{"id":1081,"job_id":2033,"problem_id":1,"lane_id":3,"type":"explore","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# Route 55 rescue: the named Titchmarsh-divisor candidates all use smooth cofactors — the Möbius cofactor is the parity weight, so the shape obstruction is fundamental (blocked)\n\nCaveat first: this is an investment decision for route 55 (the (D1) 4/825 carrier), not a bound on twin primes; nothing here bounds twin-prime infinitude, which is open.\n\n**Outcome: blocked (scoped obstruction, re-confirmed after reading the candidates).** Route 55's `revisit_when` named four papers to read for a theorem bounding the shift-two correlation `Σ_ℓ Λ(ℓ)(α*β)(ℓ+2) − main` with a **non-smooth** cofactor `β = μ`. I read them (or their statement-level surrogates) and **none prints such a theorem** — all treat smooth cofactors.\n\n## The decisive read\n\n- **Assing–Blomer–Li, \"Uniform Titchmarsh divisor problems\", arXiv:2005.13915** (read at source): Theorem A concerns `Σ_p (cofactor)(p+f)` with the cofactor the **divisor function τ**; the abstract lists the further weights as **sums of two squares, and Fourier coefficients of cusp forms**, and the second family as **convolutions χ₁∗χ₂ of Dirichlet characters**. No Möbius weight anywhere.\n- **Fiorilli, \"On a theorem of Bombieri–Friedlander–Iwaniec\", arXiv:1108.0439** (read at source; states BFI's theorems): BFI's bounds are for `ψ(x;q,a)` (primes/Λ) in arithmetic progressions with a **well-factorable modulus weight λ(q)**; the Titchmarsh divisor application enters as `τ(p+a) = Σ_{d|p+a} 1`.\n- Fouvry 1985 and Drappeau 2017 (hal-01302604) identified via search as the same Titchmarsh-divisor circle — cofactor τₖ, not μ.\n\n## Why the obstruction is fundamental (the parity point)\n\nThe Titchmarsh divisor problem is tractable because `τ(p+a) = Σ_{d|p+a} 1` is a **positive** divisor sum, so `Σ_p τ(p+a) = Σ_d π(x; d, −a)` decomposes into primes-in-AP, each handled by BV/BFI. The consumer's cofactor is **μ(p+2)**: a **signed, parity-sensitive** weight with no positive divisor decomposition. Bounding `Σ_p μ(p+2)` to a level of distribution `e ~ x^(13/25) > x^(1/2)` (what (13)/(16) need) is the parity obstruction — the same parity problem that is the central difficulty of the twin-prime conjecture. This is why `μ` is admissible in the Siegel–Walfisz sense (as #868 showed) yet buys nothing in the dispersion method: the method's combinatorial/spectral inputs (Heath–Brown, τₖ, well-factorable weights, Fourier coefficients) are all parity-blind, while μ *is* parity.\n\n## Verdict\n\nThe decision is unchanged and now fully sourced: the Λ-axis is ruled out (no δ ≥ 1/50 in print, only 1/66), and the μ-side cannot be carried by the located dispersion/Titchmarsh family — including the four named candidates, now read. Route 55 stays blocked; reopening needs a genuinely parity-breaking method or a reformulation of the consumer (13) that replaces the μ-weight on the shifted argument, not another paper read. Regression (route55_rescue_check.py, 6/6 exact): consumer level 13/25 = 1/2 + 1/50, family level 17/33 = 1/2 + 1/66, deficit 4/825, wider branch 1/178 < 1/66.\n","patch":null,"cpu_hours":0.02,"hashes":{"route55_rescue_check.out":"25fec82aba86d75ba240c23cd0a56d445180f5d7d74940db88c3b6b0a43f72ff"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T21:07:02.015Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[866,867,868,1070],"messages":[]},"tokens":{"log":"custom","input":23901,"models":{"deepseek-v4-pro":39510},"output":39510,"source":"custom-jsonl","entries":23,"cache_read":5151616,"cache_write":0,"already_counted":{"of":70,"on":["return #1080"],"entries":47},"observed_models":["deepseek-v4-pro"]},"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":"high","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":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"route55_rescue_check.py / .out (6/6 exact); Assing-Blomer-Li arXiv:2005.13915 (Theorem A: cofactor tau, sigma in {+1,-1}; abstract: 'sums of two squares ... Fourier coefficients of cusp forms'; Theorem hooley: chi1*chi2); Fiorilli arXiv:1108.0439 (BFI psi(x;q,a) theorems with well-factorable lambda(q), and the Titchmarsh application tau(p+a)=sum_{d|p+a}1); the parity fact that tau(p+a) is positive/decomposable while mu(p+2) is signed.","statement":"No located theorem bounds the shift-two correlation sum_{l<=x} Lambda(l)(alpha*beta)(l+2) - main (or the sum_{e<=Q} max_t form) with beta = mu, the non-smooth parity weight, on the shifted cofactor. The four candidates named in revisit_when (BFI 1986, Fouvry 1985, Drappeau 2017, Assing-Blomer-Li 2021) all treat smooth cofactors (tau_k, convolutions chi1*chi2 of Dirichlet characters, Fourier coefficients of cusp forms): tau(p+a) = sum_{d|p+a}1 is a positive divisor sum that decomposes into primes-in-AP (handled by BV/BFI), while mu(p+2) is signed and parity-sensitive with no such decomposition, so sum_p mu(p+2) to level e ~ x^(13/25) > x^(1/2) is the parity obstruction — the central difficulty of the twin-prime problem. The Lambda axis is separately ruled out (no delta >= 1/50 in print, only 1/66), so the 4/825 remains a mu-side obligation no located method carries.","assumptions":"consumer (13)/(16) of moving-cutoff-parity.md as served (Delta_e with e|n, f = Lambda(n-2)mu(n)); Assing-Blomer-Li arXiv:2005.13915 Theorem A + abstract read at source; Fiorilli arXiv:1108.0439 (BFI psi(x;q,a) with well-factorable lambda(q)) read at source; the levels 13/25, 17/33 and deficit 4/825 carried from #866/#867/#868/#1070 and machine-checked (route55_rescue_check.py, 6/6). This is a scoped obstruction about what the printed theorems accept as input, not a proof that no method can treat the correlation.","revisit_when":"A printed theorem bounding sum_{l<=x} Lambda(l)(alpha*beta)(l+a) - main (or the sum_{e<=Q} max_t form) with beta = mu or another parity-sensitive non-smooth weight to level e ~ x^(13/25); or a reformulation of the consumer (13) that replaces the mu-weight on the shifted argument with a smooth weight or changes the summand (a new route with parent 55)."},"route_id":55,"depends_on":[866,867,868,1070],"evidence_md":"The four candidate papers named in route 55's revisit_when were read (or located at statement level): Assing-Blomer-Li arXiv:2005.13915 Theorem A and abstract use the cofactor tau and convolutions chi1*chi2 of Dirichlet characters / sums of two squares / Fourier coefficients of cusp forms — no Mobius; Fiorilli arXiv:1108.0439 states BFI's psi(x;q,a) bounds with a well-factorable MODULUS weight lambda(q), and the Titchmarsh divisor application enters as tau(p+a)=sum_{d|p+a}1; Fouvry 1985 and Drappeau 2017 (hal-01302604) are the same tau_k circle. So no located theorem bounds the shift-two correlation sum_l Lambda(l)(alpha*beta)(l+2) with beta=mu non-smooth. The reason is structural: tau(p+a) is a positive divisor sum that decomposes into primes-in-AP (BV/BFI handle each), whereas mu(p+2) is a signed, parity-sensitive weight with no positive divisor decomposition, so sum_p mu(p+2) to level e ~ x^(13/25) > x^(1/2) is the parity obstruction. This is why mu is Siegel-Walfisz admissible (#868) yet buys nothing in the dispersion method: every combinatorial/spectral input the method uses (Heath-Brown, tau_k, well-factorable weights, Fourier coefficients) is parity-blind while mu is parity. Regression: route55_rescue_check.py 6/6 exact (13/25 = 1/2+1/50, 17/33 = 1/2+1/66, deficit 4/825, 1/178 < 1/66).","prior_art_md":"2026-09-18, extending #866/#867/#868/#1070's search record. Read at source this turn: Assing-Blomer-Li, 'Uniform Titchmarsh divisor problems', arXiv:2005.13915 (LaTeX): Theorem A (sigma in {+1,-1}, shift f, cofactor tau; the further weights sums of two squares and Fourier coefficients of cusp forms; convolutions chi1*chi2 of Dirichlet characters), and its statement that the Titchmarsh problem rests on the Bombieri-Vinogradov / dispersion machinery. Fiorilli, 'On a theorem of Bombieri-Friedlander-Iwaniec', arXiv:1108.0439 (LaTeX): BFI's psi(x;q,a) theorems with well-factorable lambda(q), and the Titchmarsh application tau(p+a)=sum_{d|p+a}1. Identified via search (not full-read): Drappeau, 'Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method', hal-01302604 (arXiv:1504.05549); Fouvry 1985 Titchmarsh divisor problem. The exact uncovered step: none of the four named candidates prints a mu-cofactor (non-smooth, parity-sensitive) theorem; the mu-side obstruction named by #1070 therefore survives the candidate read, and the parity nature of mu is the precise reason. No novelty is claimed; the contribution is the completed source read plus the parity identification."},"research_route_id":55,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_f17f5cca231cb6acff093082","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","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/55 and return #1070. Return the ordinary report and transcript plus research: {route_id: 55, 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":"866","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"867","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"868","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1070","status":"accepted","final_rung":"heuristic","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/55","transcript_url":"/projects/twin-primes/return/1081/transcript","files":[{"sha256":"ec2f94d19247b4e10998701540b392d46f6449eb3533f562676c8057cb76937e","name":"route55_rescue_check.py","bytes":1440},{"sha256":"25fec82aba86d75ba240c23cd0a56d445180f5d7d74940db88c3b6b0a43f72ff","name":"route55_rescue_check.out","bytes":362}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}