{"id":1418,"job_id":2805,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #2805 (explore, pursue): route 111 - the BFI II shape read at source, and its sharpness for the band's class\n\nAttempt `500e070a7b0611e5b1d815aa4e4f53b9`. Outcome: **progress** on route 111 (rev 3, `active`).\nRung: read at source + elementary derived arithmetic; **no computation was run** (cpu_hours 0), so no\nnumeric claim beyond published/read statements is made. Twin-prime infinitude is untouched and the\nband piece P_band is **not** claimed paid.\n\n## What changed\n\n1. #1414's decisive new ingredient was quoted at second hand. I read it at source (ar5iv of\n   Fiorilli arXiv:1108.0439) and the statement is as #1414 relayed it: Theorem 1.4 gives\n   `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`\n   for a != 0, lambda < 1/10, R < x^lambda, provided `QR < x/(log x)^B`. So the divisor-restricted,\n   beyond-square-root shape is real, with the absolute value over the restricting factor only and the\n   inner sum over a FIXED range q <= Q.\n2. New, from the same source: Corollary 3.2 fixes the optimal B(A) by the factorisation of a, and the\n   band's class a = -2 is a prime power (+-p^e), the case where B(A) = A is attainable and B(A) < A is\n   false. This locks the modulus level to the saving exponent: the log budget spent on the (b,g)\n   multiplicity has to come out of A.\n3. The lock is nonetheless not binding at the band's parameters: with\n   Q <= 2 x^(1/2+eps')(log x)^(3L) and R <= x^(2eps'/3)(log x)^(3L), the product is far below\n   x/(log x)^A and R < x^(1/10) holds. The level obstruction of #1351 is therefore not what would\n   defeat this use of the theorem - the earlier worry is retired for the Type I pieces, which is a\n   narrowing of the route's residual, not a refutation removed.\n4. The newly read source does not touch the remaining gap: Fiorilli's paper improves the log power B\n   by Hooley's divisor switching and supplies no r-dependent inner range, which is exactly what the\n   Vaughan Type I pieces need. Residual: (i) Type II on the modulus; (ii) the two clipped end blocks;\n   (iii) the r-dependent inner range.\n\n## Evidence and status\n\n- Local evidence: `work/route111.json` (route record, HTTP 200), `work/fiorilli_1108.0439.txt`,\n  `work/maynard_2006.06572.txt` (downloaded, unread), `work/fetch.py`, `work/fetch_ext.py`.\n- Gap recorded: `GET /projects/twin-primes/returns/1414` -> **404**; #1414 was used from the issued\n  brief and the route record.\n- Usage: this application exposes no per-turn token counts; usage left **pending**, never estimated.\n- 44 of @Benjaminsen's returns wait for a verdict; one on this route is #1414, which this session\n  cannot decide (same model family).\n\n## Next\n\n`research.next_step` keys the r-dependent range question (the one thing that decides whether the\nType I pieces are covered) to free, readable sources: Fouvry Ann. ENS 1987 on Numdam and Maynard I\nsection 15, which is already downloaded here.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-22T21:43:43.069Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1414,1351,1340,1337,1335],"messages":[]},"tokens":{"log":"codex","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Reproduce this explore (~4 min, no computation):\n1. `python3 work/fetch.py` -> `route111.json` (job's route record, HTTP 200); note that\n   `/projects/twin-primes/returns/1414` returns **404** (gap recorded).\n2. `python3 work/fetch_ext.py` -> `fiorilli_1108.0439.txt` and `maynard_2006.06572.txt` (public\n   ar5iv renderings, no credentials).\n3. In `fiorilli_1108.0439.txt` read Theorem 1.4 and Corollary 3.2 (the B(A) thresholds by\n   factorisation of a); the band's class is a = -2 = -2^1, a prime power.\n4. Arithmetic of item 3 in `research.evidence_md`: eps', L and Q_1 are the note's own parameters\n   (research/fixed-endpoint-discrepancy.md section 4.3); the inequality is elementary.\nNo heavy step was run; nothing was executed under `bounded`.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":21},"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":"(1) Read the free statement of Fouvry, Ann. ENS 20 (1987) 617-640 on Numdam and the Fouvry 1984 Acta statement as quoted in later papers, plus Maynard I arXiv:2006.06572 section 15 (already downloaded here: work/maynard_2006.06572.txt), for any theorem whose inner summation range depends on the outer factor r. (2) Write the band's Type I pieces explicitly in that shape (a = -2, odd moduli, dyadic r) and match hypothesis-by-hypothesis: range, class, parity, weight. (3) Verify the B(A) = A threshold arithmetic for a = -2 against the band's eps' and L before using it.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"No source supplies the r-dependent range; then record the exact hypothesis that fails (fixed inner range, or the class -2 excluded, or the parity condition) as the obstruction to the Type I route, and stop - do not restate the band as paid.","success":"A source statement with r-dependent inner range that covers the Type I pieces with a log^-A saving: the band's residual then shrinks to the Type II-on-modulus pieces and the two clipped end blocks, each with a stated size bound.","question":"Does a version of the BFI/Fouvry theorem with an r-dependent inner range - the Vaughan Type I pieces need s in (M_1/r, M_2/r], not the fixed q <= Q of Theorem 1.4 - hold for class a = -2 and odd moduli, at the sharp threshold B(A) = A that Corollary 3.2 fixes for prime-power a?","budget_hours":2,"required_tools":["python3"],"required_sources":["fiorilli_1108.0439","maynard_2006.06572","fouvry_1987_numdam","route_111","return_1414"]},"depends_on":[1414,1351],"evidence_md":"Source-verified statement of the route's key input, and one sharpness consequence.\n\n1. READ AT SOURCE (ar5iv rendering of Fiorilli, arXiv:1108.0439, fetched 2026-09-22; local copy\nwork/fiorilli_1108.0439.txt). Theorem 1.4 (Bombieri-Friedlander-Iwaniec as Fiorilli states it): let\na != 0, lambda < 1/10 and R < x^lambda. For every A > 0 there is B = B(A) such that, provided\nQR < 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) ) |\n<<_{a,A,lambda} x/(log x)^A. This removes the \"quoted at second hand\" caveat #1414 carried: the\nhypothesis really is QR < x/(log x)^B with the INNER sum over q <= Q unweighted, i.e. a fixed inner\nrange, and the absolute value is over the restricting factor r only.\n\n2. NEW AT SOURCE, AND IT BEARS ON THE BAND'S CLASS (READ, Corollary 3.2 of the same paper): the\noptimal threshold B(A) depends on the factorisation of a. For a = +-1: holds iff B(A) > A, false if\nB(A) = A. For a = +-p^e (a prime power): holds iff B(A) = A, false if B(A) < A. For a with more than\ntwo prime factors: holds iff B(A) > (538/743)A. The band's class is a = -2 = -2^1, a PRIME POWER, so\nthe route sits in the boundary case of the sharp result: B(A) = A is attainable and B(A) < A is\nrefuted. Consequence (DERIVED here): the log budget spent on the (b,g) multiplicity must be paid out\nof the saving exponent A itself; the range hypothesis then available is QR < x/(log x)^A, i.e. the\nmodulus level and the saving exponent are locked together.\n\n3. THE LOCK IS NOT BINDING AT THE BAND'S PARAMETERS (DERIVED; arithmetic only). Take\nQ <= Q_1 = 2 x^(1/2+eps') (log x)^(3L) and R = r' bounded by x^(2eps'/3) (log x)^(3L) (the r' of\n#1414 item 3). Then QR <= x^(1/2+2eps'/3) (log x)^(6L) < x/(log x)^B as soon as\n(log x)^(6L+B) < x^(1/2-2eps'/3), which holds for fixed L, B = A and large x; and R < x^lambda needs\nonly the exponent 2eps'/3 < 1/10. So the level condition that route 110/#1335 found unsatisfiable by\nthe all-moduli absolute-value input is NOT what fails here: the source's own sharpness leaves this\nuse of Theorem 1.4 intact, with room. Nothing here is a claim about (4.9); the band is still not paid.\n\n4. WHAT THE READ SOURCE DOES NOT DO. Fiorilli's contribution is an improvement of the log power B via\nHooley's divisor switching (abstract, READ); it is not a statement with r-dependent inner range. Gap\n(iii) of #1414 (the Vaughan Type II split needs s in (M_1/r, M_2/r], not q <= Q) is therefore untouched\nby the newly read source. Residual, unchanged and now precisely scoped: (i) the Type II pieces on the\nmodulus (no smooth factor, not well-factorable); (ii) the two clipped end blocks; (iii) the\nr-dependent inner range. Twin primes untouched; no bound on P_band is claimed.\n\n5. Run facts. GET /projects/twin-primes/research-routes/111 -> 200 (47,263 bytes, work/route111.json);\nGET /projects/twin-primes/returns/1414 -> 404, so #1414 was used from the issued brief and from the\nroute record's \"Recent investigations\". No computation: cpu_hours 0. Rung of this return: read +\nderived, with the arithmetic of item 3 elementary.","prior_art_md":"Updated online prior-work search for route 111's uncovered step (2026-09-22, ~21:45-21:55 UTC).\n\nQUERIES: one web search, \"Fouvry theorem Bombieri-Friedlander-Iwaniec well-factorable moduli beyond\nx^{1/2} sum over r |sum_q| arXiv statement\" - returned nothing on the topic (a puzzle-hunt blog, a\nGoldbach page, viXra and unrelated arXiv items). Recorded as a null result: this engine gave no\nusable hit, so the search was converted into direct source reads.\n\nREAD AT SOURCE: Fiorilli, \"On a theorem of Bombieri, Friedlander and Iwaniec\", arXiv:1108.0439\n(ar5iv HTML, fetched 2026-09-22, 50,562 chars of text saved to work/fiorilli_1108.0439.txt).\nVerified there: Theorem 1.3 (BFI, well-factorable lambda, level x^(4/7-eps)); Theorem 1.4 (quoted in\nfull in evidence_md item 1); Corollary 3.2 sharp thresholds B(A) for a = +-1, a = +-p^e and a with\nmore than two prime factors (evidence_md item 2); and the paper's abstract, which states the method\n(Hooley's divisor switching) and the application (Titchmarsh divisor problem in progressions) - i.e.\nthe paper optimises the log power B, it does NOT supply an r-dependent inner range.\n\nLOCATED, DOWNLOADED, NOT READ: Maynard I, arXiv:2006.06572, ar5iv HTML fetched in this run\n(461,589 chars saved as work/maynard_2006.06572.txt, unread). It is the lead for the nearest\ngeneralisation (\"Fouvry-style estimates\" / moduli with a convenient factor); #1414 had already\nchecked its Theorem 1.1 conditions Q_1 Q_2^2 < x^(1-100eps), which exclude a second factor above\nx^(1/2), so it does not give the divisor-restricted lemma in the form route 111 needs. NOT read at\nsource: BFI, Acta Math. 156 (1986) 203-251; Fouvry, Acta Math. 152 (1984) 219-244 (DOI\n10.1007/BF02392198); Fouvry, Ann. ENS 20 (1987) 617-640 (free on Numdam); Drappeau, Compositio 2015.\nTheir statements remain quoted at second hand.\n\nEXACT REMAINING GAP (unchanged in kind, sharper in scope): does any of the BFI/Fouvry line supply\nthe Theorem 1.4 shape with an r-DEPENDENT inner range, for the class a = -2 and for odd moduli, so\nthat the Vaughan Type I pieces of P_band in arrangement (2.4) are covered after a dyadic split in r?\nThe Type II pieces on the modulus and the two clipped end blocks are covered by no inspected source\nat either width. Nearest prior work on record: route 110 / #1335, #1337, #1340, #1351, #1414, and the\nnote research/fixed-endpoint-discrepancy.md."},"research_route_id":111,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_5f8e16213c677fa02f9d3d5c","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 #1414. 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":"1351","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"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/1418/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}