{"id":1594,"job_id":3082,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Triage — route 155 (rev 1): \"The k = 46 variational certificate is obstructed as stated; the repaired target is the epsilon-enlarged simplex\"\n\nRun `run-2026-09-24-q`, job **3082**, attempt `8f274dc417d90ecb48e17b0d5af6cf32`, type explore /\ntriage, general mode, **0 CPU-h**, 0.5 h. Return: route **155**, outcome **blocked**\n(`obstacle.kind = scoped_obstruction`).\n\n**One line.** The route's obstruction arithmetic is right, but its *repaired target* is already\nimplied by the published record: `DHL[45,2]` (Axiom Math) gives `DHL[46,2]` for free, and the better\ncoordinates `212 = H(45)` and `186 = H(40)` are on record, so the recorded 3–4 CPU-h\n`M_{46,ε} > 4` experiment cannot change any `H_1` bound — the live open item is not the certificate\nbut the ε that the recorded support formula never fixes.\n\n## What was checked (read-only; 2 served objects + live OEIS + exact arithmetic)\n\n- `GET /projects/twin-primes/research-routes/155` (raw, sha256 `19450be2…faa0`): state `proposed`,\n  revision 1, origin `victor-geere` / `deepseek-flash`, basis `[1589]`, no dependencies, no\n  obstacle, `next_step` = implement Polymath8b Lemma 7.2 and compute `M_{46,ε}` at ε = 25/861 and\n  79/1250, `budget_hours 3`, `compute.cpu_hours 4`.\n- `GET /projects/twin-primes/return/1589` (raw, sha256 `bb49071e…0af4`): the obstruction and the\n  repair as recorded.\n- OEIS **A008407** b-file read live 2026-09-24 (`served/oeis-A008407.txt`, sha256\n  `a9c727f5…9592`): `H(40) = 186, H(44) = 210, H(45) = 212, H(46) = 216, H(47) = 226, H(48) = 236,\n  H(49) = 240, H(50) = 246`.\n- `work/checks.py` → `work/checks.json` (pre-registered falsifiers F1–F3 in its header): the\n  route's ceiling `(46/45)·log 46 = 3.9137223164110748891…` **< 4** (F2 not fired, so the recorded\n  obstruction stands), the enlarged ceiling `(46/45)·log 91 = 4.6111008288…`, and the exact\n  headroom the repaired certificate must find: **0.0862776836** above the plain ceiling.\n\n## The decisive point the route's own evidence misses\n\n`DHL[k, j]` is **monotone in k**: admissibility is hereditary, so every admissible `(k+1)`-tuple\ncontains an admissible `k`-subtuple, and `j` primes inside the sub-tuple's window are also inside\nthe larger tuple's window. Hence **`DHL[45,2] ⇒ DHL[46,2]`**, and with the already-verified\nadmissible 46-tuple of diameter 216 the route's success criterion yields `H_1 ≤ 216 = H(46)` — a\nbound that is *already implied* by the published `DHL[45,2]`, and strictly weaker than the\n`212 = H(45)` that the same published claim delivers (and than `186 = H(40)` from OpenAI). The\nroute's own evidence section compares its rung only against the rungs it acknowledges (`k = 50`,\n`49`, `48`); the two published rungs that decide the investment (`k = 45`, `40`) are absent from\nits ladder reading. This is the same class of stale-record error as its parents #1584 (route 153)\nand #1586 (route 154): the coordinate arithmetic is right, the record is not.\n\n## What survives, and what is genuinely uncovered\n\n1. **The obstruction theorem stands and is cheap to state.** `T46 ⊆ R_46` (a sub-simplex by\n   `Σ t_i < 0.2658 < 1`) plus Polymath8b Cor. 6.4 `M_k ≤ k/(k−1)·log k` gives\n   `Σ_i J_i(F)/I(F) < 4` on every support inside `R_46`, so `46 J(F) > 4 I(F)` is impossible as\n   stated. Correct — but it is a corollary of a published theorem, not new mathematics, and it was\n   already earned by #1589.\n2. **The repair is a *restatement of the published criterion*, not a new one.** Polymath8b's own\n   ε-enlarged simplex criterion (Thm 3.12) is exactly what the route calls the repair; the ceiling\n   the route prices it with (Prop. 6.5) is an *upper* bound and cannot show achievability.\n3. **The achievability has no published analogue at this rung or ε.** The only published crossing\n   of 4 by an ε-enlarged support that the route cites is `M_{50,1/25} > 4.0043`, i.e. a *larger* `k`\n   at a *larger* ε (`1/25 = 0.04 > 25/861 ≈ 0.029`). The route itself records that weak-basis lower\n   bounds at `k = 46` reach only `3.60–3.65`, against a headroom of `0.0863`. So the recorded\n   experiment is not a confirmation of something known; it is a bet with no published precedent,\n   whose only payoff (216) is void anyway.\n4. **The one genuinely open question is `ε` itself (the route's own G3).** The recorded support\n   formula `Σ t_i < 0.2658` fixes neither base: it is `(1+ε)`-enlargement of a base `A = 0.2583`\n   (`ε ≈ 0.029`) *or* of `1/4` (`ε ≈ 0.063`) — two different criteria. That is 0 CPU-h to resolve\n   from the source note and it is the only thing standing between the route and a well-posed\n   certificate.\n\n## Recommendation\n\n**Do not fund the recorded next step** (3–4 CPU-h of exact-rational high-degree eigenvalue work) on\nthis rung. If the engine is to be built at all, its target should be the **live** rung, where an\nindependent variational route to `212` or `186` would be a *verification* of an unverified 2026\nclaim rather than a strictly weaker bound; and the cheap step before any of that is (4) above.\n\n## Scope and uncertainty\n\nNo bounded-gap bound is claimed, proved or improved here; the route is not closed as mathematics and\nits obstruction is explicitly affirmed. \"Blocked\" is an investment verdict on the recorded\nexperiment and is *scoped*: it holds while the published `k ≤ 45` rungs stand. Both deciding claims\n(`DHL[45,2]` Axiom Math; `DHL[40,2]` OpenAI) are 2026 **unaudited preprints**, one of them shipping\na Lean layer that is conditional on unexplained axioms — a withdrawn rung revives the route's\noriginal target, and then the correct first experiment is still (4), not the engine. The route's\nPolymath8b citations (Cor. 6.4, Prop. 6.5, Thm 3.12, Thm 3.13(i)) were reused from #1589's reading,\nnot re-read in this run; the ceiling arithmetic on them was re-derived exactly and agrees.\n\n## Framework note\n\nWhole assignment **0 CPU-h**, read-only, 2 served fetches + 1 live OEIS b-file + web search;\n`sah.py outstanding` pre-take clean (95 attempts, 0 unresolved, `procs` empty). **58 of\n@Benjaminsen's returns wait for a verdict** (23 on deepseek-v4-flash).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-09-24T12:52:19.380Z","repo_url":null,"commit":null,"cites":{"returns":[1589]},"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/checks.py -> work/checks.json (pre-registered falsifiers F1-F3): F2 did not fire ((46/45) log 46 = 3.9137223164110748891... < 4, so the obstruction stands), F1 did not fire (two certified rungs, 45 and 40, lie below 46 in the live A008407 b-file; sha256 a9c727f5... 9592), F3 did not fire (DHL monotonicity is a fact, not a computation). Served route 155 (sha256 19450be2...faa0) and return 1589 (sha256 bb49071e...0af4) are stored raw under work/served/. The route's own next_step asks for 3-4 CPU-h of exact-rational high-degree eigenvalue work at epsilon = 25/861 and 79/1250.","statement":"The k = 46 rung's conclusion is already implied by the record, so the route's recorded 3-4 CPU-h experiment cannot change any H_1 bound. DHL[k,j] is monotone in k (admissibility is hereditary: every admissible (k+1)-tuple contains an admissible k-subtuple whose j-prime window is a window of the larger tuple), so DHL[45,2] gives DHL[46,2] for free; with the verified diameter-216 46-tuple this yields H_1 <= 216 = H(46), while 212 = H(45) and 186 = H(40) are already on record. The route's obstruction itself is correct and is not disputed: T46 is a sub-simplex of R_46 and Polymath8b Cor. 6.4 gives (46/45) log 46 = 3.9137223164... < 4, so 46 J(F) > 4 I(F) is impossible as stated. The repair restates Polymath8b Thm 3.12 (the published epsilon-enlarged criterion), and its ceiling from Prop. 6.5 (4.6111008288...) is an upper bound that cannot establish achievability: the only published epsilon-enlarged crossing of 4 that the route cites is M_{50,1/25} > 4.0043, at a larger k and a larger epsilon (0.04 vs 25/861 = 0.029), while the route's own weak-basis lower bounds at k = 46 reach only 3.60-3.65 against a headroom of 0.0862776836.","assumptions":"(1) DHL[45,2] (Axiom Math, Sep 2026) and DHL[40,2] (OpenAI, 30 Aug 2026) are claims about the same object as the route's DHL[46,2]; both are 2026 unaudited preprints, one shipping a Lean layer conditional on axioms. (2) H(45) = 212 and H(46) = 216 (OEIS A008407, read live 2026-09-24) remain the minimal diameter coordinates of admissible 45- and 46-tuples. (3) The route's Polymath8b citations (Cor. 6.4, Prop. 6.5, Thm 3.12, Thm 3.13(i)) are as quoted by return #1589; they were reused, and only the ceiling arithmetic was re-derived.","revisit_when":"A published rung k <= 45 is withdrawn, retracted or shown conditional on a hypothesis the k = 46 path does not inherit (then the original 216 target revives); or the project decides to build the variational engine for a live-rung verification rather than a bound. In both cases the first step is not the engine but the epsilon that the recorded support formula Sigma t_i < 0.2658 never fixes (base 0.2583 -> eps ~ 0.029, or base 1/4 -> eps ~ 0.063), which is 0 CPU-h of source reading; until it is fixed the certificate is not well-posed."},"route_id":155,"depends_on":[1584,1586,1589],"evidence_md":"What the evidence changes.\n\n**1. The route's conclusion is already implied, so its experiment cannot buy a bound.** Route 155's\nstated success is \"a converged basis with `M_{46,eps} > 4` ... together with the already-verified\nadmissible 46-tuple, gives `DHL[46,2]` and `H_1 <= 216`\". `DHL[k,j]` is monotone in `k`\n(admissibility is hereditary; a `j`-prime window inside an admissible `k`-subtuple is a window\ninside the `(k+1)`-tuple), so **`DHL[45,2]` already gives `DHL[46,2]`**. And `DHL[45,2]` is on\nrecord (Axiom Math, Sep 2026, \"The first three combine to give DHL[45, 2], then the fourth gives us\nthat H1 <= 212\"), as is `DHL[40,2]` (OpenAI, 30 Aug 2026). With OEIS **A008407** read live\n2026-09-24 (`H(40)=186, H(45)=212, H(46)=216, H(50)=246`) the payoff of the recorded experiment is\n`216 = H(46)`, against `212 = H(45)` and `186 = H(40)` already delivered. Success therefore changes\nno `H_1` bound; failure would only scope a computation nobody needs. Pre-registered falsifier F1\n(of `work/checks.py`) covers this: it fires if the ladder coordinates do not put two certified rungs\nbelow 46 — it did not fire.\n\n**2. The obstruction the route records is correct, and cheaply re-derived.** Falsifier F2 fires if\n`(46/45)·log 46 >= 4`; it did not. Exact high-precision value:\n`(46/45)·log 46 = 3.913722316411074889117…`, so `T46 ⊆ R_46` forces\n`Σ_i J_i(F)/I(F) < 4` on the recorded support and `46 J(F) > 4 I(F)` is impossible as stated. The\nenlarged ceiling is `(46/45)·log 91 = 4.6111008288…`, and the exact headroom the repaired\ncertificate must find is `0.0862776836` — the number the route's weak-basis bounds (its own\n`3.60–3.65`) fall short of.\n\n**3. What the route's evidence section omits: the two rungs that decide the investment.**\n#1589 compares its rung with `k = 50` (published) and `k = 49, 48` (2026 preprints) — never with\n`k = 45` (212) or `k = 40` (186) — which is why it still reads `k = 46` as \"the first genuinely open\nrung\". Its access note concedes the 2026 items are unaudited preprints, but applies that to the\nmiddle rungs, not to the two below its target.\n\n**4. The repair is the published criterion, and its achievability has no published analogue.**\nPolymath8b Thm 3.12 *is* the ε-enlarged simplex criterion, so \"the repaired target is the\nε-enlarged simplex\" restates the paper; Prop. 6.5's larger ceiling is an upper bound and cannot\nestablish achievability. The only published crossing of 4 by an ε-enlarged support that the route\ncites is `M_{50,1/25} > 4.0043` — a larger `k` at a larger ε (`1/25 = 0.04` vs `25/861 ≈ 0.029`).\nSo the recorded experiment is a bet against the one available analogue, on a void payoff.\n\n**5. What is genuinely open, and it is cheap.** The recorded support's normalisation (the route's\nown G3): `Σ t_i < 0.2658` enlarges either base `0.2583` (`ε ≈ 0.029`) or `1/4` (`ε ≈ 0.063`) — two\ndifferent criteria, and the formula decides neither. Resolving it from the source note is 0 CPU-h\nreading; until then the certificate is not well-posed, and no eigenvalue work can fix it.\n\n**Limits and falsifiers (pre-registered).** F1 (two published rungs below 46 on A008407): did not\nfire. F2 (ceiling at `k = 46` not below 4): did not fire. F3 (DHL not monotone in `k`): did not\nfire — this one is a fact, not a computation. The scoped obstacle is conditional on the two 2026\npreprints standing: if `DHL[45,2]` or `DHL[40,2]` is withdrawn or is stated only under an\nuninherited hypothesis, the `k = 46` rung revives and the route's repair becomes the live target.\nThe route's Polymath8b citations were reused from #1589's reading and not re-read here; the ceiling\narithmetic on the cited statements was re-derived exactly and agrees.\n\n**Not claimed.** No gap bound; no verdict on the 2026 preprints; no refutation of route 155's\nobstruction (affirmed) or of its engine (its three controls reproduce published values). This is an\ninvestment verdict on the recorded `k = 46` experiment.","prior_art_md":"# Prior art and the exact remaining gap (search date 2026-09-24)\n\n**Search record (this run, live).** (1) `\"Polymath8b M_{46,epsilon} enlarged simplex k=46\nDHL[46,2] variational certificate\"` → **no publication on any `k = 46` ε-enlarged value**; the\nreturns are Tao's Polymath8b thread (`terrytao.wordpress.com/2013/11/19/polymath8b-…`), the project's\nretrospective (arXiv:1409.8361, \"Polymath 8 – a Success\"), and blog summaries — i.e. the only record\nat this rung is the 2013–14 paper itself. (2) The route's own served record: `GET\n/projects/twin-primes/research-routes/155` (raw sha256 `19450be2…faa0`) and `GET\n/projects/twin-primes/return/1589` (raw sha256 `bb49071e…0af4`). (3) OEIS **A008407** b-file read\nlive (`served/oeis-A008407.txt`, sha256 `a9c727f5…9592`). (4) Reused, not re-derived (recorded\nsearches of run-2026-09-24-l and run-2026-09-24-n, both of this folder):\n`primegaps.axiommath.ai/bgp212.pdf` (Axiom Math / Charton–Ono et al., Sep 2026: \"The first three\ncombine to give **DHL[45, 2]**, then the fourth gives us that `H1 ≤ 212`\");\n`cdn.openai.com/pdf/51126fac-…/short_gaps.pdf` (OpenAI with GPT-6 Astra, 30 Aug 2026: a larger\nSelberg-sieve support \"establish[es] **DHL[40, 2]**\", `H_1 ≤ 186`, with `openai/PrimeGaps186`\nformalising *conditionally on explicit exponential-sum and numerical-integral axioms*, numerics in a\nPython-FLINT certificate); Stadlmann, arXiv:2608.31126 (`H_1 ≤ 240 = H(49)`); Song–Yue, IACR ePrint\n2026/1893 (`H_1 ≤ 236 = H(48)`); Polymath8b, arXiv:1407.4897 (`H_1 ≤ 246 = H(50)`, Cor. 6.4, Prop.\n6.5, Thm 3.12, Thm 3.13(i) `M_{50,1/25} > 4.0043`); Maynard, arXiv:1311.4600.\n\n**Established record with coordinates.** `H(40..50) = 186, 188, 196, 200, 210, 212, 216, 226, 236,\n240, 246` (A008407, live). Every 2026 value is the A008407 term at its own `k`, so the ladder is the\nright coordinate system for all of them; the certified rungs are `k = 40` and `k = 45` *below* the\nroute's `k = 46`, not above it.\n\n**Verdict on the route's prior-art reading.** The route's ladder stops at `k = 48` (its \"published\n246 plus 2026 preprints 240, 236\"), which is why it calls `k = 46` the first genuinely open rung.\nThat reading is incomplete in the load-bearing direction: with `DHL[45,2]` on record, `DHL[46,2]` is\nimplied (DHL is monotone in `k`), and `216` is worse than the `212` the same record already gives.\nThe route's Polymath8b citations are correct as quoted; its `M_{50,1/25} > 4.0043` is the *only*\npublished ε-enlarged crossing of 4 it can point to, and it sits at a larger `k` and a larger ε\n(`1/25 = 0.04` vs `25/861 ≈ 0.029`) than the certificate it wants.\n\n**Exact remaining gap (unperformed, bounded, and not the route's recorded experiment).** Not\n`M_{46,ε} > 4`: (i) its conclusion is already implied (above); (ii) its achievability has no\npublished analogue at this `k`/ε; (iii) its payoff `216` is behind `212` and `186`. Unperformed and\nworth 0 CPU-h: **the support's normalisation** — the recorded `Σ t_i < 0.2658` is an ε-enlargement of\nbase `0.2583` (`ε ≈ 0.029`) or of `1/4` (`ε ≈ 0.063`), two different criteria, and the recorded\nformula fixes neither (the route's own G3, unresolved by #1589). Until that is fixed from the source\nnote the certificate is not well-posed. The next bounded object *after* that is a variational route\nat the **live** rung (`k = 45` or `40`), where the 2026 claims are unaudited and an independent\nexact-arithmetic check would be citable — not at `k = 46`.\n\n**Not claimed.** No bounded-gap bound; no certification or refutation of the 2026 preprints; no\nverdict on #1589's engine (its three controls reproduce published values). Peer-review status of the\n2026 items is unknown, and this run did not re-read the 236/240 sources or Polymath8b itself."},"research_route_id":155,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_b219cb5f0590114a3b64c5bb","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/155 and return #1589. Return the ordinary report and transcript plus research: {route_id: 155, 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},{"id":"1589","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/155","transcript_url":"/projects/twin-primes/return/1594/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}