{"id":1893,"job_id":4264,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4264: route 155 step check (restricted k = 46 quadratic-form specification)\n\n**Outcome: known.** The step's typed support/operator specification, its scaling and marginal controls, and its list of analytic assumptions are already on record. They come from #1625, #1631, #1633, #1641, #1755, #1758 and #1606, across routes 155, 157, 159 and 164. The certificate's value is still open. #1641's negative sign was computed on a path that #1755 rejected. No corrected capped k = 46 quotient exists yet, and no prime-gap bound follows.\n\n**The specification on record.** Normalized coordinates u = t/A, A = 2583/10000. Support: s < U = 886/861, and the sum of coordinates above d = 40/861 is at most c_r, where r is the number of such coordinates (c1 = c2 = 500/861, c_m = 1600/2583 for m >= 3). Target:\n\nc^T(M2^cap - (1/A) M1^cap) c > 0, with 1/A = 10000/2583.\n\n- M2^cap = J_cap = k * integral over s(x) <= ell = 836/861 of (integral from 0 to h(x) of F)^2. The fiber limit is h = min(U - s, max(d, c_{r+1} - R)) on legal bases and 0 on illegal ones.\n- The C_r and D_r corrections need the cutoff s <= ell (#1631), and the all-small region C_0 is included (#1758).\n- M1^cap = I_0 - Delta, where Delta is taken over the full support with no base cutoff. The sufficient route replaces M1^cap by I_0.\n\nThe physical form equals the normalized form times A^46 (#1625). The unrestricted M_{46,25/861} > 1/A (#1606) is a benchmark, not this certificate.\n\n**This check.** check4264.py (stdlib, public reads, 1 s under run-limited) confirms:\n- the constants scale exactly and are printed identically across the returns;\n- the F = 1 Jacobian and k-count identities hold at k = 2, 4, 46;\n- #1641 uses J_sum = k J_one (J_0/I_0 = 3.9014 on the 1/A scale).\n\nRung: measured (exact rational identities and record reads). Nothing from the step's own experiment was run, and no eigenvalue was computed.\n\n35 returns wait for a verdict.\n\n## Sources\n- Route 155 record (revision 3); returns #1589, #1594, #1625, #1606, #1631, #1633, #1641, #1755, #1758 (`GET <project base>/return/<id>`).\n- Candidate note (althofer.de/H1_216_candidate.pdf, sections 8 and 8.1) and eprint 2026/1893 (Lemmas 4.19-4.22, Appendix B): these are cited through #1625 and #1631 and were not re-read here.\n- Files: check4264.py (a5553d83...), check4264.json (3a218696...).\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":{"check4264.py":"a5553d836d719a9671e0806669253310af3e5f0751ce7a275331567aefc825bf","check4264.json":"3a21869698f89786e3681fafe8b53bd39d65c67d4c1d3e5826d997bbcb8b7f59"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T21:36:38.106Z","repo_url":null,"commit":null,"cites":{"files":["a5553d836d719a9671e0806669253310af3e5f0751ce7a275331567aefc825bf","3a21869698f89786e3681fafe8b53bd39d65c67d4c1d3e5826d997bbcb8b7f59"],"handles":[],"returns":[1625,1631,1633,1641,1755,1758,1606],"messages":[]},"tokens":{"log":"claude-code","input":90,"models":{"claude-opus-5-5":25531},"output":25531,"source":"claude-jsonl","entries":45,"cache_read":3864774,"cache_write":112055,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"python3 check4264.py > out.json (stdlib, reads <project base>/return/{1625,1631,1641,1758}); compare with check4264.json (sha256 3a21869698f89786e3681fafe8b53bd39d65c67d4c1d3e5826d997bbcb8b7f59). About 1 s.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.020833333333333332,"omitted":1,"outputs":48},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T21:38:15.708Z","file_notes":null,"research":{"outcome":"known","route_id":155,"depends_on":[1625,1631,1633,1641,1755,1758,1606],"evidence_md":"**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.\n\n**What each return settles.**\n- #1625 (accepted, measured): physical-to-normalized scaling t = A u, A = 2583/10000. I_t = A^46 I_u and J_t = A^47 J_u, so the physical target 46J > I becomes a normalized ratio above 1/A = 10000/2583. The normalized cutoffs are U = 886/861, d = 40/861, c1 = c2 = 500/861, c_m = 1600/2583 (m >= 3). The marginal radius is 0.2508 physical (836/861), and a centre of 1/4 gives a different operator. It also ran a separable Jacobian control (k = 2, 3, 46).\n- #1631: the typed operator. Support T46 = {s < U, sum of coordinates above d <= c_#}. Capped numerator J_cap = k * integral over s(x) <= ell of (integral from 0 to h(x) of F dt)^2 dx, with h = min(H, max(d, c_{r+1} - R)) on legal bases and 0 on illegal ones (eprint 2026/1893, Lemma 4.20). Denominator I_0 = integral over s < U of F^2, using the sufficient route of Lemma 4.22. It also proves that the source's Appendix-B premise u - d < ell fails here (u - delta - ell = 3/1000), so C_r and D_r need the cutoff s <= ell.\n- #1633: a feasible point in that band. It also fixes the convention that k counts once (J_sum = k J_one), with the F = 1 control 2k/(k+1), and shows that a failed J_cap/I_0 test does not decide J_cap/I_cap.\n- #1641: J_cap = J_0 + k(E_C + E_D + E_E) and I_cap = I_0 - Delta, where Delta is taken over s < U with R_r > c_r and has no base cutoff. Its form is c^T(M2^cap - (1/A) M1^cap)c. Its numbers are unverified (#1755), but the specification stands.\n- #1755 (accepted): the occurrence path matches brute force of the physical definition to 0.19% (k = 3, candidate geometry), and the radial path is rejected. #1758 (accepted, verified): the all-small C_0 region, S < 386/861, with an exact k = 2 primitive, -8572960/273547449.\n- #1606 lists the analytic assumptions: Stadlmann Prop. 1 hypotheses (2)-(3) and the S_Z/Zhang-type equidistribution on the 1/4-to-A annulus, as cited and open inputs. #1589 lists the 20 scalar conditions with omega >= 0.\n\n**Check (check4264.py, stdlib, public reads, 1 s).** The constants scale exactly, and #1625/#1631/#1641/#1758 print the same values. The F = 1 Jacobian identities hold at k = 2, 4, 46, and u - delta - ell = 3/1000. #1641's J_0/I_0 = 3.9014 is on the 1/A scale, so it uses J_sum; a J_one reading would give 0.0848.\n\n**Open, and not part of this step:** the corrected capped value (route 164's driver run, then re-optimisation on route 160) and the analytic inputs.","prior_art_md":"Record search 2026-09-26: route 155's returns #1589, #1594 and #1625. Also every return listed with the assignment: #1627, #1631, #1633, #1634, #1635, #1641, #1642, #1755, #1758, #1869, #1886 and #1887, plus #1606, which #1641 and #1633 cite. The step asks for an extraction internal to the project's record: the candidate note section 8.1 (read in #1625, sha 6f4b6e8e...) and eprint 2026/1893 Lemmas 4.19-4.22 and Appendix B (read in #1631). No new online search was run. #1625's search (2026-09-24: candidate title and parameters, Polymath8b Thm 3.12 and Cor. 6.4) and #1633's search (exact polytope integration, De Loera et al. arXiv:1108.0117) cover it. The eprint PDF was not re-read here. #1633 records a 403 on that PDF, and the region formulas are attributed to #1631."},"research_route_id":155,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_4034392de13941ccc0aa0300","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #155's next experiment was set by return #1625, 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\":\"Use primary candidate sections8/8.1 and its cited GPY/Stadlmann definitions. Write the full physical I and J domains, then scale t=A u with A2583/10000. Retain eta25/861,threshold10000/2583,delta/A40/861,B1/B2/A500/861 and Bm>=3/A1600/2583. Distinguish the unrestricted Polymath benchmark from the actual hybrid operator. Check Jacobian factors and marginal limits on a small exact separable test before any high-degree work. Itemize analytic distribution assumptions and unresolved proof gaps; do not equate formal input axioms with mathematical refutation.\",\"compute\":{\"ram_gb\":0.25,\"disk_gb\":0.1,\"cpu_hours\":0.01},\"failure\":\"If a required marginal domain or hybrid term is unspecified or inconsistent, record it and stop before eigenvalue computation. Passing the unrestricted benchmark must not be reported as a restricted certificate or a new prime-gap bound.\",\"success\":\"A self-contained typed support/operator specification with matching physical and normalized Rayleigh quotients and explicit analytic assumptions, or a precise primary-source gap preventing that identification.\",\"question\":\"What is the exact dimensionless restricted quadratic-form certificate corresponding to the candidate's physical46J>I, including every marginal domain and B cutoff?\",\"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 #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- 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\nThe route's own returns: #1589, #1594, #1625 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 155, 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":"1606","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1625","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1631","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1633","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1641","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1755","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1758","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/155","transcript_url":"/projects/twin-primes/return/1893/transcript","files":[{"sha256":"a5553d836d719a9671e0806669253310af3e5f0751ce7a275331567aefc825bf","name":"check4264.py","bytes":3875},{"sha256":"3a21869698f89786e3681fafe8b53bd39d65c67d4c1d3e5826d997bbcb8b7f59","name":"check4264.json","bytes":1527}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}