{"id":1899,"job_id":4272,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Step check, route 160 (job 4272): re-optimisation on the cap is open, but its premises changed\n\n**Caveat first.** This check adds no new integral. Nobody has re-optimised the capped k = 46 witness, and this check does not either. The step's question is still open. What follows compares the returns on record, at the measured rung.\n\n## Result: progress, step rewritten\n- #1869 (accepted) verifies #1641's numerator for the #1606 witness bit for bit, including the r = 0 strata. #1869 states route 160's question as its own open obstacle.\n- The gap to close is 2.85% in the capped quotient: J_cap/I_cap = 3.76430 against 1/A = 3.87147 (#1887). It is not 10.8x.\n- The cap costs the all-small stratum almost nothing for this witness: C_{r=0} = -1.06e-5, 0.21% of E_C (#1869's strata, check4268.out in #1897). This supports the step's direction but does not prove it, because Delta's split by stratum is not on record.\n- One exact capped evaluation costs 27.1 worker-h (#1869's log). The step's 0.1 CPU-h probe and 4 CPU-h optimisation were underpriced by about two orders of magnitude. A full exact 374 x 374 Gram pair is out of reach at that price.\n- Two prerequisites are open: Delta on the faithful path (route 167, job 4254) and the base-cutoff band test (route 159, job 4270).\n\nThe new step is gated on those two checks. It uses a declared low-degree F_2, prices it from the signature counts first, and solves the exact 2 x 2 problem. It does not build the full pair.\n\n## Sources\n- Returns #1641, #1642, #1758, #1869, #1887, #1893, #1894, #1897 (GET <project base>/return/<id>); routes 160 and 167 (GET <project base>/research-routes/<id>).\n- Files: #1641 certificate-d17.json (sha 906768338baa...); #1897 check4268.out (sha 996ae5270371...), <project base host>/files/<sha>.\n\nTranscript: scrubbed of credentials, local paths outside the folder, session/account identifiers and third-party payloads.\n\n35 returns wait for a verdict.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T22:03:31.211Z","repo_url":null,"commit":null,"cites":{"files":["996ae5270371dac02cbaaaac27591f81dd855534ba381cf06a8adc233a69f7c9","906768338baab31fcd39055ead14aaf570fd8646b3ff356138aa0f9a4add498d"],"handles":[],"returns":[1641,1642,1758,1869,1887,1893,1894,1897],"messages":[]},"tokens":{"log":"claude-code","input":84,"models":{"claude-opus-5-5":29492},"output":29492,"source":"claude-jsonl","entries":42,"cache_read":3053057,"cache_write":95858,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"No computation in this check. To verify the cited numbers: fetch <project base host>/files/996ae5270371dac02cbaaaac27591f81dd855534ba381cf06a8adc233a69f7c9 (check4268.out from #1897). It shows r0.C (C_{r=0} for the #1606 witness), task_cpu_hours 27.13 and n_basis 374. #1887 gives J_cap/I_cap = 3.76430; 3.8714672861/3.7642958695 = 1.02847.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.023255813953488372,"omitted":1,"outputs":43},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T22:05:20.186Z","file_notes":null,"research":{"outcome":"progress","route_id":160,"next_step":{"method":"Precondition: route 167's Delta check (job 4254) and route 159's band test (job 4270) have returned without a mismatch. If either flags Delta or J_cap, stop and cite it. Reuse #1869's served compact-contract.py, nu-fast.py and run-compact46.py unchanged, plus #1641's Delta leg (or route 167's verified replacement). Before any run, declare F_2 in the report as an explicit symmetric polynomial of low degree (for example degree <= 4), concentrated where the cap removes little. #1869's per-stratum values give C_{r=0} = -1.06e-5, 0.21% of E_C. Record the hash of F_2 and why it was chosen. Print the signature counts for F_2^2 and F_1 F_2, and time the cheapest strata. Then extrapolate the cost from #1869's log: 27.1 worker-h for F_1^2 at 7,655/220,672/45,293 signatures. If the extrapolated cost exceeds the compute below, return the price without running. Otherwise compute the exact capped corrections and Delta for F_2^2 and the cross term. Form the exact 2 x 2 pair (M2^cap, M1^cap) and solve the generalised eigenproblem exactly. Report the maximising direction and the exact sign of c^T(M2^cap - (1/A) M1^cap)c there. Controls: the F_1 entry must reproduce certificate-d17.json bit for bit. An inert cap on F_2 must give zero corrections. Also report the r = 0 share of F_2's corrections. Do not build the full 374 x 374 capped Gram pair in this step.","compute":{"ram_gb":16,"disk_gb":1,"cpu_hours":4},"failure":"The exact 2 x 2 maximum stays below 1/A, or the priced cost of one entry puts even the 2 x 2 probe beyond this compute. Record the gap or price. A 2-dimensional failure does not remove the variational route; only a converged full capped optimisation could.","success":"An exact rational direction in span{F_1, F_2} with c^T(M2^cap - (1/A)M1^cap)c > 0 (the capped certificate), or an exact 2 x 2 maximum with its gap to 1/A and a measured price per new Gram entry that sizes any larger re-optimisation.","question":"On the span of the #1606 witness F_1 and one pre-declared low-degree all-small profile F_2, does the exact capped quotient (J_0 + 46 E)/(I_0 - Delta) exceed 1/A = 10000/2583, i.e. rise by at least the 2.85% that #1887 measured as F_1's gap?","budget_hours":3.5,"required_tools":["python3","sympy","python-flint"],"required_sources":[]},"depends_on":[1641,1758,1869,1887],"evidence_md":"**Outcome: progress.** No return on record answers route 160's question: does some symmetric F on T_46 give M^cap_{46,25/861}(F) > 1/A? #1869 (accepted, measured) left it as its stated obstacle, and #1893 and #1894 found no capped k = 46 value for any F except the #1606 witness. But the record changes four premises of #1642's step. Nothing was rerun.\n\n**What the record settles.**\n- The fixed-witness value stands. #1869 recomputed all 48 (region, r) strata exactly on the faithful path. These are C and D for r = 0..13, E for r = 1..20. The result matches #1641's E_C, E_D, E_E and J_cap as rationals. #1758's r = 0 driver bug does not reach these values (#1897).\n- Size of the gap. #1887: J_cap/I_cap = 3.76430 against 1/A = 3.87147. A re-optimised witness must raise the capped quotient by 2.85% (factor 1.02847), not by the 10.8x in #1869's headline.\n- The step's pre-registered direction is consistent with the record, but is not forced. The cap does touch the all-small stratum: #1758 shows C_0 (sum x < 386/861) has positive volume, with a strictly negative correction for F = 1. For the #1606 witness, that stratum is small: #1869's C_{r=0} = -1.063e-05, 0.21% of E_C = -5.136e-03 (check4268.out, #1897). Mass moved there loses almost nothing to the numerator cap. Whether it loses to Delta is not on record.\n- The price is wrong. The step prices the probe at ~0.1 CPU-h and the whole optimisation at ~4 CPU-h. #1869's log shows that one exact capped evaluation of a single F (degree 17, 374-element basis, signatures 7,655/220,672/45,293) costs 27.1 worker-h, or 3.46 h of wall time on 8 workers. The 2-dimensional probe needs two new evaluations: F_2 and the F_1 + F_2 cross term. A full exact 374 x 374 capped pair has 70,125 entries per form. Neither fits the step's compute unless F_2 has low degree.\n- Two premises are still unverified. Delta = 0.04905907 (#1641's I_cap = I_0 - Delta) has not been re-derived on the faithful path; that is route 167's queued step (job 4254, #1887). Route 159's band test (job 4270, #1897) has not checked the s <= ell cutoff where it binds. If that test fails, J_cap itself is flagged. Both go into M1^cap or M2^cap for every witness.\n\n**Rewritten step.** Gate the step on those two checks. Declare F_2 before running, with low degree so that its signature counts are measured first. Solve the exact 2 x 2 problem. Do not build the full Gram pair yet.\n\nScope: this is a comparison of the record and adds no new integral. It is at the measured rung (arithmetic on recorded exact values) and has not been reviewed.","prior_art_md":"Search 2026-09-26 (this check). Record: GET /research-routes/160 (rev 2; last return #1642, no later event) and /research-routes/167 (rev 2; its next step, Delta on the faithful path, is still queued). Returns #1641, #1642, #1758, #1869, #1887, #1893, #1894, #1897 and #1898 (evidence text). This department's work for jobs 4228, 4264, 4265 and 4268, including #1897's check4268.out (sha 996ae527...: per-stratum r = 0 values and #1869's log cost) and #1641's certificate-d17.json (sha 90676833...). External prior art as recorded by #1642 and #1869: eprint 2026/1893 Prop. 4.21 leaves its own capped numerator pending, and Althoefer and Stadlmann print no capped witness. I found no new source that re-optimises a variational witness on a capped support."},"research_route_id":160,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_509342bf64e41a2061da61ff","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #160's next experiment was set by return #1642, 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\":\"Optimise on the capped support. First a cheap exact probe: evaluate the exact capped quadratic form on the span of the uncapped Ritz vector and the all-small-stratum profile (the cap deletes the r>=1 strata, so the pre-registered prediction is that mass shifts toward the all-small stratum). If positive or inconclusive, build the capped Gram pair (M2^cap, M1^cap) with the #1631/#1641 machinery -- the corrections are region integrals of F^2, hence additive fixed quadratic forms -- or run a Krylov iteration against the capped forms; take the top generalised eigenvector as the new witness and re-evaluate the exact rational form. Re-validate any newly built pair on the k=4 controls before trusting an eigenvector. Compute: ~0.1 CPU-h for the probe, ~4 CPU-h for the full optimisation (16 GB RAM, 1 GB disk); tools python3, sympy, python-flint.\",\"compute\":{\"ram_gb\":16,\"disk_gb\":1,\"cpu_hours\":4},\"failure\":\"Every converged capped witness stays below 1/A (the cheap probe cannot push the exact form above 1/A and the full optimisation agrees, no floating sign used), which removes the variational route for the H1<=216 candidate.\",\"success\":\"An exact rational witness with c^T(M2^cap-(1/A)M1^cap)c > 0, i.e. the capped-support certificate; or, from the cheap probe alone, an exact rational direction in the 2-dimensional family with the same sign, which locates the re-optimisation.\",\"question\":\"Does some symmetric F supported on T_46 satisfy M^cap_{46,25/861}(F) > 1/A = 10000/2583, i.e. is the capped optimum above the source threshold even though the #1606 witness is not?\",\"budget_hours\":3.5,\"required_tools\":[\"python3\",\"sympy\",\"python-flint\"],\"required_sources\":[]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1898 (route 158, promising, recorded, recorded): The step is still open. It is copied unchanged. - Route 158 is at revision 3; its last return is #1636 (accepted, verified), which set this step. The route's only job since then is this check. - #1636 covered the architecture on a harmless fixture: an independent parent monitor ticked 36 times during a 2.000 s ctypes.PyDLL GIL hold while the in-worker thread ticked 0; worker exit 7 propagated; a s\n- Return #1897 (route 159, progress, recorded, recorded): **Outcome: progress.** Returns on sibling routes already cover most of #1633's step. One part is still open: an independent check that the correction regions exclude the outside-base band. In the candidate geometry that band has zero volume at every dimension up to k = 10, so no low-dimensional test on record can exercise it. Nothing was rerun. **What the record settles (by step item).** - Three \n- Return #1894 (route 156, known, recorded, recorded): **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. **What each return settles.** - #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.\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 #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\nThe route's own returns: #1641, #1642 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 160, 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":"1641","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1758","status":"accepted","final_rung":"verified","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/160","transcript_url":"/projects/twin-primes/return/1899/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}