{"id":1927,"job_id":4293,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4293 (pursue route 143): at x = 17 and 19, dim 2, the Selberg minorant of the window at bandwidth 2/h certifies exactly one grid step above the true exponent. That is the pre-registered success branch.\n\n**Caveats first.**\n- These are exact full-period computations on Z/q, q = x# with x ≤ 19, in float64 (FFT). KMAX = 64, and the grid is #1823's, h = round(x²·2^(j/2)) with j = −10..12 and h > 4. Rung: **measured**. No asymptotic θ is measurable at this scale, and nothing here proves the route's lemma or its TPC-strength input.\n- \"Band-limited\" does not mean \"few fractions\". The minorant's spectrum is |ξ|/q < 2/h, which still leaves 2q/h frequencies: 5,005 (x = 17, h = 204) and 76,076 (x = 19, h = 255). Every reduced denominator d | q occurs among them. The result removes the high frequencies, not the large denominators.\n- I took the minorant's Fourier transform from Vaaler's formula (Vaaler 1985; Montgomery, *Ten Lectures*, ch. 1), from memory; I did not read the paper in this session. Gate g2 below checks it numerically against Beurling's function, computed independently via trigamma.\n- Scale for the success/failure test: one grid step is log√2/log x = 0.122 (x = 17) and 0.118 (x = 19).\n\n## Setup\n- m = Selberg minorant of I = [−½, h−½], whose integer points are exactly 0..h−1, at bandwidth δ = 2/h. δh = 2 is an integer, so the minorant is extremal. Its mass is h − 1/δ = h/2, so mean_m = V·h/2.\n- m̂(t) = Ĵ(t/δ)·χ̂_I(t) − (1/2δ)(1 − |t|/δ)(e(−ta) + e(−tb)) for |t| < δ, and 0 otherwise, with Ĵ(τ) = πτ(1−|τ|)cot(πτ) + |τ|.\n- Since δ < ½, the DFT of m periodised to Z/q is exactly m̂(r/q). So M is band-limited **by construction**.\n- S_m(N) = Σ_i t(N+i) m(i), with spectrum T̂·conj(M). X_m = S_m − mean_m, with the DC term zeroed as in #1823.\n- Certificate: Σ_N (|X_m|/mean_m)^(2k) < 1 for some k ≤ 64. Then S_m > 0, and m ≤ 1_[0,h) with t ≥ 0 gives S_h ≥ S_m > 0: every length-h window meets the sifted set.\n- Fejér control F = IFFT(T̂|D̂_h|²/h), normalised by μ (#1911). A certified F makes every window of 2h−1 consecutive integers nonempty.\n- L* = exact least empty-free window length (longest cyclic zero run of t, plus 1).\n\n## Gates (every h of all 8 (x, dim) cells)\n- **g1: m ≤ 1_[0,h) pointwise on all of Z/q.** Worst excess ≤ 2.2e-16, i.e. float noise; the extremal minorant touches 0 at points spaced h/2 apart.\n- **g2: exact-spectrum m against the spatial form −½[B(δ(a−y)) + B(δ(y−b))]**, with Beurling's B via trigamma, summed over |j| ≤ 100 periods on i ∈ [−2h, 3h), wherever q ≥ 20h. Max difference ≤ 5.1e-7, which is the check's own truncation of the periodisation. This verifies the m̂ formula.\n- **g3: FFT route against a direct O(q²) correlation** for S_m at x = 11, dim 2, h = 60/121/242: max difference 6.2e-15.\n- **`true` reproduces #1823's record:** dim 2 h_true = 60/84/144/180, and the corrected dim-1 values 1.129/1.326/1.150/1.293.\n\n## Result: pre-registered test SUCCESS (measured)\n| x | dim | L* | exp_true (h, k) | exp_m (h, k) | m − true (grid steps) | h_F, k (certified length 2h_F−1, its exp) |\n|---|---|---|---|---|---|---|\n| 11 | 1 | 14 | 1.129 (15, 6) | 1.270 (21, 9) | +0.141 (1) | 8, 8 (15, 1.129) |\n| 13 | 1 | 22 | 1.326 (30, 5) | 1.326 (30, 13) | 0 | 15, 6 (29, 1.313) |\n| 17 | 1 | 26 | 1.150 (26, 12) | 1.388 (51, 7) | +0.238 (2) | 18, 9 (35, 1.255) |\n| 19 | 1 | 34 | 1.293 (45, 9) | 1.412 (64, 10) | +0.119 (1) | 23, 10 (45, 1.293) |\n| 11 | 2 | 42 | 1.707 (60, 7) | 1.707 (60, 37) | 0 | 30, 7 (59, 1.700) |\n| 13 | 2 | 66 | 1.727 (84, 10) | 2.000 (169, 9) | +0.273 (2) | 60, 8 (119, 1.863) |\n| 17 | 2 | 108 | 1.754 (144, 11) | **1.877 (204, 15)** | **+0.123 (1)** | 72, 18 (143, 1.752) |\n| 19 | 2 | 150 | 1.764 (180, 13) | **1.882 (255, 15)** | **+0.118 (1)** | 90, 28 (179, 1.762) |\n\n- **Pre-registered test.**\n  - Success needed ≤ 0.15 at x = 17 and 19, dim 2. Observed: +0.123 and +0.118, one grid step each.\n  - Failure needed ≥ +0.3 at x = 19, dim 2, or no certificate anywhere. It does not fire.\n  - In every column the certified set of grid h is upward-closed.\n- **Over all 8 cells.** m lies 0–2 grid steps above the truth (≤ +0.273). The two 2-step cells are x = 13 dim 2 and x = 17 dim 1; neither was pre-registered, and both are intermediate by #1921's note (b).\n- **The one-step loss is a sup loss of the minorant itself.** At h = h_true, min S_m/mean_m = −0.164 (x = 17) and −0.205 (x = 19), dim 2: the signed minorant count is negative somewhere, so no k can certify, and KMAX is not binding.\n  - At h_m the minimum is +0.156 and +0.204.\n  - This is not the drift of B/abs in #1823, which were +0.37..0.59 and +0.86..1.18 above the truth at x = 17 and 19.\n- **Fejér control (gate passes in all 8 cells).** h_F ≤ h_true (Jensen, #1911) and 2h_F − 1 ≥ L* hold everywhere.\n  - The certified length 2h_F − 1 lands at h_true − 1 or h_true in 6 of 8 cells (e.g. 143 vs 144, 179 vs 180). This is a measurement, not a derivation.\n\n## What this changes\n- The route's finite-scale certificate survives after all frequencies |ξ|/q ≥ 2/h are removed, at a cost of one √2 step in h at x = 17 and 19 (dim 2). The quantity a proof must control is then S_m − mean_m = (1/q) Σ_{0<|r|<2q/h} T̂(r)·conj(m̂(r/q))·e(rN/q). The kernel m̂ is explicit, signed, and supported below 2/h.\n- It is still a large sum (≈ 2q/h terms, with all denominators present). The next step asks whether a small-denominator part of it carries the certificate.\n- Combined with #1823 (arithmetic phases carry the certificate; kernel phases are dispensable), the target is now: sup-norm cancellation in T̂ restricted to |r|/q < 2/h, weighted by an explicit band-limited kernel.\n\n## Not done\n- No bandwidth other than δ = 2/h.\n- No denominator split.\n- Dims only 1 and 2, x ≤ 19 (x = 23 would need q ≈ 2.2e8 points).\n\n## Sources\n- Route 143 returns #1823 (results2851.json, sha256 0636c65d…, which holds the exp_true values and grid), #1911 (step and Fejér inequality) and #1921 (step check, notes a–c). Earlier: #1457, #1463.\n- J. D. Vaaler, \"Some extremal functions in Fourier analysis\", Bull. AMS 12 (1985) 183–216. H. L. Montgomery, *Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis* (CBMS 84, 1994), ch. 1. Formula used as recalled, checked numerically by gate g2.\n\nFiles: minorant4293.py, table4293.py, table4293.out, results4293.json. CPU ≈ 0.08 h (sum of single-process wall times).\n\n40 of @Benjaminsen's returns wait for a verdict.\n\nTranscript: removed the API token, session/account identifiers and local absolute paths outside the working folder. Housekeeping at the start: reconciled delayed usage of the department's previous return (#1926), re-ran the readiness selftest.\n","patch":null,"cpu_hours":0.08,"hashes":{"table4293.out":"3fc5aa0f9a1248265e4943f74922efb0854c38cb02a3ea7cbb658c6c6bd298ae","minorant4293.py":"a3677a6f22716afcf0467e134907914d307003b3359c6c850ae26ad2bbdd8122","results4293.json":"13d44a4825930a4718325f6c5690a919675d0a571b3f5193ab01bbd76bfc4b42"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-09-27T02:04:10.732Z","repo_url":null,"commit":null,"cites":{"files":["0636c65d9497d9136dedff08867d23c4ffb4ddaf553aff62e69309de170bf929"],"handles":[],"returns":[1823,1911,1921,1457,1463],"messages":[]},"tokens":{"log":"claude-code","input":122,"models":{"claude-opus-5-5":59664},"output":59664,"source":"claude-jsonl","entries":61,"cache_read":7658024,"cache_write":176403,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Setup: python3 >= 3.9 with numpy and scipy (tested numpy 2.0.2, scipy 1.13.1, py 3.9.6, macOS arm64). Scripts from <project base> /files/<sha256> listed in `files`.\n1. `for x in 11 13 17 19; do for d in 1 2; do python3 minorant4293.py $x $d > mn_x${x}_dim${d}.json; done; done` (defaults KMAX=64, j=-10..12; ~0.08 CPU-h total; x=19 ~100-150 s per dim and ~1.5 GB RAM).\n2. `python3 table4293.py > table4293.out`.\nExpected: table4293.out's table as in the report (sha256 of our table4293.out: 3fc5aa0f9a1248265e4943f74922efb0854c38cb02a3ea7cbb658c6c6bd298ae). Comparison rule: every `summary` h_cert/exp/k_at_cert, every row's k fields and the gate verdicts (g1 <= 1e-12, g2 <= 1e-6, F_gate true) must match results4293.json; floats rounded to 4 decimals and gate magnitudes are float noise and may differ across FFT builds, so compare fields, not file hashes. Cross-check: `true` h_cert must equal #1823 results2851.json (dim 2: 60/84/144/180).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.07936507936507936,"omitted":5,"outputs":63},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-27T02:06:24.998Z","file_notes":null,"research":{"outcome":"result","route_id":143,"next_step":{"method":"Extend minorant4293.py: per h on the #1823/#4293 grid and for D over the divisors of q (sorted), compute X_{<=D} by inverse FFT of the in-band terms with d <= D, its sup, and the exact L1 tail over in-band d > D. Report the least D (and log D / log x) for which the hybrid certifies at h_m and at one grid step above, x = 11..19, dim 1 and 2. Also report the in-band term count with d <= D.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"At x = 19, dim 2, no D < q/x certifies within two grid steps of h_m, because the in-band L1 tail over large denominators exceeds mean_m. Then large denominators are load-bearing inside the band and the phase target cannot be truncated to small d at this scale. Record that as a scoped obstruction.","success":"At x = 17 and 19, dim 2, the hybrid certifies at h_m or one step above with D <= h (log D/log x <= exp_m). Then the certificate is carried by fractions a/d with d <= h and |a/d| < 2/h up to an explicit L1 remainder, a major-arc form the route's lemma can target.","question":"Does a small-denominator part of the band-limited minorant certificate carry it? Split the in-band spectrum of X_m (|r|/q < 2/h; the Selberg minorant at bandwidth 2/h from job 4293's return, minorant4293.py) by reduced denominator d = q/gcd(r,q) into d <= D and d > D. Does the hybrid certificate max|X_{<=D}| + (1/q) sum_{d>D, in band} |T-hat(r)||M(r)| < mean_m hold at h_m with D well below q?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[1823,1911],"evidence_md":"Measured (float64, exact full period Z/x#, x = 11..19, dim 1, 2, KMAX 64, #1823 grid j = -10..12).\n\nObject: Selberg minorant m of [-1/2, h-1/2] at bandwidth 2/h (mass h/2), periodised; spectrum exactly m-hat(r/q), zero for |r|/q >= 2/h. Certificate sum (|X_m|/mean_m)^(2k) < 1 proves S_h > 0 since m <= 1_[0,h).\n\nGates, every cell: m <= 1_[0,h) pointwise (excess <= 2.2e-16); m-hat formula vs Beurling-function spatial form <= 5.1e-7 (check truncation); FFT vs direct correlation 6e-15; true reproduces #1823 (dim 2 h_true 60/84/144/180).\n\nPre-registered SUCCESS: dim 2, exp_m - exp_true = +0.123 (x=17: h 204 vs 144) and +0.118 (x=19: h 255 vs 180), one grid step each. Failure branch (>= +0.3 at x=19 or no certificate) not fired. All 8 cells: 0-2 steps (max +0.273 at x=13 dim 2; x=17 dim 1 +0.238). Certified sets upward-closed.\n\nThe one-step loss is the minorant's own sup: at h_true, min S_m/mean = -0.164 (x=17), -0.205 (x=19), dim 2, so no k certifies (KMAX not binding). Compare #1823's B/abs drift +0.37..1.18.\n\nFejer control gate passes in all 8 cells (h_F <= h_true, 2h_F - 1 >= exact L*); 2h_F - 1 lands at h_true or h_true - 1 in 6/8 cells (measurement).\n\nWhat changes: the certificate is carried by frequencies |xi|/q < 2/h with an explicit signed kernel, at one sqrt(2) step cost at x = 17, 19. The band still has ~2q/h terms (76,076 at x=19, h=255) with all denominators d | q, so the next step asks whether small denominators carry it.\n\nNot done: other bandwidths, denominator split, x >= 23. Rung: measured. No asymptotic claim.","prior_art_md":"Search updated 2026-09-27 (~02:00 UTC), reusing route 143's record (2026-09-22/23/26: Kuperberg arXiv:2210.09775, 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/#1921's 2026-09-26/27 queries (Selberg/Vaaler extremal-function theory: arXiv:1702.04579, 1704.00837, 1410.3366; Tao 254A notes 4).\n\nNew queries (web): \"Selberg minorant large sieve short intervals sifted set nonempty Jacobsthal function exponential sum positivity\"; \"band-limited minorant interval Jacobsthal OR reduced residues gaps Fourier certificate\"; \"Beurling-Selberg minorant lower bound number of sifted integers in short interval exponential sum over rationals a/q small denominators\".\n\nInspected: arXiv:2609.25879 (O. Ramare, \"The large sieve through Parseval, Large sifted sets are regular\", 2026-09-22), abstract via export.arxiv.org API only: sifted sets are small or their Fourier polynomial is very regular near the origin, with N int_{-delta}^{delta} |S(beta)|^2 >= (1 - 2/sqrt(delta N)) |S(0)|^2. This is the nearest new source: it concerns low-frequency L2 mass of a sifted set, not sup-norm positivity of a minorant-smoothed window count. Other hits (Carruth-Goncalves-Kelly box minorant, Carneiro-Littmann-Vaaler Gaussian subordination arXiv:1008.4969, Carneiro survey) are extremal-function theory, titles/snippets only. Selberg's classical use of the majorant is the large sieve inequality (upper bounds).\n\nNot found: any use of a band-limited minorant of the window to certify that every short window of a sifted set on Z/x# is nonempty, or any measurement of the Jacobsthal/twin-sieve gap from low frequencies |xi| < 2/h. Access gaps: no paper read in full; Vaaler 1985 and Montgomery ch. 1 used from recall, checked numerically (gate g2). An empty search is not evidence of novelty.\n\nExact remaining gap: sup-norm cancellation of sum_{0<|r|<2q/h} T-hat(r) conj(m-hat(r/q)) e(rN/q) at k ~ x/log x; and whether denominators d <= D << q carry it (next step)."},"research_route_id":143,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-27T02:04:10.732Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_8947c6ce2bfb2d65d3fb559c","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/143 and return #1911. Return the ordinary report and transcript plus research: {route_id: 143, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.\n\nStep check: return #1921 compared this step with the returns on record and found it still open. Build on what it read; do not redo it.\n\nThe 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 de","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/1927/transcript","files":[{"sha256":"a3677a6f22716afcf0467e134907914d307003b3359c6c850ae26ad2bbdd8122","name":"minorant4293.py","bytes":9044},{"sha256":"6d6b52cd87b59ebf1298e777d82048d67a5e379ecb77cabdc7f01e04c14fa9b2","name":"table4293.py","bytes":1585},{"sha256":"3fc5aa0f9a1248265e4943f74922efb0854c38cb02a3ea7cbb658c6c6bd298ae","name":"table4293.out","bytes":3021},{"sha256":"13d44a4825930a4718325f6c5690a919675d0a571b3f5193ab01bbd76bfc4b42","name":"results4293.json","bytes":69765}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}