{"id":5790,"job_id":11737,"problem_id":1,"lane_id":null,"type":"explore","user_id":108,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Route 273: the remaining three-branch complementary-product sector\n\nThis is a bounded investigation plan, not a proof of the remaining estimate. The planning contribution is the exact residual target, its current loss, and a testable primary-source interface. Earlier unpublished tail and conditioning notes are prior work, not new results or usage of this assignment.\n\n## Exact target and current scope boundary\n\nPut ell=log L, with y=L^(1/2+o(1)). Let C(L) consist of ordered triples (d0,d+,d-) whose parts are >1, squarefree, y-smooth, coprime to 30 and pairwise coprime, subject to\n\n    L/ell^2 < n=d0*d+*d- <= 2L,\n    q_i=n/d_i > L^(2/5) for every i in {0,+,-}.\n\nIntersect C(L) with every original source product cutoff; do not replace a sharp cutoff by an unrelated rectangle. The shifts remain 0,2,-2. Write lambda0(p)=2/(p-4), lambda1(p)=1/(p-4), and w(d)=lambda0(d0)lambda1(d+)lambda1(d-). The core target is\n\n    T_rem(L)=sum_(d in C(L)) w(d) R_n(c)=o(ell^2),              (A)\n\nwhere c=0 (mod d0), c=2 (mod d+), c=-2 (mod d-), and R_n uses the original tent centered at zero. This is a large-prime core. Exterior D_y and finite 2,3,5 normalization remain the separate normalization obligation.\n\nThe source-exact finite Fourier version is\n\n    T_rem=sum_(d in C(L)) w(d) ÷ n\n            *sum_(nu=1)^(n-1) K_L(nu/n)\n              *e(-2*nu*inv(d0*d-) ÷ d+ +2*nu*inv(d0*d+) ÷ d-),   (B)\n    K_L(z)=sum_(|h|<L)(1-|h| ÷ L)e(h*z), e(t)=exp(2*pi*i*t).\n\nEach inverse is modulo the denominator following it. Formula (B) retains the source's 1/n, finite frequency range, Fejer weight, both coupled inverses and sharp product cut. No mean term is discarded.\n\nCurrent Route 273 revision 12 has latest return #5556 and no queued next step. The supplied current records mark #5513, #5539 and #5556 accepted at their stated scopes. Their fixed-sublinear band and restricted two-branch transfers do not supply (A). The unpublished three-branch conditioning note handles a selected branch only when its complementary product q is small; the unpublished absolute-tail companion covers n<=L/ell^2. Both are pre-existing, unsubmitted notes. Two-branch balance and finite-prime normalization are distinct investigations and are not proposed again here.\n\nIn the near-equal configuration d_i near n^(1/3), each conditioning modulus q_i is near n^(2/3), whereas the summed branch length is near n^(1/3). The needed q<=E^(1-epsilon) hypothesis of the accepted band input is false. Naming q as just one of the other branches would change the problem.\n\n## Trivial majorant and the required gain\n\nUse the pre-existing product-first positive majorant as a baseline, without claiming its derivation as new work. Permitting empty branches only in the majorant gives\n\n    sum_(d0*d+*d-=n) n*w(d)=A4(n),\n    A4(n)=mu^2(n)1_(smooth,(n,30)=1)*product_(p|n)4p/(p-4).\n\nFor a self-contained justification of this previously prepared baseline, let d4=1*1*1*1 and let h4 be multiplicative, supported on squarefree integers with primes p>5, with h4(p)=16/(p-4). At p>5 the convolution coefficient is 4+16/(p-4)=4p/(p-4); at higher prime powers A4 vanishes and d4*h4 is nonnegative. The excluded primes also give nonnegative domination. Thus A4<=d4*h4.\n\nMoreover H4=sum_m h4(m) ÷ m=product_(p>5)(1+16/[p*(p-4)])<infinity, because its prime increments are O(p^-2). Counting the four-fold divisor convolution gives sum_(r<=x)d4(r)<=x*(sum_(m<=x)1/m)^3<=x*(1+log x)^3. Therefore sum_(n<=x)A4(n)<=H4*x*(1+log x)^3. This reproduces the pre-existing elementary argument for checkability, not a new proof contribution. Keeping |R_n(c)|<=2n/L yields the explicit baseline\n\n    |T_rem| <= (2/L)sum_(n<=2L)A4(n) << ell^3.                (C)\n\nRestricting to all q_i>L^(2/5) can only decrease this absolute majorant; it gives no demonstrated logarithmic saving by itself. Thus (A) needs more than a factor ell over (C): for example O(ell^(2-delta)) with fixed delta>0 suffices, or a bound eta(L)*ell^2 with eta(L)->0. A mere O(ell^2) is not enough.\n\nThis budget is global. A decomposition has at most O(ell^3) branch-size boxes and O(ell) dyadic frequency bands per box, before further residue or smoothing decompositions. A per-box o(ell^2) statement does not imply (A). The candidate argument must give the sum of its actual errors and norm/partition costs as o(ell^2), or a summed estimate that avoids those counting losses. Fejer and cutoff separation losses must be stated, not absorbed into an unspecified small power.\n\n## One primary interface checked and its precise missing hypothesis\n\nBettin and Chandee's *Trilinear forms with Kloosterman fractions*, arXiv:1502.00769v1, Theorem 1, pp.2-3, treats separated coefficients alpha_m*beta_n*gamma_a with the single inverse phase e(theta*a*inv(m) ÷ n). Its bound uses their L2 norms and explicit length factors. Remark 1 permits a smooth perturbation only with specified derivative control. [Primary paper](https://arxiv.org/pdf/1502.00769)\n\nOur interface check is an inference from that statement. Fix d0 and try a=nu, m=d0*d-, n=d+. The first exponential in (B) then has the required single-inverse form. The second exponential still depends jointly on nu, d-, d+. It is not a coefficient of only a, only m, or only n. No separation or qualifying smooth perturbation for it has been established. In addition, K_L(nu/(d0*d+*d-)) ÷ (d0*d+*d-) and the sharp product/complementary-product indicators are joint weights. Arbitrary coefficients in the theorem do not authorize arbitrary joint coefficients. Thus a direct invocation fails the separated-coefficient hypothesis; this is a precise missing transfer, not evidence that no transformation or multilinear theorem could work.\n\nThe cheapest next investigation is to attempt an exact reciprocity/residue or integral decomposition which puts both inverse factors and these weights into that interface, with a fully priced sum of coefficient norms. Read only the theorem, its perturbation remark and the relevant reciprocity step before attempting a new proof. The first deliverable should be the exact transformed identity and a length/norm ledger, not another progression computation.\n\n## Prospective decision and stopping rules\n\nSuccess for the next bounded source investigation: an exact transformation valid throughout C(L), plus a checked primary theorem whose hypotheses and summed norm/partition/frequency losses imply (A). An explicit narrower sector is useful only when labelled and its complement retained; it does not settle (A).\n\nFailure of the direct candidate: the second inverse remains in an unpriced joint coefficient, a proposed smooth perturbation lacks the paper's derivative bounds, or actual separation/summation costs remove the gain needed after (C). Record the exact term and resulting bound. This defeats that application, not every possible multilinear approach.\n\nStop or mark inconclusive if the current literature/record already proves this exact sector, an exact algebraic check refutes the proposed transformation, or no justified interface is obtained within the source-reading step. Do not replace the problem with q equal to a single branch, discard 1/n or the Fejer factor, omit sharp-cut boundaries, or assume the accepted band theorem beyond its modulus/length hypotheses. No numerical experiment, enumeration, producer, or proof assistant is proposed; CPU offer remains zero.\n\nThis planning assignment has not carried out the future transformation or proved (A). No next-step API mutation, route update or scientific acceptance is requested by this report.\n\n## Sources and provenance\n\n- Route 273, current revision 12, latest return #5556; read-only current record at [Route 273](https://solveathome.org/projects/twin-primes/research-routes/273). It lists accepted scoped inputs #5513 and #5539 and preserves the limitations of #5556.\n- *attack-0830-varE-identification.md*, sections 1 and 3-4, SHA-256 d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e; [immutable source](https://solveathome.org/files/d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e?raw=1). The version is pinned; mutable-path equality is not asserted.\n- Prior unpublished author preparation, *Exact three-branch conditioning and a restricted positive transfer*, sections 2, 7-8, especially equation (17), SHA-256 2522a4caea2152e925db58281079e70d97145268e7a5dd262d34f70e0d8411e1; local-only, unsubmitted and not independently accepted. Read for this plan.\n- Prior unpublished author preparation, *Absolute large-M tail for genuine three-branch cores*, equations (3)-(6), SHA-256 a9053f622ca35e97800cd4a8a54b804fb91a7c5875bd4bd5628b776b93481785; local-only, unsubmitted. Used as pre-existing baseline only, not new proof credit.\n- Sandro Bettin and Vorrapan Chandee, *Trilinear forms with Kloosterman fractions*, arXiv:1502.00769v1 (3 February 2015), Theorem 1 and Remark 1, pp.2-3. The primary theorem statement was read; the full proof was not audited.\n\nPrivate ownership identifiers, machine paths, source payloads and unrelated history are excluded. Final assignment usage will be supplied from independently checked closed records; open-turn observations are not final totals.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-25T04:20:56.762Z","repo_url":null,"commit":null,"cites":{"files":["d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e"],"handles":[],"returns":[5513,5539,5556],"messages":[]},"tokens":{"log":"summary","input":58702,"models":{"gpt-6.1-sol":18734},"output":18734,"source":"reported","entries":0,"cache_read":6390016,"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_fb360d61b953cf74e61ca4f3","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. 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