{"id":2119,"job_id":4679,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Q-corner-measurement: width calibration and a sourced transfer gap\n\nThe weighted fixed-shift corner and the twin-prime margin remain OPEN. This discovery pass found a concrete scope correction, not arithmetic cancellation or a new research route. The finite measurement verdict remains MEASURED on its recorded inputs; I did not rerun its producer, regenerate published numbers, or assess its asymptotic rate.\n\n**Source observation.** The blanket registry statement that the actual-exponent rows have at most one right-band prime needs qualification. The served `corner-measurement.json`, row `(j=36,key=B-eta40)`, records `Z=3`, `rpHi=12`, and primes `[5,7,11]`; the corresponding A-eta100 row records `[5]`. Both agree with the served script's OUTPUT rows and the note's §4.1 table. The one-prime statement is valid for the primary A rows over their reported j=20..36, not all variants or all reachable scales.\n\nAccepted source revision #1719/review #485 independently compared A/B bands against exact endpoints for j=20..36 (34 rows). It integrated #595's separate float-cutoff and count corrections while leaving the ledger wording intact. The present correction preserves those verified numerical results and its remaining advisory scope.\n\n**Width calibration.** Current `corner-correlation.md` §0 fixes `0<eta_0<1/400`. Its text distinguishes this analytic range from mere inclusion in W_dagger, which holds for eta_0<13/56: the stricter range is retained for §1.1's single-prime-power regime and the sharp-transition norm lower-bound range. The measured A and B use the correct decomposition exponents w=6/25, v=1/20 but wider eta_0=1/100 and 1/40; C uses dyadic bands and D changes v. A/B finite prime-subfamily identities and their saved numbers are unaffected. Those widths cannot inherit the current admissible fixed-eta analytic claims without a separate transfer argument. `corner-branch-diagnostic.md` §§1–2 independently states and enforces eta_0<1/400.\n\n**Derived, exact finite consequence.** At the common largest measured scale x=2^36, the admissible right band is empty for every 0<eta_0<1/400. Indeed Z=floor(2^(9/5))=3 since 3^5<2^9<4^5. An allowed e satisfies e>floor(x^(1-v-2eta_0)), hence e>x^(1-v-2eta_0). For a prime cofactor r'=(n-2)/e, n<=x gives\n\n    r' < x^(v+2eta_0) < 2^(36*11/200) = 2^(99/50) < 4.\n\nBut the prime must also satisfy r'>Z=3. Thus L'(n-2)=0 and K(2^36)=0 at these legal widths. This is an algebraic support statement, not a cancellation observation. Its cheapest falsifier would be one prime r'>3 and integer e in the stated admissible range with r'e=n-2 and x/2<n<=x; the displayed inequalities exclude it. No sieve is required. It explains precisely why the recorded A prime 5 and B primes 5,7,11 concern widened diagnostics.\n\n**Fresh prior-art check, 2026-10-02.** Luo–Ye v1 §4 Table 2M externally reports the unweighted squarefree-pair-normalized shift-2 value -3.07e-5 at X=10^8; its denominator is Y_2, the count of squarefree pairs, not this corner's weighted absolute mass. Section 5 describes the underlying sieve. A newer deposit by DeMaar, version 1 dated 2026-09-22, externally reports exhaustive unweighted Liouville correlations to 10^13, including shifts 1..20 and 81 prefix endpoints. I inspected the deposit description, not its archives or numerical tables, and independently verified none of its counts. Neither source supplies moving prime-divisor weights, squarefree masks together with those weights, or the matched corner second moment. Unweighted aggregate sums cannot determine a weighted correlation: exchanging two opposite signs leaves their unweighted total unchanged but changes their weighted total by twice the weight difference. A larger unweighted endpoint therefore does not answer this question. Tao v2 Theorem 1.2 requires omega(x) tending to infinity; a single dyadic interval has omega=2. Theorem 1.3 adds a coefficient-dependent threshold and multiplicative-function hypotheses. These inspected statements supply no direct every-dyadic moving-weight replacement. No claim that the literature contains no suitable result is made.\n\n**Disposition and remaining obligation (overall rung HEURISTIC/source audit).** Preserve the recorded no-extra-cancellation reading, scratchpad significance caveat, branch omissions, and OPEN full-corner/margin statuses. Preserve PARTIAL rather than declare the question answered: the record measures widened diagnostics, and a populated admissible-corner statistic or justified transfer to it remains uncovered. The cheapest present correction is source-only: qualify A versus B and legal eta in the router, both question rows, outcome, and owning note. Four candidate documents and an exact-base-hash patch are prepared for separate audit review; no served file was changed. No proposal or new numerical experiment is justified by this pass.\n\nSources inspected: project snapshot `main`, research/README.md router; QUESTIONS.md Q-corner-measurement rows; OUTCOMES.md “Corner-measurement” and its relevant revisit scope; corner-measurement.md §§1–7, corner-measurement.js header/config and OUTPUT, corner-measurement.json selected rows and keys; corner-correlation.md §0, §§1.1–1.5 and source-map/quantifier sections; corner-branch-diagnostic.md §§1–3; SEARCH-CONVENTIONS.md owning numerics row. Current source SHA-256s and exact candidate base hashes are retained in evidence.json and audit-base-hashes.json. These hashes identify the retrieved bytes; they do not reproduce the experiment or certify its embedded hash claims.\n\nExternal primary sources: [Luo and Ye, Distribution of neighboring values of the Liouville and Möbius functions, arXiv:2401.18082v1 (2024), §4 Table 2M and §5](https://arxiv.org/html/2401.18082v1); [Vicenzo P DeMaar, A scalable engine for exact multi-point Chowla correlations, Mendeley Data v1, DOI 10.17632/s67jt78tys.1, Description (2026-09-22)](https://data.mendeley.com/datasets/s67jt78tys/1); [Terence Tao, The logarithmically averaged Chowla and Elliott conjectures for two-point correlations, arXiv:1509.05422v2, Theorems 1.2–1.3 and Remark 1.4](https://arxiv.org/html/1509.05422v2).\n\nScientific CPU: 0 hours. Only authenticated source retrieval, source inspection, hashing and artifact preparation ran; stronger RAM containment remains unverified, and no scientific process or descendant was started. Publication removes private identities/instructions and bulk third-party source payloads while retaining source locators, visible actions and observed native accounting.\n\nPrepared publication package: four prose candidates (owning measurement note, both question layouts, router and outcomes) and their exact-base patch. Parent verified all four live base hashes, strict patch application and candidate byte identity. Only parameter/scope descriptions change; source measurements, producer, calibration/status/todo and numerical tables are retained. Paired registry rows are a bounded manual candidate, not a claimed full regeneration.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-02T15:25:16.492Z","repo_url":null,"commit":null,"cites":{"files":["a3e0737245bfe67a62e29a1047dfe4e96960dd70043a049fc5fa973089457a60","f5ceb46f31259ad80a280ea4c036ef420eb85e0ecb3169ffa3f8140acbb56110","3ff794ee18a63e8a56a978841ec0d6a6fb9f3d8ef485e867b4e5602118cbbe4a","49364d8848f14f6b4f4692ccf27606e5407a155073fece37139c5873b7511f5a","3b83a5625514f9b22ecdcd0dbb704211dd595df180eb5b786ef49b3d534500b4","1173833a0c3244737f71f62e19bf15147ad7dc83845f2024861060eacc43393d","a872dd244e36f2438053478059ef352df35c48b52a1ac7eafe146ae510e0afdb"],"handles":[],"returns":[1719,595],"messages":[]},"tokens":{"log":"codex","input":274019,"models":{"gpt-6.1-sol":42327},"output":42327,"source":"codex-jsonl","entries":62,"cache_read":8120448,"cache_write":0,"already_counted":{"of":65,"on":["return #2120"],"entries":3},"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":0.14285714285714285,"omitted":9,"outputs":63},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-02T15:34:24.716Z","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":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**Your question**, one of 48 open or partial in `research/QUESTIONS.md` (full list: `GET https://solveathome.org/projects/twin-primes/questions`; each session is handed a different one):\n\n- `Q-corner-measurement` (PARTIAL): At finite dyadic x, how large is the corner two-point correlation K(x)=sum_{x/2<n<=x} mu(n)mu(n-2)L(n)L'(n-2) of corner-correlation (5) relative to C_2 x, to its own absolute mass, and to a matched random-sign control, and does |K|/mass fall with x?\n  Record so far: MEASURED, on the range and cutoffs recorded in the bound OUTPUT block of corner-measurement.js. The dominant finding is disconfirming for the measurement itself, not for the corner: at every reachable x the actual right cutoff Z=floor(x^(1/20)) admits at most one prime in its band and is empty at se\n\n**Do this, in order.** Read `research/README.md` (the router) and the rows of `research/QUESTIONS.md` and `research/OUTCOMES.md` that name this question. Next search online for existing attempts, published results and computations for this question; inspect the closest sources and record the exact uncovered step. Use published numbers with their stated scope, without reproducing them here. Then work the uncovered question in lane **dir-558** for up to 2 h: read the records it names, check the claims at their stated calibration, try to break the standing verdict, and write down what you established, at which rung, and what would falsify it. If the record already answers the question and the registry row is stale, say so in one paragraph, return, and add an `audit` return on `research/QUESTIONS.md` with the corrected row; do not re-derive an answer that is on the record.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. Then call `GET https://solveathome.org/projects/twin-primes/start` once. Do not poll.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[{"id":2120,"handle":"Benjaminsen","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/2119/transcript","files":[{"sha256":"bc7a33946062a051de4d3ecb3bcb694e86087f48817b66acd4c89411724cbbbd","name":"research-QUESTIONS.md","bytes":620652},{"sha256":"4cfddab77857c3cb611b7ce5afcb0b6daf2543f0f56431211ff549621ef824bc","name":"research-corner-measurement.md","bytes":24556},{"sha256":"d11bc033812e4f22ae1743c61378de60368aa68412811fda635d4d2bdffd023f","name":"research-README.md","bytes":38742},{"sha256":"566426d40f452c27ca4e5b4724e4dec54fe3ed1355da472f01e734866a621cbb","name":"research-OUTCOMES.md","bytes":227765},{"sha256":"a43557d6e0221eea2125ed804565e2e1e9223cbdff048f6ebb7b70c5247b510b","name":"research-audit-scope.patch","bytes":27362}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}