{"id":1814,"job_id":2799,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2799 (pursue route 46 rev 3): all four named d-edge inputs priced at w = 6/25, 1/5, 1/8. None of them forces w >= 6/25, and none holds on the d-edge's hard band nu >= 9/20 at any w. The binding hypothesis is the Möbius weight on the modulus e, not the split.\n\n**Caveats first.**\n- This is a reading of theorem hypotheses plus exact arithmetic. No estimate is proved or improved, and nothing about the twin-prime margin changes.\n- Section 3 (the nu < 9/20 sub-strip is in classical BV range at every fixed w) is a derivation sketch at the heuristic rung. I have not checked it against the corpus's subtracted density on that strip; that check is the next step.\n- The Deshouillers–Iwaniec paper itself was not reached (no arXiv copy). Its Theorem 12 is read as the q = 1 case of Drappeau's Theorem 2.1, which Drappeau states it extends.\n- Route locator: \"Pascadi arXiv:2404.04239 Lemmas 3.2–3.3\" mixes two papers. arXiv:2404.04239v3 has no Lemma 3.x. The corpus's Lemmas 3.2–3.3 are in Pascadi, *Non-Abelian Amplification and Bilinear Forms with Kloosterman Sums*, GAFA, doi:10.1007/s00039-026-00746-0 (open access). I priced both.\n\n## 1. The object and what each input needs\n\nThe d-edge object is reachability-coverage.md §3.1 (3): Σ_Q λ_s(Q) Σ_{r prime ~ x^w} (log r) C_{Y,Z}(Qr − 2), with Q ~ x^{1−w}. By prime-detection-spec.md §4, β_Z(t) = log t − Σ_{ℓ|t, ℓ≤Z} Λ(ℓ). So\n\nC = μ_{(Y,E0]} * L − μ_{(Y,E0]} * Λ_{≤Z} * 1.\n\nThe modulus-side variable e ~ x^ν (residual-coverage.md §1: ν is e's exponent, ν ∈ [1/20, 19/20]) carries μ(e). Only the complementary variable t (or m) is smooth. Every named input needs smooth or unweighted variables in specific slots.\n\n| input (read at source) | deciding hypothesis | at w = 6/25, 1/5, 1/8 | d(price)/dw |\n|---|---|---|---|\n| Bettin–Chandee 1502.00769v1 Cor. 1 (1.4), register reading (22max+17min)/20 (**control**) | smooth f, g on m1, m2 | 20.8/20, 21/20, 21.375/20: fail (reproduced) | −1/4: worsens as w falls |\n| Drappeau 1504.05549v4 Thm 1.1 (engine: Thm 5.1) | (5.5): x^η ≤ N ≤ Q^{2/3−η}, x^{1/4} ≤ Q ≤ x^{1/2+δ}. (5.6) sums over q with **no weight on q** | short-variable window w < 1/3 holds with margin 7/75, 2/15, 5/24. But μ(e) on the modulus is outside (5.6), so the input is silent beyond ν = 1/2 | 0 (δ depends only on η) |\n| Drappeau Lemma 5.2 (= Iwaniec–Kowalski Thm 17.4, level 1/2) | Σ_q max_a \\|·\\|, so weighted moduli are allowed; q ≤ x^{1/2}/R | holds for ν < 9/20 (ν < 1/2 for the μ*L piece) | 0 |\n| Deshouillers–Iwaniec Thm 12 (= Drappeau Thm 2.1, q = 1) | g smooth in c and d (2.2) | silent: the modulus side carries μ(e) | none |\n| Topacogullari 1605.02364v1 Thm 1.1 (engine: Lemma 2.1, (2.3)) | every weight v_1..v_k smooth | silent: λ_s(Q), primes r and μ(e) are not smooth | none |\n| Pascadi GAFA Lemmas 3.2 (Ramanujan), 3.3 (Weil) | complete sums, no size parameters | w-free. The size-dependent DI analogue in 2404.04239v3 is Cor. 18, with smooth c, d in Φ: silent, as for DI | none |\n| corpus dispersion row (§3.4): trivial completion needs e1e2 ≤ Q | ν ≤ (1−w)/2 | 19/50, 2/5, 7/16, all inside ν < 9/20 | −1/2 |\n\n`price2799.py` (stdlib, exact rationals) produces `price2799.json` and checks the control: 104/5, 21, 171/8 = 20.8, 21.0, 21.375 over 20. The script also checks the register identity (17/20)(A+B) + max/4 = (22max+17min)/20, Drappeau's window at all three splits, and the dispersion range lying inside the BV range.\n\n## 2. Answer to the route's question\n\n- **w = 6/25 is not forced from below by the named class.** No hypothesis in any of the four names the split. Drappeau's only size condition on the short variable (N ≤ Q^{2/3}, so w < 1/3 at Q = x^{1/2}) is met with more room as w falls.\n- **The failure clause does not fire as stated.** Only the register reading of Bettin–Chandee worsens as w falls. The other inputs are w-neutral or silent.\n- **The class cannot be priced \"hold\" on the hard band at any w.** For ν ≥ 1/2 the modulus is e > x^{1/2} and carries μ(e). Drappeau's beyond-1/2 statement admits only unweighted moduli. DI, Topacogullari and Pascadi's Kloosterman engines need smooth variables in the modulus slots. This is the known parity-sensitive input: Möbius-weighted moduli beyond x^{1/2} (Murty–Vatwani, JNT 2017; Maynard's beyond-1/2 results need well-factorable weights, which μ(e) is not).\n- So the split axis is **region-favourable (#1408's T(w)) and input-neutral** for this class. Lowering w costs no named input anything, and buys nothing on the hard band.\n\n## 3. By-product (heuristic, needs the check in next_step)\n\nOn the d-edge, both pieces of C are bilinear convolutions (λ_s(Q), primes r) in residue classes 2 mod e or eℓ, with a smooth complementary variable. The large sieve at level 1/2 (Lemma 5.2 row; Siegel–Walfisz from the primes r) therefore controls the discrepancy for eℓ ≤ x^{1/2−ε}, that is ν < 9/20, at every fixed w > 0. That is 4/9 of the ν-range [1/20, 19/20]. In reachability-coverage.md §4.2, (19/25, 1/20) and (19/25, 9/20) are priced at γ_req = 11/10 and 3/2 by the moment instrument. On this reading they are BV-range points: an instrument limit, as that note itself allows (\"not a lower bound on the actual sum\"). If this holds, the true d-edge band is ν ∈ [9/20, 19/20], at every w.\n\n## Sources\n- arXiv:1502.00769v1 (Bettin–Chandee) p.4 (1.4); #737 excerpt. arXiv:1504.05549v4 (Drappeau): Thm 1.1 (§1), Thm 2.1 (2.2)–(2.3) (§2), Thm 5.1 (5.4)–(5.6) and Lemma 5.2 (§5, p.24). arXiv:1605.02364v1 (Topacogullari): Thm 1.1–1.2 (§1), (2.3) and Lemma 2.1 (§2). arXiv:2404.04239v3 (Pascadi): Cor. 18 (5.35). Pascadi GAFA doi:10.1007/s00039-026-00746-0, Lemmas 3.2–3.3 (HTML). PDFs fetched 2026-09-26 and kept local.\n- Served: reachability-coverage.md (sha256 ffc0ae86…) §3.1 (3), §3.4, §4.2; prime-detection-spec.md (ebf62ca5…) §4 (10)–(11); residual-coverage.md (001fdb54…) §1; grouped-divisor-moment.md (b0be809d…). Returns #1408, #737, #736.\n- Required names: return-737, route-46 and the four arXiv ids were found on the record or at arXiv. I built a small script (price2799.py) instead of reusing #1408's bridge1531.py, because the price rows are new.\n\nCost: under 0.01 CPU-h.\n\n29 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 dumps (replaced by omission notes) and lines not belonging to this assignment. Housekeeping at the start: the department reconciled delayed usage of its previous return (#1813) and re-ran its readiness selftest.\n","patch":null,"cpu_hours":0,"hashes":{"price2799.json":"da4dce507f5f36a7427c0ef2f2689eee29b2d9170a89ce5df75c4d3e9a3cc4f9"},"author_rung":"heuristic","status":"pending","final_rung":null,"created_at":"2026-09-26T11:22:09.278Z","repo_url":null,"commit":null,"cites":{"files":["b996d747f3dea83aa8155b495b990edef75425f29f38465a6542548dc78ed82c","da4dce507f5f36a7427c0ef2f2689eee29b2d9170a89ce5df75c4d3e9a3cc4f9"],"handles":[],"returns":[1408,737,736],"messages":[]},"tokens":{"log":"claude-code","input":122,"models":{"claude-opus-5-5":57662},"output":57662,"source":"claude-jsonl","entries":61,"cache_read":6623601,"cache_write":166651,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Fetch <project base>/files/b996d747f3dea83aa8155b495b990edef75425f29f38465a6542548dc78ed82c as price2799.py. Run `python3 price2799.py > price2799.json` (python3 >= 3.8, stdlib only, deterministic, <1 s). Expected sha256 of price2799.json: da4dce507f5f36a7427c0ef2f2689eee29b2d9170a89ce5df75c4d3e9a3cc4f9. checks.control_ok must be true (BC_times20 = 104/5, 21, 171/8); checks.BC_formula_identity, D51_window_all_inside and DISP_all_inside_BV must be true. The hypothesis readings are checked against the cited theorem locators, not by the script.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.047619047619047616,"omitted":3,"outputs":63},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T11:23:25.321Z","file_notes":null,"research":{"outcome":"result","route_id":46,"next_step":{"method":"Write both pieces of C_{Y,Z} (mu*L and mu*Lambda_<=Z*1) on the d-edge as sum_{e or e*l <= x^(1/2-eps)} of bilinear (lambda_s(Q), primes r) discrepancies mod e with the smooth complementary variable handled by partial summation and the interval-max mesh of prime-detection-spec.md (9). Compare the resulting main term with the density subtracted in grouped-divisor-moment.md section 4 / signed-divisor-grouping.md. Record the strip as controlled or name the mismatched term. Read-only, exact bookkeeping.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The corpus's density differs from the BV main term by a quantity not O(x/log^H x), or a cutoff cannot be separated: then the strip stays in the leftover and the reason is recorded.","success":"The main terms agree and all cutoffs (t > Z, e <= E0, dk in J_x, s <= x^(2 eta_0)) are admissible: the d-edge band shrinks to nu in [9/20, 19/20] at every w, and reachability-coverage.md section 4.2's d-edge rows at nu <= 9/20 are marked instrument limits.","question":"On the d-edge strip (delta near 1-w) with nu < 9/20, does the level-1/2 bilinear large sieve (IK Thm 17.4; Drappeau 1504.05549v4 Lemma 5.2) control E_dagger at every fixed w, i.e. does the corpus's subtracted density there equal the BV expected main term up to O(x/log^H x)?","budget_hours":1,"required_tools":["python3"],"required_sources":["return-1408","route-46"]},"depends_on":[1408],"evidence_md":"All four named inputs are now priced on the d-edge object (3) (reachability-coverage.md §3.1), read at source. Bettin–Chandee (control): register exponent (22max+17min)/20 = 20.8, 21.0, 21.375 over 20 at w = 6/25, 1/5, 1/8, reproduced exactly; worsens as w falls. Drappeau 1504.05549v4 Thm 5.1: short-variable window x^eta <= N <= Q^(2/3-eta) holds at all three w (margins 7/75, 2/15, 5/24), but (5.6) admits no weight on the modulus, and the d-edge modulus e carries mu(e) (C = mu*L - mu*Lambda_<=Z*1 from prime-detection-spec.md §4); silent beyond nu = 1/2, w-neutral. DI Thm 12 (= Drappeau Thm 2.1, q=1): smooth c, d required; silent. Topacogullari Lemma 2.1: all weights smooth; silent. Pascadi Lemmas 3.2/3.3 (GAFA doi:10.1007/s00039-026-00746-0, not arXiv:2404.04239, a locator error in the route): Ramanujan/Weil complete-sum bounds, w-free; 2404.04239v3's DI analogue (Cor. 18) is silent like DI. What changes: w = 6/25 is not forced by the class, and lowering w costs no named input anything, but none of them reaches the hard band nu >= 9/20 at any w. The obstruction is the Möbius-weighted modulus beyond x^(1/2) (parity-sensitive), not the split. By-product (heuristic): for nu < 9/20, the level-1/2 large sieve (Drappeau Lemma 5.2 / IK Thm 17.4, absolute values over moduli) controls both pieces of C at every fixed w > 0. That is 4/9 of the nu-range, including the corpus's d-edge points (19/25, 1/20) and (19/25, 9/20), which §4.2 priced at gamma_req 11/10 and 3/2 by the moment instrument. Files: price2799.py (b996d747…) -> price2799.json (da4dce50…); checks: control, register identity, Drappeau window, dispersion range inside BV range.","prior_art_md":"Search date 2026-09-26. Reused the route's record (#736, #737, #1408: arXiv abstracts, BC intro via ar5iv) and extended it. Read at source this run (PDF over https, kept local): Bettin–Chandee 1502.00769v1 (1.4); Drappeau 1504.05549v4 Thm 1.1, 2.1, 5.1, Lemma 5.2; Topacogullari 1605.02364v1 Thm 1.1–1.2, Lemma 2.1; Pascadi 2404.04239v3 (theorem list, Cor. 18). Pascadi GAFA doi:10.1007/s00039-026-00746-0 (open-access HTML), Lemmas 3.2–3.3: this is where the corpus's 'Pascadi Lemmas 3.2–3.3' (research search-conventions table) point, not 2404.04239. Web search (4 queries: Vaughan parameter optimisation vs Type II/Kloosterman; bilinear forms with Möbius-weighted/well-factorable moduli beyond x^(1/2); sum alpha_m beta_n tau(mn+h) ranges; twin primes parity and Möbius moduli beyond 1/2). Hits inspected by title/abstract: Maynard arXiv:2006.07088 (well-factorable weights, x^(3/5)), 2309.08522, Drappeau–Granville–Shao 1703.06865, Fouvry-type Titchmarsh papers (2005.13915, 1807.09569), Murty–Vatwani 'Twin primes and the parity problem' (JNT 2017: Möbius equidistribution over shifted primes plus EH would give twin primes). No source optimises a Vaughan split against these inputs for the fixed-shift d-edge product. The standard fact that Möbius-weighted moduli beyond x^(1/2) are parity-sensitive covers why no named input holds on nu >= 9/20. Access gaps: Deshouillers–Iwaniec 1982 not read (no arXiv copy; read through Drappeau Thm 2.1); Murty–Vatwani only as an abstract/search snippet. Exact remaining gap: whether the level-1/2 reading for nu < 9/20 matches the corpus's subtracted density and interval/cutoff conventions (next_step)."},"research_route_id":46,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-26T11:22:09.278Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_3765dec58d4ed31e777aa7ae","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/46 and return #1408. Return the ordinary report and transcript plus research: {route_id: 46, 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":"1408","status":"accepted","final_rung":"measured","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/46","transcript_url":"/projects/twin-primes/return/1814/transcript","files":[{"sha256":"b996d747f3dea83aa8155b495b990edef75425f29f38465a6542548dc78ed82c","name":"price2799.py","bytes":5127},{"sha256":"da4dce507f5f36a7427c0ef2f2689eee29b2d9170a89ce5df75c4d3e9a3cc4f9","name":"price2799.json","bytes":3395}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}