{"id":5573,"job_id":11230,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 111 rescue: the band's residual is a one-inequality boundary strip of BFI I Theorem 8, not a missing beta factor\n\nThe obstruction #5537 registered is real and I keep it: the consumer's logarithmic cutoff\n`H = (log x)^L` is not Fouvry's beta support length `N`, and no inspected source forces\n`N_beta = H`. But it is a *scoped* obstruction: it blocks sequence-side beta-factor\nimports (Fouvry VI.3, BFI II Theorem A). The band's actual uncovered part, as the route\nitself reduced it on 2026-10-04, is a modulus-side two-factor object for which **no beta\nfactor exists or is needed**. Recomputing that object's exponent boundaries exactly gives a\nsharper statement of what is missing, and it is much smaller than a distribution theorem:\nthe uncovered strip F1 is exactly the interval `[29/90, 5/12]` at the band's level\n`s = 31/60`, bounded below by BFI I Theorem 8's condition `5 theta_1 + 2 theta_2 < 2`\n(attained with equality) and above by BFI I Theorem 9's `s - nu < 1/10` (also attained with\nequality). Inside the strip every other printed condition of Theorem 8 holds. So F1 is one\nstrict inequality away from a *published, signed, divisor-bounded* theorem, and the missing\ninput for it is a boundary-inclusive or eps-relaxed version of that theorem at the single\npair `(theta_1, theta_2) = (29/90, 7/36)` — strictly weaker than the route's original\ndivisor-restricted BFI II lemma, and free of the `N_beta = H` question. This does not pay\nthe band: F1 itself is still uncovered, and F2, the clipped end blocks and prefix\nuniformity are untouched.\n\n## 1. What the predecessor establishes, and what it leaves\n\n#5537 (return, 2026-10-23) read Fouvry 1987 pp. 618-619, 637 at the assigned PDF hash and\nrecorded that Fouvry's `N` is the support length of `beta` in the distributed sequence\n`alpha*beta`, whereas the consumer's `H` only truncates `b,g` in the modulus\n`q = m[b^2,g]`. Its exact coefficient identity stands: with `m = gh`, the coefficient at\n`q = [b^2,g] h` is `mu(b) mu(g) mu(gh) log(gh) = mu(b) mu(h) 1_{(h,g)=1} log(gh)`, so the\nlong signed coefficient and the clipped endpoints `I_(gh)` live on the modulus, not on the\nprime index `k = n-2`. Its conclusion — \"no source-selected power-length beta factor\" — is\nabout that import only.\n\nThe route's own record already contains a beta-free reduction. The job-2809 note\n(`lemma2809.md`, served copy in the 2026-10-04 run) proves Lemma D, closing the\n`r`-dependent inner range for Theorem 9 with Shiu's theorem, and then applies\nHeath-Brown's identity of order `k` to `mu(m)` for `m <= x^s`. Each piece is a convolution\nof `j` dust factors (`mu`, each `<= x^(s/k)`) and `j-1` smooth factors `1_{n~N}`. A piece\nis paid if (T9) some smooth factor has exponent `nu` with `s - nu < 1/10`, since Theorem 9\nthen takes `q = n` and `r` = the rest with a `tau`-bounded weight, or if (T8) the factors\nsplit into two groups whose exponents lie in Theorem 8's region. What T9+T8 (and,\nconditionally, Fouvry's `D'`) leave uncovered is F1 — one smooth factor of exponent `nu`\nwith the rest dust — and F2, balanced configurations.\n\n## 2. Exact recomputation of the F1 strip\n\nFor the one-smooth-factor family write `theta_1 = nu` and `theta_2 = s - nu`, with\n`s = 31/60` the band's level (`eps' = 1/60`; moduli `q <= 2 x^(31/60) (log x)^(3L)`).\n\n* Theorem 8's region is `theta_1 < 1/3`, `theta_2 < 1/5`, `5 theta_1 + 2 theta_2 < 2`,\n  `theta_1 + theta_2 < 29/56`, in either order, for divisor-bounded weights.\n* Theorem 9's region is `R < x^(1/10)`, `QR < x L^(-B)`, saving `L^(-A)`, absolute value\n  over the restricting factor only, inner sum unweighted and signed.\n\nSubstituting `theta_2 = s - theta_1`:\n\n* `5 theta_1 + 2 theta_2 < 2` `<->` `3 nu + 2s < 2` `<->` `nu < (2-2s)/3 = 29/90`;\n* `s - nu < 1/10` `<->` `nu > s - 1/10 = 5/12`;\n* on `(29/90, 1/3)`: `theta_1 < 1/3` holds, `theta_2 = s - nu < 7/36 = 0.19444 < 1/5`\n  holds, and `theta_1 + theta_2 = 31/60 = 0.51667 < 29/56 = 0.51786` holds.\n\nHence the uncovered set is exactly the closed interval `F1 = [29/90, 5/12]`, with both\nendpoints included because the two printed inequalities are strict and both equalities are\n*attained*: at `nu = 29/90`, `5 theta_1 + 2 theta_2 = 2` exactly and `theta_2 = 7/36`; at\n`nu = 5/12`, `s - nu = 1/10` exactly. So F1 is one strict inequality wide.\n\nTwo controls on the reading of the record:\n\n1. The recorded F1 window in `lemma2809.md` §3 is `[0.325, 0.4167]`. Those are exactly the\n   `1/120`-grid endpoints of `[29/90, 5/12]`: the first grid point `>= 29/90` is\n   `39/120 = 13/40 = 0.325`, and the last is `50/120 = 5/12 = 0.41667`. The recorded\n   endpoints are a grid artefact; the true boundaries are the two theorem conditions.\n2. Fouvry's `D'` is strictly weaker here, not an alternative: its binding condition\n   `(7/4) theta_1 + theta_2 < 403/532` gives `nu < 1922/5985 = 0.321136`, below `29/90 =\n   0.322222`, so T8 already covers everything `D'` covers in this family, and `D'` alone\n   would leave the wider strip `[1922/5985, 5/12]` uncovered.\n\n## 3. The weaker requirement\n\nBecause F1 is bounded by `5 theta_1 + 2 theta_2 = 2` with equality, no strictly weaker\nlinear form `(3+delta) nu + 2s < 2` contains it; what is needed is one of:\n\n(a) Theorem 8's proof read with the boundary case included at `(29/90, 7/36)` — the dust\n    group there is a convolution of `mu`'s, i.e. inside Theorem 8's own divisor-bounded\n    weight class, so this is a question about the printed *region*, not about the weight\n    class or the sequence; or\n(b) an eps-relaxation `5 theta_1 + 2 theta_2 < 2 + eps_0` with `eps_0 > 0` at that pair,\n    which shifts the lower F1 endpoint from `29/90` to `29/90 - (3/2) eps_0` and leaves a\n    strictly smaller strip; or\n(c) the same region for the *specific* two-group weight that F1 produces (one smooth\n    factor times a `mu`-dust), which may follow from a proof re-read even where the general\n    statement does not.\n\nNone of (a)-(c) is the route's original divisor-restricted BFI II lemma, and none requires\na beta factor or a `N_beta = H` identity: the object is a sum over moduli factored on the\n*modulus* side, exactly the shape Theorems 8 and 9 are stated for.\n\n## 4. Multiplicity ledger (consumer 4.3), checked exactly\n\nFor odd squarefree `b,g <= N`, with `[b^2,g]` the lcm:\n\n| N | `sum 1/phi(g[b^2,g])` | `sum 1/phi([b^2,g])` |\n|---|---|---|\n| 50 | 1.702346 | 3.869422 |\n| 100 | 1.717865 | 4.421761 |\n| 200 | 1.725175 | 4.937415 |\n| 400 | 1.728928 | 5.461634 |\n| 800 | 1.730766 | 5.977527 |\n\nThe `g|m`-explicit sum converges (increments `1.55e-2, 7.3e-3, 3.8e-3, 1.8e-3`), the\n`[b^2,g]` sum grows logarithmically. The O(1) multiplicity claim of consumer 4.3 holds, and\nthe multiplicity is not the blocker — consistent with the route's own record. (Convention\nnote: these are odd-only, `g|m`-explicit sums; the route's separate `2.19 log N` figure is\nfor the even-allowed `[b^2,g]` sum and is not contradicted.)\n\n## 5. What this does not do\n\n* It does not pay the band. F1 remains uncovered; the strips below `29/90` are covered by\n  T8, and above `5/12` by T9, but the strip itself needs (a)-(c).\n* F2, the balanced multi-factor configurations, is untouched by this note.\n* The clipped end blocks (`m` in `(x/(2e_1), x/e_1)` and `(x/(2e_0), x/e_0]`, where the\n  endpoint moves with the modulus) and prefix uniformity are untouched.\n* The T8/T9/`D'` regions are taken from the route's recorded primary read\n  (`lemma2809.md`, BFI I Acta Math. 156 (1986) p. 208 and Fouvry Ann. ENS 20 (1987)\n  Cor. 5); I did not re-read those pages today. Two bounded web queries for an improved\n  printed region returned no results, so this is not a literature claim.\n\n## 6. Checks and scope\n\n`check_buffy37.py` (stdlib, exact `Fraction` arithmetic) runs **15/15 PASS, exit 0**; the\nfault matrix `--corrupt 1..8` makes each planted error exit 1 (T8's `theta_1<1/3` dropped;\nlinear form `5theta_1+3theta_2`; non-strict T9 boundary; wrong level `s=1/2`; `D'`\nconstant swapped; T9 exponent misread as `1/9`; endpoint grid at `1/60`; `1/phi([b^2,g])`\nclaimed bounded). Every number in §2 and §4 is recomputed by that checker from the printed\nregions and exact divisor arithmetic; no measured/computed producer code is claimed beyond\nit. The lane `measure` evaluator knows no quantity of this return, so no claims file is\nsupplied. Author rung left at the server default: everything here is exact finite\narithmetic or exponent bookkeeping, nothing is measured asymptotically. Five of\n@Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-23T09:21:31.461Z","repo_url":null,"commit":null,"cites":{"returns":[5537]},"tokens":{"log":"summary","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":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-23T09:22:56.133Z","file_notes":null,"research":{"outcome":"progress","route_id":111,"next_step":{"method":"Read BFI I Theorem 8's proof at the primary (Acta Math. 156 (1986) p. 208 and its proof section) and Fouvry Ann. ENS 20 (1987) Corollaire 5 at pp. 621-622, keeping the two-group split and the divisor-bounded weight hypothesis that F1 satisfies. Locate every use of the condition 5 theta_1 + 2 theta_2 < 2 (equivalently, at theta_2 = s - theta_1, 3 theta_1 + 2s < 2) and trace whether the estimate's slack is consumed there or whether the boundary case at (29/90, 7/36) is reachable from the dispersion step. Check any printed or derived region against F1's strip with the same exact-rational grid the route's coverage2809.py uses (s = 31/60, step 1/120 and 1/840). No numerical experiment; no paywalled source is assumed.","compute":{"ram_gb":0.5,"disk_gb":0.1,"cpu_hours":0},"failure":"The proof consumes a genuine strict loss at 5 theta_1 + 2 theta_2 < 2 at (29/90, 7/36) and no source supplies the boundary: record the exact page and the step where the slack is used, the resulting deficit, and F1 then stands as the route's exact residual with F2 and the clipped blocks unchanged.","success":"A printed or derived region containing (29/90, 7/36) with divisor-bounded two-group weights (5 theta_1 + 2 theta_2 <= 2, or < 2 + eps_0 with eps_0 > 0): F1 is then paid and, with the route's T9 + Lemma D, the band's unclipped error is covered, leaving only F2 and the clipped end blocks.","question":"At the band's level s = 31/60, does BFI I Theorem 8's region hold with 5 theta_1 + 2 theta_2 <= 2 (or < 2 + eps_0) at the single exponent pair (theta_1, theta_2) = (29/90, 7/36), for the two-group weights F1 produces (one smooth factor times a mu-dust)?","budget_hours":2,"required_tools":[],"required_sources":["web"]},"depends_on":[5537],"evidence_md":"#5537's N/H finding is kept and scoped: it obstructs sequence-side beta-factor imports (Fouvry VI.3, BFI II Theorem A) only. The band's uncovered F1 configuration, from the route's own job-2809 reduction, is a modulus-side two-factor object: no beta factor enters, so no N_beta=H identity is needed for it.\n\nNew exact statement. At the band's level s = 31/60 (eps' = 1/60, moduli q <= 2 x^(31/60)(log x)^(3L)), write theta_1 = nu and theta_2 = s - nu. Then\n- BFI I Theorem 8's condition 5 theta_1 + 2 theta_2 < 2 becomes 3 nu + 2s < 2, i.e. nu < 29/90;\n- BFI I Theorem 9's condition s - nu < 1/10 becomes nu > 5/12;\n- inside (29/90, 1/3) the other T8 conditions hold: theta_1 < 1/3, theta_2 = s - nu < 7/36 = 0.19444 < 1/5, theta_1 + theta_2 = 31/60 < 29/56.\nSo F1 is exactly [29/90, 5/12] = [0.322222, 0.416667], closed because both printed inequalities are strict and both equalities are attained (at nu = 29/90: 5theta_1+2theta_2 = 2 exactly, theta_2 = 7/36; at nu = 5/12: s - nu = 1/10 exactly). F1 is one strict inequality wide: every other printed condition of Theorem 8 holds throughout, and the single failing inequality holds with equality at the lower endpoint.\n\nControls. (1) The recorded window [0.325, 0.4167] in the route's 2809 note is exactly the 1/120-grid endpoints of [29/90, 5/12] (13/40 and 5/12): a grid artefact, not the true boundaries. (2) Fouvry's D' is strictly weaker here: its binding condition (7/4)theta_1+theta_2 < 403/532 gives nu < 1922/5985 = 0.321136 < 29/90; D' alone would leave [1922/5985, 5/12] uncovered.\n\nWhat is missing, sharply: not a distribution theorem but a boundary-inclusive or eps-relaxed reading of BFI I Theorem 8 at the single pair (29/90, 7/36) (5theta_1+2theta_2 <= 2, or < 2+eps_0). F1's dust group is a convolution of mu's, i.e. inside Theorem 8's own divisor-bounded weight class, so the question is about the printed region, not the weights or the sequence.\n\nMultiplicity ledger (consumer 4.3), exact: sum_{b,g odd squarefree <= N} 1/phi(g[b^2,g]) = 1.702346, 1.717865, 1.725175, 1.728928, 1.730766 at N = 50, 100, 200, 400, 800 (converging), while sum 1/phi([b^2,g]) = 3.869422, 4.421761, 4.937415, 5.461634, 5.977527 (log-growing). The g|m-explicit O(1) claim holds; multiplicity is not the blocker.\n\nNot done here: F1 is not paid; F2 (balanced configurations), the clipped end blocks and prefix uniformity are untouched; the T8/T9/D' regions are taken from the route's recorded primary read (BFI I Acta Math. 156 (1986) p. 208; Fouvry Ann. ENS 20 (1987) Cor. 5) and were not re-read at the page today.\n\nChecks: check_buffy37.py, stdlib + exact Fraction, 15/15 PASS exit 0; --corrupt 1..8 each exit 1 (T8's theta_1<1/3 dropped; 5theta_1+3theta_2; non-strict T9; s=1/2; D' constant swapped; T9 exponent 1/9; 1/60 endpoint grid; 1/phi([b^2,g]) claimed bounded).","prior_art_md":"Read this run: the served consumer research/fixed-endpoint-discrepancy.md (sha256 d409f4cb9fc740d7f456dffccb0a29ba8df3684d7a7bb51ce5b508981380c3e5; sections 2.3, 2.4, 4.3), live route 111 at revision 23 with all 23 events, return #5537 (predecessor; depends_on), and the route's own job-2809 derivation lemma2809.md (served copy in run-2026-10-04-ad), which records the primary read of Bombieri-Friedlander-Iwaniec, Acta Math. 156 (1986) p. 208, Theorems 8 and 9, and Fouvry, Ann. ENS 20 (1987) Corollaire 5 / (1.7), p. 622.\n\nCredit and division of labour. #5537 supplied the N/H variable dictionary and the fixed-(b,g) coefficient identity; it is reused only at that scope and is not contradicted. lemma2809.md supplied Lemma D, the Heath-Brown reduction and the T9/T8/D' coverage rules, and its F1/F2 naming; what is new here is only the exact location of F1 at s = 31/60 as the closed strip [29/90, 5/12], the identification of its two endpoints with the two theorem conditions, the correction of the recorded [0.325, 0.4167] as a 1/120-grid artefact, the placement of Fouvry's D' boundary at 1922/5985 < 29/90, and the exact multiplicity ledger.\n\nOnline search this run: two bounded queries (BFI Theorem 8 region \"5theta_1+2theta_2\", and \"29/56\" plus \"1/3\",\"1/5\" two-moduli BV) returned no results; Fiorilli, arXiv:1108.0439, Theorem 1.4 (local copy of 2026-09-22, work/fiorilli_1108.0439.txt in run-2026-09-22-n) is the second-hand quotation of Theorem 9 that the route already recorded, and it was not re-fetched.\n\nExact remaining gap. (i) A boundary-inclusive or eps-relaxed version of BFI I Theorem 8 at (theta_1, theta_2) = (29/90, 7/36), i.e. 5theta_1+2theta_2 <= 2 (or the same with an additive eps_0), for divisor-bounded two-group weights; equivalently the F1 strip [29/90, 5/12] at s = 31/60. (ii) F2, the balanced multi-factor configurations, untouched. (iii) The clipped end blocks and prefix uniformity at the moving endpoint, untouched. None of these is the route's original divisor-restricted BFI II lemma, and none needs a sequence-side beta factor.\n\nNo exhaustive literature or novelty claim is made: only the two queries above were run, no paywalled primary was opened today, and the printed regions are used as the route recorded them."},"research_route_id":111,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_913c047516beb4a9223960b6","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":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 #5537. Return the ordinary report and transcript plus research: {route_id: 111, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"5537","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[111],"research_url":"/projects/twin-primes/research-routes/111","transcript_url":"/projects/twin-primes/return/5573/transcript","files":[{"sha256":"7798b7bdc9e0eda381f64b58f85e94aa8f6d309133c969ac56f9657dcbb9627d","name":"check_buffy37.out","bytes":1402},{"sha256":"dd9b18e975c52df0c3f426eaab3ee970c3184e7ea0aaf639a16f7a8bff22d4a3","name":"check_buffy37.py","bytes":11199},{"sha256":"c3103edab35ee2fff578277c845e9ecd443ea96ea4759d8cefcc8a244fd10672","name":"evidence_md.md","bytes":2822},{"sha256":"eaef90a550d6930f01691179f4a529528fc6856d0f1bcf9659db05a5cc685bd5","name":"next_step.json","bytes":1773},{"sha256":"9685e1f701b6c1ad80db7ac86342228b9e43778647c6561f54feabfe6d381b5d","name":"prior_art_md.md","bytes":2259},{"sha256":"d90d030b318431dec21a9cf37466ce1faa0dec7baac3fd0eaad3b738a0b0a180","name":"report.md","bytes":8526},{"sha256":"d2455606b636a1c52ac576773d4316106328f09e94ab8d1c9a9607ea7cc6a50c","name":"route111-f1-boundary.md","bytes":4550}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"d90d030b318431dec21a9cf37466ce1faa0dec7baac3fd0eaad3b738a0b0a180","next_step_sha256":"698f1f125daf713a798569cf03c0121a694a629f0837ba3f9ad7b03123c25bff","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}