{"id":1818,"job_id":2809,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2809 (pursue route 111 rev 4): the r-dependent inner range is not an obstruction; the band's Type I pieces are paid on the unclipped range, and BFI I Theorem 8 covers part of the rest\n\n**Caveats first.**\n- Nothing here bounds P_band. The clipped end blocks (gap ii of #1414) are untouched. Two explicit Heath-Brown configuration families remain uncovered (F1, F2 below).\n- Lemma D is a short derivation from BFI I Theorem 9 and Shiu's theorem. It has not been reviewed.\n- The Theorem 8 coverage needs a standard Perron separation of the cutoff m in (M_1, M_2], which is described but not written out.\n- Fouvry's Corollaire 5 is stated for 1-bounded weights. Its use here for divisor-bounded weights is conditional, and it is marked C5 wherever it appears.\n\n## What was asked, and the answer\nThe question: does a version of the BFI/Fouvry theorem hold with an r-dependent inner range s in (M_1/r, M_2/r], for class -2 and odd moduli?\n- **At the primary** (BFI I, Acta Math. 156 (1986) p. 208, READ): Theorem 9 has a fixed inner range q <= Q, unweighted, with R < x^(1/10-eps) and QR < x L^-B. This confirms #1414's and #1418's second-hand quotation (Fiorilli's Thm 1.4). BFI's version has no Lambda(a) term.\n- **No source states the r-dependent version**, and it is not needed. Lemma D (lemma2809.md section 2, DERIVED) proves E(M) = sum_{r<=R} c(r) sum_{s<=M/r, s odd} Delta(y; rs) << x L^-A, uniformly for x^(1/2)L^-C <= M <= x^(31/60)L^C and c(r) << tau(r)^B. The proof:\n  - r <= L^(2D): apply Theorem 9 separately to each r.\n  - Larger r: cut into intervals (rho, rho(1+delta)] with delta = L^-D, and fix the inner range at M/(rho(1+delta)). Apply Theorem 9 after Cauchy-Schwarz for the tau-weights.\n  - The tail s in (M/(rho(1+delta)), M/r] is short in both variables. Trivially it costs x L^O(1) delta^2 per block, using Shiu's short-interval bound for tau^B/phi. That is x L^O(1) delta over the O(L/delta) blocks.\n  - Fiorilli (arXiv:1108.0439, section 6, READ) uses the same split into a fixed range plus a tail near x. There the tail needs divisor switching. At moduli ~ x^(1/2) it is trivial.\n- **Consequence** (DERIVED; this is #1414 item 3 made unconditional on the range): the note's Vaughan Type I pieces c(r)1(s) on the unclipped m-range [x/e_1, x/(2e_0)] are O(x L^-A). Their modulus is r' s with r' = r[b^2,g] <= x^(1/90) L^(3L). The class is -2, and the moduli are odd, as Theorem 9 requires. The log(rs) weight and the interval J are handled by partial summation and differencing.\n\n## New ingredient for gap (i): signed factorable weights\n- BFI I **Theorem 8** (READ at the primary) is signed. It allows gamma_{q1} << tau(q1)^B and delta_{q2} << tau(q2)^B, for theta_1 < 1/3, theta_2 < 1/5, 5theta_1 + 2theta_2 < 2 and theta_1 + theta_2 < 29/56.\n- Fouvry 1987 (READ, pp. 617-622, Numdam) enlarges the region to D' in Corollaire 5, for 1-bounded weights.\n- The band's level is 31/60 < 29/56 (eps' = 1/60, note section 4.3). The band is signed, and the note uses the sign only in the main term.\n- Neither theorem is in the note's source matrix. The matrix lists only BFI I Theorem 10 and Maynard II for factorable weights.\n- Vaughan's Type II piece mu_{>U} * gamma_V is not factorable in the needed way. Heath-Brown's identity of order k >= 100 on mu(m) is: dust factors <= x^(s/k), plus smooth factors of any size.\n- **Coverage** (MEASURED, exact rationals; coverage2809.py, <= 4 smooth factors, grid 1/120): a configuration is covered by T9 (Theorem 9 + Lemma D) or by T8 (Theorem 8 on some grouping). At s = 31/60 the Theorem 8 window for the smaller group is only (0.1944, 0.2000).\n- The uncovered configurations:\n  - **F1**: one smooth modulus factor x^nu with nu in [0.325, 0.4167] at s = 31/60 ([1/3, 0.4] at s = 1/2), and a mu-cofactor between x^(1/10) and x^(s-1/3). Theorem 9 needs the cofactor below x^(1/10). Theorem 8 needs both groups below x^(1/3). F1 persists with C5.\n  - **F2**: near-balanced multi-smooth configurations. Examples: (0.2, 0.2) + dust at s = 1/2 without C5, and (1/4, 1/4) or (0.25, 0.2583) even with C5.\n  - Counts at s = 31/60: 8,463 of 35,919 grid configurations are uncovered by T9+T8, and 3,756 are uncovered with C5.\n\nRungs: source statements READ at the primary; Lemma D DERIVED; the reduction of gap (i) to F1 and F2 DERIVED modulo the Perron step; the coverage map MEASURED. Nothing bears on twin primes directly. The D-margin still needs the signed Type II statement of #151.\n\n## Names the proposer used\n- fiorilli_1108.0439: fetched from arXiv and read (Thms 1.4 and 3.1, section 6).\n- maynard_2006.06572: fetched. Read: Thm 1.1 (convenient factor; (1.3) excludes a second factor above x^(1/2)) and Prop. 12.1 (\"Fouvry-style\", a prime-side factorisation).\n- fouvry_1987_numdam: fetched from Numdam and read, pp. 617-622.\n- route_111 and return_1414: read from the record.\n- python3: used, stdlib only.\n- In addition, BFI I itself was read (Tsinghua archive mirror). It was previously recorded as blocked (projecteuclid).\n\n## Sources\n- Bombieri, Friedlander, Iwaniec, Primes in arithmetic progressions to large moduli, Acta Math. 156 (1986) 203-251, doi:10.1007/BF02399204, Theorems 8, 9, 10 (p. 208) and section 16 (p. 248). Read via https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/6385-11511_2006_Article_BF02399204.pdf.\n- Fouvry, Autour du théorème de Bombieri-Vinogradov II, Ann. Sci. ENS (4) 20 (1987) 617-640, doi:10.24033/asens.1547: Théorème and Corollaires 1, 2, 5, (1.7), pp. 619-622.\n- Fiorilli, On a theorem of Bombieri, Friedlander and Iwaniec, arXiv:1108.0439: Thm 1.4, Thm 3.1, section 6.\n- Maynard, Primes in arithmetic progressions to large moduli I, arXiv:2006.06572: Thm 1.1 and Prop. 12.1.\n- Shiu, A Brun-Titchmarsh theorem for multiplicative functions, J. reine angew. Math. 313 (1980): cited from memory, not re-read this run.\n- Served: research/fixed-endpoint-discrepancy.md (x-content-sha256 f6858860...), sections 2.1-2.4 and 4.3; route 111 rev 4; returns #1414, #1418, #1351.\n\nFiles: lemma2809.md (the derivation), coverage2809.py and cov2809.out (sha256 fc7c06ce...). The script reproduced byte for byte on a second run. Cost: under 0.02 CPU-h.\n\n31 of @Benjaminsen's returns wait for a verdict.\n\nTranscript: removed the API token, session/account identifiers, local absolute paths outside the working folder, third-party PDF text and page images (omission notes), and lines not belonging to this assignment. Housekeeping at the start: the department reconciled delayed usage of its previous return (#1816) and re-ran its readiness selftest.\n","patch":null,"cpu_hours":0.02,"hashes":{"cov2809.out":"fc7c06cef9772d5685d408fe923ab317ac8108daf9d62fc1fe2368519ccfaa15"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T11:46:56.443Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1414,1418,1351,1340],"messages":[]},"tokens":{"log":"claude-code","input":114,"models":{"claude-opus-5-5":74009},"output":74009,"source":"claude-jsonl","entries":57,"cache_read":7041297,"cache_write":187481,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Fetch <project base>/files/<sha256> for coverage2809.py (d37f2169a0b21f58388793fdf1f1a4e3dc19f76d34a5e32332b1f3d1fef87a6f). Run `python3 coverage2809.py 120 4 > cov2809.out` (python3 >= 3.8, stdlib only; ~35 s, one core; deterministic) -> sha256 fc7c06cef9772d5685d408fe923ab317ac8108daf9d62fc1fe2368519ccfaa15. Check at s=31/60: the T8 window is (0.1944, 0.2000), the one-factor uncovered interval is [0.3250, 0.4167], and 8463 / 3756 configurations are uncovered. Cheapest credible check of the derivation: read lemma2809.md (b18fbf67c41edd236821717b28c92d2541ddaa711dca04d261bcdf1c887bb5ee) section 2 against BFI I Thm 9 (Acta 156, p. 208). The windows follow in two lines from Thm 8's inequalities at theta_1 + theta_2 = s.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.14754098360655737,"omitted":9,"outputs":61},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T11:48:48.186Z","file_notes":null,"research":{"outcome":"progress","route_id":111,"next_step":{"method":"Read Fouvry 1987 sections III-VI (downloaded, Numdam) for the exact roles of R, S, N in C.1-C.5 and the proof of Cor. 5. Translate each condition into the (s, smooth exponents, dust) model of coverage2809.py as an extra rule, and rerun the grid to list what stays uncovered. Check the order-K hypothesis in Cor. 5's proof. If F1 remains, state it as the exact input needed: Thm 9 with R up to x^(0.19), smooth inner factor >= x^(1/3).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"No condition of the main Theoreme applies to F1 (e.g. it needs N >= x^delta on the prime side in a range the band's Lambda(n-2) cannot supply). Record that condition as the obstruction and stop.","success":"An added rule, with its source locator, removes F1 or F2 from the uncovered list at every s in [1/2, 31/60]; or Cor. 5 is confirmed for order-K weights.","question":"Do Fouvry's 1987 main Theoreme (conditions C.1-C.5; order-K modulus weights (gamma)*(xi) on [R,2R]x[S,2S] against (alpha)*(beta)) or Fouvry 1985 (Crelle 357, Titchmarsh) cover the band's residual Heath-Brown configurations: F1 (a smooth modulus factor x^nu, nu in [1/3, s-1/10], with a mu-cofactor in [x^(1/10), x^(s-1/3)], s <= 31/60) and the balanced family F2 (e.g. two smooth factors ~ x^(1/4))? Does Cor. 5 hold for order-K weights?","budget_hours":2,"required_tools":["python3"],"required_sources":["fouvry_1987_numdam","bfi_1986_acta156","lemma2809","coverage2809"]},"depends_on":[1414],"evidence_md":"Gap (iii) of #1414 closed without a new source. BFI I Thm 9 (READ at the primary, Acta 156 p. 208: R < x^(1/10-eps), fixed inner q <= Q, QR < x L^-B, saving L^-A) plus Lemma D (DERIVED, lemma2809.md) handle the r-dependent inner range s <= M/r. For r <= L^(2D) the lemma takes each r singly. Larger r are cut into blocks (rho, rho(1+delta)] with the inner range fixed at M/(rho(1+delta)). The short tails in r and s cost x L^O(1) delta^2 per block (Shiu). This is the split Fiorilli uses in section 6, where the tail near x needs divisor switching; at x^(1/2) it is trivial. Consequence: the note's Vaughan Type I pieces (modulus r[b^2,g]s, r <= x^(1/90)) on the unclipped m-range are O(x L^-A).\n\nNew for gap (i): BFI I Thm 8 (READ: signed, tau^B-bounded factorable weights, theta_1 < 1/3, theta_2 < 1/5, 5theta_1 + 2theta_2 < 2, theta_1 + theta_2 < 29/56) and Fouvry 1987 Cor. 5 (READ; region D', 1-bounded weights, so used conditionally). The band's level 31/60 < 29/56, and the band is signed. Neither theorem is in the note's matrix. With Heath-Brown's identity (k >= 100) on mu(m), coverage2809.py (MEASURED, exact) finds the configurations T9+T8 leave uncovered:\n- F1: one smooth factor x^nu, nu in [0.325, 0.4167] at s = 31/60, with the mu-cofactor in [x^(1/10), x^(s-1/3)].\n- F2: near-balanced multi-smooth configurations, e.g. (1/4, 1/4) survives C5.\nThe band's error term is therefore no longer \"all Type II on the modulus\". It is F1 + F2 + the two clipped end blocks. The Perron separation for T8 is standard and is not written out.","prior_art_md":"Search updated 2026-09-26 (~11:35-11:45 UTC), reusing #1414/#1418's record.\nWebSearch: (1) 'Bombieri Friedlander Iwaniec Theorem 8 factorable weights q1 q2 level 29/56 primes arithmetic progressions fixed residue'. This found the BFI I full text on a Tsinghua archive mirror, previously recorded as blocked. (2) 'Mobius weighted moduli beyond square root Bombieri-Vinogradov ... Heath-Brown identity factorable'.\nREAD AT THE PRIMARY: BFI I (Acta 156, doi:10.1007/BF02399204), Thms 8, 9, 10 (p. 208) and section 16. Fouvry, Ann. ENS 20 (1987) 617-640 (Numdam, doi:10.24033/asens.1547), pp. 617-622: main Théorème (C.1-C.5, order-K weights (gamma)*(xi) on moduli), Cor. 5 (region D'), and his remark that BFI's Thm 8 extends to D*. Fiorilli arXiv:1108.0439: Thms 1.4 and 3.1, and section 6 (the fixed-range + tail split).\nREAD IN PART: Maynard I arXiv:2006.06572, Thm 1.1 and Prop. 12.1. Section 15 lemma statements (Deshouillers-Iwaniec; exponential sums near x^(1/7)) are prime-side tools, not modulus-weight theorems.\nSCREENED, NOT RELEVANT: arXiv:2608.13299 (convolution-type BV with well-factorable weights, level 5/8 via Pascadi) and arXiv:2607.09110 (Mobius-twisted EH under GRH), abstracts only.\nNOT READ: Fouvry 1984 Acta 152, Fouvry 1985 Crelle 357, Drappeau 2015, Shiu 1980 (cited).\nNo inspected source states Thm 9 with an r-dependent inner range; Lemma D supplies it at moduli <= x^(31/60). No inspected source covers F1 (a smooth modulus factor >= x^(1/3) with a mu-cofactor in [x^(1/10), x^(0.19)]) or the balanced family F2. Exact remaining gap: F1, F2, and the clipped end blocks. Candidates: Fouvry's main Théorème (C.1-C.5) and Fouvry 1985 (Titchmarsh range for R)."},"research_route_id":111,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_deeeca351dbe23573afec825","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/111 and return #1418. Return the ordinary report and transcript plus research: {route_id: 111, 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>, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1414","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/111","transcript_url":"/projects/twin-primes/return/1818/transcript","files":[{"sha256":"b18fbf67c41edd236821717b28c92d2541ddaa711dca04d261bcdf1c887bb5ee","name":"lemma2809.md","bytes":7016},{"sha256":"d37f2169a0b21f58388793fdf1f1a4e3dc19f76d34a5e32332b1f3d1fef87a6f","name":"coverage2809.py","bytes":5751},{"sha256":"fc7c06cef9772d5685d408fe923ab317ac8108daf9d62fc1fe2368519ccfaa15","name":"cov2809.out","bytes":6086}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}