{"id":1588,"job_id":3060,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Triage report — route 153 (job 3060, exploration/triage, general mode)\n\n**Question.** Is one bounded next experiment on route 153 (\"Chen's 248 TP gap and the path to 46 and\nbelow\") justified, and is it the one the route records?\n\n**Answer.** A bounded experiment is justified, but **not the recorded one**. The route's recorded\nnext step (implement the hybrid-support Maynard variational problem, control it at a published\n`M_k`, then price the rungs `k = 47, 46` giving `H_1 ≤ 226` and `≤ 216`) was superseded 21 days\n**before** the route was authored: published 2026 work already prices the whole ladder past those\nrungs. The live record on 2026-09-24 is `H_1 ≤ 186` (OpenAI, 3 Sep 2026), with `≤ 212`\n(Axiom Math, Sep 2026) and `≤ 240` (Stadlmann, 31 Aug 2026) behind it.\n\n**What the evidence changes.**\n\n1. **The route's prior-art verdict on the record is wrong in a way that changes the investment.**\n   Route 153 treats `212` and `186` as \"unverified 2026 claims of the local handoff\" and states that\n   \"246 remains the established record\". They are, instead, the current published record: an Axiom\n   Math paper (Charton, Hong, Lau, Ono, Remy, Siu et al., `primegaps.axiommath.ai/bgp212.pdf`,\n   `H_1 ≤ 212`) and an OpenAI paper released with GPT-6 Astra on 3 Sep 2026 (`H_1 ≤ 186`,\n   abstract read live), both reported by Science News on 11 Sep 2026. Song–Yue (IACR ePrint\n   2026/1893, approved 9 Sep 2026) state `H_1 ≤ 236` and are therefore already behind the record.\n2. **The route's own falsifier fired.** Its uncertainty G2 states: \"a new ingredient could certify\n   `M_46 > 4` tomorrow, and the proposition is falsified the day one is published.\" Bounds of 212\n   and 186 correspond to the `k = 45` and `k = 40` rungs of the route's own ladder (verified below),\n   i.e. published work has priced rungs strictly below `k = 47` and `k = 46`.\n3. **The route's arithmetic is correct — that is why the finding is decisive.** Every row of the\n   route's priced ladder equals OEIS A008407 at the route's own index (`work/ladder_check.py`,\n   `work/ladder_check.json`): `H(40..50) = 186, 188, 196, 200, 210, 212, 216, 226, 236, 240, 246`.\n   And every 2026 record value is an A008407 term: `240 = H(49)`, `236 = H(48)`, `212 = H(45)`,\n   `186 = H(40)`. So the ladder is the right coordinate system for the *values*, while the rung\n   that achieves them moved from `k = 49` to `k = 40` through the level-of-distribution and\n   sieve-weight side — not by pricing `M_47`/`M_46`.\n4. **Consequence for the proposed experiment.** Even a fully successful run of the recorded next\n   step yields `H_1 ≤ 226` or `≤ 216`, both weaker than the published `212` and `186`. It cannot\n   advance the record; its only residual value would be independent reproduction of a table that is\n   not the frontier. **Do not fund it as recorded.**\n\n**What is *not* covered, and what I am not claiming.** I did not verify the route's quoted\nPolymath8b thresholds (`M_105 > 4`, `M_54 > 4.00238`, `M_{50,1/25} > 4.0043`, `M_{51,1/50} >\n4.00156`) against the paper; they are cited, not reproduced. I did not read the Axiom or OpenAI\npapers in full (the Axiom PDF is not text-extractable by this client; the OpenAI paper was read at\nits abstract). The route's contributions 1 and 2 — the portable instrument and the Lean layer —\nare untouched by this finding and remain unverified by any third party. This triage makes no\nmathematical claim about any gap bound: it is a statement about the record as of 2026-09-24.\n\n**Cost.** 0 CPU-h. Two served fetches (route 153, return #1584), five public fetches. No\ncomputation was rerun: per the task, published numbers are cited, not reproduced.\n\n**Recorded next step.** A distinct, bounded step on the *record* side, not the `M_k` side — see\n`next_step` in this return.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-09-24T11:35:43.688Z","repo_url":null,"commit":null,"cites":{"returns":[1584]},"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":"promising","route_id":153,"next_step":{"method":"Read-only, literature only, 0 CPU-h. (1) Fetch each of the four papers (arXiv abs+HTML/PDF, the Axiom PDF, the OpenAI PDF, the IACR PDF) and extract the paper's own statement of the mechanism: the number of forms k, the level of distribution or theta/varpi it assumes, the ratio threshold, and the final bound. (2) Tabulate the quadruple per paper and compare the bound against the live OEIS A008407 term at the paper's stated k (a(40)=186, a(45)=212, a(48)=236, a(49)=240 already read live on 2026-09-24). (3) Compare the stated thresholds against the thresholds the route quotes from Polymath8b (M_105>4; M_54>4.00238 under BV; M_{50,1/25}>4.0043; M_{51,1/50}>4.00156), reading them out of arXiv:1407.4897 rather than trusting the route's quotes. (4) Pre-registered falsifier: a paper whose claimed bound is not an A008407 term for any k, or which states no tuple length at all, refutes 'the ladder coordinates the record' and requires route 153's priced-ladder contribution to be withdrawn rather than corrected. (5) Negative control: also record Polymath8b's own 246 = a(50) and Maynard's 600, which must come out of the same table.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"A paper's bound is not an A008407 term, or a paper states no tuple length: the ladder is then the wrong coordinate system for the record, contribution 3 is withdrawn rather than corrected, and the route's framing must be re-derived from the papers' own coordinates.","success":"A single table giving, per 2026 result, (k, theta/varpi, threshold, bound) with each bound matched to its A008407 index, plus a one-line correction of route 153's contribution 3 stating which rungs are priced as of the search date and by whom; the route's remaining uncovered surface (independent verification of contributions 1-2) is stated separately.","question":"For each of the four published 2026 bounded-gap results (Stadlmann H_1<=240, arXiv:2608.31126; Axiom Math H_1<=212, bgp212; OpenAI H_1<=186, short_gaps; Song-Yue H_1<=236, IACR 2026/1893), what (tuple length k, level of distribution / theta, required ratio threshold) does the paper itself state, and does the claimed bound equal OEIS A008407's a(k) at that stated k -- i.e. is the route's (k, ratio) ladder the right coordinate system for the record, or only for its values?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[1584],"evidence_md":"# Evidence — what this triage changes\n\nThe route's recorded next experiment is **superseded by published work that predates the route**,\nand the route's own pre-registered falsifier fires. Concretely:\n\n1. **The record moved past the route's whole priced ladder, before the route existed.** Route 153's\n   ladder is `k: 50, 49, 48, 47, 46, 45, 44, …, 40` with binding bounds `246, 240, 236, 226, 216,\n   212, 210, …, 186` (served `research-routes/153`, sha256 `b2fb5b195eba09a8b536a048bc37223e6d\n   d4cabdcf8be1c8a8033c63718a7baf`, `created_at 2026-09-24T10:11:07Z`). Live, published 2026\n   results are:\n   - **Stadlmann, arXiv:2608.31126 (submitted 31 Aug 2026): `H_1 ≤ 240`** — Bombieri–Vinogradov\n     combined with newer equidistribution estimates for smooth moduli;\n   - **Axiom Math, Charton–Hong–Lau–Ono–Remy–Siu et al., `primegaps.axiommath.ai/bgp212.pdf`\n     (Sep 2026): `H_1 ≤ 212`**;\n   - **OpenAI, \"Improved short gaps between primes\" (3 Sep 2026, with GPT-6 Astra), abstract:\n     `liminf (p_{n+1} − p_n) ≤ 186`**;\n   - **Song & Yue, IACR ePrint 2026/1893 (approved 9 Sep 2026): `H_1 ≤ 236`** — already behind the\n     record.\n   Science News (D. Mackenzie, 11 Sep 2026) reports the sequence within three days of Stadlmann's\n   preprint: 246 → 240, then Axiom to 212, then OpenAI to 186, and describes the mechanism as\n   choosing optimal sieve weights by piecing together regions of high-dimensional space where the\n   different approaches prevail — i.e. exactly the hybrid-support optimisation the route proposes to\n   implement, on the level-of-distribution side.\n\n2. **The route's second constant reading is refuted in the same way.** Its uncertainty G5 says \"the\n   2026 `k = 49` (240) and `k = 48` (236) results are 2026 and not peer-reviewed at the search\n   date; 246 remains the established record\", and its prior-art verdict treats \"212 or 186\" as\n   unverified handoff claims. They are the current published record. So the route's contribution 3\n   (the \"priced ladder\" as a description of where the frontier is) must be **corrected, not\n   extended**: as of 2026-09-24 the frontier rung is `k = 40` (`H_1 ≤ 186`), and the route's next\n   two rungs (`k = 47` → 226, `k = 46` → 216) are history.\n\n3. **The route's arithmetic is nevertheless correct, which is what makes the finding decisive**\n   (`work/ladder_check.py` → `work/ladder_check.json`, 0 CPU-h, pre-registered falsifier\n   \"a claimed bound that is not an A008407 term\" did **not** fire):\n   - rows `k = 40..50` of OEIS A008407 read live from `https://oeis.org/A008407/list` (page stamped\n     \"Last modified September 24 07:30 EDT 2026\"): `186, 188, 196, 200, 210, 212, 216, 226, 236,\n     240, 246` — every route ladder row equals `a(k)` at the route's own `k`;\n   - every 2026 record value is an A008407 term: `240 = H(49)`, `236 = H(48)`, `212 = H(45)`,\n     `186 = H(40)`.\n\n4. **Therefore the recorded experiment cannot advance the record.** Even a fully successful run\n   gives `H_1 ≤ 226` or `≤ 216`: strictly weaker than the published `212` and `186`. Its only\n   residual value is independent reproduction of a non-frontier table. The route's framing (\"the obstruction at each rung is the analytic\n   ratio `M_k > 4`\") is also the wrong lever as stated: the record moved by changing the level of\n   distribution and the sieve weights, and by combining regions of the variational space.\n\n**Scope and limits.** 0 CPU-h, read-only, no published computation rerun (published numbers are\ncited, per the task). Not verified here: the route's quoted Polymath8b thresholds (`M_105 > 4`,\n`M_54 > 4.00238`, `M_{50,1/25} > 4.0043`, `M_{51,1/50} > 4.00156`) against the paper; the full text\nof the Axiom and OpenAI papers (abstract/title only for OpenAI, `application/pdf` unsupported by\nthis client for Axiom); the peer-review status of the four 2026 preprints. Route contributions 1\nand 2 (the portable instrument, the Lean layer) are untouched by this finding and still have no\nthird-party verification.","prior_art_md":"# Prior art and exact remaining gap (search date 2026-09-24)\n\n**Search record.** Served: `GET /research-routes/153` and `GET /return/1584` (raw, sha256 in\n`work/served/`); #1584's own `prior-art-findings.md`, `proposal-*.md` and instrument files are store\nobjects it declares. `GET /docs/research/0020/` → **404 \"not found: research/0020/\"**, so the \"local\nhandoff\" the route leans on is not in the served snapshot.\nOnline, in order: (1) \"Polymath8b `M_k` computation code\" → Tao's Polymath8b III thread (read live:\n`M_k` is the Selberg-sieve variational quantity; the best lower bounds came from numerical quadratic\nprogramming, and `M_k` is close to the theoretical limit of that sieve at the Bombieri–Vinogradov\nlevel of distribution); (2) \"Stadlmann 2026 prime gaps\" → arXiv:2608.31126 and the Science News\nreport of 11 Sep 2026; (3) OEIS A008407 (terms read live); (4) the Axiom Math and OpenAI results;\n(5) IACR ePrint 2026/1893.\n\n**The 2026 record, with sources.**\n- `H_1 ≤ 240` — J. Stadlmann, *Bounded gaps between primes*, arXiv:2608.31126v1, 31 Aug 2026\n  (Bombieri–Vinogradov + newer equidistribution estimates for smooth moduli).\n- `H_1 ≤ 212` — F. Charton, L. Hong, K. Lau, K. Ono, G. Remy, H. C. Siu et al. (Axiom Math),\n  *A new bound for small gaps between primes*, `primegaps.axiommath.ai/bgp212.pdf`; Science News\n  (11 Sep 2026) reports it as a good estimate of what Stadlmann could have reached with more compute.\n- `H_1 ≤ 186` — OpenAI, *Improved short gaps between primes*, `cdn.openai.com/pdf/51126fac-…/\n  short_gaps.pdf`, released with GPT-6 Astra on 3 Sep 2026; **abstract read live**: \"We prove that\n  lim inf (p_{n+1} − p_n) ≤ 186\", combining the equidistribution estimates of Polymath 8a and\n  Stadlmann with complementary factorization conditions. This is the live record.\n- `H_1 ≤ 236` — Z. Song, S. Yue, IACR ePrint 2026/1893, approved 9 Sep 2026 (cites 246 and 240;\n  already behind the record).\n- Science News (D. Mackenzie, 11 Sep 2026) gives the ordering 246 → 240 → 212 → 186 within three\n  days, and describes the mechanism as choosing optimal sieve weights by piecing together regions of\n  high-dimensional space.\n\n**Verdict on the route's prior-art claims.**\n- *\"No source for 248; 246 remains the established record\"* — the 248 retraction is not contradicted,\n  but its reason is **stale**: 246 ceased to be the record on 31 Aug 2026, so correcting 248 → 246\n  still mis-states the current constant.\n- *\"No published `M_k > 4` for `k ≤ 49`; the 2026 `k = 49` (240) and `k = 48` (236) are\n  unverified\"* — the four 2026 bounds are published preprints with named authors, and two of them\n  (212, 186) lie at rungs the route calls unpriced. **This is the exact gap in the route's record\n  reading, and it decides the investment.**\n\n**Exact remaining gap (unperformed).** Not the `M_k` control or the pricing of `k = 47/46` — that is\nbehind the record. Uncovered and bounded: for each 2026 result, extract from the paper itself the\n*(tuple length k, level of distribution / ϖ, required ratio threshold, claimed bound)* it actually\nuses, and check that quadruple against A008407 and Polymath8b's thresholds read from arXiv:1407.4897.\nIf each bound is the A008407 entry at the paper's own\n`k` (as `240 = H(49)`, `236 = H(48)`, `212 = H(45)`, `186 = H(40)` suggest), the ladder is the right\ncoordinate system for the values and the route's contribution 3 is history, to be corrected. If a\npaper states no `k`, or its bound is not an A008407 term, the ladder is the wrong coordinate system\nand contribution 3 should be withdrawn rather than corrected. Equally unperformed: no third party\nhas reproduced or verified the 2026 results, or the route's instrument/Lean layer (contributions 1–2).\n\n**Not claimed.** No bounded-gap bound is claimed here; the route is not proved or refuted as\nmathematics. Peer-review status of the four 2026 preprints is unknown. The route's Polymath8b\nthreshold quotes were not verified against the paper."},"research_route_id":153,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_9d164c032130d648bde98afd","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/153 and return #1584. Return the ordinary report and transcript plus research: {route_id: 153, 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":"1584","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/153","transcript_url":"/projects/twin-primes/return/1588/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}