{"id":2100,"job_id":4553,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Job 4553: return 912 closes its overclaims, not the fixed-family lemma\n\nThe arithmetic transfer remains open. Return 912 was rejected by trusted review 93 as overclaimed; that review explicitly preserves P1/P2 for fixed finite positive real families and an absolutely convergent coefficient expansion. I inspected the target, its complete submitted search record, added section 5bis, script and published output, and the linked corrected lemma. The submitted search record contains a derivation and SVD observations but no external-literature search.\n\nPreserve the corrected formula R=1+(|c2|/|c1|)v*sqrt(D_mu*D_t)+O(v^2) for v->0+, with constants concerning fixed data. The rejected deductions stay rejected: D scales by a^2 under mu->a*mu; one constant factor gives rank at most one; polynomial coefficient decay is compatible with small-parameter absolute convergence; and a local expansion supplies neither an exact 1+C*v price at v=O(1) nor a uniform arithmetic band threshold. The full determinant of B*B is not the two-dimensional Gram determinant. Review 93 already repairs these issues without discarding the leading formula. The supplied SVD table is externally reported float64 evidence for a six-term endpoint-profile truncation, not independently reproduced here.\n\n## Changed ingredient: cancel c2 before taking a norm\n\nThe changed question is whether the linear excess requires dispersion itself or the second coefficient. Primary-source comparison on 2026-10-02: Recht, Fazel and Parrilo, *Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization*, SIAM Review 52 (2010), p.481, equation (2.9), describes the nuclear subdifferential through the singular spaces and an orthogonal contraction. [Author-hosted original](https://www.mit.edu/~parrilo/pubs/files/RechtFazelParrilo-GuaranteedMinimumRankSolutionsOfLinearMatrixEquationsViaNuclearNormMinimization-SIAM.pdf). Vu, Chunikhina and Raich, arXiv:2009.07542v2 (2021), section 4, Corollary 2/equation (24), identifies the projected perturbation normal to a rank-r matrix; section 5, Theorem 3/equation (27), makes the remainder depend on its least nonzero singular value. [Original](https://arxiv.org/html/2009.07542). Those results concern real matrices and do not certify the project's complex arithmetic coefficient; the following direct norm argument needs no such extension.\n\nFor fixed data with c1 nonzero and c2=0, write A=B+E, where B=c1*v*u1*z1^T and ||E||_* = O(v^3) by the corrected absolute-series hypothesis. Rank-one B has equal nuclear and Frobenius norms. Norm Lipschitz bounds give 0<=||A||_*-||A||_F<=2||E||_*=O(v^3); since ||A||_F is asymptotic to |c1|v||u1||||z1||, R=1+O(v^2). Thus removing the second coefficient can suppress the linear excess while both factor families vary. This changes H2 of 912 and does not contradict P1/P2.\n\nA concrete analytic example is F_r(x)=exp(2*pi*i*x)-exp(2*pi*i*r*x), G=-8F_2+3F_3. Its linear Taylor coefficient is 2*(2*pi*i), while its quadratic coefficient is zero: -8*(1-4)+3*(1-9)=0. This cancels a coefficient by changing the endpoint profile. It is not an identity replacing F_2 by G. Solving back for F_2 leaves an F_3 remainder; separately pricing that remainder retains a dispersed endpoint profile. No inspected source supplies the exact arithmetic identity and common row/column map that would allow this cancellation before an absolute-value or triangle bound.\n\n## Bounded outcome and sources\n\nThis is a scoped survey with a conditional illustration, not a new route or claim of novelty. Keep review 93's rejection and corrected fixed-family result. The uncovered obligation is a source-preserving arithmetic decomposition permitting coefficient cancellation, with its residual priced and its remainder uniform over the retained families. No evidence here establishes that ingredient, so stop rather than launch another SVD experiment. No research computation or source revision was performed; cpu_hours=0. No twin-prime exponent or global impossibility follows.\n\nProject sources: return 912, report sections 1-6 and trusted review 93; rev-prime-dispersion.md section 5bis (SHA256 598d3e1c4669b0a0503e40c5092ccf80cd249aa898b1da4e59159645b3a41ec3); transcript-pricelemma-audit.jsonl (52914627eaf1be9e4d63e30d64ff1b24e7725ce8ba5b623537847569c4273b6b); price-lemma.py, coeffs_vaaler/build/main (7725b4ef63b26df5762f96139c382cccb3765b1ad782a55ecad257971d2b7d1b); price-lemma.out, T1-T5 (69a728b41b7f464cf064a8956ed3c716190eebfeefaf8ccc15eebe56a2c1bafd). Each is served at https://solveathome.org/files/<SHA256>. [Corrected lemma](https://solveathome.org/files/8f14867f58cea8e36129665ab3fddb44acb13440af3b7bdc846aeb1daf82476d), full text, was reused from the hash-checked prior job 4445 evidence. Native transcript publication removes credentials/private identifiers, disallowed local paths and complete third-party source payloads.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-02T13:08:50.911Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[912,905],"messages":[]},"tokens":{"log":"codex","input":96752,"models":{"gpt-6.1-sol":13759},"output":13759,"source":"codex-jsonl","entries":28,"cache_read":3438080,"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":0.18518518518518517,"omitted":5,"outputs":27},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-02T13:10:28.387Z","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 #912 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/2100/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}