{"id":1414,"job_id":2801,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job 2801: rescue of route 111. The band's Type I part escapes #1351's per-block obstruction via BFI's smooth-inner-sum theorem; the Type II part and the end blocks remain\n\nCaveat first: the key theorem is quoted at second hand (Fiorilli, arXiv:1108.0439, Thm 1.4, citing Bombieri-Friedlander-Iwaniec, Acta Math. 156 (1986)). Its primary was not read, and one hypothesis (fixed versus r-dependent inner range) is unchecked. Rungs: item 1 measured/derived; items 2-3 cited plus derived (heuristic as a match); nothing on twin primes.\n\n## What changes\n1. **#1351 confirmed.** At level x^(1/2+eta), BFI II saves (log y/log x)^2 = 4 eta^2, a constant. The block sum grows like (4 eps'^3/(3 log 2)) log x (ratio 0.9997 at log2 x = 1e5; blocksum2801.out). No all-moduli absolute-value input pays the band.\n2. **Different ingredient.** BFI (Fiorilli Thm 1.4): sum_{r<=R} |sum_{q<=Q} (psi(x;qr,a) - x/phi(qr))| << x/log^A x for fixed a != 0, R < x^(1/10) and QR < x/log^B x. The absolute value is only over the restricting factor. This is route 111's divisor-restricted lemma with unweighted m, with a log^-A saving rather than a constant.\n3. **Match.** With the note's Vaughan cutoffs U = V = x^(eps'/3), the Type I pieces of mu(m) in the modulus arrangement (2.4) have modulus r[b^2,g] s, with r[b^2,g] <= x^(2eps'/3+o(1)) < x^(1/10) and s unweighted apart from log m. On the unclipped m-range (I_m = J) this is the theorem's shape at a = -2. Any fixed log power of multiplicity is absorbed by A.\n4. **Remaining obstruction.** (i) Vaughan Type II on the modulus (two arbitrary factors > x^(eps'/3)); (ii) the two clipped end blocks, whose endpoint depends on m; (iii) the r-dependent inner range, to be checked in the primary.\n\n## Derivation (item 3)\n(2.4): P_band = -sum_m mu(m) log m sum_{b,g} mu(b)mu(g) sum_{n in I_m, m[b^2,g] | n} Lambda(n-2) + atom. Vaughan gives mu(m) = sum_{rs=m, r<=UV} c(r) + Type II, with |c(r)| <= tau(r). The Type I error is sum_{r,b,g} c(r)mu(b)mu(g) sum_{s in (M_1/r, M_2/r]} log(rs) Delta_{r[b^2,g]s}(x, x/2; -2). Bounding it by sum_{r'} tau(r')^3 |sum_s ...| with r' = r[b^2,g] < x^(1/10), and applying partial summation for log s, gives the item 2 shape.\n\n## Sources\n- Fiorilli, arXiv:1108.0439, Thms 1.3, 1.4, 3.1 (read at ar5iv).\n- BFI, Acta Math. 156 (1986); Fouvry, Acta Math. 152 (1984): located, not read.\n- research/fixed-endpoint-discrepancy.md, sections 2.1, 2.3, 3 (matrix), 4.3 (read, served docs).\n- Route 111 record and return #1351 (read).\n\n## Files\nblocksum2801.mjs (sha256 6b7544881f0eedfdc9d372a6fbbbdeef4b188550ee745fb7138f3ca13468ff3a) and its stdout blocksum2801.out (7b0d2282a59070d9c147e658f1eff8e69bf4076d75ed43f25ff58823d15056b8); node, under 1 s.\n\nTranscript: removed credentials, session/account/launch ids and private local paths (automated scrubber). Housekeeping: 44 returns wait for a verdict.","patch":null,"cpu_hours":0.001,"hashes":{"blocksum2801.out":"7b0d2282a59070d9c147e658f1eff8e69bf4076d75ed43f25ff58823d15056b8"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-22T21:35:10.547Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1351,1340],"messages":[]},"tokens":{"log":"claude-code","input":68,"models":{"claude-opus-5-5":30772},"output":30772,"source":"claude-jsonl","entries":34,"cache_read":2692822,"cache_write":124068,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"node blocksum2801.mjs > blocksum2801.out; sha256sum blocksum2801.out must equal 7b0d2282a59070d9c147e658f1eff8e69bf4076d75ed43f25ff58823d15056b8 (deterministic, <1 s, Node >= 18). The literature step: read BFI Acta Math. 156 (1986) for the theorem Fiorilli numbers 1.4 and check its range hypotheses against section 3 of this report.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":35},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":111,"next_step":{"method":"Read the BFI I and Fouvry 1984 primaries for the exact statement and hypotheses. Match them line by line to the Type I pieces on the unclipped m-range. Then write the residual (Type II on the modulus plus the two clipped end blocks) as the precise remaining input, with its own size bound.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The primary requires a fixed inner range or excludes the needed class/moduli in a way no splitting repairs; record that hypothesis as the obstruction.","success":"The theorem as stated in the primary covers the Type I pieces (or does after a dyadic split in r that loses at most a log power), so the band input shrinks from (4.9) to the Type II-on-modulus pieces and the end blocks.","question":"Does the BFI/Fouvry smooth-inner-sum theorem (Fiorilli Thm 1.4; BFI Acta 156) pay the Vaughan Type I pieces of P_band in arrangement (2.4) with a log^-A saving, including the r-dependent inner range s in (M_1/r, M_2/r], class -2 and odd moduli?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[],"evidence_md":"The obstruction #1351 found is real, but it comes from the input, not from the band. It holds only where the band is paid by an absolute value over every modulus.\n\n1. #1351 item 1 CONFIRMED (DERIVED; arithmetic MEASURED by blocksum2801.mjs). At Q = x^(1/2+eta), BFI II's saving (log y/log x)^2 with y = Q^2/x equals 4 eta^2, which is a constant. Over the blocks up to x^(1/2+eps'), BLOCKSUM/((4 eps'^3/(3 log 2)) log x) = 1.134, 0.969, 0.997, 0.9997 at log2 x = 200, 1e3, 1e4, 1e5. The top block tends to 4 eps'^2 = 1.11e-3. An all-moduli absolute-value input like (4.9) cannot pay the band.\n\n2. NEW INGREDIENT (CITED at second hand: Fiorilli, arXiv:1108.0439, Thm 1.4, quoting Bombieri-Friedlander-Iwaniec, Acta Math. 156 (1986); primary not read). For fixed a != 0, lambda < 1/10, R < x^lambda and QR < x/(log x)^B: sum_{r<=R,(r,a)=1} |sum_{q<=Q,(q,a)=1} (psi(x;qr,a) - Lambda(a) - x/phi(qr))| << x/(log x)^A. This is a divisor-restricted theorem at moduli far beyond x^(1/2) with a log^-A saving, not a constant. The absolute value is over the restricting factor r only; the long factor q is summed without weights. It is a form of route 111's lemma: signed in m, unweighted m, d = r <= x^(1/10).\n\n3. MATCH TO THE BAND (DERIVED; not checked line by line). In the note's modulus arrangement (2.4) the coefficient is mu(m) log m mu(b) mu(g) on q = m[b^2,g]. Vaughan on mu(m) with U = V = x^(eps'/3), which the note already uses, gives Type I pieces m = r s with r <= UV = x^(2eps'/3) and s unweighted apart from log (partial summation). Put r' = r[b^2,g] <= x^(2eps'/3)(log x)^(3L) < x^(1/10). Then each Type I piece in the unclipped m-range, where I_m = J = (x/2,x] and there is no prefix sup, has the shape in item 2 with a = -2 and odd moduli. The sum over (b,g) and r costs (log x)^(O(L)) against an arbitrary log^-A. So the multiplicity route 111 set out to repair, and #1351's per-block deficit, both disappear for these pieces.\n\n4. WHERE THE OBSTRUCTION NOW LIVES. (i) The Vaughan Type II pieces on the modulus, alpha_a beta_b with a,b > x^(eps'/3) and arbitrary coefficients: they have no smooth factor for item 2 and are not well-factorable for BFI I Thm 10 / Maynard II. (ii) The two clipped end blocks, m in (x/(2e_1), x/e_1) and (x/(2e_0), x/e_0], where the endpoint e_0 m or e_1 m moves with the modulus. (iii) Item 2's inner range is fixed, q <= Q, while the pieces need an r-dependent range s in (M_1/r, M_2/r]. Fiorilli flags this r-dependence as a difference from Fouvry's statement, so the match in 3 is unverified until the primary is read.\n\nNothing here bears on twin primes directly. The D-margin still needs the signed Type II statement.","prior_art_md":"Search 2026-09-22 (about 21:35 UTC; 2 WebSearch queries, 2 page fetches).\nQueries: (a) Fouvry \"Autour du theoreme de Bombieri-Vinogradov\" moduli rs smooth factor level beyond 1/2, fixed residue, absolute value over r; (b) Bombieri-Vinogradov beyond x^(1/2), moduli with a smooth factor, Fouvry, Titchmarsh divisor problem, q = rs, log^-A.\nREAD (ar5iv rendering of Fiorilli, \"On a theorem of Bombieri, Friedlander and Iwaniec\", arXiv:1108.0439, Canad. J. Math.): Theorem 1.3 (BFI, well-factorable lambda, level x^(4/7-eps)); Theorem 1.4 (BFI, absolute value over r < x^lambda with lambda < 1/10, inner q <= Q, QR < x/log^B x, saving x/log^A x); Theorem 3.1 (Fiorilli, r-dependent inner range q <= x/(rM) with M <= (log x)^A). These are quoted as Fiorilli states them. The BFI and Fouvry primaries were NOT read, so their exact hypotheses are cited, not verified.\nLOCATED, NOT READ: Fouvry, Acta Math. 152 (1984) 219-244 (DOI 10.1007/BF02392198); Fouvry, Ann. ENS 20 (1987) 617-640 (Numdam, free); Fouvry, J. reine angew. Math. 357 (1985); BFI, Acta Math. 156 (1986) 203-251; Maynard I (arXiv:2006.06572) section 15, \"Fouvry-style estimates\", which generalises Fouvry to moduli with a convenient factor; Drappeau (Compositio 2015) for the friable case.\nRELATION TO THE RECORD. The note's source matrix (section 3) marks \"Fouvry 1985 primaries\" UNREAD. Its BFI I Thm 10 row rejects the Type I piece 1_{r|m} 1_{m~Q} as not well-factorable, which is correct for Thm 10. Neither the matrix, route 110/#1337, #1340 nor #1351 considers the smooth-inner-sum theorem (Thm 1.4 above), which needs no well-factorability for exactly that shape.\nEXACT REMAINING GAP. (a) Read BFI I (Acta 156) and Fouvry (Acta 152) for the theorem behind Thm 1.4. Check the r-dependent inner range, the class a = -2 and the odd-moduli restriction, and whether psi(x) - psi(x/2) is admissible. (b) The Vaughan Type II pieces on the modulus and the two clipped end blocks remain uncovered by any inspected source."},"research_route_id":111,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_da55f23c995cabb5136f4e91","run_id":"run_d49357432886d26fb306d436","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/111 and return #1351. 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":[],"research_url":"/projects/twin-primes/research-routes/111","transcript_url":"/projects/twin-primes/return/1414/transcript","files":[{"sha256":"6b7544881f0eedfdc9d372a6fbbbdeef4b188550ee745fb7138f3ca13468ff3a","name":"blocksum2801.mjs","bytes":840},{"sha256":"7b0d2282a59070d9c147e658f1eff8e69bf4076d75ed43f25ff58823d15056b8","name":"blocksum2801.out","bytes":356}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}