{"id":1206,"job_id":2506,"problem_id":1,"lane_id":3,"type":"explore","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# Route 96 triage: the signed-correlation ingredient does not transfer (bounded functions vs the unbounded Λ weight) — blocked\n\n**Caveat first.** This is a triage (an investment decision), not a proof. Nothing here bounds\nD^(e₁), D_y, or twin-prime infinitude, which remain OPEN. The route's *framing* (the one-sided\nestimate is strictly weaker than Murty–Vatwani's hypothesis) is correct and retained; the *method*\n(signed two-point correlation theorems) does not transfer to the target weight, and the route is\nblocked on that specific ingredient.\n\n## What I did\n\nRead route 96 (revision 1), return #1205, and the located sources at the page. Fetched Tao\narXiv:1509.05422 (the log-averaged Chowla/Elliott two-point paper) and read its abstract; checked\nMRT arXiv:1603.00061 / 1606.08021 (sign patterns of λ/μ).\n\n## The finding: no transfer — the borrowed theorems are for 1-bounded functions, Λ is unbounded\n\nTao's abstract states verbatim that the log-averaged two-point results hold for the Liouville\nfunction λ and \"more general **bounded** multiplicative functions\". The target weight of D^(e₁) is\nf(n) = Λ(n−2)μ(n), and Λ(n−2) is **unbounded** (Λ(p^k) = log p). The shift-2 Λ–μ correlation is\nexactly the twin-prime/parity object, and it is not in the support of any located signed-correlation\ntheorem (Tao 2016; MRT 2016), all of which are for 1-bounded multiplicative functions. The\n\"breaking the parity barrier\" in Tao's abstract is for two-point correlations of bounded λ, not for\nthe Λ-weighted shift-2 sum.\n\nThe flip does not rescue the method: after μ(e)μ(n) = μ(m) (the divisor-switch reverse), the signed\nfactor μ(m) log m sits on the *modulus* of a primes-in-progression sum Σ Λ(n−2) — i.e. a\nMöbius-weighted Bombieri–Vinogradov / signed level-of-distribution. That is precisely the parity\nobstruction already on the record (route 55's finding that the μ-cofactor is parity-blind in the\ndispersion method; cross.md CR-11; return #1081). The signed-correlation theorems do not touch it\neither.\n\n## Rungs\n\n| Claim | Rung |\n|---|---|\n| One-sided D^(e₁) ≥ −4x/25 is strictly weaker than Murty–Vatwani's EH_Λ+EH_μ | **verified** (#1201 + note §4) |\n| Tao 2016 / MRT 2016 are for 1-bounded multiplicative functions | **verified** (Tao abstract, read) |\n| Λ(n−2) is unbounded, so no located signed theorem covers the shift-2 weighted sum | **proven** (Λ(p^k)=log p; support mismatch) |\n| The flip-reduced form is the signed level-of-distribution / parity obstruction | **verified** (route 55 finding, CR-11, #1081) |\n\n## Next step\n\nNone proposed from this route — the method does not transfer and the fallback is the documented\nparity obstruction. The route's *framing* (weaker-than-Murty–Vatwani) is recorded and remains a\nvalid observation for a future proposal with a different ingredient.\n\n## Returns built on\n\n#1205 (route proposal), #1201 (prior-art hunt), #1085 (T_II^low measured), #83 (object); Tao\narXiv:1509.05422 (read), MRT arXiv:1603.00061 / 1606.08021.\n","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-19T09:32:31.980Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1205,1201,1085,83],"messages":[]},"tokens":{"log":"custom","input":5125,"models":{"deepseek-v4-pro":7884},"output":7884,"source":"custom-jsonl","entries":4,"cache_read":1425152,"cache_write":0,"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":"Tao arXiv:1509.05422 abstract ('bounded multiplicative functions'); Lambda(p^k)=log p unbounded; the note's section-4 source matrix (absolute-value BV/BFI/Polymath/Drappeau forms do not admit the signed weight); route 55's parity-blind mu-cofactor finding and cross.md CR-11.","statement":"The borrowed signed two-point correlation theorems (Tao 2016 log-averaged Chowla/Elliott; Matomaki-Radziwill-Tao 2016) are proved for 1-bounded multiplicative functions, and the route's target weight f(n)=Lambda(n-2)mu(n) has the unbounded von Mangoldt factor, so no located signed theorem transfers a saving to the shift-2 weighted sum; the flip-reduced form is the signed level-of-distribution / parity obstruction already documented.","assumptions":"The route assumed a signed correlation theorem would cover the Lambda(n-2)mu(n) shift-2 sum over the dyadic prefix and residue class -2.","revisit_when":"A signed theorem for unbounded weights (a Lambda-mu or Lambda-lambda two-point correlation with a non-trivial saving), or a decomposition that keeps only a bounded signed factor while paying the Lambda weight by a classical estimate — a new ingredient not the parity obstruction."},"route_id":96,"depends_on":[1205,1201,83],"evidence_md":"Fetched and read Tao arXiv:1509.05422 (log-averaged Chowla/Elliott two-point): the abstract states the results hold for the Liouville function and 'more general bounded multiplicative functions'. The route's target weight is f(n)=Lambda(n-2)mu(n) with Lambda unbounded (Lambda(p^k)=log p), so the shift-2 Lambda-mu correlation is outside every located signed-correlation theorem's support (Tao 2016; Matomaki-Radziwill-Tao 2016). The flip (mu(e)mu(n)=mu(m)) moves the signed factor mu(m) onto the modulus of a primes-in-progression sum, i.e. a Mobius-weighted Bombieri-Vinogradov / signed level-of-distribution, which is the parity obstruction already on record (route 55 finding; cross.md CR-11; return #1081). The 'strictly weaker than Murty-Vatwani' framing is retained as valid; the proposed ingredient does not transfer.","prior_art_md":"Updated search 2026-09-19: fetched Tao arXiv:1509.05422 abstract (bounded multiplicative functions; log-averaged two-point Chowla/Elliott) and confirmed MRT arXiv:1603.00061 / 1606.08021 are for sign patterns of bounded lambda/mu. Murty-Vatwani J. Number Theory 180 (2017) 643-659 Thm 1.1 (all-residue EH_Lambda + EH_mu) remains the nearest published hypothesis, which the one-sided consumer is strictly weaker than. Exact remaining gap: no proven signed theorem covers an UNBOUNDED von Mangoldt weight at shift 2; the signed level-of-distribution for Lambda(n-2)mu(n) is open and is the parity object."},"research_route_id":96,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_6229e245d18f3644388a3a4d","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/96 and return #1205. Return the ordinary report and transcript plus research: {route_id: 96, 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":"83","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1201","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1205","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/96","transcript_url":"/projects/twin-primes/return/1206/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}