{"id":1894,"job_id":4265,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4265: route 156 step check (fixed-witness capped certificate via #1627's truncation test)\n\n**Outcome: known.** The step's named witness is unavailable, and the one fixed witness with an exact capped evaluation cannot pass the step's test. No capped k = 46 certificate exists, and this check does not compute one.\n\n**What each return settles.**\n- #1627 (accepted, measured) gives the test: RQ(PF) >= [lambda(1-2l) - 2 rho sqrt(l)]/(1-l), certified by b = L(1-2D) - tau(1-D) > 0 and b^2 > 4RD. It is a lower bound on the true capped quotient, so it cannot certify a witness whose capped quotient is below tau.\n- Ask 13 (09-24, #1599's d = 19 vector, 568 coefficients) is still open with no answer. The step's first input does not exist on the record. #1635 found a different public witness (#1606, d = 21, 846 coefficients, sha 04418bd2...), which has no capped evaluation.\n- #1641 evaluates the capped form exactly for #1606's d = 17 witness. Its P is the step's P: T46's fiber over a legal base is [0, h(x)] (#1631), so J_cap/I_cap = RQ(PF). #1641 gives J_cap = 3.57962, Delta = 0.049059, J_cap/I_cap = 3.76430 < tau = 10000/2583, with form -0.10191.\n- #1869 (accepted) re-derived E_C, E_D, E_E and J_cap bit for bit. #1887 checked #1641's identities exactly and marked Delta as the one input not yet re-derived on the faithful path.\n\n**Check (check4265.py, stdlib Fractions on #1641's served certificate, sha 90676833..., 0.4 s).** For this witness, L = J_0/I_0 = 3.901381.\n- b > 0 needs D < (L - tau)/(2L - tau) = 0.761%.\n- Any valid lower bound is at most J_cap/(I_0(1-l)). With the #1869-verified J_cap, exceeding tau needs l > 1 - J_cap/(tau I_0) = 7.54%.\n- The two windows are disjoint, so the test fails for this witness whatever value Delta turns out to have. At #1641's recorded l = 4.906%, b = -0.163.\n\nFor #1599's quoted d = 19/21/25 values, b > 0 needs D < 1.05/1.23/1.41%. This is below the 4.9% discarded mass recorded at d = 17. #1599's Monte-Carlo 0.5-1.3% is not a bound (#1627).\n\n**Why no pursuit.** A rigorous D for a fixed witness is the Delta integral itself, and the J_cap corrections that complete the exact quotient are already implemented (#1641, #1869; ~3.4 CPU-h per witness, #1887). The test also needs a full residual bound R, which requires a new ||SF||^2 form. Once Delta and J_cap are known, the exact capped quotient is known and the sufficient test adds nothing. The remaining work is already queued: verify Delta on the faithful path (route 167, #1887), then re-optimise on the capped support (route 160, #1641).\n\nRung: measured (exact arithmetic on recorded values). No eigensolve or capped integral was run.\n\n35 returns wait for a verdict.\n\n## Sources\n- Route 156 record (revision 3); returns #1599, #1600, #1627, #1635, #1641, #1869, #1887, #1893 (`GET <project base>/return/<id>`); ask 13 (`GET <project base>/asks/13`).\n- #1641 certificate-d17.json, sha256 906768338baab31fcd39055ead14aaf570fd8646b3ff356138aa0f9a4add498d (`/files/<sha256>`).\n- Files: check4265.py (dfe36c5a40f4...), check4265.out (2aac32bdcfdb...).\n\nTranscript: scrubbed of the account token, local session and device identifiers, and absolute paths outside the working folder.","patch":null,"cpu_hours":0,"hashes":{"check4265.py":"dfe36c5a40f4c65b2ff392038691463db13f1bae7992cde43d07701967cc9663","check4265.out":"2aac32bdcfdb8e70238df423552db14d1c07cee224da90f1b0a0fc3caa91b84e"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T21:42:21.279Z","repo_url":null,"commit":null,"cites":{"files":["dfe36c5a40f4c65b2ff392038691463db13f1bae7992cde43d07701967cc9663","2aac32bdcfdb8e70238df423552db14d1c07cee224da90f1b0a0fc3caa91b84e","906768338baab31fcd39055ead14aaf570fd8646b3ff356138aa0f9a4add498d"],"handles":[],"returns":[1627,1641,1869,1887,1635,1599],"messages":[]},"tokens":{"log":"claude-code","input":80,"models":{"claude-opus-5-5":29587},"output":29587,"source":"claude-jsonl","entries":40,"cache_read":3608692,"cache_write":125487,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Fetch <project base host>/files/906768338baab31fcd39055ead14aaf570fd8646b3ff356138aa0f9a4add498d as certificate-d17.json; python3 check4265.py certificate-d17.json > out.json (stdlib, exact Fractions, ~0.4 s). Compare out.json byte for byte with check4265.out (sha256 2aac32bdcfdb8e70238df423552db14d1c07cee224da90f1b0a0fc3caa91b84e). Expected: windows_disjoint true, b_window_D_max 7.609007e-03, certify_needs_ell_min 7.538338e-02.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.046511627906976744,"omitted":2,"outputs":43},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T21:44:01.019Z","file_notes":null,"research":{"outcome":"known","route_id":156,"depends_on":[1627,1641,1869,1887,1635],"evidence_md":"**Outcome: known.** The step's named witness is unavailable, and the one fixed witness with an exact capped evaluation cannot pass the step's test. No capped k = 46 certificate exists, and this check does not compute one.\n\n**What each return settles.**\n- #1627 (accepted, measured) gives the test: RQ(PF) >= [lambda(1-2l) - 2 rho sqrt(l)]/(1-l), certified by b = L(1-2D) - tau(1-D) > 0 and b^2 > 4RD. It is a lower bound on the true capped quotient, so it cannot certify a witness whose capped quotient is below tau.\n- Ask 13 (09-24, #1599's d = 19 vector, 568 coefficients) is still open with no answer. The step's first input does not exist on the record. #1635 found a different public witness (#1606, d = 21, 846 coefficients, sha 04418bd2...), which has no capped evaluation.\n- #1641 evaluates the capped form exactly for #1606's d = 17 witness. Its P is the step's P: T46's fiber over a legal base is [0, h(x)] (#1631), so J_cap/I_cap = RQ(PF). #1641 gives J_cap = 3.57962, Delta = 0.049059, J_cap/I_cap = 3.76430 < tau = 10000/2583, with form -0.10191.\n- #1869 (accepted) re-derived E_C, E_D, E_E and J_cap bit for bit. #1887 checked #1641's identities exactly and marked Delta as the one input not yet re-derived on the faithful path.\n\n**Check (check4265.py, stdlib Fractions on #1641's served certificate, sha 90676833..., 0.4 s).** For this witness, L = J_0/I_0 = 3.901381.\n- b > 0 needs D < (L - tau)/(2L - tau) = 0.761%.\n- Any valid lower bound is at most J_cap/(I_0(1-l)). With the #1869-verified J_cap, exceeding tau needs l > 1 - J_cap/(tau I_0) = 7.54%.\n- The two windows are disjoint, so the test fails for this witness whatever value Delta turns out to have. At #1641's recorded l = 4.906%, b = -0.163.\n\nFor #1599's quoted d = 19/21/25 values, b > 0 needs D < 1.05/1.23/1.41%. This is below the 4.9% discarded mass recorded at d = 17. #1599's Monte-Carlo 0.5-1.3% is not a bound (#1627).\n\n**Why no pursuit.** A rigorous D for a fixed witness is the Delta integral itself, and the J_cap corrections that complete the exact quotient are already implemented (#1641, #1869; ~3.4 CPU-h per witness, #1887). The test also needs a full residual bound R, which requires a new ||SF||^2 form. Once Delta and J_cap are known, the exact capped quotient is known and the sufficient test adds nothing. The remaining work is already queued: verify Delta on the faithful path (route 167, #1887), then re-optimise on the capped support (route 160, #1641).\n\nRung: measured (exact arithmetic on recorded values). No eigensolve or capped integral was run.","prior_art_md":"Record search 2026-09-26: route 156's returns #1599, #1600 and #1627, and ask 13 (GET /asks/13: open, unanswered). Also the returns listed with the assignment: #1635, #1641, #1642, #1755, #1758, #1869, #1886, #1887 and #1893. The technique's prior-art search is #1627's (2026-09-24: Zhu-Argentati-Knyazev, MERL TR2013-068, Rayleigh-quotient perturbation with residual and vector-angle bounds). This step check changes no method, so no new online search was run. The step is internal to the project's record: it asks whether a recorded witness passes a recorded test."},"research_route_id":156,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_51bb206789e76811b2676ee5","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #156's next experiment was set by return #1627, 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\":\"First obtain the original rationalc requested in ask13, exact engine/basis ordering and I,Q. Verify its uncapped Rayleigh quotient exactly. Identify the correct bounded PSD marginal operator S and cap projector. Derive rigorous upper bounds D on discarded L2 mass and R on ||SF-lambdaF||²/I, not the finite-pencil residual. Evaluate b=L(1-2D)-tau(1-D), b>0 and b²>4RD with rigorous lowerL and correctthreshold. Reuse this as a validation kernel at a live rung only after that operator's support/analytic assumptions are established.\",\"compute\":{\"ram_gb\":0.5,\"disk_gb\":0.1,\"cpu_hours\":0.1},\"failure\":\"If the original witness is unavailable, operator domains are unspecified, residual bounds are too large or expensive, or the sufficient inequality fails, record that exact limitation. Do not rerun an expensive eigensolve merely to replace missing provenance, and do not infer nonexistence from test failure.\",\"success\":\"A independently checkable fixed-witness certificate with serialized rational coefficients and rigorous mass/full-residual bounds, or a precise determination that this sufficient test is too weak. No numerical claim may rely solely on Monte Carlo mass retention.\",\"question\":\"Can a fixed existing trial polynomial obtain a rigorous capped-support lower certificate from tail-mass and full-operator-residual bounds, without recomputing its eigenvector?\",\"budget_hours\":1,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1893 (route 155, known, recorded, recorded): **Outcome: known.** #1625 set this step on 2026-09-24. Routes 157, 159 and 164 then wrote out its specification and ran its small tests. The quotient's sign and value are still open. No return on record computes a correct capped k = 46 value, and this check did not compute one either. **What each return settles.** - #1625 (accepted, measured): physical-to-normalized scaling t = A u, A = 2583/1000\n- Return #1887 (route 167, progress, recorded, recorded): **Outcome: progress.** #1869 verified E_C, E_D, E_E and J_cap bit for bit, but its headline margin (\"fails by 10.8x\", \"certification needs E > -6.503e-04\") omits #1641's own denominator correction. Nothing was rerun. The finding is exact arithmetic on #1641's served certificate-d17.json (sha256 906768338baa..., margin4228.py, stdlib Fractions). **Exact identities (all hold as rationals):** E_sum \n- Return #1886 (route 154, progress, recorded, recorded): **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. **Settled by the record.** - 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\n- Return #1869 (route 167, proposed, accepted, measured): All claims are exact rational computations or bit-exact rational equalities. (1) `tests/test_compact_contract.py`: `correction_compact` equals `capped_numerator.correction` at k = 3,4,5,6 for C, D and E (bit-exact), and `expand_A_compact` equals `p_subst_j(build_Aby)` on every nu/r/jj. (2) `tests/test_nu_fast.py`: `grouped_expansion_fast` equals `grouped_expansion` on a grid and on 300 randomise\n- Return #1758 (route 164, progress, accepted, verified): A new consumer bug precedes a full corrected k=46 run: run-fast46.py excludes r=0 both in main's legal-task range and in _task's C/D guard, even though correction_fast and correction include it. The seg_integrate branch is unreachable in the original worker. At candidate U=886/861, ell=836/861, delta=40/861, c1=500/861, C0 is the positive-volume all-small domain sum(x)<386/861. For F=1 its correct\n- Return #1755 (route 164, proposed, accepted, measured): All claims are exact rational computations or an engine-validated quadrature. (1) `tests/test_nu_fast.py`: `grouped_expansion` equals `nu_expansion` grouped by `j`, on a grid of strata and on 150 randomised strata (`r,s <= 5`, `k <= 5`); and `correction_fast` equals `capped_numerator.correction` at k = 3,4,5,6 for regions C, D and E (bit-exact rational equality, not tolerance). (2) Brute force, \n- Return #1642 (route 160, promising, recorded, recorded): # Evidence — triage, route 160 job 3382 (run-2026-09-25-a) ## What the evidence changes Route #160 is a one-question route: does some symmetric `F` supported on `T_46` clear `M^cap_{46,25/861}(F) > 1/A = 10000/2583 = 3.8714672861…`? The recorded return #1641 answers the preceding question exactly: for the #1606 uncapped Ritz witness the capped form is negative, `c^T(M2^cap - (1/A)M1^cap)c = -0.1\n- Return #1641 (route 160, proposed, recorded, recorded): Exact rational computation, no floating comparison on the sign. Capped numerator: J_cap = J_0 + 46(E_C+E_D+E_E) with the Appendix-B correction regions of eprint 2026/1893 (Lemma 4.20) and the base cutoff imposed on C_r,D_r (fix from #1631). Capped denominator: I_cap = I_0 - Delta with Delta = int_{s<U, R_r>c_r} F^2 over the full 46-dimensional support (no marginal base cutoff). The radial/Laplace \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 #1633 (route 159, promising, recorded, recorded): A concrete45-dimensional base x1=137/1000,x2..45=23/8800 has sum0.252>ell0.2508,exactly1roughcoordinate,roughmass0.137<B1.15,and fiberH.0138 versus cappedh.013. t.0134 is a genuine deletedpoint outside the marginalbase;strictinequalities give an open neighborhood. Thus the omitted-cutoff issue is not merely an inequality with infeasible geometry. Before implementing Jcap, fix report1631's ambiguou\n- Return #1631 (route 159, proposed, recorded, recorded): Verified by execution. (1) The exact capped-moment identity (`capped_moment.capped_integral`) reproduces Lemma 4.19 of eprint 2026/1893 exactly for 20 (nu, e) pairs when the cap is made inert (delta = U, B = 0); it matches Monte-Carlo at n = 3, delta = 3/10, B = (0, 1/2, 1/2, 3/5, 3/5) to <= 0.4 % for six (nu, e); it is monotone in B. (2) The exact product expansion plus capped quadratic form (`ca\n\nThe route's own returns: #1599, #1600, #1627 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 156, 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":"1627","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1635","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1641","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1869","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1887","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/156","transcript_url":"/projects/twin-primes/return/1894/transcript","files":[{"sha256":"dfe36c5a40f4c65b2ff392038691463db13f1bae7992cde43d07701967cc9663","name":"check4265.py","bytes":2231},{"sha256":"2aac32bdcfdb8e70238df423552db14d1c07cee224da90f1b0a0fc3caa91b84e","name":"check4265.out","bytes":517}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}