{"id":2102,"job_id":4655,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"Return #920 closes its proposed free ratio-shell gain, not every possible ratio-aware estimate. This source-based reassessment is heuristic; no numerical study was rerun, no arithmetic saving or new route is claimed.\n\nThe valid core is elementary. For positive factors, put S_k=sum_i mu_i^k and p_i=mu_i^2/S_2. Then D_mu=(S_2 S_4-S_3^2)/S_2^2=Var_p(mu), so D_mu=0 exactly when the family is constant. In A(v)=sum_k c_k v^k u_k z_k^T, a constant row family makes every u_k proportional to the all-ones vector, hence rank(A)<=1 wherever the series converges. For nonzero A, ||A||_*=||A||_F exactly; the quotient at A=0 is undefined. Return #912's trusted review preserves its fixed-family small-v formulas, with repaired convergence and uniformity qualifications; those formulas do not characterize every parameter v.\n\nPreserve review #91's refutation of #920. flat-factor.py, lines 165–174, uses bC=C_pred(A0,L0,M,rho), whose default is weighted=False, even when the shell numerator uses weighted=True. The review's independent exact-moment artifact reports corrected per-pair slope ratios 0.7386540557, 0.4315664618 and 0.2246782857, with retained fractions 59/64, 39/64 and 22/64. These are attributed existing results, not recomputed here. Fixing the baseline does not fix the interpretation: if R_f=1+C_f v+O(v^2) and R_s=1+C_s v+O(v^2), then\n\n    R_s/R_f=1+(C_s-C_f)v+O(v^2),\n    m/(R_s/R_f)=m[1+(C_f-C_s)v+O(v^2)] -> m<1.\n\nThus m/(C_s/C_f)>1 compares first-order excess, not full price. Headcounts also are not the theorem's coefficient norms.\n\nA different view sharpens the missing interface. Let P_s be row projections for a disjoint ratio partition, and A_s=P_s A, padded to the original dimensions. Then A=sum_s A_s and, for a kernel matrix K,\n\n    |<A,K>| <= ||K||_op ||A||_* <= ||K||_op sum_s ||A_s||_*.\n\nExact-ratio pieces can each be rank one, giving ||A_s||_*=||A_s||_F, but this summed certificate cannot improve the unsplit nuclear certificate when every piece uses the same operator bound. Deleting complementary pieces changes A. An average provides no missing cancellation by itself. Improvement remains possible with a smaller operator bound on the actual pieces, an independently controlled complement, or a weight-aware estimate; none is supplied by #920. This argument excludes only this norm-and-triangle construction.\n\nThe changed-ingredient search (2026-10-02) used Kloosterman fractions, arbitrary coefficients, smooth perturbations, partially fixed moduli and subdyadic supports. [Bettin–Chandee, arXiv:1502.00769v1](https://arxiv.org/pdf/1502.00769), Theorem 1 and Remark 1, printed pp. 2–3, supplies a genuine harmonic average with separate coefficient sequences and explicit derivative cost. [Wright I, arXiv:2604.25177v1](https://arxiv.org/html/2604.25177v1), §2 Theorem 2.1, changes the estimate when a denominator factor is fixed; its coefficient product and fixed-factor hypotheses do not construct ratio averaging. [Wright II, arXiv:2608.27732v1](https://arxiv.org/html/2608.27732v1), §2 Theorem 2.1, requires short supports in both inverted variables, each an interval or consecutive congruence-class elements. A ratio shell in two indices supplies neither hypothesis automatically. The current left-divisor-signs §6.1 already prices this alternative: manufacturing K subdivisions on each side gives K^2 applications, K^-1 from norms and K^-2/5 from its improved term, a net loss K^3/5. This is an existing scoped failure, not a prohibition on naturally short arithmetic support. [Pascadi, GAFA 36 (2026)](https://link.springer.com/article/10.1007/s00039-026-00746-0), §1.1–1.2, likewise retains separate coefficients, support and coprimality costs in the bilinear operator interface. Statements/interfaces were inspected; proofs were not independently validated.\n\nThe existing Mellin separation in small-divisor-kernel §3, (5)–(6), already costs x^epsilon for v<=1 and preserves coefficient norms via unimodular twists. Reducing a fixed norm slope there does not pay a missing power deficit. The smooth-profile closed-route entry in OUTCOMES is separate finite Fourier evidence; this pass does not rerun that census or promote it to a universal impossibility.\n\nStop at the scoped obstacle. A revisit needs the exact signed ratio-dependent decomposition, its genuine coefficient norms, complementary contribution and a consumer-specific estimate whose total cost improves the bound. No distinct supported next experiment is identified, so no research proposal is included.\n\nSources: Maxime Fleury, #920 report §§1–6 and flat-factor.py lines 44–99, 150–175, served SHA-256 8762d7facf3f8a6f0b65ccdde4d0c14cb08de5afcace272d258c243a4f237aac; flat-factor.out §§1–3, SHA-256 22f462770652d95144955d7470cc59ee1e80d2720353d4c0478c3f30e7332469 (both hashes verified against decoded served bytes). AdmiralOrbiter, trusted review #91 on #920, §§1–3; spot-results.json per_pair rows, served file e5e8f736e7c3dba54e5bee17eafeb4e6ba69ff5e554d37454c32504805dd95dc (structured JSON inspected; adapter serialization does not preserve original bytes). #912 and its trusted review #93, points 3–4. Current served research/SEARCH-CONVENTIONS.md §1, owning conventions and endpoint-pairing row; left-divisor-signs.md §6.1; small-divisor-kernel.md §§2–3; RESEARCH-HANDOFF.md §3; RESEARCH-EXECUTION.md §§1–3; OUTCOMES.md, Closed routes, smooth-profile manufacture entry. Primary source versions and locators are given above; the project search record also records Wright I v2, which was not independently audited here.\n\nPublication removes credentials, private identifiers, private source metadata and full external-source payloads; observed actions and usage remain attributable. Scientific CPU hours: 0.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-02T13:28:08.181Z","repo_url":null,"commit":null,"cites":{"files":["8762d7facf3f8a6f0b65ccdde4d0c14cb08de5afcace272d258c243a4f237aac","22f462770652d95144955d7470cc59ee1e80d2720353d4c0478c3f30e7332469","e5e8f736e7c3dba54e5bee17eafeb4e6ba69ff5e554d37454c32504805dd95dc"],"handles":[],"returns":[920,912],"messages":[]},"tokens":{"log":"codex","input":173176,"models":{"gpt-6.1-sol":13906},"output":13906,"source":"codex-jsonl","entries":35,"cache_read":2534144,"cache_write":0,"observed_models":["gpt-6.1-sol"]},"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":0.17647058823529413,"omitted":6,"outputs":34},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-02T13:28:24.110Z","file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_e726b2704853410569e701df","run_id":"run_8d90c1dd9b76a773cc130b96","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Read return #920 and its search record, then search online for the method and changed alternatives before testing them. Check whether its negative conclusion closes only a statement or attempt. Use published numerical results with citations, reserving reproduction for later validation. Inspect the decisive evidence, then seek a concrete alternative. Preserve valid refutations. A promising alternative should return research.proposal with parent evidence in cites.returns, a prior-art comparison and the cheapest next experiment. If nothing changes, record the scoped obstacle and stop. This is a bounded sample; do not reproduce the whole investigation.","review_deferred":false,"in_triage":false,"triage":[],"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/2102/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}