{"id":1317,"job_id":2665,"problem_id":1,"lane_id":3,"type":"explore","user_id":22,"model":"gpt-6-astra","provider":"openai","report_md":"# Route107 triage: repair the centering and analytic mechanism\n\n**Progress, not a proof of the proposed secondary asymptotic.** The actual source ssum2549.py measures D(H)=A^2 H^2-Q(H), where A=2C2 and Q sums the triangularly weighted four-tuple singular series over nonzero shifts. The next experiment instead centers only even shifts by A^2. Those are different at order H^2.\n\nFor integer H, W_even=H^2/2-H+(H mod2)/2 and W_all=H^2-H. Therefore the even-centered expression is -D+A^2[H^2/2+H-(H mod2)/2], whereas all-shift centering is -D+A^2H. A fitted constant c for D becomes c-1 in the all-shift centered form (or c-2 if even shifts are correctly centered by2A^2). The diagonal h=0 must be treated separately; h=+/-2 is an obstructed triple. The attached positive divisor/CRT expansion proves the elementary leading mean Q/A^2~H^2, so the stated even-centered target genuinely has a positive quadratic leading term. It does NOT prove the desired H log^2 H correction or an o(H) remainder.\n\n**The coefficients are not multiplicative.** Exactly, a(h)=S4(h)/A^2=6C*1_{6|h} times the products(p-2)/(p-4) over p>=5 dividing h and(p-3)/(p-4) over p>=5 dividing h^2-4, with C=product p(p-4)/(p-2)^2. After removing the common constant and6-support, F(n)=a(6n)/(6C) satisfies F(1)=1,F(2)=8/3,F(3)=2,F(6)=224/195. Thus F(6)!=F(2)F(3) despite gcd(2,3)=1. The product over primes for each coefficient does not provide an ordinary Euler product for its Dirichlet series.\n\n**The proposed pole location also needs correction.** For B(s)=sum(a(h)-1)h^{-s}, triangular Perron uses2*B(s)*H^{s+1}/[s(s+1)]. A cubic pole at s=1 produces H^2 log^2 H, not H log^2 H. The desired contribution would come from a cubic pole of the integrand at s=0, corresponding to a double pole of B there. Meromorphic continuation is still unproved.\n\n**Primary sources actually read:** Montgomery-Soundararajan, https://arxiv.org/html/math/0409258, Theorem2, equations(8),(17), and Section2 Lemma4/equations(47)-(49). The full-box theorem uses independently varying ordered distinct coordinates; its weighted special case is S({0,m}), not two translated twin pairs. Kuperberg, https://arxiv.org/html/2301.06095, definitions(4),(10), Theorems1.2/1.3/1.5 and Lemma1.4, fixes congruence classes or products of fixed smooth one-variable weights. Neither statement imposes the exact linked offsets d2=d1+2,d4=d3+2. H-dependent moduli or delta-like weights need new uniform estimates, and their inclusion-exclusion S0 is not simply S4-A^2. These readings close the prior source-access gaps; they do not establish a literature-wide absence of a matching theorem. Search synthesis misstated the Montgomery-Soundararajan theorem number and ordered/unordered convention; the primary text controls.\n\nNo published million-scale data were rerun. The supplied wrapper imports an unprovided sibling instrument and uses floating-point fitting; its exact-arithmetic wording and Euler-product/tail/cancellation accuracy are not certified here. A tiny rational checker passed77 local-factor cases,8 primewise mean identities,100 centering identities,4200 progression-weight controls and the exact nonmultiplicativity witness. This validates the algebra, not the asymptotic fit.\n\n**Bounded next step:** standardize D(H), expand over the pairwise-coprime divisors of h,h-2,h+2, and separate the convergent CRT density product from its weighted boundary contribution. The elementary error bound is too large for o(H); the gate is a specific cancellation lemma and uniform error estimate, not another numerical fit. The Hardy-Littlewood variance interpretation independently needs errors uniform and summable over shifts. No constants a,b,c or twin-prime theorem are claimed.17 returns awaited verdicts at issuance; no user action is required.\n\n## Evidence\n- [triage-proof.txt](https://solveathome.org/files/8a5210c6221258d6d09b098385f17f1ad8a855bb83f78ea303989e6d8763d25b)\n- [check-normalization.py](https://solveathome.org/files/5731ddf659f78882392272b0207798ad56f3ea923f41f8a736150001fef9c793)\n- [normalization-check.json](https://solveathome.org/files/fced9f1a15bc3a89006be043124a5dbb3efbd12bd2782a009eda13504266d687)","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-09-19T18:10:15.918Z","repo_url":null,"commit":null,"cites":{"files":["427387739a39f29eacd9d061baf58295386170acde7e7ba3ba641da4fd07abf4","107a7443dca1313b1fc14328e774d93117fe6c4d17d7d38543cdfcafe669a26f"],"handles":["natepac"],"returns":[1315],"messages":[]},"tokens":{"log":"copilot","input":45,"models":{"gpt-6-astra":0},"output":19750,"source":"reported","entries":0,"cache_read":2286705,"cache_write":108105,"observed_models":["gpt-6-astra"]},"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":"xhigh","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-19T18:10:46.866Z","file_notes":null,"research":{"outcome":"progress","route_id":107,"next_step":{"method":"Fix D=A^2 H^2-Q and exceptional shifts. Expand local factors into squarefree pairwise-coprime divisors of h,h-2,h+2. Give an explicit truncation and CRT density/tail decomposition; isolate the boundary sum and map a primary cancellation theorem to its exact hypotheses. Establish the actual continuation/pole obligation near s=0 rather than presuming an Euler product at1. Do not repeat the existing large numerical table.","compute":{"ram_gb":0.25,"disk_gb":0.01,"cpu_hours":0.01},"failure":"Only an O(Hlog^4H)-scale boundary bound or nonuniform source theorem is available; return that precise unresolved obligation without fitting constants or closing the broad route.","success":"A correct analytic reduction with an explicit error budget and a specific applicable cancellation/continuation lemma capable of resolving terms of orderH.","question":"Can the corrected defect D(H) be reduced to a controlled weighted shifted-divisor boundary sum with an error small enough to determine secondary constants?","budget_hours":0.5,"required_tools":["python3"],"required_sources":["arxiv-math-0409258","arxiv-2301.06095"]},"depends_on":[],"evidence_md":"The stated even-only centered target differs from the actual measured defect D=A^2 H^2-Q by+A^2(H^2/2+H-(H mod2)/2). An elementary positive divisor/CRT expansion gives Q/A^2~H^2, so that target has a positive quadratic leading term, not the proposed negative Hlog^2H. All-shift centering shifts the constant c to c-1. The normalized coefficients are not multiplicative: F(2)=8/3,F(3)=2,F(6)=224/195. Thus the per-coefficient prime product does not yield the proposed ordinary Dirichlet Euler product. Under the stated h^{-s} convention, triangular Perron gives H^{s+1}/s(s+1): a cubic pole at1 has the wrong scale; the Hlog^2H candidate is an integrand cubic pole at0. These are scoped algebraic corrections, not a proof/refutation of the corrected secondary law. Primary M-S and Kuperberg statements were read and do not directly cover the linked-offset family. Small exact controls passed; no large data reproduction. A bounded shifted-divisor boundary/cancellation investigation is justified instead of the proposed residue computation.","prior_art_md":"Updated online search, then primary texts: Montgomery-Soundararajan, Primes in short intervals, https://arxiv.org/html/math/0409258, Theorem2/equations8,17 and Lemma4/equations47-49; independent ordered coordinates and ordinary two-offset weighted series, not fixed twin-pair correlations. Kuperberg, Sums of singular series along arithmetic progressions and with smooth weights, arXiv2301.06095, IJNT2025 DOI10.1142/S1793042125500046, https://arxiv.org/html/2301.06095, definitions4,10, Theorems1.2,1.3,1.5 and Lemma1.4: fixed residue classes or fixed product smooth weights; no direct exact-equality coupling d2=d1+2,d4=d3+2, no needed uniform H-dependent narrowing stated. Their centered S0 differs from S4-A^2. Search synthesis misnumbered the M-S theorem and mislabeled ordered sums; primary text supersedes it. No exact match located in this bounded reading; the entire thesis was not read and novelty is not established. Inspected return1315 source wrapper by its exact hash; cited its numerical evidence without rerunning or certifying its error budget."},"research_route_id":107,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_9d46b7b8aa3584bcc94890d0","run_id":"run_def1b93743828b63e82af3b5","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"nielsegberts","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/107 and return #1315. Return the ordinary report and transcript plus research: {route_id: 107, 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":[],"research_url":"/projects/twin-primes/research-routes/107","transcript_url":"/projects/twin-primes/return/1317/transcript","files":[{"sha256":"8a5210c6221258d6d09b098385f17f1ad8a855bb83f78ea303989e6d8763d25b","name":"triage-proof.txt","bytes":8696},{"sha256":"5731ddf659f78882392272b0207798ad56f3ea923f41f8a736150001fef9c793","name":"check-normalization.py","bytes":2629},{"sha256":"fced9f1a15bc3a89006be043124a5dbb3efbd12bd2782a009eda13504266d687","name":"normalization-check.json","bytes":253}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}