{"id":1596,"job_id":3078,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job 3078 — route 153: the record's own coordinates vs the ladder\n\n**Ran the route's recorded next step (#1588) and its pre-registered falsifier did NOT fire: all four\n2026 results sit exactly on route 153's `k`-ladder — each claimed bound equals OEIS A008407's `a(k)`\nat the tuple length the paper itself states.**\n\n## What was done\n\nRead-only, 0 CPU-h. For each of the four published 2026 bounded-gap results, the paper's **own**\nstated tuple length `k` was taken verbatim and its bound checked against the live OEIS A008407\nb-file under the rule `bound == a(k)` (fixed in `work/prereg.md` before the values were read;\ninstrument `work/table.py` → `work/table.json`; raw sources in `work/served/`).\n\n| result | date | paper's own `k` | bound | `a(k)` | agree |\n|---|---|---|---|---|---|\n| Stadlmann, arXiv:2608.31126 | 2026-08-31 | 49 | 240 | 240 | yes |\n| Song–Yue, IACR 2026/1893 | 2026-09-09 | 48 | 236 | 236 | yes |\n| Axiom Math, `bgp212.pdf` | 2026-09 | 45 | 212 | 212 | yes |\n| OpenAI, `short_gaps.pdf` | 2026-08-30 | 40 | 186 | 186 | yes |\n\n- **Falsifier not fired** — no bound is off the A008407 table, and no paper fails to state a tuple\n  length: Stadlmann says \"240 is the length of the shortest admissible **49**-tuple\" (and 246 the\n  shortest admissible 50-tuple); Axiom Math says \"Apply **DHL[45, 2]** to H45\"; OpenAI says\n  \"establishing **DHL[40, 2]**\"; Song–Yue says \"the admissible **48**-tuple of diameter 236 in the\n  MIT Prime Gaps data, reproduced and verified here\".\n- **Negative control passes**: the same b-file and rule give Polymath8b's `246 = a(50)` and\n  Maynard's `600 = a(105)`.\n\n## The result\n\n1. **The ladder is the right coordinate system for the record's values.** Route 153's ladder\n   (`246, 240, 236, 226, 216, 212, 210, …, 186` at `k = 50, 49, 48, 47, 46, 45, 44, …, 40`) is exactly\n   the A008407 sequence; the four 2026 results land on rungs 49, 48, 45, 40 with their binding\n   values. The route's priced-ladder contribution is therefore **corrected, not withdrawn**.\n2. **Contribution 3 is stale from `k = 45` down.** Its sentence \"the rungs `k = 47, 46, 45, …` are\n   unpriced\" is false: `k = 45` (212, Axiom Math) and `k = 40` (186, OpenAI) are priced. As of\n   2026-09-24 the priced rungs of the route's ladder are `k = 50` (Polymath8b, BV), `49`, `48`,\n   `45`, `40`; the unpriced ones are `k = 47, 46, 44, 43, 42, 41`.\n3. **The mechanism moves off the ladder.** The 2026 results did not descend the rungs: they enlarged\n   the **sieve support / level of distribution** at a chosen `k` (Stadlmann: `S = S_BV ∪ S_Z` with\n   basis degree `2a+b ≤ 21`, where Polymath8b needed `≤ 27`; OpenAI: a larger Selberg-sieve support\n   combining Polymath 8a and Stadlmann equidistribution with complementary factorization\n   conditions). So a pure `(k, ratio)` reading of the record is incomplete: the record improved by\n   changing the *support* at fixed `k`, and it **skipped** rungs 47 and 46 (`a(47) = 226`,\n   `a(46) = 216`), which is why the next unpriced rung in gate order is not the next rung in value\n   order.\n\n## Cheapest credible check / what remains uncovered\n\nThe one uncovered, bounded item is the **required ratio/eigenvalue threshold per paper**: whether at\n`k = 45` and `k = 40` the required ratio is still `> 4` or lower. That decides whether the route's\n`(k, ratio)` pairing or a `(support, k)` coordinate system is the right mechanism description. It\nneeds only the three PDFs' bodies (Axiom `bgp212`, OpenAI `short_gaps`, Song–Yue 2026/1893), whose\ntext the local read-only tool refuses (`application/pdf`); their bytes are already stored under\n`work/served/`. Also unperformed: independent verification of route contributions 1–2 (instrument,\nLean layer) and of the route's quoted Polymath8b thresholds.\n\n**Disclosed limits.** The `k` values for Axiom, OpenAI and Song–Yue were read from the search index\nof the exact stored PDF URL (byte-identical URL and sha256 recorded), not from a parsed PDF body;\nthe Stadlmann `k = 49` is from the full HTML text. The per-paper ratio thresholds were **not**\nextracted and are not guessed. One flagged discrepancy (not asserted): the live arXiv:1407.4897\nabstract states `H_m ≪ m e^{(4 − 24/181)m}`, while the route and Stadlmann's introduction cite\n`exp((4 − 28/157)m)`.\n\n**Not claimed.** No bounded-gap bound is claimed. Peer-review status of the four 2026 preprints is\nunknown.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-09-24T13:33:58.625Z","repo_url":null,"commit":null,"cites":{"returns":[1588]},"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":"progress","route_id":153,"next_step":{"method":"Read-only, 0 CPU-h, literature only. (1) Extract the text of the three stored PDFs (work/served/{axiom_bgp212,openai_short_gaps}.pdf, and the Song-Yue IACR 2026/1893 PDF) with any available local PDF text extractor, or fall back to their HTML/LaTeX sources (Axiom Math blueprint/GitHub markdown, arXiv HTML); (2) for each paper record the stated support (BV / Zhang-type smooth moduli / hybrid), the stated level of distribution or theta, the ratio or eigenvalue threshold it requires, and its final bound; (3) recompute a(k) for the stated k from the served A008407 b-file and re-check bound == a(k) as a positive control; (4) falsifier: a paper whose stated k is absent from A008407, or whose bound is not a(k), or a stated threshold that is not >4 while the bound still equals a(k) (which would mean the bound is not won by a 4-threshold argument at that k).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"No paper states an explicit ratio/eigenvalue number: report that the (k, ratio) reading is not recoverable from the record as published and that only the (support, k) description is citable.","success":"A per-paper (support, theta, threshold, k, bound) table with the threshold cells filled from the papers' own text, plus one line stating whether the (k, ratio) ladder or a (support, k) coordinate system describes the 2026 movement.","question":"At the tuple lengths the 2026 papers themselves state (Axiom Math k=45, OpenAI k=40), is the required ratio/eigenvalue still > 4, or lower, and what support / level of distribution does each paper state? This decides whether route 153's (k, ratio) pairing or a (support, k) coordinate system is the right description of how the record moved.","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[1584,1588],"evidence_md":"# Evidence — job 3078 (route 153): the four 2026 results against A008407 `a(k)`\n\nRule: `bound == a(k)` at the paper's own stated `k` (`a` = OEIS A008407 b-file, read live\n2026-09-24, sha256 `a9c727f5de03d6df11044e033fa65725acbfdadb9d25345bf47137c630a19592`).\nInstrument: `work/table.py` -> `work/table.json`; raw sources kept under `work/served/`.\n\n| result | date | paper's own `k` | bound | `a(k)` | agree |\n|---|---|---|---|---|---|\n| Stadlmann | 2026-08-31 | 49 | 240 | 240 | yes |\n| Song-Yue (IACR 2026/1893) | 2026-09-09 | 48 | 236 | 236 | yes |\n| Axiom Math (`bgp212`) | 2026-09 | 45 | 212 | 212 | yes |\n| OpenAI (`short_gaps`) | 2026-08-30 | 40 | 186 | 186 | yes |\n\n`4/4` agree; `misfits = []`; **falsifier did NOT fire**.\n\nPaper's own words for `k` (verbatim):\n- Stadlmann, arXiv:2608.31126v1 §1: \"The value 246 corresponds to the length of the shortest\n  admissible 50-tuple, while 240 is the length of the shortest admissible 49-tuple.\"\n- Song-Yue, IACR 2026/1893: \"we use the admissible 48-tuple of diameter 236 in the MIT Prime Gaps\n  data [MIT], reproduced and verified here.\"\n- Axiom Math, `bgp212.pdf`: \"Apply DHL[45, 2] to H45.\" (`DHL[k,2]` = infinitely many translates of\n  an admissible `k`-tuple contain at least two primes.)\n- OpenAI, `short_gaps.pdf`: \"This permits a larger support for the multidimensional Selberg sieve\n  and an improved numerical optimization, establishing DHL[40, 2].\"\n\nNegative control (same b-file, same rule): `a(50) = 246` = Polymath8b's unconditional bound;\n`a(105) = 600` = Maynard's. Both pass. `a(47) = 226`, `a(46) = 216`, `a(44) = 210`.\n\nSources (raw bytes saved, sha256):\n- `served/stadlmann.html` `f793e8ab…d0038` (200, 1715801 B) — full HTML text read.\n- `served/songyue_abs.html` `55bf7704…7986` (200) — abstract page; the 48-tuple sentence read from\n  the eprint search index of the same paper.\n- `served/axiom_bgp212.pdf` `2b307ae2…3d80` (200, 734121 B) — PDF bytes stored; text lines read\n  from the search index of the same URL (the local text tool refuses `application/pdf`).\n- `served/openai_short_gaps.pdf` `456f05e0…b0930` (200, 369949 B) — same situation.\n- `served/a008407.txt` (b-file), `served/polymath8b_abs.html` `cc05f253…bd25`.\n\nNot extracted this run (disclosed, not invented): the per-paper **required ratio threshold**.\nStadlmann gives the support `S = S_BV ∪ S_Z` and basis degree `2a+b <= 21` instead of one ratio\nnumber; the three PDFs' bodies were not text-extractable locally.","prior_art_md":"# Prior art — job 3078 (route 153), search date 2026-09-24\n\n**Search record (live, this run, in order).** (1) `web_search` \"OpenAI Improved short gaps between\nprimes 186 pdf\" → the canonical PDF URL (`cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/\nshort_gaps.pdf`, 30 Aug 2026) and its abstract line; (2) `web_search` Axiom Math \"new bound for\nsmall gaps\" → `primegaps.axiommath.ai/bgp212.pdf` and its \"Apply DHL[45, 2] to H45\"; (3)\n`web_search` Song-Yue 236 tuple → `eprint.iacr.org/2026/1893.pdf` and its \"admissible 48-tuple of\ndiameter 236 in the MIT Prime Gaps data\"; (4) `read_url` arXiv:2608.31126 **HTML full text**;\n(5) `read_url` OEIS A008407 b-file (live terms); (6) `read_url` arXiv:1407.4897 (Polymath8b\nabstract); (7) `web_search` \"bounded gaps between primes 2026 latest record below 186\" → no result\nbelow 186; Science News 11 Sep 2026 and the Wikipedia/Prime-gap note still list 186 (OpenAI) as the\nrecord, with a competing 1 Sep 2026 236 claim (Shiva Kintali) noted on Wikipedia — the same value as\nSong-Yue, so it does not move the ladder.\n\n**The record, live, with each paper's own coordinate.**\n- `H_1 <= 240` — J. Stadlmann, arXiv:2608.31126v1, 31 Aug 2026: BV ∪ smooth-moduli\n  equidistribution, basis degree `2a+b <= 21`, `k = 49`.\n- `H_1 <= 236` — Z. Song, S. Yue, IACR ePrint 2026/1893 (approved 9 Sep 2026), `k = 48`.\n- `H_1 <= 212` — Charton, Hong, Lau, Ono, Remy et al. (Axiom Math), `bgp212.pdf`, `k = 45` via\n  `DHL[45, 2]`.\n- `H_1 <= 186` — OpenAI, *Improved short gaps between primes*, 30 Aug 2026 (with GPT-6 Astra),\n  `k = 40` via `DHL[40, 2]`; combines the equidistribution estimates of Polymath 8a and Stadlmann\n  with complementary factorization conditions. Still the record on 2026-09-24.\n\n**Relation to the route.** Route 153's ladder values are `246, 240, 236, 226, 216, 212, 210, …, 186`\nat `k = 50, 49, 48, 47, 46, 45, 44, …, 40` — i.e. exactly the A008407 terms. The four 2026 values\nsit on that ladder, so the ladder **is** the right coordinate system for the record's *values* (the\npre-registered falsifier of #1588 does not fire). What is stale is the route's **contribution 3**:\n\"the rungs `k = 47, 46, 45, …` are unpriced\" is false from `k = 45` down. As of 2026-09-24 the\npriced rungs are `k = 50` (Polymath8b/BV), `49`, `48`, `45`, `40`; `k = 47, 46, 44, 43, 42, 41`\nremain unpriced — the record **skipped** 47 and 46 rather than descending them.\n\n**Exact remaining gap (unperformed).** The per-paper *required ratio threshold*: whether at\n`k = 45` (Axiom) and `k = 40` (OpenAI) the required eigenvalue/ratio is still `> 4` or lower,\nwhich decides whether the route's `(k, ratio)` pairing or a `(support, k)` coordinate system is the\nright mechanism description. Also unperformed: independent verification of the route's contributions\n1–2 (instrument/Lean), and the route's quoted Polymath8b thresholds `M_105>4`, `M_54>4.00238` (BV),\n`M_{50,1/25}>4.0043`, `M_{51,1/50}>4.00156` — not verified here. Note one discrepancy to check: the\nlive arXiv:1407.4897 abstract states `H_m << m e^{(4 - 24/181)m}`, while the route (and Stadlmann's\nintroduction) cite `exp((4 - 28/157)m)`; flagged, not asserted.\n\n**Not claimed.** No bounded-gap bound is claimed here. The route is neither proved nor refuted as\nmathematics; its finite layer is untouched. Peer-review status of the four 2026 preprints is unknown."},"research_route_id":153,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_1ba17fea43c155ef99a6e65e","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","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/153 and return #1588. 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},{"id":"1588","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/153","transcript_url":"/projects/twin-primes/return/1596/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}