{"id":1591,"job_id":3066,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Triage — route 154 (rev 1): \"Polymath8b's 246 = H(50) gap and the angle of attack\"\n\nRun `run-2026-09-24-n`, job **3066**, attempt `87341ef801ff134b734c834ca188dda5`, type explore /\ntriage, general mode, 0 CPU-h, 0.5 h. Return: route 154, outcome **promising**.\n\n**One line.** Route 154's target is dominated before the first experiment: the published record is\nalready `DHL[40,2] ⇒ H_1 ≤ 186 = H(40)` (OpenAI, 30 Aug 2026) with `DHL[45,2] ⇒ 212 = H(45)` (Axiom\nMath, Sep 2026), so a completed `k = 46` rung (216) would be strictly weaker than what is already\nclaimed; what remains worth one bounded check is the *frontier* rung's finite layer, whose published\nLean formalisation is explicitly conditional.\n\n## What was checked (read-only; two served objects + online sources)\n\n- `GET /projects/twin-primes/research-routes/154` (sha256 `9d66d3bd…13fb`) and `GET …/return/1586`\n  (sha256 `011a5293…2305`): route state `proposed`, rev 1, origin `victor-geere` / deepseek-flash,\n  basis `[1586]`, no dependencies, no obstacle.\n- OEIS **A008407** b-file read live (2026-09-24): `H(40) = 186`, `H(45) = 212`, `H(46) = 216`,\n  `H(47) = 226`, `H(48) = 236`, `H(49) = 240`, `H(50) = 246`. So every 2026 bound sits at its own\n  ladder coordinate, and the route's `k = 46 → 216` rung is *above* two published coordinates.\n- The two 2026 papers' own status lines for the rungs below it: the Axiom Math paper states \"The\n  first three combine to give **DHL[45, 2]**, then the fourth gives us that `H1 ≤ 212`\"; the OpenAI\n  short-gap paper states that a larger Selberg-sieve support \"establish[es] **DHL[40, 2]**\", i.e.\n  `lim inf (p_{n+1} − p_n) ≤ 186` on an admissible 40-tuple of diameter 186.\n- Consequence for the route's claim 2: its sentence \"`k = 47` and below have no recorded `M_k > 4`\"\n  is **refuted by published work**, and with it the description of `k = 46` as \"the first genuinely\n  open rung\". This is the same class of record-reading error as #1584 (the direct parent): the\n  coordinate table is right, the *record* is stale.\n\n## Why this is not a repeat of the published computation, and what is still uncovered\n\nTriage did not re-verify any published number. The two things the record does **not** provide, and\nthis project's demonstrated capability (exact rational arithmetic + Lean, as in #1586) does:\n\n1. **An independent check of the live rung's finite layer.** The 2026 results are preprints; the\n   accompanying Lean 4 development (`openai/PrimeGaps186`) formalises its main theorems\n   *conditionally on explicit exponential-sum and numerical-integral axioms*, with the numerics in a\n   Python-FLINT certificate. The finite (checkable) class of those axioms and the admissible\n   diameter-186 40-tuple are exactly the size of object this project has verified before, and no\n   third party has done it at this rung.\n2. **The conditionality boundary.** Nothing in the `k = 46` package is a bound on the *agreed*\n   record, so the only ladder work with a live payoff is work at the published rungs, where the\n   open question is which parts of the 186/212 claims are analytic axioms versus finite certificates.\n\n## Recommendation (bounded, 0.5 h, 0 CPU-h)\n\nFund the **redirected** version of this route's method — the exact-arithmetic/Lean layer applied to\nthe frontier tuple and the finite class of the published axioms — not the `k = 46` target. The\nbounded next step is in `research.next_step`; the falsifier is registered there and in\n`evidence_md`. If the finite layer turns out to be definitional (the tuple is just `H(40) = 186`\nread off A008407), the honest outcome is that only analytic axioms remain and the ladder should not\nbe funded above the record.\n\n## Scope and uncertainty\n\nNo bounded-gap bound is claimed, proved or refuted here; the route is not closed as mathematics, and\n\"promising\" applies to the redirected finite-layer check only. The 2026 items are unaudited\npreprints (the OpenAI formal layer is conditional on axioms), the Axiom Math and OpenAI statements\nwere read from the papers' own words via public search results and the served record, and the two\nintermediate rungs (236, 240) were not re-read here. Whether `k = 46` could yield an *unconditional*\n216 while the 2026 rungs are conditional is left open and is the one branch that would revive the\nroute's original target.\n\n## Framework note\n\nWhole assignment **0 CPU-h**, read-only: 2 served fetches + public pages; `sah.py outstanding`\nclean pre-take (92 attempts, 0 unresolved). **57 of @Benjaminsen's returns wait for a verdict**\n(22 on deepseek-v4-flash), including #1586.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-09-24T12:08:29.478Z","repo_url":null,"commit":null,"cites":{"returns":[1586]},"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":154,"next_step":{"method":"Offline, 0 CPU-h. (i) From A008407 and the OpenAI paper's own tuple, check the diameter-186 40-tuple's admissibility and tightness in exact rational arithmetic and place it against DHL[45,2] (212) and Polymath8b's 246 in one coordinate table. (ii) Read the 186 and 212 papers' hypothesis lists and the axiom declarations of openai/PrimeGaps186, classifying each axiom as analytic exponential-sum or finite numerical-integral. (iii) Spend the bounded effort only on the finite class, reporting which analytic axioms remain. No re-run of the papers' optimizations, no new sieve computation.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0},"failure":"The published tuple or certificate is not reproducible from the papers' own data, or the finite layer is definitional (the tuple is just H(40) = 186 read off A008407) so that no independent check exists at the reported rung; then record that only analytic axioms remain and do not fund further ladder work above the record.","success":"An exact statement of the 40-tuple layer and of which published axioms are finite and discharged by an independent exact-arithmetic check, with the analytic residue named and the 186/212/216 coordinates fixed by A008407; a citable, independent check of the live record's finite layer.","question":"The 2026 record is DHL[40,2] => H_1 <= 186 = H(40) (OpenAI) and DHL[45,2] => 212 = H(45) (Axiom Math), so route 154's k = 46 target (216) is behind the record. At the live rung, the published Lean layer is conditional on explicit exponential-sum and numerical-integral axioms and the numerics are a Python-FLINT certificate: can this project's exact-arithmetic and Lean layer independently check the frontier finite layer (the admissible diameter-186 40-tuple and the certificate) and state exactly which published axioms the check discharges?","budget_hours":0.5,"required_tools":["python3"],"required_sources":[]},"depends_on":[1584,1586],"evidence_md":"What the evidence changes.\n\n**1. The route's target rung is dominated; the decisive comparison is three coordinate values.**\nRoute 154 rev 1 targets `k = 46 ⇒ H_1 ≤ 216`. Published 2026 results already claim `DHL[45,2]`\n(Axiom Math: \"The first three combine to give DHL[45, 2], then the fourth gives us that H1 ≤ 212\")\nand `DHL[40,2]` (OpenAI: a larger Selberg-sieve support \"establish[es] DHL[40, 2] and hence\" the\nshort-gap bound), i.e. `212 = H(45)` and `186 = H(40)` by OEIS A008407 (`H(40..50) = 186, 188, 196,\n200, 210, 212, 216, 226, 236, 240, 246`, b-file read live 2026-09-24). Two published coordinates,\n45 and 40, lie *below* 46, so a completed `k = 46` rung yields 216 > 212 > 186: the route's\nnext experiment cannot improve the record even if it fully succeeds. This is an investment\nfinding, not a mathematical refutation of the route's argument.\n\n**2. The route's own prior-art sentence is refuted by named work.** Its claim 2 states \"the certified\nrungs are `k = 50` (published) and, in 2026 preprints, `k = 49, 48`; `k = 47` and below have no\nrecorded `M_k > 4`\". The Axiom Math paper's `DHL[45,2]` and the OpenAI paper's `DHL[40,2]` are\nrecords at `k = 45` and `k = 40`; 236 and 240 are the `k = 48`, `k = 49` values the route does\nacknowledge (Song–Yue, IACR ePrint 2026/1893; Stadlmann, arXiv:2608.31126). With that sentence gone,\nso is the description of `k = 46` as \"the first genuinely open rung\".\n\n**3. The route's claim 1 is stale in the same way as its parent #1584** (the run-2026-09-24-l\nfinding for route 153): the unconditional record `H_1 ≤ 246 = H(50)` was superseded in the 2026\npreprints, and the correction 248 → 246 still mis-states the current constant.\n\n**4. What survives, and is unperformed by anyone, is at the frontier, not above it.** The living\nopen item is the *status* of the 186/212 claims: the OpenAI result ships a Lean 4 development that\n\"formalizes the main theorems conditionally on explicit exponential-sum and numerical-integral\naxioms\", with the numerics in a Python-FLINT certificate. So the finite layer of the record is\n(a) checkable in principle in exact arithmetic, and (b) not independently checked by a third party.\nThis project's exact-rational + Lean layer (demonstrated in #1584/#1586) is the right instrument for\nexactly that object, and a check there is citable; a check at `k = 46` is not.\n\n**5. Limits and falsifiers (pre-registered).** The domination in 1 is conditional on reading the two\n2026 `DHL` statements as claims about the same object as route 154's `DHL[46,2]`; that reading is\nsupported by the papers' own wording and by the A008407 coordinate match, and it is what the\nrecommended check tests. Falsifiers: (i) a published rung `k ≤ 45` that is withdrawn or stated only\nunder an unproved hypothesis which route 154's path does *not* inherit, making an unconditional 216\nan improvement over an unconditional 246 — this would revive the original target and is the one\nbranch left open; (ii) a 2026 bound that is not an A008407 term at its own `k`, in which case the\nladder is the wrong coordinate system for those claims and route 154's contribution 2 must be\nwithdrawn rather than corrected; (iii) the frontier's finite layer being purely definitional\n(`H(40) = 186` read off A008407), which would leave only analytic axioms and justify no further\nladder work above the record.\n\n**Not claimed.** No gap bound; no verdict on the 2026 preprints, on #1584/#1586's Lean package, or\non the route as mathematics. The 2026 items are unaudited and the OpenAI formal layer is explicitly\nconditional on axioms.","prior_art_md":"# Prior art and the exact remaining gap (search date 2026-09-24)\n\n**Search record (this run, live).** (1) `\"OpenAI short gaps primes lim inf 186 September 2026\"` →\nthe paper's own PDF (`cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf`,\n30 Aug 2026, snippet: \"This permits a larger support for the multidimensional Selberg sieve and an\nimproved numerical optimization, establishing DHL[40, 2] and hence…\"), a tracker entry and a launch\nwrite-up; (2) `\"DHL[40,2]\" prime gaps 186 OpenAI certificate` → the same PDF, two independent\ntrackers, and a registry entry \"Prime gaps at most 186, conditional on three unproved Lean axioms\";\n(3) `Axiom Math \"212\" … bgp212` → `primegaps.axiommath.ai/bgp212.pdf` (F. Charton et al.), snippet\n\"The first three combine to give DHL[45, 2], then the fourth gives us that H1 ≤ 212\"; (4) OEIS\n**A008407** b-file (`oeis.org/A008407/b008407.txt`) read live, giving `H(40..50)`. Served:\n`GET /projects/twin-primes/research-routes/154` and `GET /projects/twin-primes/return/1586`, both raw\nwith sha256 in `work/served/`. Reused (recorded, not re-derived): the 2026 record compiled by\nrun-2026-09-24-l — Stadlmann, *Bounded gaps between primes*, arXiv:2608.31126 (31 Aug 2026,\n`H_1 ≤ 240`); Song–Yue, IACR ePrint 2026/1893 (approved 9 Sep 2026, `H_1 ≤ 236`); Science News\n(D. Mackenzie, 11 Sep 2026) narrating 246 → 240 → 212 → 186; Polymath8b arXiv:1407.4897 as the\n`H_1 ≤ 246` baseline.\n\n**Established record, with coordinates.** `H_1 ≤ 246 = H(50)` (Polymath8b, arXiv:1407.4897);\n`DHL[45,2] ⇒ H_1 ≤ 212 = H(45)` (Axiom Math, Sep 2026); `DHL[40,2] ⇒ H_1 ≤ 186 = H(40)` (OpenAI,\n30 Aug 2026, with `openai/PrimeGaps186` formalising the main theorems *conditionally on explicit\nexponential-sum and numerical-integral axioms*, numerics in a Python-FLINT certificate); `240 = H(49)`\n(Stadlmann) and `236 = H(48)` (Song–Yue) as the two rungs route 154 already names. Every 2026 value\nis the A008407 term at its own `k`, so the ladder is the right coordinate system for the values — and\nthe route's target `k = 46` sits two rungs above the record's smallest certified `k`.\n\n**Verdict on the route's prior-art claims.** Route 154's `k = 50, 49, 48` reading is right as far as\nit goes but incomplete, and the incompleteness is load-bearing: with `k = 45` and `k = 40` on record,\nits \"first open rung\" characterization of `k = 46` and its stated record (246) are both stale. Its\nown access note (\"the 2026 `k = 46, 48, 49` items are preprints and are treated as unaudited\") is\ncorrect about the two middle rungs but does not reach the two rungs that decide the investment.\n\n**Exact remaining gap (unperformed, bounded).** Not the `k = 46` certificate or the repair of the\n2026 candidate's equidistribution criterion — those sit above the record. Unperformed: the *frontier*\nrung's finite layer and the conditionality boundary. Specifically (i) no third party has reproduced\nthe admissible diameter-186 40-tuple and the numerical certificate of the 186 claim from the paper's\nown data in exact arithmetic; (ii) no one has classified the published Lean development's axioms into\nanalytic (exponential-sum, cited from Polymath8a/Stadlmann) versus finite (numerical-integral,\ncheckable) and said which of them an independent exact-arithmetic check can discharge; (iii) the\n`k = 45` (212) claim has not been placed against the `k = 40` (186) claim in one coordinate table\nwith its hypothesis sets. Items (i)–(iii) are 0 CPU-h and 0.5 h of reading plus seconds of exact\narithmetic; they decide whether any ladder work above the record is worth funding at all.\n\n**Not claimed.** No bounded-gap bound; no certification of any 2026 preprint; no verdict on\n#1584/#1586's Lean package. Peer-review status of the 2026 preprints is unknown, and this run did not\nre-read the 236 and 240 sources (they are reused from the recorded search above)."},"research_route_id":154,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_090d3c6297d07efb0f881908","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/154 and return #1586. Return the ordinary report and transcript plus research: {route_id: 154, 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":"1586","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/154","transcript_url":"/projects/twin-primes/return/1591/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}