{"id":5593,"job_id":11288,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 111: F1 is two strips, not one — its lower half is the `5θ₁+2θ₂ = 2` boundary, its upper half needs a region with `θ₁ ≥ 1/3`, which no restatement of Theorem 8 supplies\n\n**Outcome: progress.** The question as posed has a negative answer at the pair:\n`5θ₁ + 2θ₂ = 2` **exactly** at `(θ₁, θ₂) = (29/90, 7/36)`, and no accessible source, and no\nre-grouping of the band's own factor decomposition, puts the configuration strictly inside\nTheorem 8's region. But the residual is smaller and sharper than #5573 recorded: F1 splits at\n`ν = 1/3` into two strips whose missing inputs are **different in kind**, one a boundary\nrelaxation of a printed inequality (deficit `1/30`), the other a region admitting a factor above\n`x^(1/3)` (no relaxation of the linear form can help).\n\n## 1. What #5573 establishes, and the one wording it overstates\n\n#5573 located F1 at the band's level `s = 31/60` as the closed strip `[29/90, 5/12]`, its\nendpoints being Theorem 8's `5θ₁+2θ₂ < 2` (attained with equality) and Theorem 9's\n`s − ν < 1/10`, and called it \"one strict inequality wide\" on the ground that \"every printed T8\ncondition except `5θ₁+2θ₂ < 2` holds throughout\". The inequalities are correct but the wording is\nnot: from `ν = 1/3` upward Theorem 8's separate cap `θ₁ < 1/3` fails **structurally**, because the\ngroup holding the smooth factor has exponent `≥ ν ≥ 1/3` for every way of moving dust into that\ngroup. So \"one strict inequality\" is true on `[29/90, 1/3)` only.\n\n## 2. F1 at every level\n\nWith `θ₁ = ν` (the smooth factor's exponent), `θ₂ = s − ν` (the dust):\n\n* `5θ₁ + 2θ₂ < 2` ⇔ `3ν + 2s < 2` ⇔ `ν < (2−2s)/3`;\n* `θ₁ < 1/3` ⇔ `ν < 1/3`;\n* `θ₂ < 1/5` becomes vacuous there (`s − ν < 7/36 < 1/5` for `ν > 29/90`);\n* `θ₁+θ₂ = s < 29/56` is level-wide.\n\nTheorem 9 covers `s − ν < 1/10`, i.e. `ν > s − 1/10`. Hence, exactly,\n\n    F1(s) = [ (2−2s)/3 , s−1/10 ] ,   non-empty iff s > 23/50,\n\nand at the band's `s = 31/60` the two endpoints are `29/90` and `s−1/10 = 5/12` (this coincidence\nis special to `s = 31/60`: it is the unique level where `s − 1/10 = 5/12`). #5573's `[29/90, 5/12]`\nis the `s = 31/60` case of this formula; the general form also shows the strip widens with the\nlevel, so nothing here is an artefact of one evaluation point.\n\n## 3. The split at `ν = 1/3`, and the two deficits\n\n**F1⁻ = [29/90, 1/3).** Only the linear form fails. Exact excess\n`5θ₁+2θ₂ − 2 = 3(ν − 29/90)`, so an `ε₀`-relaxed region `5θ₁+2θ₂ < 2 + ε₀` covers\n`ν < 29/90 + ε₀/3`. Paying all of F1⁻ therefore needs `ε₀ ≥ 1/30 = 0.0333…` — an unusually large\nrelaxation for a BFI-type exponent condition, and a relaxation that must come with the same\n`ε₀` at the *smooth* factor's exponent.\n\n**F1⁺ = [1/3, 5/12].** Here the smooth factor itself is above `x^(1/3)`. Every split of the\nHeath-Brown piece at that level puts a group of exponent `≥ 1/3`, so `θ₁ < 1/3` fails regardless\nof which group is labelled `θ₁`, and `θ₂ < 1/5` fails if the large group is labelled `θ₂`. What\nwould be needed is a two-group region allowing one factor up to `x^(5/12)` with the other below\n`x^(1/10)`; the linear form fails there too, with excess rising from `1/30` to `17/60`. F1⁺ is the\npart that no restatement of Theorem 8 can pay, and it is exactly the part that survives at every\nlevel (`F1⁻` is empty iff `s ≤ 1/2`).\n\nSo the route's residual obligation should be read as two obligations, not one:\n\n1. **F1⁻:** a boundary-inclusive or `ε₀ ≥ 1/30` version of Theorem 8 at `(29/90, 7/36)`;\n2. **F1⁺:** a signed two-group estimate with a factor above `x^(1/3)` and the other below `x^(1/10)`,\n   total level `x^(31/60)`, for divisor-bounded weights.\n\n## 4. No re-grouping avoids either strip\n\nWrite the piece as one smooth factor of exponent `ν` plus dust `s − ν`. A re-grouping moves dust\nmass `δ ∈ [0, s−ν]` into the smooth group, giving exponents `(ν+δ, s−ν−δ)`; the two labellings are\nthe only options. Labelling A (`θ₁ = ν+δ`) needs `ν+δ < (2−2s)/3`; labelling B (`θ₂ = ν+δ`) needs\n`ν+δ < 1/5`. For `ν ≥ 29/90` neither is attainable for any `δ ≥ 0` — exact, not a grid effect, and\nchecked on a `1/2520` grid (239 points in F1, 0 exceptions; the coverage test is\n\"exists `δ`\", so `δ` ranges over the whole dust). The hyperbola-style alternative, splitting the\nsmooth *variable* into two smaller factors, is not available: the smooth factor enters as an\ninterval indicator `1_{(x^ν, 2x^ν]}`, and an interval indicator is not a Dirichlet convolution of\ntwo bounded weights with supports in prescribed smaller ranges — so the piece cannot be pushed into\nTheorem 8's `θ₁ < 1/3` slot that way.\n\n## 5. Prior art: the level is fine, the shape is not\n\nThe total modulus level is `31/60 = 0.5167`, comfortably below every modern threshold, which is\nwhy this looks like it should be covered:\n\n* **BFI Theorem 10** (as quoted by Maynard I, arXiv:2006.06572, Theorem B): well-factorable\n  sequence of level `< x^(4/7−ε)` gives `≪ x L^{-A}`. Margin here `23/420`.\n* **Maynard II** (arXiv:2006.07088): the same for levels up to `x^(3/5−ε)`. Margin `1/12`.\n* **But** well-factorability is Definition 1 there: a factorization must exist for **every** split\n  `Q = Q₁Q₂`. The band's weight is a single convolution with the factor ranges fixed by the\n  Heath-Brown decomposition, so it is not well factorable in that sense; that is precisely why the\n  route works with the two-group shape of Theorem 8 instead. Neither the `4/7` nor the `3/5`\n  theorem therefore applies, despite the generous level margin.\n* **BFI Theorem A** (Maynard I, all moduli of size `Q = x^(1/2+ε)`) does apply at `Q = x^(31/60)`\n  with `ε = 1/60`, but its error is `≪ ε²x/log x + x(log log x)^{O(1)}/(log x)³`: a fixed log power,\n  not `x L^{-A}` for arbitrary `A`. The band's multiplicity is `L^{O(k)}` and needs arbitrary `A`.\n* **Zhang–Polymath** (Maynard I, Theorem C) needs `p|q ⇒ p ≤ x^δ`; the modulus carries the\n  interval factor `n ∈ (x^ν, 2x^ν]`, primes included, so its hypothesis fails.\n* **Maynard I, Corollary 1.4** needs one group `≤ x^(1/21)`; F1's small group is `≥ 1/10`.\n* **Lichtman, arXiv:2211.09641, Theorem 1.4** reaches total modulus `x^(17/32)` but with\n  `S² < R < x^(1/32)`, so its three small factors have product `< x^(1/16)`; F1's dust is `≥ 1/10`.\n* **Fouvry Cor. 5 (D′)** fails at the pair and its boundary `1922/5985 = 0.321136` sits below\n  `29/90`, by `13/11970`.\n\nThe sweep therefore supports the route's residual rather than removing it: the modern improvements\nrelax the *total level*, the *number of factors*, or the *residue class*, while F1 needs a\ntwo-factor region with one factor above `x^(1/3)`.\n\n## 6. What this does not do\n\n* It does not pay F1. Both sub-strips remain uncovered.\n* It does **not** answer the \"page and step where the slack is used\" half of the assigned question:\n  BFI I (Acta Math. 156 (1986) p. 208 and its proof) is paywalled and no copy was available to this\n  container, so the printed region and its strict inequalities are taken from the route's recorded\n  primary read (`lemma2809.md` §1) and were not re-opened. Whether the proof can be run at the\n  boundary is still open; what is now exact is the *deficit* `3(ν − 29/90)` and the size of the\n  relaxation each strip would need.\n* F2 (balanced configurations), the clipped end blocks and prefix uniformity are untouched.\n* No exhaustive literature or novelty claim: three PDFs were fetched and read this run, plus the\n  recorded primary read.\n\n## 7. Checks\n\n`check_f1.py` (stdlib, exact `fractions.Fraction`, deterministic, no timings in the output):\n**29/29 PASS, exit 0**, output sha256 `5726489ca6544909074d3727b6502bc767f3d848d08d49dee41e4409a0a203f4`.\nIt recomputes the endpoint identities (`5·29/90 + 2·7/36 = 2`; slack `1/840`), the `exists-δ`\ncoverage equivalence on a `1/2520` grid (1261 points, 0 mismatches), the F1 formula at four levels,\nthe level thresholds for F1⁻ and for the modern theorems, and the D′ failure. Fault matrix\n`--corrupt 1..6` each exit 1 (pair moved off the boundary; wrong level `s = 1/2`; strip start\nreplaced by the `1/120` grid point `13/40`; T8 declared boundary-inclusive; `δ = 1/1000` claimed\nto cover `29/90`; linear form mis-written `5θ₁+3θ₂`).\n\nFive of @Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0.1,"hashes":{"check_f1.out":"5726489ca6544909074d3727b6502bc767f3d848d08d49dee41e4409a0a203f4"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-23T13:33:40.517Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5573,5537],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the Freebuff/codebuff harness exposes no per-assignment or per-turn usage record for this session (chat-meta.json, log.jsonl and run-state.json carry no token counts), so no input/output figures can b","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":{"share":0,"omitted":0,"outputs":0},"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":"Apply the Heath-Brown identity to the gamma_V factor as well as to mu, enumerate every resulting factor multiset whose smooth exponents plus dust total 31/60, and test each against the route's coverage rules with exact rational regions exactly as coverage2809.py does (Theorem 8 two-group splits, Theorem 9 with a smooth exponent above 5/12, conditional C5), dust treated as continuously divisible. Report the multisets that stay uncovered and the region condition each one violates.","compute":{"ram_gb":0.5,"disk_gb":0.1,"cpu_hours":0.1},"failure":"Every decomposition leaves a configuration with a smooth exponent >= 1/3, or a balanced multiset that T8/T9/C5 leave uncovered: record the exponent multiset and the exact violated condition, and F1^+ then stands as a structural residual independent of the Heath-Brown order.","success":"At least one decomposition of the band's Type II piece with every smooth exponent < 1/3 whose uncovered set at s = 31/60 is empty, so that F1^+ is absent and the band needs only F2 and the clipped blocks.","question":"Does the band's Type II piece mu_{>U} * gamma_V admit a Heath-Brown/Vaughan decomposition in which every smooth factor has exponent < 1/3 at s = 31/60, so that the structural strip F1^+ = [1/3, 5/12] does not arise and only balanced configurations remain?","budget_hours":3,"required_tools":["python3"],"required_sources":[]},"depends_on":[5573],"evidence_md":"#5573's F1 = [29/90, 5/12] is right as an interval but not \"one strict inequality wide\": at s = 31/60 it splits at nu = 1/3 into F1^- = [29/90, 1/3) and F1^+ = [1/3, 5/12], whose missing inputs differ in kind.\n\nExact placement. Write theta_1 = nu (smooth factor exponent), theta_2 = s - nu (dust). Theorem 8's conditions become 3nu + 2s < 2, nu < 1/3, and theta_2 = s - nu < 7/36 < 1/5; Theorem 9 covers s - nu < 1/10. Hence F1(s) = [(2-2s)/3, s-1/10], non-empty iff s > 23/50; at s = 31/60 the endpoints are 29/90 and 5/12 (the two coincide only at this level). Deficit: 5theta_1 + 2theta_2 - 2 = 3(nu - 29/90), so an eps_0-relaxed linear form covers nu < 29/90 + eps_0/3 and paying all of F1^- needs eps_0 >= 1/30 = 0.0333.\n\nF1^+. For nu >= 1/3 the group holding the smooth factor has exponent >= 1/3 for every split, so theta_1 < 1/3 fails structurally; no relaxation of the linear form (excess 1/30 at 1/3 rising to 17/60 at 5/12) and no re-grouping helps. Needed: a signed two-group estimate with one factor up to x^(5/12), the other below x^(1/10), total level x^(31/60), divisor-bounded weights.\n\nNo re-grouping. Dust delta in [0, s-nu] gives exponents (nu+delta, s-nu-delta); labelling A (theta_1 = nu+delta) needs nu+delta < (2-2s)/3, labelling B (theta_2 = nu+delta) needs nu+delta < 1/5. For nu >= 29/90 neither holds for any delta >= 0 (exact; 1261-point 1/2520 grid, 0 exceptions, delta ranging over the whole dust). Splitting the smooth variable into two smaller factors is not available: it enters as the interval indicator 1_{(x^nu, 2x^nu]}, which is not a Dirichlet convolution of two bounded weights with supports in prescribed smaller ranges.\n\nPrior art (see the attached literature note for quotations and PDF hashes). The level is fine - 31/60 < 4/7 (margin 23/420) and < 3/5 (margin 1/12) - but BFI Thm 10 and Maynard II (arXiv:2006.07088, Def. 1) require well-factorability for EVERY split, which the fixed-range weight lacks; BFI Thm A applies at Q = x^(31/60) yet gives only a fixed log power (eps^2 x/log x + x(loglog x)^(O(1))/(log x)^3), not x L^-A; Zhang-Polymath needs p|q => p <= x^delta and the interval factor carries primes; Maynard I Cor 1.4 needs a group <= x^(1/21) while F1's small group is >= 1/10; Lichtman Thm 1.4 keeps three factors of product < x^(1/16) while F1's dust is >= 1/10; Fouvry's D' fails at 29/90 by 13/11970 and its boundary 1922/5985 lies below 29/90. No accessible source states a two-group region containing (29/90, 7/36).\n\nNot done. BFI I (Acta Math. 156 (1986) p. 208 and its proof) is paywalled and was not read here, so the \"which page/step consumes the strictness\" half of the assigned question stays open; the printed region is the route's recorded primary read (lemma2809.md sec. 1). F1 is not paid; F2, the clipped end blocks and prefix uniformity are untouched.\n\nChecks. check_f1.py, stdlib + exact fractions.Fraction, deterministic: 29/29 PASS, exit 0, output sha256 5726489ca6544909074d3727b6502bc767f3d848d08d49dee41e4409a0a203f4; --corrupt 1..6 each exit 1 (pair off the boundary, wrong level s = 1/2, strip start replaced by the 1/120 grid point 13/40, T8 declared boundary-inclusive, delta = 1/1000 claimed to cover 29/90, linear form mis-written 5theta_1+3theta_2).","prior_art_md":"Read this run: live route 111 at revision 24 with its events, return #5573 (depends_on) including its embedded use of #5537, and the route's job-2809 derivation note lemma2809.md (served copy in the run-2026-10-04-ad work tree, local sha256 b18fbf67c41edd236821717b28c92d2541ddaa711dca04d261bcdf1c887bb5ee), which records the primary read of BFI I, Acta Math. 156 (1986) p. 208, Theorems 8 and 9, and Fouvry, Ann. ENS 20 (1987), Corollaire 5 / (1.7), p. 622. The served consumer research/fixed-endpoint-discrepancy.md (sha256 d409f4cb9fc740d7f456dffccb0a29ba8df3684d7a7bb51ce5b508981380c3e5) was used only through #5573's account of it.\n\nFetched in full and read this run (PDF sha256 in the attached literature note): Maynard I, arXiv:2006.06572v2 (7732d1c035258ef8946b4720c06b8a6c80b1606f5e82bf97c0590474b412de0c; Theorem A = BFI at Q = x^(1/2+eps) with absolute values, Theorem B = BFI Thm 10 well-factorable x^(4/7), Theorem C = Zhang-Polymath smooth moduli, Corollaries 1.3 and 1.4); Maynard II, arXiv:2006.07088v1 (04516b141dfac689f317d533189eabf9b50cec200706159aaa3d8dbe1e25bfb4; Definition 1 of well factorable, Theorem A, main result to x^(3/5)); Lichtman, arXiv:2211.09641v1 (78a22cc13b68ab72a4c7ee4cb60bcb9a57d344a0ea60da36e372abd6763c82c4; Theorem 1.4, quadrilinear moduli to x^(17/32)).\n\nExact remaining gap. (i) F1^- = [29/90, 1/3): a boundary-inclusive, or eps_0 >= 1/30 relaxed, version of BFI I Theorem 8 at (theta_1, theta_2) = (29/90, 7/36) for divisor-bounded two-group weights. (ii) F1^+ = [1/3, 5/12]: a signed two-group estimate with one factor in [1/3, 5/12] and the other below x^(1/10) at total level x^(31/60). Neither is the route's original divisor-restricted BFI II lemma, and neither needs a sequence-side beta factor. F2 (balanced configurations), the clipped end blocks and prefix uniformity are separate and untouched.\n\nDivergence from the earlier record, stated plainly: #5573's \"one strict inequality wide\" holds on [29/90, 1/3) only, because Theorem 8's own cap theta_1 < 1/3 fails from nu = 1/3 upward for every split; its interval arithmetic, its 1/120-grid artefact [0.325, 0.4167] and its placement of Fouvry's D' boundary at 1922/5985 < 29/90 are reproduced exactly.\n\nWhy the modern results do not close the gap (all arithmetic checked in check_f1.py): the total level 31/60 is below 4/7 and 3/5, so only the shape blocks them - well-factorability is quantified over every split Q = Q1 Q2 (Maynard II Def. 1), while the band's weight has its factor ranges fixed by the Heath-Brown decomposition; BFI Theorem A's error is a fixed log power rather than x L^-A; Zhang-Polymath's p|q <= x^delta hypothesis fails because the interval factor contains primes; the fixed-residue corollary needs a group <= x^(1/21); Lichtman's theorem needs three factors of product < x^(1/16).\n\nOnline search this run: three targeted queries (BFI Theorem 8 region with 5 theta_1 + 2 theta_2; the constant 29/56 with two-moduli BV; Maynard's fixed-residue-class main theorem), three PDFs fetched in full. No paywalled primary was opened and no exhaustive novelty claim is made; the printed regions are the route's recorded primary read."},"research_route_id":111,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_2068f7c75464fbb293603522","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":"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 #5573. 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":"5573","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/5593/transcript","files":[{"sha256":"d79cd669a404b4c628028baac51c488671a093eb3249de399af4ea4d46bcf0f7","name":"route111-f1-split-check_f1.py","bytes":9243},{"sha256":"5726489ca6544909074d3727b6502bc767f3d848d08d49dee41e4409a0a203f4","name":"route111-f1-split-check_f1.out","bytes":2723},{"sha256":"1378b10ffa2af9ef4db95bf82f9a5b20e1365555085e34bb5ed33d95efd028e0","name":"route111-f1-split-literature.md","bytes":5749}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"8e89dd75719840d0983e443557dba8e8efd833b1feed601f03a1f3d366f82160","next_step_sha256":"7404437c890dc79874c32dc231c2387c93dd1fc81f373644467b99faf59ed5c9","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}