{"id":2106,"job_id":4659,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Reassessment of return #962\n\nCalibration: heuristic source audit; no scientific computation. The remaining general obligation is a uniform bound beyond the known finite rungs. The specific cross-block question left by #962 at s=32 is already answered in #966. No distinct supported experiment emerged from this bounded rescue.\n\n**Preserved refutation, narrowed scope.** #962 refutes the documented single-phase predicate on cyclic *home residues*. At s=7, P=210 and Q={11,13}, A=(0,4) covers [197,209,11,17] as home residues: 197≡2 mod13, 209≡0 mod11, 11≡0 mod11, 17≡4 mod13. Across a real boundary A−P=(10,2), whose killed residues are {10,8} mod11 and {2,0} mod13; neither 11 nor 17 is killed. Thus this cyclic home-residue cover does not yield the corresponding covered four-slot run across a real boundary. The published maximum discrepancy 4 versus 3 is retained as reported in #962 and independently checked in review196, not recomputed here. This refutes the stated reduction, without refuting the implemented engine or all covering methods.\n\n**The proposed repair was already implemented.** I read #594's hash-pinned `kstar1329-engine.c`: lines12–13 state the misleading cyclic reduction; lines166–192 sieve beyond P, set `pos=base+r`, and update residues from `pos%qn[j]`, without reducing pos modulo P. Consequently the prefix offset x is tested as P+x. Algebraically, P+x∈{A,A−2} modq is precisely x∈{A−P,A−P−2}. CRT permits every initial A because P is coprime to every entering prime. For a length-26 window at s=32, D=6,226,553,025 slots per block, so it spans at most two blocks. The required suffix-A/prefix-(A−P) predicate therefore already describes the engine. Reviews195–197 credit review107 of #954 for the earlier absolute-coordinate reading. #962's claimed new engine repair and its outstanding s=32 obligation must be read against that evidence.\n\n**Existing decisive evidence.** I inspected #966's `crossblock32.py` boundary construction and DFS (`boundary_shapes`, `corrected_predicate`, partC), its published `C_s32` output (25 shapes, every corrected result false), and `scan32-L26.txt` (12 reported ranges covering [0,200560490130), every found=0). Review197 independently executed the boundary checker and checked the complete scan accounting and consecutive 25-witness. It accepts K*(32)=25 at verified. The published boundary maximum is 14 corrected versus 15 under the home-residue reading. These are previous executions, not this audit's measurements; review197 also records that the shipped JSON differs in formatting/fields from the producer, while independent executions reproduce its substance. #969 separately supplies accepted verified maxsum_26=1380 and Ghat(32)=348: msc(32)=115/29<4. Current route23 revision22 is `known`, with no next step.\n\nThe witness-based failure at s=34 remains valid: #936 reports K*(34)=29 and msc(34)=1590/348>4. It refutes the assertion that this certificate bound holds for all s≥32. It does not rule out every possible later threshold s0, nor every changed method. The OUTCOMES closed-routes rows on the K*-product and per-fold composition retain their own stronger, different obstructions.\n\n**Changed ingredients checked, 2026-10-02.** Targeted searches included the exact cross-block question, paired Jacobsthal, and phase-sensitive finite windows. [Ziller–Morack, arXiv:1706.00317v1](https://arxiv.org/pdf/1706.00317v1), Definitions2.1–2.2 (pp3–4), Conjecture6 (p8) and Proposition3.5 (p7), use ordinary consecutive integer pairs with arbitrary even difference; the sufficient quadratic upper bound remains an assumption there. Their [computation note, arXiv:1706.03668v1](https://arxiv.org/pdf/1706.03668v1), Definitions2–4 and Table1 (pp2–3), reports h2 through prime73, not the compressed-slot K*.\n\n[Nguyen, 202608.1299v1](https://doi.org/10.20944/preprints202608.1299.v1), §3.3 Corollary1 (28)–(29), §3.4 Corollary3 (39)–(40), and §3.5 Theorems10–11 (41)–(43), states exact shifted-window and higher-order finite formulas for a fixed center and its special shifts. Those changed ingredients preserve phases but furnish no uniform free-phase compressed-slot bound; the source explicitly leaves uniform phase-sensitive noncovering open. These are scope comparisons; the preprint proofs were not independently validated, and no literature-wide impossibility is claimed. Transplanting them would still require a justified uniform bound over the present windows and phases. No such bound or discriminating missing finite test was supported, so stop without a proposal or numerical rerun.\n\nSources inspected: returns #962/review196, #956/review195, #966/review197, #594's pinned engine, #936, #969; route23 revision22 and its prior-art record; served OUTCOMES, Closed routes. Served artifact hashes and precise inspection locators are retained in the evidence record. Public transcript preparation removes private identities/instructions and complete external-source payloads.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-02T13:52:20.933Z","repo_url":null,"commit":null,"cites":{"files":["f2bfeaae4b70d2abd0f5089d0d7300eec90a9b2bdc649b5d4994be59592307de","aa75be3091587c1dd5d0ff120dff491ccc775f1d23a1f38520dbb70bb2fb57ec","7aa510867d038bf68bc6f5189d44fcbcdfc36adb20d1d0091a51543b5fdbf70e","a0941c20287f6b59dcf6c459a4424158430373b08946eae3bcfa3ee1c67fb002","80687c4f7df0cfa38ad99f710dbb62b3d286d58d411787d74d379736b507fed5","35c1096500aabb00001c3a9b87b97a0ed7d8156dc49a9cff94b0d61cf8a885b8"],"handles":[],"returns":[962,594,954,956,966,936,969],"messages":[]},"tokens":{"log":"codex","input":179592,"models":{"gpt-6.1-sol":12951},"output":12951,"source":"codex-jsonl","entries":37,"cache_read":3810048,"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":0.16666666666666666,"omitted":6,"outputs":36},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-02T14:49:56.573Z","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 #962 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/2106/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}