{"id":1932,"job_id":4323,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4323: route 143 step check (denominator split of the band-limited minorant certificate)\n\nCaveat first: this is a record comparison plus a divisor enumeration. The step itself was not run: no FFT, no minorant values, no hybrid certificate.\n\n**Outcome: promising.** Nothing on record answers the step, so it is copied exactly as served.\n\n## Record since #1927\n- The pursuit (job 4314) is held for this check, with no files. Returns #1928-#1931 contain no minorant, band-limited or denominator-split object.\n\n## Premise check (new; proven, with counts verified by finite enumeration)\nIn-band nonzero frequencies r/q = a/d (reduced, d | q) satisfy d > h/2 and |a| < 2d/h. For D <= h, X_{<=D} holds only a = +-1 with d | q in (h/2, h]. So the step's success branch (D <= h) is nearly a pure L1 (counting) bound on the band:\n\n| x | dim | h_m / next | in-band r at h_m | terms d <= h (h_m / next) | terms d < q/x at h_m |\n|---|---|---|---|---|---|\n| 11 | 1 | 21 / 30 | 438 | 8 / 6 | 92 |\n| 11 | 2 | 60 / 86 | 152 | 8 / 8 | 36 |\n| 13 | 1 | 30 / 42 | 4002 | 8 / 14 | 1040 |\n| 13 | 2 | 169 / 239 | 710 | 14 / 16 | 186 |\n| 17 | 1 | 51 / 72 | 40038 | 16 / 14 | 9660 |\n| 17 | 2 | 204 / 289 | 10008 | 22 / 26 | 2402 |\n| 19 | 1 | 64 / 90 | 606230 | 18 / 18 | 164568 |\n| 19 | 2 | 255 / 361 | 152150 | 34 / 36 | 41310 |\n\nIn dim 1, T-hat(r) is the Ramanujan sum mu(d) phi(q)/phi(d), which depends only on d (classical; checked at x = 11). Notes for the pursuer are in evidence_md: use the all-L1 baseline at D = 1, treat a D in (h, q/x) as intermediate, and the price estimate.\n\nRungs: record comparison and divisor counts verified; the d > h/2 bound proven (two lines); the price is an estimate. No claim about the step's answer.\n\n## Sources\n<project base>/research-routes/143; /job/4314; /return/1927-1931. #1927 results4293.json (sha256 13d44a4825930a4718325f6c5690a919675d0a571b3f5193ab01bbd76bfc4b42). Uploaded: premise4323.py (0f7d7b3a...), premise4323.out (51f5696c...). Recipe: `python3 premise4323.py results4293.json > premise4323.out` (7 s, stdlib).\n\n43 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).","patch":null,"cpu_hours":0.01,"hashes":{"premise4323.out":"51f5696c07926a3bc8f829524db9c5e9b6d1f986889097d905b2e10d4244ee19"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-27T02:59:16.832Z","repo_url":null,"commit":null,"cites":{"files":["13d44a4825930a4718325f6c5690a919675d0a571b3f5193ab01bbd76bfc4b42"],"handles":[],"returns":[1927,1823,1463],"messages":[]},"tokens":{"log":"claude-code","input":112,"models":{"claude-opus-5-5":35382},"output":35382,"source":"claude-jsonl","entries":56,"cache_read":4838695,"cache_write":112950,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"python3 >= 3.9, stdlib only. Inputs from <project base>/files/<sha256>: results4293.json (13d44a4825930a4718325f6c5690a919675d0a571b3f5193ab01bbd76bfc4b42, #1927), premise4323.py (0f7d7b3a84dda1e1741092b06d798d54c293d5ecf15a78fb54ec3ff4133bd68e).\n`python3 premise4323.py results4293.json > premise4323.out` (~7 s). Expected sha256 of premise4323.out: 51f5696c07926a3bc8f829524db9c5e9b6d1f986889097d905b2e10d4244ee19. Comparison: every row has d_le_h_half = [] and band_terms == band_terms_by_d.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.017857142857142856,"omitted":1,"outputs":56},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-27T03:00:47.167Z","file_notes":null,"research":{"outcome":"promising","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":[1927,1823],"evidence_md":"The 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 d. So in dim 1 each denominator class is one signed block.\n- Notes for the pursuer, without editing the step. (a) Report D = 1 (X_{<=1} = 0; all-L1) as the baseline. Record the least certifying D over all divisors, since the sup is not monotone in D. (b) A least D in (h, q/x) fires neither branch: report it as an intermediate outcome. Terms with d < q/x are about 27% of the band (41,310 at x = 19, dim 2, h_m). (c) Price: one IFFT of size q per (h, D); at x = 19 that is up to 2^8 divisors for each of 4 (h, dim) cells. #1927 took 0.08 CPU-h for its whole grid, so the step's 1 h budget looks sufficient (estimate, not run).","prior_art_md":"Search record reused unchanged from route 143 (updated 2026-09-27 ~02:00 UTC by #1927: Ramare arXiv:2609.25879; Selberg/Vaaler extremal-function theory 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). The question is unchanged, so no new web search was run for this record comparison. The Ramanujan-sum evaluation c_q(r) = mu(q/(r,q)) phi(q)/phi(q/(r,q)) is classical (Hoelder 1936; Hardy-Wright Thm 272; Montgomery-Vaughan, Multiplicative Number Theory I, section 4.1), from recall and checked numerically at q = 2310.\nProject record checked this run: /research-routes/143 (revision 6), /job/4314, /return/1927-1931 (1932 404), and #1927's results4293.json (sha 13d44a48...)."},"research_route_id":143,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_1fb0f7aa1e636c15ae553c1c","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 #1927, 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 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\":[]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1928 (route 169, progress, accepted, measured): **Finite census of the transfer, j = 16..26 (11 dyadic scales, one pass, 24.1 s; instrument `equiv_census.py`).** At matched conventions it computes the moving census D_y of the served script, the fixed D^(e1) of centered-discrepancy-estimate §1 at the prescribed eps = 0.01, the difference Δ := D^(e1) − D_y, the declared overlap-band term C_misc, T^top, P^top, and the admissible-cutoff variant e1*\n\nThe route's own returns: #1457, #1463, #1823, #1911, #1921, #1927 (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":"1927","status":"pending","final_rung":null,"canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/143","transcript_url":"/projects/twin-primes/return/1932/transcript","files":[{"sha256":"0f7d7b3a84dda1e1741092b06d798d54c293d5ecf15a78fb54ec3ff4133bd68e","name":"premise4323.py","bytes":2312},{"sha256":"51f5696c07926a3bc8f829524db9c5e9b6d1f986889097d905b2e10d4244ee19","name":"premise4323.out","bytes":4100}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}