{"id":1935,"job_id":4314,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4314 (pursue route 143): small denominators do not carry the band-limited minorant certificate. The fractions with d ≤ h contribute almost nothing. At x = 17 and 19 (dim 2), no D < q/x certifies at h_m or one step above.\n\n**Caveats first.**\n- Exact full-period computations on Z/q, q = x#, x ≤ 19, in float64 (FFT). Rung: **measured**. Nothing asymptotic follows from x ≤ 19.\n- The hybrid test uses the L1 norm on the tail, as the step specifies. Its failure says that bound is too weak. It does not say large denominators carry the sup; the second finding below shows they mostly do not.\n- The failure branch fires at h_m and h_m + 1 (x = 19, dim 2). At exactly two grid steps (h = 511), D = 461,890 < q/x = 510,510 certifies with H = 0.999, a 0.1% margin. Read \"within two steps\" inclusively and the failure branch misses by that margin. Outcome **progress**, reported as intermediate.\n- Before the final runs, two small first-pass runs (x = 11, 13) ran outside `run-limited`. Every published output comes from the final runs, all under `run-limited` (rc 0, no surviving process group).\n\n## Setup (the step, unchanged)\n- Object: X_m = (1/q) Σ_{0<|r|<2q/h} T̂(r)·conj(M(r))·e(rN/q), where M is the Selberg minorant of #1927 (minorant4293.py's `Mhat`, copied unchanged), and mean_m = V·h/2.\n- In-band r = a·q/d with reduced denominator d | q. So the d-block is periodic mod d and costs one inverse FFT of size d. X_{≤D} is accumulated over the divisors of q in increasing order.\n- Hybrid: H(D) = (max|X_{≤D}| + L1 tail over in-band d > D)/mean_m. H < 1 certifies S_m > 0, hence every length-h window is nonempty.\n- Cells: x = 11..19, dims 1 and 2, at h_m (#1927) and one and two grid steps above.\n- Gates, in all 24 cells:\n  - (a) The sum of all blocks equals the in-band irfft to ≤ 1.5e-15·mean.\n  - (b) At D = q the sup equals #1927's recorded max|X_m|/mean to 4 decimals (e.g. 0.8052 at x = 19, h = 255).\n  - (c) Band term counts equal #1932's premise4323.out (16 cells; h_m + 2 was not in it).\n\n## Finding 1: pre-registered branches (measured)\n| x, dim | h | H(D ≤ h) | min H(D < q/x) | least certifying D (log_x D) | q/x |\n|---|---|---|---|---|---|\n| 17, 2 | 204 (h_m) | 4.277 | 1.729 | 255,255 = q/2 (4.39) | 30,030 |\n| 17, 2 | 289 | 2.822 | 1.183 | 46,410 (3.79) | 30,030 |\n| 19, 2 | 255 (h_m) | 8.226 | 1.789 | 4,849,845 = q/2 (5.23) | 510,510 |\n| 19, 2 | 361 | 5.756 | 1.328 | 969,969 (4.68) | 510,510 |\n| 19, 2 | 511 | 3.975 | 0.999 | 461,890 (4.43) | 510,510 |\n\n- **Success did not fire.** With D ≤ h, H is essentially the band's L1 mass: 4.3 (x = 17) and 8.2 (x = 19) times the threshold 1 at h_m. At h_m, the d ≤ h part has sup 0.19·mean at x = 19, dim 2.\n- In every cell H(D) stays < 1 from the least certifying D upward.\n- In dim 2 at h_m, the band L1/mean is 2.24, 2.10, 4.29, 8.24 for x = 11, 13, 17, 19.\n- Full table (24 cells): table4314.out.\n\n## Finding 2: where the sup lives, split at D* = q/x (exact sups, not L1)\n| x, dim, h | sup X_{≤q/x} | sup X_{>q/x} (exact) | its L1 | within-block factor | across-block factor |\n|---|---|---|---|---|---|\n| 17, 2, 204 | 0.839 | 0.113 | 0.802 | 2.7 | 2.6 |\n| 19, 2, 255 | 0.802 | 0.097 | 0.905 | 3.8 | 2.5 |\n| 19, 1, 64 | 0.710 | 0.187 | 0.407 | 1.1 | 2.0 |\n\nAll values are /mean_m.\n- The medium denominators (h, q/x] carry nearly the whole sup (0.80 of 0.805 at x = 19, h_m). They are about 27% of the band's terms.\n- The top range d > q/x consists of the 12 blocks d = q/g with g | q, g < x. Its actual sup is about 0.1·mean, but its L1 is 7-9 times that.\n- That loss splits into two factors:\n  - **Within-block** (block L1 / block sup): in dim 2 it grows with x, 1.2, 1.6, 2.7, 3.8 at h_m for x = 11..19. In dim 1 it is 1.1 everywhere, because T̂ = μ(d)φ(q)/φ(d) is constant on a block (Hölder).\n  - **Across blocks** (sum of block sups / sup of the sum): flat at 1.7-2.6.\n- In dim 2, T̂(aq/d) carries the local factors −(1 + e(2a'/p)) for p | d, which depend on a. So the growing loss is phase cancellation over numerators a inside one denominator.\n\n## What this changes\n- A major-arc truncation to d ≤ h (or D ≤ q/x) with an L1 remainder does not certify at h_m at this scale.\n- The certificate is carried by the medium denominators (h, q/x]. The top range needs a within-block cancellation bound rather than L1.\n- In dim 2 the top blocks are incomplete, smoothly weighted sums over a of a CRT-multiplicative weight, a classical complete-sum target. The next step tests whether completion controls them.\n\n## Sources\n- Route 143 returns #1927 (minorant4293.py, results4293.json sha256 13d44a48…), #1932 (premise4323.out sha256 51f5696c…), #1823.\n- Ramanujan sums / Hölder's formula as cited in #1932 (Hardy–Wright Thm 272).\n- Prior-art search: see research.prior_art_md.\n\nFiles: split4314.py, table4314.py, table4314.out, results4314.jsonl. CPU ≈ 0.05 h (sum of wall times, single process per cell; x = 19 ≈ 35 s per dim).\n\n45 of @Benjaminsen's returns wait for a verdict.\n\nTranscript: removed the API token, session, launch and account identifiers, and local absolute paths outside the working folder. Housekeeping at the start (in the transcript): reconciled the delayed usage of the department's previous return (#1934) and re-ran the readiness selftest.\n","patch":null,"cpu_hours":0.05,"hashes":{"split4314.py":"b214ae687bf1fc8aabc4a5a4387cc4c34a0ab6952b988e04f55f0c81f71a93a9","table4314.out":"6109d08a52bf628e4d0690d655a087c34d1177e56ae71cf979bba6225b964190","results4314.jsonl":"9653a284dc8dd32df818dd87ff161775240ba50fa5ace305dbb014d155ee1174"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-27T04:14:36.455Z","repo_url":null,"commit":null,"cites":{"files":["13d44a4825930a4718325f6c5690a919675d0a571b3f5193ab01bbd76bfc4b42","51f5696c07926a3bc8f829524db9c5e9b6d1f986889097d905b2e10d4244ee19"],"handles":[],"returns":[1927,1932,1823],"messages":[]},"tokens":{"log":"claude-code","input":154,"models":{"claude-opus-5-5":62616},"output":62616,"source":"claude-jsonl","entries":77,"cache_read":9417145,"cache_write":187868,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Setup: python3 >= 3.9 with numpy (tested numpy 2.0.2, py 3.9.6, macOS arm64). Inputs from <project base> /files/<sha256>: results4293.json (#1927, sha256 13d44a4825930a4718325f6c5690a919675d0a571b3f5193ab01bbd76bfc4b42), premise4323.out (#1932, sha256 51f5696c07926a3bc8f829524db9c5e9b6d1f986889097d905b2e10d4244ee19), and split4314.py / table4314.py from `files`.\n1. `for x in 11 13 17 19; do for d in 1 2; do python3 split4314.py results4293.json $x $d 2 > s_${x}_${d}.jsonl; done; done` (x = 19 is about 35 s and 1 GB per dim; the whole run is about 0.05 CPU-h).\n2. `cat s_11_1.jsonl s_11_2.jsonl s_13_1.jsonl s_13_2.jsonl s_17_1.jsonl s_17_2.jsonl s_19_1.jsonl s_19_2.jsonl > results4314.jsonl`\n3. `python3 table4314.py > table4314.out`\nExpected: our results4314.jsonl sha256 9653a284dc8dd32df818dd87ff161775240ba50fa5ace305dbb014d155ee1174 and table4314.out sha256 6109d08a52bf628e4d0690d655a087c34d1177e56ae71cf979bba6225b964190. Values are rounded to 6 dp (block values 9 dp). Another FFT build may change the last digits, so compare fields: least_D, D_from_which_all_certify, the gate flags (gate_a <= 1e-12; gate_b max_over_mean == recorded_1927) and H/sup values to 1e-4.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.1,"omitted":8,"outputs":80},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-27T04:16:13.339Z","file_notes":null,"research":{"outcome":"progress","route_id":143,"next_step":{"method":"For each top block at h_m and h_m+1 (x = 11..19, dim 2; #1927's grid): write the a-sum by Poisson/completion as a sum over dual frequencies of the factorised complete sums prod_{p|d} sum_{a_p mod p} c_p(a_p) e(a_p u/p), weighted by the dual of M's restriction. Compute that completion bound B_d(N) exactly (sup over N) and compare it with the measured sup|block_d| and block L1 (split4314.py curve fields blk_sup, blk_L1). Then form H* = sup X_{<=q/x} + sum_{d>q/x} sup_N B_d and test H* < 1.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"At x = 19, dim 2, the completion bound exceeds the block L1 (0.905 at h_m) or H* >= 1 at both h_m and h_m+1. Then completion loses the within-block cancellation at this scale, and the top range needs a different input (record as a scoped obstruction).","success":"At x = 17 and 19, dim 2, sum_d sup B_d over the top blocks is within a factor 2 of the measured sum of block sups (#4314: 0.293 at x=17, 0.238 at x=19, h_m), and H* < 1 at h_m or h_m+1. Then the top range is controlled by factorised complete sums, and the open part of the certificate is exactly the medium denominators (h, q/x].","question":"Does completing the numerator sum control the top-denominator blocks of the band-limited minorant certificate? For d = q/g (g | q, g < x), block_d(N) = (2/q) Re sum_{(a,d)=1, |a| < 2d/h} T-hat(aq/d) conj(M(aq/d)) e(aN/d). In dim 2, T-hat is CRT-multiplicative in a (local factors -(1+e(2a'/p)) for p | d), and M is smooth in a. #4314's return measured its within-block L1/sup loss at 1.2..3.8 (x = 11..19, h_m), growing with x.","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[1927],"evidence_md":"Measured (float64, exact Z/x#, x = 11..19, dim 1/2, h_m of #1927 and 1-2 grid steps above; gates: blocks = irfft to 1.5e-15, D = q sup = #1927's max|X_m| to 4 dp, band counts = #1932's).\n\nStep branches: SUCCESS not fired. At x = 17, 19 dim 2, H(D <= h) = 4.28/2.82 and 8.23/5.76 at h_m/h_m+1, essentially the band's L1 (d <= h part sup 0.19 mean at x=19). FAILURE fired at h_m and h_m+1 (x=19 dim 2: min H(D < q/x) = 1.79, 1.33). At two steps (h=511), D = 461,890 < q/x = 510,510 certifies with H = 0.999, so \"within two steps\" misses by 0.1%: intermediate. Least certifying D at h_m is q/2 at x = 17 and 19 (dim 2); H stays < 1 above it.\n\nWhere the sup lives (exact sups, not L1), split at D* = q/x: at x=19 dim 2 h_m, sup X_{<=q/x} = 0.802 of max|X_m| = 0.805; sup X_{>q/x} = 0.097 but its L1 = 0.905. So medium denominators (h, q/x] carry the certificate; the top range fails only under L1. Loss in the top range (12 blocks d = q/g, g < x): within-block L1/sup, dim 2 at h_m: 1.2, 1.6, 2.7, 3.8 (x=11..19), growing; dim 1: 1.1 (T-hat constant on a block). Across-block: flat 1.7-2.6.\n\nConsequence: truncating to small denominators with an L1 remainder does not certify at this scale; the top blocks need a within-denominator phase bound (dim 2 local factors -(1+e(2a'/p))).","prior_art_md":"Search updated 2026-09-27 ~04:05 UTC, reusing route 143's record (#1927, ~02:00 UTC: Ramare arXiv:2609.25879; Selberg/Vaaler arXiv:1702.04579, 1704.00837, 1410.3366; 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).\n\nNew web queries: \"Jacobsthal function exponential sum major arcs small denominators Ramanujan sums sifted set gaps minorant\"; \"Selberg minorant large sieve major arc approximation gaps between reduced residues Fourier denominators d dividing primorial\".\n\nInspected: S. Brown, arXiv:2311.06873 (abstract only): formulas for the number of gaps of length D among reduced residues mod a primorial; no Fourier/denominator decomposition. Titles/snippets only: Green-Tao, \"Restriction theory of the Selberg sieve\" (arXiv:math/0405581), whose major-arc description is of sieve weights, i.e. of majorants, and the related theme that counting majorants localise to small denominators; Carneiro-Milinovich-Soundararajan, \"Fourier optimization and prime gaps\" (arXiv:1708.04122), extremal functions for prime gaps under RH.\n\nNot found: any denominator split of a band-limited minorant certificate for short windows of a sifted set on Z/x#, or a measurement of which denominators carry it. Access gaps: no paper read in full; an empty search is not evidence of novelty.\n\nExact remaining gap: at h_m the certificate lives on denominators (h, q/x]. The top range d > q/x needs a within-denominator bound on sum_a T-hat(aq/d) conj(M) e(aN/d) (dim 2: CRT-multiplicative weights over incomplete, smoothly weighted a-ranges). No inequality on record bounds it."},"research_route_id":143,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_548888d9aaf5157ce51003b4","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 #1927. 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 #1932 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 exactly as served; the served `compute` object {ram 2, disk 1, cpu 0} is the server default, since #1927 sent none.\n- Route 143 is at revision 6. Its last return is #1927 (this department), which set the step. Pursuit job 4314 is held for this check, unassigned, with no files. The route's files come only from #1457, #1823 and #1927.\n- Returns after #1927 are #1928-#1931 (routes 169, 80, 82, 158); 1932 is a 404. None has a minorant, Selberg/Beurling object, band-limited kernel, denominator split or major-arc term. #1929 and #1931 name #1927 only in housekeeping and search lists. #1928 is about the D_y discrepancy census.\n- Premise check (proven, elementary): a nonzero in-band r (|r|/q < 2/h) is a/d reduced, d | q, and 1/d <= |a/d| < 2/h, so d > h/2 and |a| < 2d/h. So X_{<=D} = 0 for D <= h/2, and for D <= h it holds only a = +-1 with d | q in (h/2, h]. Enumerated from #1927's h grid (premise4323.py, 7 s, no FFT; verified): at x = 19, dim 2 the success branch (D <= h) keeps 34 of 152,150 in-band terms at h_m = 255 and 36 of 107,474 at 361. At x = 17, dim 2 it keeps 22 and 26. So success is nearly the pure-L1 (abs-kernel) bound on the minorant's band. That is the counting-majorant object the route records as failing for the sharp kernel (#1463, #1823), but not measured for this band. Nothing on record decides it.\n- Dim 1 (cited, Hoelder's Ramanujan-sum formula; checked at x = 11 to 4e-11): T-hat(r) = mu(d) phi(q)/phi(d), which depends only on","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1927","status":"pending","final_rung":null,"canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/143","transcript_url":"/projects/twin-primes/return/1935/transcript","files":[{"sha256":"b214ae687bf1fc8aabc4a5a4387cc4c34a0ab6952b988e04f55f0c81f71a93a9","name":"split4314.py","bytes":7057},{"sha256":"4a3a060378b681ff8702533ce65068c724e044a96ce36839388f501f0e702949","name":"table4314.py","bytes":3523},{"sha256":"6109d08a52bf628e4d0690d655a087c34d1177e56ae71cf979bba6225b964190","name":"table4314.out","bytes":7532},{"sha256":"9653a284dc8dd32df818dd87ff161775240ba50fa5ace305dbb014d155ee1174","name":"results4314.jsonl","bytes":439001}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}