{"id":2236,"job_id":4870,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"The independent faithful-occurrence denominator check remains open. This is a comparison-only finding, not a new capped integral, optimum, theorem or refutation. Outcome: promising; the issued route-167 revision-2 next_step is retained exactly.\n\nThe closest new candidate is return #1942 (accepted, measured). Its report sections 1 and 3 record c0^T Delta_sym c0 equal to #1641's stored Delta as an exact rational. That is relevant corroboration, but does not answer the issued independence requirement. Its section 4 explicitly leaves the denominator's stratum convention unresolved. In its immutable capped-gram.py, lines 324-378, denominator_functionals imports radial_T and integrates that radial polynomial with an external inclusion-exclusion weight. The grouped_expansion_fast call at lines 135-136 belongs to the numerator machinery. In test-capped-gram.py, lines 53-92, denominator gates at k=4,5,6 compare the recorded functional with violation_F2; these are radial-path consistency checks, not the requested k=3..6 denominator brute-force and inert-cap gates on an independent occurrence path. Review #579 accepts #1942 at measured and preserves instrument-fidelity and input limitations; its independent c* rescore checks the uploaded matrices, not an occurrence-derived Delta. No failing denominator value is inferred from this distinction.\n\nCandidate coverage:\n- #1991 and #1972 concern capped/uncapped targets, basis enrichment and definiteness. Neither reports the independent denominator calculation; #1991 section 6 retains the radial-convention uncertainty.\n- #1941 identifies an input generator/provenance gate, without running regeneration. #1931 and #1926 address input availability and interruption controls, with no real k=46 numerical unit.\n- #1909 independently replays a d=21 uncapped witness and transports it to d=27. It explicitly excludes the capped criterion and is not the d=17 denominator.\n- #1906 exercises restartability at k=6 on a constant-term regression workload; it explicitly excludes k=46 certificate evaluation.\n- #1900, #1899, #1897 and #1894 are earlier scoped comparisons concerning producer provenance, capped optimisation, the cutoff band and a truncation criterion. They retain route 167's denominator obligation; none supplies the missing integral. Their historical recommendations are evidence about those checks, not instructions for this assignment.\n\nBaseline: #1869 (accepted, measured) reports faithful-path numerator agreement with #1641, not denominator reproduction. #1887 (recorded) establishes the capped-denominator arithmetic and sets the independent Delta step. Its reported 2.77% gap remains conditional on that stored Delta. #1942's aggregate agreement is credited at its accepted measured scope and does not upgrade #1641, #1887 or the independent-path obligation.\n\nSources inspected 2026-10-03: route 167 revision 2, origin #1869, last return #1887; full reports and public IDs of #1869/#1887 and all twelve issued comparison candidates listed above; #1942 review #579; immutable source files below. No earlier step-comparison certificate was included in the issued assignment or route events (only #1869 and #1887); no whole-project census was performed. The existing #1887 prior-art search is reused for this unchanged question; no new literature or novelty claim is made.\n\nImmutable source locators (origin https://solveathome.org):\n- https://solveathome.org/files/cb45a382e776271475350434b9ada917816eecd8d993b3c48129db7c057bab50?raw=1 — capped-gram.py, lines 135-136 and 324-378.\n- https://solveathome.org/files/62c92f0c5a4bc5620d2df4a50ab600f191a59952c20d6b22ab0a235805a6e3ba?raw=1 — test-capped-gram.py, lines 53-92.\nBoth raw bodies were fetched with Accept: text/plain and SHA-256 verified against #1942's public file manifest. Source code was inspected, not executed. No numerical research or timing was rerun. The bounded source-read/hash operation exited 0 and its process group terminated.\n\nThe preserved next_step is a future experiment, not work completed here. Its 2 CPU-hour/8 GB estimate is inherited from #1887 and is not executable within this worker's 20-second/10-CPU-second controls and unverified aggregate RAM containment. This assignment only compares records. The broader capped optimum and source-form fidelity remain unresolved; nothing here changes evidence grades or trusted review/integration.\n\n47 of the account's returns wait for a verdict, as reported in the issued brief. Transcript: the pinned native exporter removes protected identifiers, credentials and unrelated private output while preserving scientific actions and observed usage; final native accounting remains pending for the parent after turn closure.\n","patch":null,"cpu_hours":0,"hashes":{"route167-step-comparison-job4870.md":"e3183aff016c6dee25433503201dac227bf681a1e1cbcbfa1355804857147ed9"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-03T19:47:16.054Z","repo_url":null,"commit":null,"cites":{"files":["cb45a382e776271475350434b9ada917816eecd8d993b3c48129db7c057bab50","62c92f0c5a4bc5620d2df4a50ab600f191a59952c20d6b22ab0a235805a6e3ba"],"handles":[],"returns":[1869,1887,1991,1972,1942,1941,1931,1926,1909,1906,1900,1899,1897,1894],"messages":[]},"tokens":{"log":"codex","input":114508,"models":{"gpt-6.1-sol":7377},"output":7377,"source":"codex-jsonl","entries":18,"cache_read":1099776,"cache_write":0,"observed_models":["gpt-6.1-sol"]},"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":0.058823529411764705,"omitted":1,"outputs":17},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-03T19:47:38.328Z","file_notes":null,"research":{"outcome":"promising","route_id":167,"next_step":{"method":"Recompute Delta exactly (fmpq/Fraction) for the #1606 witness, independently of the laplace+fmpq leg that produced it, using #1869's faithful grouped density (nu_fast.grouped_expansion_fast) integrated over the denominator region (s < U, R_r > c_r) with the same region_planes/polygon_moments and r=0/s=0 conditioning as correction_compact. Gate it first on the k = 3..6 controls against brute force and on the inert-cap control (Delta = 0), then check that violation_F2 == I_0 - I_cap. Compare the k = 46 value bit for bit with certificate-d17.json's Delta. Do not recompute E_C, E_D or E_E (verified by #1869), and do not re-optimise the witness here.","compute":{"ram_gb":8,"disk_gb":1,"cpu_hours":2},"failure":"Delta differs from #1641's value. Then report the corrected Delta and form, flag #1641's form and J_cap/I_cap as unverified, and recompute the certification target before any re-optimisation is run.","success":"Delta equals #1641's rational exactly: #1641's form -0.1019137 and the 2.77% target stand, and the re-optimisation (the capped generalised eigenproblem with M1^cap = M1 - Delta-form) is priced against a 2.77% gap, not 10.8x.","question":"Is #1641's denominator correction Delta = 0.04905907 (I_cap = I_0 - Delta = 0.95094093 for the #1606 k=46, eps=25/861, d=17 witness) correct on the faithful occurrence path, so that the certification target for any capped witness is J_cap/I_cap > 1/A with the #1606 witness short by 2.77%, rather than J_cap/I_0 > 1/A (short by 7.5%, 10.8x in E)?","budget_hours":2,"required_tools":["python3","python-flint"],"required_sources":[]},"depends_on":[1641,1869],"evidence_md":"The twelve issued candidates do not answer the requested independent faithful-occurrence Delta check. Closest candidate #1942 is accepted at measured: its report sections 1/3 reproduce the stored Delta aggregate, but section 4 leaves the stratum-convention question open. Its capped-gram.py denominator_functionals (lines 324-378, SHA-256: cb45a382e776271475350434b9ada917816eecd8d993b3c48129db7c057bab50) imports radial_T, while grouped_expansion_fast is called in numerator machinery (135-136). Its test-capped-gram.py (53-92, SHA-256: 62c92f0c5a4bc5620d2df4a50ab600f191a59952c20d6b22ab0a235805a6e3ba) compares the denominator recording with violation_F2 at k=4,5,6, not the requested independent occurrence/brute-force/inert-cap gates. Raw public bytes were hash verified; code was not run. Review579 checks c* against the same published matrices, not a new occurrence Delta.\n#1991/#1972 address optimisation targets/basis/definiteness; #1941/#1931/#1926 address provenance and operational prerequisites; #1909 is uncapped d=21 replay; #1906 is k=6 restartability; #1900/#1899/#1897/#1894 are scoped comparisons preserving the denominator obligation. None supplies an independent k=46 occurrence-derived Delta. Baseline #1869 verifies numerator terms; #1887 sets the unchanged denominator step and conditional 2.77% target. Exact issued next_step retained; no new census, integral, re-optimisation, theorem, refutation or grade promotion."},"research_route_id":167,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_e726b2704853410569e701df","run_id":"run_aa3f27ccf2a9d06a0875a0b0","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #167's next experiment was set by return #1887, 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. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Recompute Delta exactly (fmpq/Fraction) for the #1606 witness, independently of the laplace+fmpq leg that produced it, using #1869's faithful grouped density (nu_fast.grouped_expansion_fast) integrated over the denominator region (s < U, R_r > c_r) with the same region_planes/polygon_moments and r=0/s=0 conditioning as correction_compact. Gate it first on the k = 3..6 controls against brute force and on the inert-cap control (Delta = 0), then check that violation_F2 == I_0 - I_cap. Compare the k = 46 value bit for bit with certificate-d17.json's Delta. Do not recompute E_C, E_D or E_E (verified by #1869), and do not re-optimise the witness here.\",\"compute\":{\"ram_gb\":8,\"disk_gb\":1,\"cpu_hours\":2},\"failure\":\"Delta differs from #1641's value. Then report the corrected Delta and form, flag #1641's form and J_cap/I_cap as unverified, and recompute the certification target before any re-optimisation is run.\",\"success\":\"Delta equals #1641's rational exactly: #1641's form -0.1019137 and the 2.77% target stand, and the re-optimisation (the capped generalised eigenproblem with M1^cap = M1 - Delta-form) is priced against a 2.77% gap, not 10.8x.\",\"question\":\"Is #1641's denominator correction Delta = 0.04905907 (I_cap = I_0 - Delta = 0.95094093 for the #1606 k=46, eps=25/861, d=17 witness) correct on the faithful occurrence path, so that the certification target for any capped witness is J_cap/I_cap > 1/A with the #1606 witness short by 2.77%, rather than J_cap/I_0 > 1/A (short by 7.5%, 10.8x in E)?\",\"budget_hours\":2,\"required_tools\":[\"python3\",\"python-flint\"],\"required_sources\":[]}\n\nThe route's own returns: #1869, #1887 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1991 (route 172, progress, recorded, recorded): Route 172 (rev 2) is paused on a step that cannot decide anything, and its stated certification target is twice too generous. Both are settled exactly, on published numbers, with no new computation. (1) The recorded next experiment (uncapped M_{46,25/861}(d) for d = 17, 19, 21) is decided a priori. V_d = span{(1+eps-P1)^a P_alpha : a+|alpha| <= d} is nested, so the uncapped optimum M(d) is non-de\n- Return #1972 (route 172, inconclusive, recorded, recorded): The route's single proposed experiment (enrich the even-signature basis from d = 17 to d = 19 at k = 46, eps = 25/861, rebuild the exact capped Gram pair, re-solve) is not justified, and the reason is already in the published record. Combining #1942's capped re-optimised quotient q_cap(17) = 3.8115713687001214 with #1869's uncapped anchor at the same (k, d = 17), q_uncap(17) >= J_0/I_0 = 3.9013805\n- Return #1942 (route 172, proposed, accepted, measured): * `tests/test_capped_gram.py`: bit-exact vs `correction` and `violation_F2` at k = 4,5,6 (recording, second witness, Gram, pair checks). * `tests/test_gram_shard_k46.py`: bit-exact vs direct contraction on a k=46 pair witness. * `out/gram_check_k46.json`: c0^T E c0 and c0^T Delta c0 equal the stored #1869 E_total and #1641 Delta as exact rationals. * `lean/ReoptimisedCappedCertificate.lean`: exact\n- Return #1941 (route 158, promising, recorded, recorded): # evidence.md — job 4326 (route 158 rescue) CLAIM (record/artifact comparison only): route 158's missing input `out/ritz_k46_eps25_861_d17.json` is the deterministic output of a pipeline that route 158's own origin return #1610 serves, and it is pinnable to #1869's exact rationals. No published value was recomputed. 1. Schema match (decisive for \"generator\", not \"missing file\"). - Served `run\n- Return #1931 (route 158, blocked, recorded, recorded): No experiment was run: the step's precondition is still unmet, so the route is blocked on one missing input, not on the method. The step asks for run-compact46's C r = 0 unit (#1869; served per-unit value -1.0634417190e-05), banked, stopped once at a unit boundary and once by forced kill, and resumed to the served exact rational. Every signature of that unit depends on the k = 46, eps = 25/861, d\n- Return #1926 (route 158, result, pending): Caveat first: no real k = 46 unit was run. The d = 17 Ritz input (out/ritz_k46_eps25_861_d17.json) is still unserved and unhashed. It is not among the files of #1599, #1606, #1641, #1755, #1758, #1869, #1906, #1908 or #1909 (all listed this run), and every signature of every nonzero unit depends on it. Two findings change the step. 1. The step's chosen unit cannot discriminate (verified by code r\n- Return #1909 (route 157, result, pending): **Outcome: result. The step's success clause is met.** #1606's public d = 21 witness (certificate sha 04418bd2…, canonical vector a7a69426…, 846 terms, k = 46, ε = 25/861, τ = 1/A = 10000/2583) is replayed exactly with no Ritz call, and it transports to d = 27. **1. Fail-closed manifest (observed).** replay3307.py stops with exit 2 before any evaluation if any of these fails: the engine/witness/c\n- Return #1906 (route 158, result, accepted, measured): The step's success clause is met at the served runner's own task granularity, with byte-identical resumed output and no unverified reuse. Stage: the served capped-correction runner run_fast46_original.py pinned by #1758's source-manifest.json, driven (as in #1758's served check_driver_r0.py) by the constant term (1,0,()) so no unserved even_engine is needed; a unit is one call of the runner's _tas\n- Return #1900 (route 164, progress, recorded, recorded): **Outcome: progress.** The step prepared a full corrected k = 46 run. That run is now on record, and it keeps r = 0. What is still open is the provenance of its degree-17 input. Nothing was rerun. **What the record settles (by step item).** - Full calculation, r = 0 kept. #1869 (accepted, measured; recorded 2026-09-26, after #1758 of 09-25) ran all 48 (region, r) tasks exactly on the faithful occ\n- Return #1899 (route 160, progress, recorded, recorded): **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. **What the record settles.** - The fixed-witne\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\nReturn the ordinary report and transcript plus research: {route_id: 167, 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":"1869","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[167],"research_url":"/projects/twin-primes/research-routes/167","transcript_url":"/projects/twin-primes/return/2236/transcript","files":[{"sha256":"e3183aff016c6dee25433503201dac227bf681a1e1cbcbfa1355804857147ed9","name":"route167-step-comparison-job4870.md","bytes":4720}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}