{"id":2212,"job_id":4207,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Job4207: finite phase-feasibility pilot on the old 31# lattice\n\nNo full-block upper bound is established. On the preregistered 10,000 starts, the new bounded engine reports no covering at any L=38..42, and no node-cap exhaustion. This is a finite pilot measurement, not K*<=41, certificate survival, or a theorem about twin primes.\n\n## Inputs, sampling and observed checks\n\nP=200560490130 (31#); Q={37,41,43,47,53,59,61,67,71,73}. prereg4207.json was saved before the pilot: seed4207, 100 disjoint equal integer-coordinate subranges, 100 midpoint anchors per subrange with seeded +/-50000 jitter, first following admissible slot. This is not uniform sampling over slot ranks. Windows retain actual unwrapped positions; only 42 slots per start are generated. All 10,000 starts are unique. The 50,000 decisions cover each start once at each requested length.\n\nThe original594 engine is unsupported here (32-bit coverage mask, arrays sized for eight primes/residues below64). phase4207.c is a separate implementation with L<=63, NQ<=10, prime moduli<=73, distinct primes and increasing positions checked before array access. Fixed mask arrays and depth<=10 keep memory small. It precomputes masks for {a,a-2}; at the first uncovered slot it branches over the two possible phases for every unused prime, ordered by gain. The residual-capacity test sums each unused prime's maximum uncovered coverage, ignoring overlap. This implements the accepted search principle in901; no new search theorem is claimed. A 10,000-node cutoff returns unknown, never infeasible. None was reached.\n\nThe required608 gate was recovered via604 report section3: exact first integer75053614359224265389282351, span810, old-base start14942621. The named content-addressed certificate returned404; no hash-match of that missing file is claimed. Trial divisibility on the recovered consecutive slots confirms30 admissible slots all killed by Q(36). The new search finds feasible at L30 in30 nodes; all returned phase masks cover the slots arithmetically. Forty small cases matched exhaustive phase enumeration, and four unsupported shapes/moduli were refused. The checker independently consumes summary4207.json and reproduced subrange0; changing its L42 capacity-pass count was rejected with exit1. This checker shares the author's engine and is not independent mathematical validation of the whole implementation.\n\n## Measurements\n\n| L | windows | capacity passes | DFS nodes | maximum nodes/window | coverings |\n|---|---:|---:|---:|---:|---:|\n|38|10000|2242|140870|1813|0|\n|39|10000|1531|75911|823|0|\n|40|10000|1032|44302|611|0|\n|41|10000|665|25777|425|0|\n|42|10000|455|14643|311|0|\n\nAll100 L42 subranges have between1 and10 capacity passes per100 starts. This variation is retained in summary4207.json, not treated as uniformity. The first100-start version timed DFS only; its source and raw observations are preserved. Version2 changed only timing coverage and added end-to-end resource records. For the remaining9900 starts, all five lengths together cost42.3273s wall and35.9864s driver+child CPU; the C window kernels account for0.284689s, including0.069503s at L42. Observational timing is separate from deterministic output hashes. Peak observed driver RSS was below25MB; this observation is not enforced aggregate RAM containment.\n\nKernel-only linear projection of L42 over D31=6226553025 windows gives about12.14 CPU-hours (0.069503/9900*D31/3600), before headroom, rare difficult windows, slot generation, startup or coverage bookkeeping. It is a projection from coordinate sampling, not a worst-case bound or an impossibility result. The present one-process-per-start driver projects far more. I do not recommend a full-block certificate within the current controls. A distinct rolling-capacity implementation and contiguous-window pricing test could reduce setup costs; it must retain the phase gate and exact semantics before a larger allocation is considered.\n\n## Scope and evidence grades\n\nThe new observations are measured and unreviewed.1267 is accepted/proven for the sharper -1 transfer;1850 remains pending and reports measured m*(T37)=41. Neither is recomputed. The pilot does not supply the missing full bound, and survival remains conditional on that bound and the threshold's trusted review. Rejected609's broad finite-block consequences are not used. No unchanged dead end was reopened and no earlier profile/census was reproduced.\n\nAll research computations used the supplied bounded exec path with20s wall/10s CPU per call; recorded process groups terminated. CPU is enforced per process, aggregate RAM unverified, disk and shared core cooperative. Raw per-window observations remain local and their original hashes are in observed4207.json; the uploaded deterministic histograms, preregistered starts, producer and checker reproduce the finite claim. There is no full sieve/PARTD or full-block scan.\n\n47 handle returns await verdict, as in the brief. Publication removes credentials, private runtime identifiers, personal paths, unrelated history and hidden reasoning through the pinned native exporter. Final native usage remains pending until this turn closes.\n\n## Sources\n\nProject: route26 revision11;1854 report/research.next_step;2204 report/comparison4815;901 report and original594 engine f2bfeaae4b70d2abd0f5089d0d7300eec90a9b2bdc649b5d4994be59592307de (inspected);608 section2;604 sections2-3;1267 and1850 current grades, all at GET <project base>/return/<ID>. Exact external sources and access limits follow.\n\n2026-10-03: reused #1854/#901 search and #2204 comparison4815 certificate; no repeat census of its twelve candidates. New queries: \"Jacobsthal\" \"31\" \"phase\" covering ten primes; \"Jacobsthal\" \"paired\" algorithm Ziller 2026; \"200560490130\" \"42\" covering. Inspected Ziller, arXiv:1903.11973v2, section2.4/proposition2.9 (https://arxiv.org/html/1903.11973v2), one-class residue search; Ziller-Morack, arXiv:1706.03668v1, definitions2-4 and section2/table1 (https://arxiv.org/html/1706.03668v1), paired progressions across all even differences; Nguyen, Finite-Window Noncovering on Primorial Wheels, version1 DOI10.20944/preprints202608.1299.v1 (https://www.preprints.org/manuscript/202608.1299), abstract/introduction and sections5-6/OpenProblem2, symmetric fixed-center lift windows. These inspected sources do not give this 31# slot-filtered ten-prime L42 pilot. No claim of literature completeness or novelty. Source604 certificate hash d898dee3a6f2956d3f6115d8591d3c942495d8cb13d476c6d2224d3b1b8417ba returned404; report604 sections2-3 supply its exact first integer and span, sufficient to recover the gate. Returns608,901,1854,2204 inspected; current1267 accepted/proven and1850 pending/measured verified by ID. Exact remaining gap: full-block upper bound and trustworthy cost of a streaming implementation, plus review of1850.","patch":null,"cpu_hours":0,"hashes":{"prereg4207.json":"8b833dbfa322a15009d4b2e6eb532a930d0127e8b3ba7fb3bedaaab00c4bd142","summary4207.json":"e647f2fa2018c7b447e59be9f8b89c9f50dbb2457ccfeac159f709cff22c341a"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-03T08:35:26.087Z","repo_url":null,"commit":null,"cites":{"files":["f2bfeaae4b70d2abd0f5089d0d7300eec90a9b2bdc649b5d4994be59592307de"],"handles":[],"returns":[1854,2204,901,608,604,1267,1850],"messages":[]},"tokens":{"log":"codex","input":142665,"models":{"gpt-6.1-sol":20996},"output":20996,"source":"codex-jsonl","entries":33,"cache_read":3267072,"cache_write":0,"observed_models":["gpt-6.1-sol"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Fetch manifest sources by /files/<sha256> into a clean relative directory. Compile with cc -O2 -Wall -Wextra phase4207.c -o phase4207. Run python3 pilot4207.py gate (expected feasible30-cover,40 brute-force matches and4 shape refusals). Run python3 check4207.py all to consume summary4207.json and reproduce its integer histograms. Expected stdout: PASS sample histograms; full-block upper bound not checked. For current20s/10s local controls run each subrange separately: python3 check4207.py B for B=0..99, each a separate bounded invocation. Timing values are observational and not byte-for-byte expected. Expected deterministic summary SHA-256 e647f2fa2018c7b447e59be9f8b89c9f50dbb2457ccfeac159f709cff22c341a. No source604 certificate bytes are claimed or required; reported source integers recover its gate.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.15625,"omitted":5,"outputs":32},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-03T08:35:53.848Z","file_notes":null,"research":{"outcome":"progress","route_id":26,"next_step":{"method":"Reuse this measured pilot, its preregistration, masks and gate. Build a fixed-memory streaming filter on unwrapped consecutive31# slots with the exact ten primes37..73; update per-prime residue/phase counts across successive windows and call the existing exact DFS only when capacity>=42. Cross-check every capacity and decision against phase4207.c on 100 short disjoint contiguous ranges anchored by the published preregistration, with at most1000 windows per range. Record filtering/setup/DFS CPU separately and hardest ranges under20s wall/10s per-process controls. Stop on any witness, unknown, invariant mismatch or process limit. No full-block census; recommend it only after complete coverage design and measured worst-range cost with headroom fit an explicitly available allocation.","compute":{"ram_gb":0.1,"disk_gb":0.01,"cpu_hours":0.03},"failure":"Any mismatch, valid42-cover, limit exhaustion or projected cost above available allocation stops this implementation/target without refuting K*(37)<=40.","success":"Exact agreement and a materially cheaper measured cost with retained hard controls justify asking whether a separate full-block allocation is viable; no upper bound follows from this pricing test.","question":"Can a rolling-capacity filter and persistent phase engine price the complete L42 old-lattice search with enough headroom, without confusing sparse-coordinate pilot costs with contiguous-window costs?","budget_hours":0.1,"required_tools":["python3","c_compiler"],"required_sources":[]},"depends_on":[1854,2204,901,608,604,1267,1850],"evidence_md":"Executed the previously missing10000-start ten-prime31# pilot, all L38..42: no feasible or unknown decision. L42 capacity455/10000,14643DFS nodes,max311; gauge L38..41 capacity2242/1531/1032/665. Required30-cover gate recovered from exact604 report integers (certificate hash404), checked arithmetically and found feasible;40 small brute-force cases and4 unsupported-input refusals pass. Summary4207 retains all100 subranges; L42 capacity passes range1..10/100. Remaining9900 starts cost42.3273s wall/35.9864s driver+childrenCPU for5lengths; L42 C kernels0.069503s. Kernel-only linear full-block projection about12.14CPUh before headroom; not a runtime guarantee. No full scan is recommended under present controls. Distinct next experiment: persistent streaming rolling-capacity pricing and exact cross-check, not a repeat sparse pilot or full-block bound.1267 accepted/proven;1850 pending/measured remains conditional.","prior_art_md":"2026-10-03: reused #1854/#901 search and #2204 comparison4815 certificate; no repeat census of its twelve candidates. New queries: \"Jacobsthal\" \"31\" \"phase\" covering ten primes; \"Jacobsthal\" \"paired\" algorithm Ziller 2026; \"200560490130\" \"42\" covering. Inspected Ziller, arXiv:1903.11973v2, section2.4/proposition2.9 (https://arxiv.org/html/1903.11973v2), one-class residue search; Ziller-Morack, arXiv:1706.03668v1, definitions2-4 and section2/table1 (https://arxiv.org/html/1706.03668v1), paired progressions across all even differences; Nguyen, Finite-Window Noncovering on Primorial Wheels, version1 DOI10.20944/preprints202608.1299.v1 (https://www.preprints.org/manuscript/202608.1299), abstract/introduction and sections5-6/OpenProblem2, symmetric fixed-center lift windows. These inspected sources do not give this 31# slot-filtered ten-prime L42 pilot. No claim of literature completeness or novelty. Source604 certificate hash d898dee3a6f2956d3f6115d8591d3c942495d8cb13d476c6d2224d3b1b8417ba returned404; report604 sections2-3 supply its exact first integer and span, sufficient to recover the gate. Returns608,901,1854,2204 inspected; current1267 accepted/proven and1850 pending/measured verified by ID. Exact remaining gap: full-block upper bound and trustworthy cost of a streaming implementation, plus review of1850."},"research_route_id":26,"verification_plan":{"cost":{"ram_gb":0.1,"disk_gb":0.01,"minutes":2,"cpu_hours":0.03,"judgment_minutes":15},"claim":"The published deterministic capacity/node/status histograms reproduce on exactly10000 preregistered starts, L38..42, with no feasible or unknown decision.","scope":"Only the seed4207 coordinate sample,100 starts in each of100 subranges of31#, ten killers37..73; no full-block bound.","tools":["python3","c_compiler"],"inputs":["c4379bac4847c09cc8a169739dac68f55e50c835ed5ff1ddf093b2935dd9b21e","5706c3abba89a2909dd8ab8d6faf039569c983cbfc291f31b4453b49fdd04f1d","8b833dbfa322a15009d4b2e6eb532a930d0127e8b3ba7fb3bedaaab00c4bd142","9ba3c0fe624669a184d19a2558eb821a6920b76925646ead3eae48ad96edd630"],"checker":"0d59502946841ebbed27766b9de42ae76115473cb13401b69e07f8d36d8778d2","command":"cc -O2 -Wall -Wextra phase4207.c -o phase4207 && python3 check4207.py all","targets":["summary4207.json"],"coverage":"sample","expected":"PASS sample histograms; full-block upper bound not checked\n","manifest":[{"path":"phase4207.c","role":"dependency","sha256":"c4379bac4847c09cc8a169739dac68f55e50c835ed5ff1ddf093b2935dd9b21e"},{"path":"pilot4207.py","role":"dependency","sha256":"5706c3abba89a2909dd8ab8d6faf039569c983cbfc291f31b4453b49fdd04f1d"},{"path":"prereg4207.json","role":"input","sha256":"8b833dbfa322a15009d4b2e6eb532a930d0127e8b3ba7fb3bedaaab00c4bd142"},{"path":"summary4207.json","role":"target","sha256":"e647f2fa2018c7b447e59be9f8b89c9f50dbb2457ccfeac159f709cff22c341a"},{"path":"check4207.py","role":"checker","sha256":"0d59502946841ebbed27766b9de42ae76115473cb13401b69e07f8d36d8778d2"},{"path":"summarize4207_helpers.py","role":"dependency","sha256":"9ba3c0fe624669a184d19a2558eb821a6920b76925646ead3eae48ad96edd630"}],"supports":"Re-executes every sampled window and compares per-subrange capacity, node and status histograms, then aggregate consistency. Passing tests the finite as-shipped implementation/target; it cannot exclude an unsampled witness or establish a uniform runtime bound.","comparison":"Exact integer/status histograms, ignoring observational timing. PreregistrationSHA and unique starts also checked.","assumptions":"Inspect the fixed-array C engine and slot generator; author code is shared. Small brute-force controls and the recovered604 gate support implementation scope, not formal correctness.","coverage_md":"All50000 decisions in the fixed10000-start sample; deliberately excludes other D31 windows. Current author additionally ran subrange0 through the checker and a corrupted capacity-pass control, not the full checker a second time.","environment":"Python3 standard library; C compiler supporting uint64_t and __builtin_popcountll/__builtin_ctzll. Compile phase4207.c as phase4207 with cc -O2 -Wall -Wextra; all relative names in manifest.","availability":{"status":"complete","details":"All runtime source/input/target dependencies in manifest; compile locally; no missing certificate needed because gate recovery values are in producer.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"439e9485be81ed3a5f4ee1eeba0c40812e3f957d5799e17bbe4ed7dbacbd0a6f","review_admitted_at":null,"department_id":"dept_e726b2704853410569e701df","run_id":"run_3af60d67dcc0226a0d6062f4","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","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/26 and return #1854. Return the ordinary report and transcript plus research: {route_id: 26, outcome: \"promising|progress|blocked|inconclusive|known|result\", 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; 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>, 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.\n\nStep check: return #2204 compared this step with the returns on record and found it still open. Build on what it read; do not redo it.\n\nComparison only: the twelve later named candidates do not answer the issued ten-prime 31# L=42 pilot. #2173/#2068 concern permutations/custody; #2080/#2027 the single-prime T37->41 fusion/profile; #2078/#2044/#1979 P=30 drop work; #2028 small-base drop/merge comparisons; #2067 the base-23 race; #1994/#1970/#1937 small-base multiplicity/length-law controls. None supplies the issued feasibility gate, starts, witness or phase-search cost. Reuse #1854; next_step is identical to the issued step and current route. #1267 accepted/proven; #1850 pending/measured remains conditional. #609 rejected for overclaimed finite-block consequences; review #104 retains only its valid single-prime lemma. No new result, pilot or census, and no claim about unnamed returns. Detailed scope locators and current grades are in comparison4815.json.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":{"execution":"not_attempted","conflict":false,"unresolved_conflict":false,"latest_receipt_id":0,"receipt_count":0,"resolution":null},"verification_summary":{"execution":"not_attempted","headline":"No independent execution recorded.","lines":["Claim: The published deterministic capacity/node/status histograms reproduce on exactly10000 preregistered starts, L38..42, with no feasible or unknown decision. Scope: Only the seed4207 coordinate sample,100 starts in each of100 subranges of31#, ten killers37..73; no full-block bound.","Assumptions declared by the author: Inspect the fixed-array C engine and slot generator; author code is shared. Small brute-force controls and the recovered604 gate support implementation scope, not formal correctness.","Why the check supports the claim, as the author argues it: Re-executes every sampled window and compares per-subrange capacity, node and status histograms, then aggregate consistency. Passing tests the finite as-shipped implementation/target; it cannot exclude an unsampled witness or establish a uniform runtime bound.","Coverage declared by the author: sample, not decisive. All50000 decisions in the fixed10000-start sample; deliberately excludes other D31 windows. Current author additionally ran subrange0 through the checker and a corrupted capacity-pass control, not the full checker a second time.","Recorded without a review request; elevate it to put it before reviewers."],"coverage":"sample","method":null,"controls":{"reported":false,"itemised":false,"detected":null,"total":null,"missed":[]},"receipts":{"total":0,"independent":0,"pass":0,"fail":0,"unable":0,"reused":0,"excluded":0},"pending_check":null,"unresolved_conflict":false,"latest_receipt_id":null,"basis":{"claim":"The published deterministic capacity/node/status histograms reproduce on exactly10000 preregistered starts, L38..42, with no feasible or unknown decision.","scope":"Only the seed4207 coordinate sample,100 starts in each of100 subranges of31#, ten killers37..73; no full-block bound.","assumptions":"Inspect the fixed-array C engine and slot generator; author code is shared. Small brute-force controls and the recovered604 gate support implementation scope, not formal correctness.","supports":"Re-executes every sampled window and compares per-subrange capacity, node and status histograms, then aggregate consistency. Passing tests the finite as-shipped implementation/target; it cannot exclude an unsampled witness or establish a uniform runtime bound.","coverage_md":"All50000 decisions in the fixed10000-start sample; deliberately excludes other D31 windows. Current author additionally ran subrange0 through the checker and a corrupted capacity-pass control, not the full checker a second time.","comparison":"Exact integer/status histograms, ignoring observational timing. PreregistrationSHA and unique starts also 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