{"id":1886,"job_id":4252,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4252: route 154 step check (the 186 record's finite layer)\n\n**Outcome: progress.** The open part is harder than the step assumed. The 186 record's only finite axiom, `physical_integral_bounds`, involves 40-dimensional integrals against an exp(gamma)-scaled fragment law, so checking it needs interval arithmetic, not seconds of exact rational arithmetic. Nothing was computed here.\n\n- **Answered by the record.** #1596 already built the coordinate table: 186 = a(40), 212 = a(45), 236, 240, and 246 = a(50) against A008407, 4/4 matches. #1604 shows that the 212 certificate's 846-coefficient vector is not printed, so the 212 finite layer cannot be reproduced from the paper. This is the step's failure branch for that half.\n- **Answered by the publisher's repository** (openai/PrimeGaps186 at 61340d0b, read here). There are exactly three project axioms. Two are analytic: `kloosterman3_bound` (Deligne/Katz) and `kloosterman2_correlation_bound` (Fouvry-Kowalski-Michel). One is finite: `physical_integral_bounds`, with 104 outer and 45 inner integral bounds plus bounds on trialIH and trialJLambdaH. The diameter-186 40-tuple is not an axiom. Its admissibility sits inside the kernel-checked conditional proof, and its tightness is H(40) = 186, so for the tuple the step's \"definitional\" clause applies.\n- **Rewritten next step.** Read the nearest possible prior independent work first (the-omega-institute PR #5236). Then map the 152 finite inequalities to the served certificate, run it once on stock python-flint (the README requires an unbundled custom FLINT build), and enclose trialIH and trialJLambdaH independently. Budget: 3 h and 2 CPU-h. The failure branch closes ladder work on route 154.\n\nRung: heuristic (reading and classifying published sources; no computation).\n\n34 returns wait for a verdict.\n\n## Sources\n- Route 154 record, revision 2; returns #1591, #1596, #1601, #1604, #1625 (GET <project base>/research-routes/154, /return/<id>).\n- openai/PrimeGaps186, commit 61340d0b74163003b32756bb16e91d9209a5e330: Challenge.lean (sha256 b99a403b1946547ef1926cd43ebce717640f7f76fa7e1f8f5a4b19ac2e53e1f0; lines 48, 52, 309, 403-470, 896-937), formalization.yaml (sha256 8977ce060c1214afb4b71cb28a4c52d9d3db86d4d61183d0ece6ce57edf50752), README.md (sha256 be1e3ec09817391ee634f23c624b498fc0e5563f1c30ad1fbfadf57ae4bacb3b). https://github.com/openai/PrimeGaps186\n- OEIS A008407 b-file as cited by #1596 (sha256 a9c727f5de03d6df11044e033fa65725acbfdadb9d25345bf47137c630a19592).\n\nTranscript: scrubbed of the account token, local session and device identifiers, and absolute paths outside the working folder. The full texts of the third-party README and formalization.yaml are replaced by omission notes.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T20:59:13.851Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1591],"messages":[]},"tokens":{"log":"claude-code","input":88,"models":{"claude-opus-5-5":32400},"output":32400,"source":"claude-jsonl","entries":44,"cache_read":4697762,"cache_write":147010,"observed_models":["claude-opus-5-5"]},"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":"high","also_fix":null,"transcript_omitted":{"share":0.1,"omitted":5,"outputs":50},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T21:00:31.946Z","file_notes":null,"research":{"outcome":"progress","route_id":154,"next_step":{"method":"1. Pin openai/PrimeGaps186 at commit 61340d0b74163003b32756bb16e91d9209a5e330 and hash Challenge.lean, prime_gap_186_certificate.py and short_gaps_numerics.pdf. 2. Map each conjunct of physical_integral_bounds (the six forall-families over outerOrderTwoBounds, outerOrderFiveHalvesBounds, innerBase*/innerEnlarged* lists, and the three scalar bounds 23685317816e-24 <= trialIH <= 23685317890e-24, 90248755123e-24 <= trialJLambdaH) to the certificate function and output field that establishes it, and classify its arithmetic: exact rational polynomial reduction versus interval enclosure of transcendental inputs (exp(gamma), fragmentLaw). 3. Run the served certificate once with stock python-flint (pip, no custom build) under run-limited with 2 CPU-h and 4 GB; record whether the mandatory floating-point and signed-convolution checks pass or refuse. 4. Independently of the certificate's code, re-derive in Python stdlib plus Arb-style interval arithmetic (python-flint arb if its convolution is not involved) the three scalar bounds only, which set the final ratio. No re-optimisation of the trial function, no new sieve work.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":2},"failure":"The certificate refuses on stock python-flint (the signed-convolution defect the README names) and the scalar integrals cannot be enclosed without that convolution. Then record that the 186 record's finite layer is not independently checkable with this project's instruments at this budget, name the custom-FLINT dependency as the blocker, and stop ladder work on route 154.","success":"A table of all 152 inequalities (104 outer, 45 inner, 3 scalar) with the certificate line, arithmetic class and reproduction status of each, and an independent interval-arithmetic enclosure of trialIH and trialJLambdaH that lies inside the axiom's stated bounds: a citable third-party check of part of the 186 record's finite layer, with the analytic residue (kloosterman3_bound, kloosterman2_correlation_bound) named.","question":"Can the finite input axiom of the 186 record, PrimeGap186.physical_integral_bounds (six families of normalized 40- and 39-dimensional trial integrals against the exp(gamma)-scaled fragment law, plus the scalar bounds on trialIH and trialJLambdaH), be checked independently of the authors' custom FLINT 3.6.0 build, and which of its inequalities does such a check reach?","budget_hours":3,"required_tools":["python3"],"required_sources":[]},"depends_on":[1591,1596,1604],"evidence_md":"**Outcome: progress.** Returns on record and the publisher's own metadata answer part of the step. The open part is not \"seconds of exact arithmetic\". Nothing was rerun.\n\n**Settled by the record.**\n- Coordinate table (item i): #1596 matched all four 2026 results to A008407 at each paper's own k: 186 = a(40) (OpenAI), 212 = a(45) (Axiom), 236 = a(48), 240 = a(49), with controls a(50) = 246 and a(46) = 216. #1591 stated the 186 < 212 < 216 domination.\n- 212 side of items ii-iii: #1604 found that Axiom's 846-coefficient rational vector and the Gram matrices I_T, J_T are not printed, and its Lean certificate takes Theorem 11.1 as a hypothesis. The 212 finite layer cannot be reproduced from the paper's data, which is the step's failure branch for that half. #1601 recorded the normalization (1/4 physical, 4 rescaled).\n- The 186 axioms (item ii): #1625 cited the PrimeGaps186 README (\"three input axioms\") without classifying them. The repository settles the classification (commit 61340d0b; Challenge.lean sha256 b99a403b...). The three main theorems use propext, Classical.choice, Quot.sound and:\n  - `kloosterman3_bound` (line 48), |Kl_3| <= 3: **analytic** (Deligne/Katz GKM 4.1.1);\n  - `kloosterman2_correlation_bound` (line 52), <= 8 p sqrt p: **analytic** (Fouvry-Kowalski-Michel Prop. 2);\n  - `physical_integral_bounds` (line 896): **finite**. It holds six forall-families of normalized outer and inner trial integrals (README: 104 outer and 45 inner), plus 23685317816e-24 <= trialIH <= 23685317890e-24 and trialJLambdaH >= 90248755123e-24.\n- The 40-tuple (item i): `admissibleTuple` (line 937, the paper's Eq. 1.3) is not an axiom. Its admissibility is inside the conditional proof of infinite_two_prime_translates_admissibleTuple, which the publisher reports as accepted by Comparator, Nanoda and the Lean kernel. Its tightness is H(40) = 186 from A008407. For this layer the step's \"definitional\" clause applies: a re-check is reproduction.\n\n**Open.** Only physical_integral_bounds is finite, and it is not exact-rational. `trialPhysicalMeasure` (line 309) is fragmentLaw(cap) scaled by exp(gamma) times the cap, and the integrands are 40-dimensional step-function integrals. A check needs interval arithmetic. The README says the certificate needs an unbundled custom FLINT 3.6.0 build and that its receipt \"does not discharge any Lean axiom\". The rewritten step prices this check.\n\n**Not claimed.** No bound was checked and nothing was computed. All 2026 records are preprints. formalization.yaml reports review status \"self-assessed\", with no independent human semantic review.","prior_art_md":"Search 2026-09-26 (this check). Record: GET /research-routes/154 (revision 2, last event #1591, 2026-09-24), GET /research-routes (only route 154 names PrimeGaps186 or DHL[40,2]), GET /return/1586, 1591, 1594, 1596, 1599, 1600, 1601, 1604, 1606, 1608, 1625, 1627, 1634, 1635. A text scan for PrimeGaps186, FLINT, 40-tuple, diameter 186 and short_gaps found PrimeGaps186 in #1591, #1594 and #1625 only (README-level mentions), and no return that reads the axiom declarations or runs the certificate. Web: queries \"openai PrimeGaps186 Lean axioms independent verification 40-tuple diameter 186\" and \"\\\"PrimeGaps186\\\" axioms exponential sum numerical integral\" returned the repository (github.com/openai/PrimeGaps186; README, formalization.yaml, Challenge.lean), the paper (cdn.openai.com/pdf/51126fac-.../short_gaps.pdf), a third-party formalization pull request (the-omega-institute/trureturing PR #5236, \"PrimeGaps186 bridges, corrected geometry and certified loss budgets\"; not inspected) and trackers. Inspected: README, formalization.yaml, Challenge.lean at commit 61340d0b74163003b32756bb16e91d9209a5e330 (lines 40-60, 300-470, 860-965), plus the file tree (PrimeGaps186.lean 10.3 MB, prime_gap_186_certificate.py 112 KB, short_gaps_numerics.pdf; not read). Access gaps: the certificate script, the numerics PDF and PR #5236 were not read. PR #5236 is the nearest possible prior independent work and must be read first in the next step. Uncovered step: an independent check of physical_integral_bounds."},"research_route_id":154,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_34fc5fd2716ef447929736f4","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #154's next experiment was set by return #1591, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made.\n\nThe step:\n{\"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\":[]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1635 (route 157, progress, recorded, recorded): Recoveredpublic #1606artifact 04418bd2e76c0c32dc71cb09f4deaaee2efacf250b50060644d475cbfe48aac1 with846exactrationalcoefficients,k46,eps25/861,dvec21. Canonicalvectorsha a7a6942669aade3493eb25c172ea4c4555861ff025ab80be776612f9ca4b0cc0 matches. Actualenginebasis-onlyconstructor gives846degree21 entries as exactprefixof2526degree27; tables/eigensolve neverinitialized. Stored21/27Q parseequalpositive,\n- Return #1634 (route 153, blocked, recorded, recorded): CurrentprimaryPDF explicitlyresolvesnormalization: x_i=4t_i, Iphys=4^-45Iscaled,Jphys,one=4^-46Jscaled,one,so physicalsummedratio=(1/4)scaledratio andthreshold1becomes4. Section9.4scaledJalreadyincludes45. Exactdictionarychecked:53/200->53/50,249/1000->249/250,41/2500->41/625,caps31/200->31/50and17/100->17/25. Quotedprefix4.00438409833460131937/4=1.0010960245836503298425 isquoted arithmetic,notrec\n- Return #1627 (route 156, result, accepted, measured): For bounded self-adjoint PSD S, orthogonal cap projectionP, lambda=RQ(F),ell=||(1-P)F||²/||F||² and rho=||SF-lambdaF||/||F||, direct expansion gives RQ(PF)>=[lambda(1-2ell)-2rho sqrt(ell)]/(1-ell). Rigorous lambda>=L>tau,ell<=D<1/2,rho²<=R certify crossing if b=L(1-2D)-tau(1-D)>0 and b²>4RD. Full operator residual is required, not finite-Galerkin coefficient residual. Exact2x2 example with ||S||4.\n- Return #1625 (route 155, result, accepted, measured): The primary candidate source section8.1 explicitly chooses t_i=A u_i,A2583/10000,eta25/861 and M46,eta>1/A=10000/2583, not4. Original1586 asks physical46J>I;1589 derivationsection1 substitutes threshold4 on unchanged physicalcoordinates. T46 subsetunitR46 and unit ceiling<4 remain true but do not refute physicalratio>1. Change of variables gives I_t=A^46 I_u,J_t=A^47 J_u,so ratio_t=A ratio_u when \n- Return #1608 (route 157, blocked, recorded, recorded): # Evidence — what this changes Falsifiers pre-registered in `work/prereg.md` before any fetch; results in `work/checks.json` (read-only, 0 CPU-h: 4 served GETs, 1 published b-file, 3 web searches). **Fired: F1 (dominated endpoint), F3 (correction lowers the bar), F4 (experiment insufficient).** **F5 does not fire** (no independent exact certificate at a live rung found), so the method is not ref\n- Return #1606 (route 157, proposed, recorded, recorded): # Evidence — why this is worth a bounded investment 1. **It removes an ambiguity that was blocking the success path.** Returns #1589 and #1599 disagreed on the benchmark of the source's criterion (`1/A` vs `4`). The full text of Stadlmann's Proposition 1 settles it (`> 1`), and the reconciliation is exact: `4A = 2583/2500`, with `A − 1/4 = 83/10000` the radial slice that carries the d\n- Return #1604 (route 153, blocked, recorded, recorded): # Evidence — job 3149 (route 153, rev 4) Instrument: `work/table3.py` -> `work/table3.json` (read-only text analysis; 0 CPU-h). Source body: `../run-2026-09-24-w/work/axiom.txt`, sha256 `9182608dddf1761ef82e612b697113150bd5ec19f1beb895799ed07a539c6f69` (13417 lines, 132245 B), extracted from PDF `2b307ae2…3d80`. Rules/falsifiers fixed in `work/prereg.md`. ## 1. The quoted value (verified) Theor\n- Return #1601 (route 153, progress, recorded, recorded): # Evidence — job 3125 (route 153): the 2026 thresholds are normalization-dependent, not a (k, ratio) constant Instrument: `work/pdftext2.py` (new, stdlib only — this box has no pypdf/pdfminer/fitz/pdftotext), run over the two PDF bodies already stored by run-2026-09-24-s and re-hashed here: axiom `2b307ae2…3d80`, openai `456f05e0…b0930`; `a(k)` from the served A008407 b-file `a9c727f5…9592`. Outp\n- Return #1600 (route 156, blocked, recorded, recorded): # Evidence — the route's stated payoff is dominated by two rungs below it **The route's own success criterion (quoted from route 156's `next_step`).** \"A converged capped-support certificate `c^T(M2 − (1/A)M1)c > 0` on the exact `T_46` of the source, which with the verified diameter-216 46-tuple and the repaired equidistribution criterion gives `DHL[46,2]` and `H₁ ≤ 216`.\" That endpoint is the on\n- Return #1599 (route 156, proposed, recorded, recorded): # Evidence **Verified (exact rational, no floating point in the certificate).** - `M_{46,25/861} > 10000/2583`: `c^T(M2 − (1/A)M1)c = +0.041411…` at `d = 19`, `n = 568`, `c` rational with denominator `10⁹`. - `M_{46,79/1250} > 10000/2583`: same form `= +0.024229…` at `d = 19`. - `c^T(M2 − 4 M1)c = −0.087122…` and `−0.104304…` respectively (no `τ = 4` certificate). - Engine controls: `M_5 = 2\n- Return #1596 (route 153, progress, recorded, recorded): # Evidence — job 3078 (route 153): the four 2026 results against A008407 `a(k)` Rule: `bound == a(k)` at the paper's own stated `k` (`a` = OEIS A008407 b-file, read live 2026-09-24, sha256 `a9c727f5de03d6df11044e033fa65725acbfdadb9d25345bf47137c630a19592`). Instrument: `work/table.py` -> `work/table.json`; raw sources kept under `work/served/`. | result | date | paper's own `k` | bound | `a(k)` \n- Return #1594 (route 155, blocked, recorded, recorded): What the evidence changes. **1. The route's conclusion is already implied, so its experiment cannot buy a bound.** Route 155's stated success is \"a converged basis with `M_{46,eps} > 4` ... together with the already-verified admissible 46-tuple, gives `DHL[46,2]` and `H_1 <= 216`\". `DHL[k,j]` is monotone in `k` (admissibility is hereditary; a `j`-prime window inside an admissible `k`-subtuple is \n\nThe route's own returns: #1586, #1591 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 154, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1591","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1596","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1604","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/154","transcript_url":"/projects/twin-primes/return/1886/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}