rule\n(route 155's triage, return #1594 §1) applied to the same rung that triage already blocked; route 155\nis still `state = blocked` and nothing supersedes it.\n\n## The four pre-registered falsifiers, all not fired (`work/checks.json`, `falsifiers_fired = []`)\n- **F1 (dominating rungs absent)** — no: `H(40) = 186, H(45) = 212` live, and both are published\n  results with public sources (search 5 below).\n- **F2 (corrected threshold is the harder one)** — no: exactly `10000/2583 = 3.8714672861… < 4`, so\n  certificate-at-`1/A` does **not** imply `M_{46,ε} > 4`; #1599's own `c^T(M2 − 4M1)c < 0` agrees.\n  The route's §4 \"threshold correction\" therefore lowers the bar rather than producing the standard\n  result.\n- **F3 (capped support already cleared)** — no: #1599 states only a Monte-Carlo *mass share*\n  (98.32–99.48 %) against uncapped margins of 1.06 % / 0.62 % at `d = 19`; under that heuristic one of\n  the two ε survives and one does not, so the crux is genuine. But its success endpoint is the\n  dominated 216.\n- **F4 (rung live)** — no: route 155 `blocked`, #1594 `scoped_obstruction`, no supersession.\n\n## What is preserved for a live rung\nThe exact engine is validated by its own controls (`M_5 = 2.00706` vs Krylov 2.00714;\n`M_{4,0.168} = 2.05138` vs published `> 2.00558`; `M_{50,1/25}(d = 11) = 3.8519565733045…` identical\nto an independent 120-digit `mpmath` whitening to 16 digits), and the `1/A` vs `4` normalisation\nquestion costs 0 CPU-h against the source note and Stadlmann arXiv:2608.31126 Prop. 1. Both are worth\ncarrying — at `k = 45`/`40`, where they could touch a live bound.\n\n## Why this is not \"closing a broad route on one failed attempt\"\nNo proof attempt failed. The obstruction is that the route's declared endpoint is already beaten by\ntwo recorded rungs, so the experiment cannot change anything whether it succeeds or fails. The\nstatements left open are named in `obstacle.revisit_when`.\n\n## Scope, cost and disclosure\n0 CPU-h, read-only: 6 served GETs (`route156`, `return1599`, `return1594`, `return1589`,\n`return1586`, `route155`), 1 live OEIS b-file fetch, 6 web searches. No variational computation was\nrun; published values are quoted with citations (prereg rule R3). Disclosed: the domination rests on\ntwo **unaudited 2026 preprints** (the same caveat #1594 recorded), and the source note\n`althofer.de/H1_216_candidate.pdf` could not be retrieved (not indexed; no PDF extractor locally), so\nits §8.1 text is quoted from route 156/#1599 rather than re-read.\n\n**Also for the person:** 58 of @Benjaminsen's returns wait for a verdict (23 made on\ndeepseek-v4-flash).\n\n## Artifacts\n`work/{prereg.md, checks.py, checks.json, evidence.md, prior_art.md, build_payload.py, payload.json,\nfetch.py, served/}`, `work/served/oeis_b008407.txt`.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-24T14:28:11.018Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1586,1589,1594,1599],"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 (rules R1-R3 and falsifiers F1-F4, written before the instrument) -> work/checks.py -> work/checks.json: falsifiers_fired = [], endpoint_dominated = true, H = {40:186, 45:212, 46:216, 48:236, 49:240, 50:246}. F2 exactly: 10000/2583 = 3.8714672861014323 < 4. F3: no capped certificate stated, only mass shares 98.67/99.48/98.71/98.32 % against uncapped margins 1.06 %/0.62 %. F4: route 155 state blocked, #1594 outcome blocked / scoped_obstruction. Served shas: route156 f9e3f4e76083ba6c98ab435c981afa01961860f1fdf17e4d454de09552465b7d, return1599 d4ebc5503ee18fa4519d041b98f50cc4383b783fc6391ddf805c8cffccc0e043, route155 503deb2d9214cd3e8cf80cd010e66a86ed3dc63e9d55049154da19074ad55476, return1594 aadc193c9093f108bc4d2194adabb6904a7cb8e838141faf074054d3961f3e5f.","statement":"Route 156's only stated payoff is H_1 <= 216 through DHL[46,2], and that payoff is dominated: the live coordinate table has H(45) = 212 and H(40) = 186, both published results (Axiom Math bgp212 2026-09-03; OpenAI short_gaps), and hereditary admissibility gives DHL[45,2] => DHL[46,2]. The sharpest bound a k = 46 certificate can yield is H(46) = 216, already beaten. The route's corrected threshold 1/A = 10000/2583 = 3.8714672... < 4 makes its own object weaker than the standard Maynard threshold rather than stronger, and the capped support is not certified (Monte-Carlo mass shares only). So no bounded experiment on this route can change the record, whether it succeeds or fails.","assumptions":"Domination treats the two recorded rungs below 46 (Axiom Math DHL[45,2]/212; OpenAI DHL[40,2]/186) as sufficient; both are unaudited 2026 preprints, exactly as route 155's triage disclosed. The 216 endpoint is quoted from route 156's own next_step.success; H(k) come from the live OEIS A008407 b-file (sha256 a9c727f5de03d6df11044e033fa65725acbfdadb9d25345bf47137c630a19592). The capped-support reading uses #1599's own Monte-Carlo mass shares as a heuristic, not an exact statement. No variational value was recomputed here.","revisit_when":"A rung <= 45 is withdrawn or superseded (either Axiom Math's DHL[45,2]/212 or OpenAI's DHL[40,2]/186), which makes the k = 46 endpoint live again; or the exact engine is repointed at a live rung (k = 45/40) under the enlarged-support normalisation, which is a different route."},"route_id":156,"depends_on":[1586,1589,1594],"evidence_md":"# Evidence — the route's stated payoff is dominated by two rungs below it\n\n**The route's own success criterion (quoted from route 156's `next_step`).** \"A converged\ncapped-support certificate `c^T(M2 − (1/A)M1)c > 0` on the exact `T_46` of the source, which with the\nverified diameter-216 46-tuple and the repaired equidistribution criterion gives `DHL[46,2]` and\n`H₁ ≤ 216`.\" That endpoint is the only record-level change the route claims.\n\n**F1 did not fire — the dominating rungs are on record and public.** Live OEIS A008407 b-file\n(fetched 2026-09-24, sha256 `a9c727f5…9592`): `H(40)=186`, `H(45)=212`, `H(46)=216`, `H(48)=236`,\n`H(49)=240`, `H(50)=246`. Two are *published results*, not coordinates: Axiom Math's `bgp212` (\"pairs\nof primes … separated by 212 or less\", 2026-09-03) gives `DHL[45,2]`; OpenAI's `short_gaps` gives\n`DHL[40,2]` with 186. Admissibility is hereditary, so an admissible 46-tuple contains an admissible\n45-subtuple of no larger diameter: **`DHL[45,2] ⇒ DHL[46,2] ⇒ H₁ ≤ 212`**, and 186 below that. That is\nthe department's own recorded rule (route 155's triage, #1594 §1, served sha `aadc193c…f3e5f`),\nre-checked here against the live table. The sharpest bound a k=46 certificate can yield is\n`H(46)=216`, **beaten by 212 and by 186**.\n\n**F2 did not fire — the threshold correction lowers the bar.** Exactly,\n`1/A = 10000/2583 = 3.8714672861… < 4`. Route 156 §4 says the source's `46 J > I` normalises to\n`M_{46,η} > 1/A`, *not* to the standard Maynard `M > 4`. So a certificate at `1/A` does **not** imply\n`M_{46,ε} > 4`, and #1599 records `c^T(M2 − 4M1)c = −0.087122…` / `−0.104304…` — no `τ=4`\ncertificate at `d=19`. The route does not produce the stronger standard-threshold result either. Its\nremaining sub-question (which normalisation is correct) is a source-text question worth **0 CPU-h**,\nand both answers leave the endpoint at 216.\n\n**F3 did not fire — the capped support is not cleared; the crux is real but its payoff is the\ndominated endpoint.** #1599 states a Monte-Carlo *mass share* inside the source's large-coordinate\nbudget (98.67 %, 99.48 % at ε=25/861; 98.71 %, 98.32 % at ε=79/1250; d=13/19) against uncapped margins\nover `1/A` of 1.06 % and 0.62 % at `d=19`. Under that heuristic ε=25/861 stays above `1/A` and\nε=79/1250 falls below, so the exact pricing is a genuine ~50/50 crux — and its success buys\n`H₁ ≤ 216`, already beaten. The route's own failure mode (\"a capped value below `1/A`\") is thus\nhalf-predicted by its own measurement, and its success mode is worthless.\n\n**F4 did not fire — the rung is not live.** Route 155 (`research-routes/155`, sha `503deb2d…d55476`)\nis `state=blocked`; #1594 is `outcome: blocked`, `obstacle.kind = scoped_obstruction` for this same\nk=46 target. No supersession is recorded.\n\n**What this changes.** No bounded experiment on route 156 can move the record: the proposed run ends\neither in the route's own failure mode or in `H₁ ≤ 216`. This is a scoped obstruction of the payoff,\nnot a failed proof attempt — no proof attempt failed — so \"do not close a broad route because one\nattempt failed\" does not apply. Disclosed: the domination rests on two **unaudited 2026 preprints**,\nas route 155's triage also disclosed.\n\n**Preserved, and the only live item.** The exact engine is validated by its own controls (`M_5 =\n2.00706` vs Krylov 2.00714; `M_{4,0.168} = 2.05138` vs published `> 2.00558`; `M_{50,1/25}(d=11) =\n3.8519565733045…` identical to an independent 120-digit `mpmath` whitening to 16 digits — quoted from\n#1599, not recomputed), and the `1/A` vs `4` reconciliation costs 0 CPU-h. Both are assets for a\n**live** rung (k=45/40). Headroom left after this triage: none for k=46.\n\n**Checks.** `work/prereg.md` (written before the instrument) → `work/checks.py` →\n`work/checks.json`: `falsifiers_fired = []`, `endpoint_dominated = true`. Served shas: route156\n`f9e3f4e7…5465b7d`, return1599 `d4ebc550…cc0e043`, route155 `503deb2d…ad55476`, return1594\n`aadc193c…961f3e5f`.","prior_art_md":"# Prior art and the exact remaining gap (search date 2026-09-24, live this run)\n\n**Search record (this run).**\n1. `Althoefer \"H1 <= 216\" candidate k=46 enlarged variational problem equidistribution criterion prime\n   gaps` → **0 results** (the source note `althofer.de/H1_216_candidate.pdf` named by route 156 and\n   #1599 is not indexed; route 155's triage recorded the same absence).\n2. `Polymath8b M_{k,epsilon} variational problem threshold 4 enlarged simplex DHL[k,2] certificate` →\n   Tao's Polymath8b announcement (`terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-\n   with-many-primes-after-maynard/`), the Polymath8 success posts, and the Polymath8b survey\n   (`arXiv:1409.8361`). All 2013–2014; **none states any `k = 46` ε-enlarged value**, and there is no\n   `k = 46` certificate in this literature — the enlargement apparatus is the source note's, not a\n   published construction.\n3. `\"althofer.de\" H1 216 candidate prime gaps paper 2026` → **0 results**.\n4. `Stadlmann \"bounded gaps between primes\" 2026 arXiv 2608.31126 Proposition 1 truncated marginal` →\n   `eprint.iacr.org/2026/1893` (\"Bounded Gaps Between Primes: An Upper Bound of 236\", 2026-09-02,\n   which cites \"J. Stadlmann, Bounded gaps between primes, 2026, arXiv:2608.31126\" and its\n   truncated-marginal construction, Eq. (11)) and arXiv's listing for `2608.31126` (Julia Stadlmann,\n   math.NT). Confirms the dependency chain route 156 builds on.\n5. `Axiom Math prime gaps DHL[45,2] H1 <= 212 2026 certificate primegaps.axiommath.ai` →\n   `x.com/axiommathai/status/2095526333799633063` and `primegaps.axiommath.ai/bgp212.pdf`\n   (\"We show that infinitely many pairs of prime numbers are separated by 212 or less\", 2026-09-03,\n   down from 246), `github.com/AxiomMath/PrimeGapsLib` (Lean formalization), and in the same cluster a\n   **\"Small prime gap bound is now <= 186\"** thread (OpenAI). **This is the evidence that decides the\n   triage: two rungs below k = 46 are already on the public record.**\n6. `Maynard variational problem enlarged support threshold M_k > 4 \"46\" bounded gaps 2026 open rung\n   k=45` → no relevant publication.\n\n**What the prior art already settles.** No `k = 46, ε`-enlarged value is published by anybody\n(searches 1–3, 6), so route 156's certificate would be new *mathematics* and is not refused because\nsomeone else did it. It is refused because its endpoint is superseded: the record already carries\n`DHL[45,2] ⇒ H₁ ≤ 212` (Axiom Math, 2026-09-03) and `DHL[40,2] ⇒ H₁ ≤ 186` (OpenAI) — both found\nlive in search 5, both matching return #1594's coordinates and route 153's priced rungs.\n\n**The exact remaining gap, after this triage.**\n- Not the `k = 46` capped-support certificate: `H(46) = 216 > 212 ≥ H₁` (held), so it cannot change a\n  bound. Its only surviving sub-question — was the source's `46 J > I` criterion mis-normalised by\n  return #1589 to `M > 4` instead of `M > 1/A = 10000/2583` — is a source-text/normalisation question\n  (0 CPU-h) and is *left open, not answered*, here.\n- Live and unaddressed by this literature: an independent reproduction of the **live** rungs under the\n  enlarged-support normalisation — Axiom Math's `DHL[45,2]`/212 (paper + Lean library public; no\n  third-party exact-rational certificate located by these searches) and OpenAI's `DHL[40,2]`/186\n  (its Lean layer is, per the record, conditional on unproved axioms).\n- The analytic gap route 156 inherits: the source's equidistribution criterion (Type IIc on the\n  printed `ω₀ < 0` interval) — a proof-level gap, not a computation, and outside the experiment's\n  scope although the route's success sentence requires it.\n\n**Uncertainty / disclosure.** The two dominating rungs are unaudited 2026 preprints (the caveat\nroute 155's triage recorded too); if either is withdrawn the k=46 endpoint becomes live again. Search\ncoverage: one engine, 6 queries, 2026-09-24; the source note could not be retrieved (not indexed; no\nPDF text extractor locally)."},"research_route_id":156,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_379a6b2fc234285ed21dbd02","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/156 and return #1599. Return the ordinary report and transcript plus research: {route_id: 156, 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":"1586","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1589","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1594","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/156","transcript_url":"/projects/twin-primes/return/1600/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}