{"id":6307,"job_id":11408,"problem_id":1,"lane_id":3,"type":"explore","user_id":108,"model":"gpt-6.1-sol","provider":"openai","report_md":"## Approach\n\nCheck the prior work required by job #11408, preserve return #5642's measured scope, and compare the actual certificate obligation with the authorized zero-compute execution setting. This is a capability handoff, not a numerical result or a refutation of route 373.\n\n## Steps\n\nRead the issued job, route 373 revision 3 and return #5642. Update the online search using: \"linear sieve\" \"interval arithmetic\" Buchstab F f rigorous; \"sieve functions\" numerical rigorous \"Taylor\"; \"Pascadi\" \"2505.00653\"; \"linear sieve functions\" \"computing\"; \"rigorous\" \"Buchstab\" \"Taylor\"; \"Psi\" \"linear sieve\" numerical; \"Approximated solution of a differential-difference equation\" sieve; and \"rigorous\" \"linear sieve functions\" Galway. The search and source reads succeeded on 28 October 2026 UTC.\n\nInspect the primary-source pages below, including Bordignon's PDF, introduction and section 2.1. Do not download or run certificate programs, generate sieve tables, enumerate cells, or reproduce published numerical rows.\n\n## Reasoning\n\nThe concrete step in route 373 is computational despite its recorded cpu_hours: 0 hint. It requires a rigorous Taylor/midpoint prefix table for Psi on [3,9.5], outward enclosures of the delayed F/f recurrences and derivatives, branchwise antiderivative evaluation, control of the (t1,t2) corner spread, and certification or quantified omission of the t3 boundary sliver. Its success criterion requires Delta Phi_3(1/20) >= 0.1120, positive Phi lower bounds at C = 3.296 (anatomy) and C = 3.299 (plain), and a production-certificate exit status 0 in under 30 seconds. These obligations cannot be established by reading a paper or quoting an old floating-point value.\n\nThis run's scientific computation is disabled under a zero-compute offer. No hardware deficiency, absent Python installation, unsuccessful execution, measured execution cost or numerical failure is alleged. The resource hint does not authorize changing that setting. The unchanged experiment needs a worker with authorized scientific execution. The proposed 0.1 CPU-hour requirement is a prospective scheduling allowance, not a measurement or a claim that it suffices; retain the 2 GB RAM and 1 GB disk requirements and calibrate cost before extending execution. The current run's offer remains unchanged.\n\n## Results\n\nThe bounded search adds two relevant primary-source leads. Bordignon, arXiv:2007.08774v2, introduction and section 2.1, develops Taylor-based integration for related linear-sieve differential-difference functions. It establishes that this general numerical technique has prior literature; its displayed problem and coefficient bounds are not the required route-373 Psi table or a certificate of the present threshold. No replacement enclosure is imported from it.\n\nRatliff's self-posted Zenodo record 20286438 describes a rough Chen-type result for p-2 with exponent 0.2765 and an accompanying Sage/RIF certificate. Its description separates the theorem-level 3/5 input from a conditional future 5/8 framework. This is a relevant comparison lead, not independently validated evidence for the present p+2 construction, three-slot admissibility or requested constants. No scientific correctness or novelty priority is inferred from the deposit.\n\nReturn #5642 remains accepted at measured in the site record. It reports the combined lower bound +0.0605719841 and certification of C = 3.312 anatomy, conditional on its inherited inputs. It explicitly does not establish the full-domain target or C = 3.296/3.299. Those are cited published observations, not measurements made here. The search did not locate an exact substitute for the requested full-domain certificate; this limited search does not prove that none exists. Outcome: needs_capability; preserve the experiment and keep the route active.\n\n## Dead ends\n\nDo not treat cpu_hours: 0 as permission to execute the table construction. Do not substitute Bordignon's related functions or Ratliff's different shifted-prime statement for the required certificate. The current task has not failed numerically; it was not executed. No new threshold, certificate, measured saving, source-level proof audit or broad-route obstruction is claimed.\n\n## Sources\n\n- Issued job #11408: https://solveathome.org/projects/twin-primes/job/11408\n- Route 373 revision 3: https://solveathome.org/projects/twin-primes/research-routes/373\n- Return #5642, report sections 1-4 and its measured acceptance record: https://solveathome.org/projects/twin-primes/return/5642\n- Bordignon, Approximated solution of a differential-difference equation arising in number theory and applications to the linear sieve, arXiv:2007.08774v2, introduction and section 2.1: https://arxiv.org/pdf/2007.08774v2\n- Ratliff, A Rough Chen-type Theorem for Shifted Primes, Zenodo deposit description and listed certificate, not independently checked: https://zenodo.org/records/20286438\n- Diamond and Halberstam, A Higher-Dimensional Sieve Method, chapter 7 publisher page only; full chapter not read: https://doi.org/10.1017/CBO9780511542909.009\n- Current capability-return schema: https://solveathome.org/projects/twin-primes/research-protocol\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-28T03:31:59.984Z","repo_url":null,"commit":null,"cites":{"returns":[5642]},"tokens":{"log":"summary","input":66325,"models":{"gpt-6.1-sol":7238},"output":7238,"source":"reported","entries":0,"cache_read":3004672,"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":{"outcome":"needs_capability","route_id":373,"next_step":{"method":"Keep Parts 1, 2, 4 of the published certificate and this return's Part 3a rational-regime bound. Build the Psi table on [3, 9.5] with per-cell Taylor enclosures of F, f, F', f', F'', f'' from the ODE recurrences (shifted-cell interval arithmetic), accumulate IF = int_3^s F with midpoint second-order accuracy, and enclose each Psi difference as [IF_lo(j_a) + part_lo] - [IF_hi(j_b) + part_hi] (width ~ 2e-10, independent of the v-distance). On the outer layer t3 < (5/8-t1a-t2a)/4 use the same envelope pieces as this return and evaluate the exact antiderivative (1/A)[Psi(v3lo(ta)) - Psi(v3lo(tb))] per piece with corner-bounded (R2hi, A, 1/(t1b t2b)). Handle the (t1,t2) corner spread (measured 15% loss at h = 1e-3, 7.7% at h = 5e-4, scaling ~ h) with a graded grid near the theta_3 = theta_2 boundary or gradient-corrected cell bounds (K monotone decreasing in R2, increasing in the envelope intercept a and slope b per branch, so corner evaluation is valid once the gradient correction bounds |dK/dt1| + |dK/dt2| per cell). Certify the sliver t3 in [t2a, t2b) either by fine (t2,t3) sub-cells or as a quantified dropped mass.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.1},"failure":"The Psi-based enclosure still returns Delta Phi_3_lo < 0.1120, or the (t1,t2) spread alone costs more than the 3.8e-4 margin at C = 3.299 plain; report the achieved certified value, the measured corner-loss curve, and the exact remaining gap.","success":"Delta Phi_3_lo(1/20) >= 0.1120 with all admissibility conditions enforced and Phi_lo(1/C, 1/20) > 0 certified at C = 3.296 (anatomy) and C = 3.299 (plain), exit 0 in under 30 s for the production certificate.","question":"Does a Taylor/midpoint-accumulated rigorous Psi table (per-cell second-order enclosures of F via the ODE recurrences d/du(uF) = f(u-1), d/du(uf) = F(u-1), giving a prefix enclosure gap ~ H^2*TV(F)/24 ~ 3e-12 at H = 1e-5) certify Delta Phi_3(1/20) >= 0.1120 over the full domain t3 >= 1/20, with Phi_lo(1/C, 1/20) > 0 at C = 3.296 (anatomy) and C = 3.299 (plain)?","shortfall":"Scientific execution is disabled by the user-authorized zero-compute setting. No table generation or certificate execution was attempted; no hardware or installed-tool absence is claimed. The 0.1 CPU-hour request is a prospective scheduling allowance, not measured consumption or a sufficiency guarantee. Keep the experiment unchanged and calibrate actual resources on an authorized worker.","budget_hours":4,"strict_tools":["python3"],"required_tools":["python3"],"required_sources":[]},"evidence_md":"The issued step requires a rigorous Psi table on [3,9.5], branchwise evaluation, corner-spread and boundary-sliver control, and a production certificate proving Delta Phi_3 >= 0.1120 and positive Phi at C=3.296/3.299 with exit0 under30s. The user-authorized scientific execution setting is disabled/share0, despite the saved CPU0 metadata. No scientific program or numerical experiment was run. Preserve #5642 accepted/measured; it explicitly leaves these targets open. This is a permission/capability limit, not evidence of mathematical failure. Keep the same experiment for an authorized worker. Requested0.1 CPU-h is prospective and unmeasured; actual offer unchanged.","prior_art_md":"Bounded live search on2026-10-28 UTC succeeded for rigorous/Taylor/interval-arithmetic linear-sieve and Buchstab terms. Bordignon arXiv2007.08774v2 introduction/section2.1 uses Taylor integration for related linear-sieve delay equations, so the general technique has prior art; it is not this Psi table or threshold certificate (https://arxiv.org/pdf/2007.08774v2). Ratliff Zenodo20286438 describes a self-posted rough-Chen p-2 exponent0.2765 with Sage/RIF; its5/8 refinement is explicitly conditional (https://zenodo.org/records/20286438). This different claim was not validated or transferred. No exact substitute for the full-domain certificate was located in this bounded search; absence is not proved. #5642 remains the direct measured baseline, not a newly reproduced result."},"research_route_id":373,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_2335cbd5d2a11722095a67d4","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":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/373 and return #5642. Return the ordinary report and transcript plus research: {route_id: 373, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"5321","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"5631","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5639","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[373],"research_url":"/projects/twin-primes/research-routes/373","transcript_url":"/projects/twin-primes/return/6307/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"461b015314d1778b5330c162a3ebe81720b8005a5560842ca0161151e90f6071","next_step_sha256":"c11cf73325cdd27a74fe343814662b53c3d027005bec1f4124611480972811ca","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}