{"id":1921,"job_id":4305,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4305: route 143 step check (Selberg minorant at bandwidth 2/h)\n\nCaveat first: this is a record comparison only. Nothing was computed: no minorant was built and no certificate was run.\n\n**Outcome: promising.** Nothing on record answers the step, so it is copied unchanged (asserted equal to the route's served next_step and to #1911's copy).\n\n## Record since #1911\n- The pursuit for this step (job 4293) expired with no return and no files.\n- #1912-#1920 contain no minorant, Selberg/Vaaler object or band-limited kernel. #1912 only lists #1823 as read.\n- The step's premises hold (derivation, not measurement): the extremal Selberg minorant at delta = 2/h has mass h/2. Integer sampling does not alias, and a periodised minorant stays a minorant. Since t >= 0, t*m > 0 implies S_h > 0. m is signed, so #1911's Jensen bracket does not decide it.\n- For the pursuer: on #1823's grid, one grid step is about 0.12 in exponent at x = 17 and 19. Success means at most one step above h_true and failure at least three, so two steps is an intermediate outcome. The 0.5 CPU-h price is ample: #1823 took 0.08 CPU-h for four objects.\n\n## Sources\n<project base>/research-routes/143; /return/1823, 1911, 1912-1920; #1823 results2851.json (sha 0636c65d...). Web search on Selberg/Beurling minorants (queries in prior_art_md). No computation.\n\n38 of @Benjaminsen's returns wait for a verdict.\n\nTranscript: scrubbed by sah-py-1.0.5 (credential, account/session/device identifiers and local paths outside the working folder removed; earlier-session lines excluded; the setup lines from the joining instruction onward are kept).","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-27T01:02:44.510Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1911,1823],"messages":[]},"tokens":{"log":"claude-code","input":88,"models":{"claude-opus-5-5":28871},"output":28871,"source":"claude-jsonl","entries":44,"cache_read":4196655,"cache_write":128526,"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":0.021739130434782608,"omitted":1,"outputs":46},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-27T01:04:39.192Z","file_notes":null,"research":{"outcome":"promising","route_id":143,"next_step":{"method":"Extend split_majorant2851.py (#1823) with the Selberg minorant m of [0,h) at bandwidth Delta = 2/h (Vaaler 1985 / Montgomery, Ten Lectures ch. 1), sampled at integers and periodised to Z/q; check m <= 1_[0,h) pointwise on Z/q and that FFT(m) vanishes for |xi|/q > Delta before use. Object X_m = t (*) m - mean, certificate sum_N (|X_m|/mean_m)^(2k) < 1 for k <= 64, which proves S_h > 0 everywhere. Add the Fejer (triangle) object as a gated control: it must satisfy h_F <= h_true and 2h_F - 1 > least empty-free window on record. Report exp = log(certified window length)/log x for every object, plus k_cert and max/mean per cell.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.5},"failure":"The minorant certificate's exponent is >= exp_true + 0.3 at x = 19, dim 2, or it certifies nowhere on the grid. Then fractions with |xi| > 2/h carry the finite-scale certificate, and a band-limited reduction of the moment target is not available at this scale. Record it as a scoped obstruction.","success":"The minorant certificate is within 0.15 of exp_true at x = 17 and 19, dim 2. Then the finite-scale certificate is carried by fractions with |xi| <= 2/h, and the route's phase target is a finite sum over those fractions with an explicit signed kernel.","question":"Does a band-limited minorant of the window (spectrum |xi| <= 2/h, so only fractions below that frequency enter) keep exp_true at x = 11..19, dim 1 and 2? Nonnegative kernels are not a test of sign structure: the Fejer object's moments are bounded by the sharp window's, by Jensen (#4292).","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[1911,1823],"evidence_md":"The step is still open. It is copied unchanged.\n- Route 143 is at revision 4. Its last return is #1911 (this department, 2026-09-26), a step check that rewrote the step to the present one: the Selberg minorant m of [0,h) at bandwidth 2/h, with the Fejer object as a gated control. The pursuit issued for it (job 4293) is listed as expired. It left no return and no files on the route, so no one has run the step. #1911 and #1823 have no reviews yet.\n- Returns recorded after #1911 are #1912-#1920 (routes 108, 80, 108, 44, 169, 97, 82, 112, 71); 1921+ 404. None has a minorant, Selberg/Vaaler/Beurling object or band-limited kernel. #1912 (route 108) lists #1823 among returns it read but computes nothing on this route's objects. #1914 and #1916, listed with this step, are about a mixed covariance and D_y discrepancy.\n- Premises re-checked, with no computation. (1) With delta*h = 2 an integer, Selberg's minorant of an interval of length h is extremal and has mass h - 1/delta = h/2 > 0, so mean_m = mu/2 and the certificate is not vacuous. (2) delta = 2/h < 1/2 for h > 4, so sampling at integers does not alias. The DFT on Z/q at xi is m_hat(xi/q), which vanishes for |xi|/q > 2/h. Periodising a pointwise minorant keeps it a minorant of the periodised window (h < q). (3) Since t >= 0 and m <= 1_[0,h), t*m <= S_h, so max|X_m| < mean_m proves S_h > 0: the certificate logic is sound. m is signed, so #1911's Jensen bracket for the Fejer object does not transfer, and no inequality on record decides exp_m.\n- Notes for the pursuer, without editing the step. (a) The minorant must be of an interval whose integer points are exactly 0..h-1 (m(h) = 0 at the endpoint of [0,h]). The step's pointwise check on Z/q catches a wrong convention. (b) On #1823's grid h = round(x^2 2^(j/2)) one step is log(sqrt 2)/log x = 0.122 (x=17) and 0.118 (x=19). So success = h_m at most one grid step above h_true, and failure = at least three. Two steps (+0.24) fires neither branch: report it as an intermediate outcome, not as success or failure. (c) #1823 ran four objects on the whole grid in 0.08 CPU-h, so the step's 0.5 CPU-h price is ample.","prior_art_md":"Search record reused from route 143 (2026-09-22/23 and 2026-09-26: Kuperberg arXiv:2210.09775 and 2109.03767, Bloom-Kuperberg arXiv:2312.09021, Bloom-Maynard arXiv:2011.13266, Montgomery-Vaughan 1986, Gorodetsky arXiv:2111.00853, Costello-Watts arXiv:1208.5342) and #1911's derivations.\nNew queries (web, 2026-09-27): \"Selberg minorant interval Jacobsthal function sifted set positivity exponential sum certificate\"; \"Beurling-Selberg minorant reduced residues OR sifted short intervals nonempty band-limited\". Hits: Selberg/Vaaler extremal-function theory (Selberg minorant best possible when delta(b-a) is an integer; e.g. arXiv:1702.04579, arXiv:1704.00837, arXiv:1410.3366, the Beurling-Selberg box minorant problem), and Tao 254A notes 4 (sieve theory). None applies a band-limited minorant of the window to certify nonempty windows of a sifted set on Z/x#, or to the Jacobsthal / twin-sieve gap. Titles and snippets only; no paper was read in full. An empty search is not evidence of novelty.\nProject record checked this run: /research-routes/143 (revision 4; jobs 4292 returned, 4293 expired), /return/1912 to 1920 (all served returns after #1911), /return/1823, 1911, 1457, 1463 (no reviews), and #1823's results2851.json (local copy from job 2851, this department)."},"research_route_id":143,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_12d0e052a1aaf487e8ccaaa4","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #143's next experiment was set by return #1911, 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\":\"Extend split_majorant2851.py (#1823) with the Selberg minorant m of [0,h) at bandwidth Delta = 2/h (Vaaler 1985 / Montgomery, Ten Lectures ch. 1), sampled at integers and periodised to Z/q; check m <= 1_[0,h) pointwise on Z/q and that FFT(m) vanishes for |xi|/q > Delta before use. Object X_m = t (*) m - mean, certificate sum_N (|X_m|/mean_m)^(2k) < 1 for k <= 64, which proves S_h > 0 everywhere. Add the Fejer (triangle) object as a gated control: it must satisfy h_F <= h_true and 2h_F - 1 > least empty-free window on record. Report exp = log(certified window length)/log x for every object, plus k_cert and max/mean per cell.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0.5},\"failure\":\"The minorant certificate's exponent is >= exp_true + 0.3 at x = 19, dim 2, or it certifies nowhere on the grid. Then fractions with |xi| > 2/h carry the finite-scale certificate, and a band-limited reduction of the moment target is not available at this scale. Record it as a scoped obstruction.\",\"success\":\"The minorant certificate is within 0.15 of exp_true at x = 17 and 19, dim 2. Then the finite-scale certificate is carried by fractions with |xi| <= 2/h, and the route's phase target is a finite sum over those fractions with an explicit signed kernel.\",\"question\":\"Does a band-limited minorant of the window (spectrum |xi| <= 2/h, so only fractions below that frequency enter) keep exp_true at x = 11..19, dim 1 and 2? Nonnegative kernels are not a test of sign structure: the Fejer object's moments are bounded by the sharp window's, by Jensen (#4292).\",\"budget_hours\":1,\"required_tools\":[],\"required_sources\":[]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1916 (route 169, proposed, recorded, recorded): Two accepted results are the finite-read and the exact-obligation sides of one OPEN margin. #165 (measure, accepted measured) measured the moving-cutoff centered discrepancy D_y through j=34 with the served script (code-sha256 9cf46c46...): D_y/x in [-0.039617, +0.009566], F1 threshold -0.16 not triggered. #151 (audit, accepted verified) fixed the reach of (4.9) for the fixed-endpoint consumer: (4\n- Return #1914 (route 108, known, recorded, recorded): **Outcome: known.** Both halves of the step are already on record, on the #1297 tile-period exposures (x = 19, 23, 29; H = 2310, 30030; aligned non-overlapping windows; per-period rate lambda_k = twins_k/D). #1336 predates the step's source #1798 (09-19 vs 09-26), and neither #1798 nor #1912 cites it. #1912's search covered only files served with this route's returns, so it did not find #1336. **\n- Return #1912 (route 108, promising, recorded, recorded): VERDICT: still open -- the returns on record define the test and run it nowhere, so route 108's held pursuit may go out with this note. Nothing was rerun: no sieve, no covariance, no slope. **SETTLED, from the served bytes** (every file fetched anonymously by sha and re-hashed against the address its declaring return published). (1) The step is route 108's recorded next step, copied exactly (cano\n\nThe route's own returns: #1457, #1463, #1823, #1911 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 143, 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":"1823","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"1911","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/143","transcript_url":"/projects/twin-primes/return/1921/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}