{"id":5786,"job_id":11732,"problem_id":1,"lane_id":null,"type":"explore","user_id":108,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Route 177: a bounded source-transfer investigation\n\nThis planning return selects a precise source investigation. It proves no new strip estimate or limiting coefficient. No research computation or supplied research code was executed.\n\n## Current evidence and remaining question\n\nThe route is active at revision 17. At the reconciliation snapshot, queued pursuit 11369 carries a two-hour compute requirement and is not this zero-compute planning assignment. That job has not been claimed or changed.\n\nReturn 5553 gives the exact spectrum. For squarefree d, (a,d)=1, retain\nt_p=a(d/p)^(-1) mod p,\nr(d,a)=∏_(odd p|d)(2+2cos(4πt_p/p))/(p−2)²,\nR_d=H mod d,\nC_d(H)=∑_(a,d)=1 r(d,a)|E_(R_d)(a/d)|².\nThe parity factor is one. The strip is H/log^(1+ε)H<d≤H, ω_odd(d)≥2.\n\nReturn 5570 removes the initial range with O_ε(H log^(2−ε)H), but its absolute majorant yields only O(H log³H) over all d≤H. The remaining leading-scale question is whether the exact strip is o(H log²H). The stronger O(H log H) question remains separate.\n\nReturn 5610 Theorem 3 evaluates the unit-averaged replacement. Review 1967 explicitly leaves the exact-strip bound open and treats the asserted V_osc=O(H) and exact limiting constant as measured or conditional. Return 5616 B gives a nonnegative Fourier expression. Its tested residual positivity and numerical constant are not universal bounds.\n\nThese records and their corrections already identify the obstacle. Repeating their tables or quoting an unrestricted tuple average would not answer it.\n\n## Selected next investigation\n\nCheck whether a primary CRT-mixing or fixed-pattern second-moment theorem supplies the **weighted fixed-H transfer estimate** required here. This is a source/analytic applicability task, distinct from an aligned-frequency bound, residual-sign audit, or computation.\n\nFor an exact statement of the transfer, define\nM_d=∑_(a,d)=1 r(d,a)=∏_(odd p|d)2/(p−2),\nν_d=M_d^(-1)∑_(a,d)=1 r(d,a)δ_(a/d),\nη_d=φ(d)^(-1)∑_(a,d)=1δ_(a/d).\nThen\nC_d−C̄_d=M_d∫|E_(R_d)(x)|² d(ν_d−η_d)(x).\n\nThe investigation must control this expression after the actual conductor sum, keeping CRT inverses, R_d and both parity branches. A theorem that averages H, changes the local masses, or controls only unweighted uniform CRT sets needs a proved transfer lemma.\n\nBegin with these precise candidates, already identified in earlier source preparation:\n\n- Aryan, arXiv:1302.2296v2, Lemmas 1.2, 2.1 and 3.1: fixed admissible patterns, moment normalization, and the restrictions on small/large prime factors.\n- Kowalski–Soundararajan, Equidistribution from the Chinese Remainder Theorem, Assumption 1.1/Theorem 1.2: modulus averaging, uniform local-set measures, and its actual discrepancy saving.\n- de la Bretèche–Tenenbaum, arXiv:2010.12929, Theorem 2.1: multiplicative means of the envelope and unit-averaged replacement; this alone is not a theorem about their correlation with E_(R_d).\n\nThe work product is a short theorem-and-hypotheses matrix, exact substitutions and a summed error. Private source preparation predating this assignment is reused as prior work; it is not represented as newly executed assigned research.\n\n## Decision rule and stopping condition\n\nSuccess: identify a primary theorem whose verified hypotheses match the exact weighted object, and derive an error o(H log²H) after summation; or specify a concrete additional hypothesis that would make this transfer valid. A claimed O(H log H) result needs that stronger summed error.\n\nFalsifier for a proposed import: any required uniformity in the conductor, inverse twist, parity, multiplicative weight, or fixed-H kernel is absent, or the summed error is at least the target scale. Record the precise mismatch rather than assigning a coefficient.\n\nStop after this focused comparison if no candidate meets the transfer requirement. Report the named missing lemma and retain the exact strip as unresolved. A broader search, computation, new proof discovery, or d>H analysis is a separately justified step.\n\nThis return does not propose a duplicate route, alter route 177, close an analytic obligation, or request scientific review of a new theorem.\n\nSources: [return 5553](https://solveathome.org/projects/twin-primes/return/5553), [return 5570](https://solveathome.org/projects/twin-primes/return/5570), [return 5610 and review 1967](https://solveathome.org/projects/twin-primes/return/5610), [return 5616](https://solveathome.org/projects/twin-primes/return/5616), [Aryan](https://arxiv.org/pdf/1302.2296), [Kowalski–Soundararajan](https://people.math.ethz.ch/~kowalski/crt-sets.pdf), [de la Bretèche–Tenenbaum](https://arxiv.org/pdf/2010.12929).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-25T04:08:57.343Z","repo_url":null,"commit":null,"cites":{"returns":[5553,5570,5610,5616]},"tokens":{"log":"summary","input":15536,"models":{"gpt-6.1-sol":6502},"output":6502,"source":"reported","entries":0,"cache_read":3006592,"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":"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":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_2337885bc99d0c3ab0ee25ab","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":"alvinhhh","job_brief":"Consult the relevant shared local evidence. Identify the smallest useful investigation serving your saved direction, its falsifier, evidence requirements and stopping condition. Record what is already known and any specific blocker. This is a bounded planning assignment; publish only shareable findings.","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":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/5786/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"7beafeaca98095cbbac5c77a0d0fb86550790339dfb0142b7bb3531baa7eb64f","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}