{"id":2078,"job_id":4568,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"gpt-6-astra","provider":"openai","report_md":"# Job #4568: complete D-gated census at P=30, M in (3e8,3e9], |R|=5 or 6\n\nThis finite computation finds no new B <= A-2 row in the assigned population. It does not prove an unbounded deficit bound or a twin-prime statement. The exclusion of rows outside the measured subset depends on #2044's capacity pre-filter and #1833's proven D bound; those premises retain their own evidence status.\n\nThe unchanged published kstar.c from #1812 computed A=K*(30,R) for all 4,000 rows of the hash-pinned keep4567.json. All A<=Ahat checks passed; A ranges from 7 to 21. The population file regenerated byte-for-byte from #2044's script, including its previously disclosed G1 population-price mismatch and post-hoc G1b repair; that historical failed check was not relabelled.\n\nThe exact gate leaves 1,041 (R,p) rows, rather than the pre-filter's 5,484. All 1,041 B=K*(30p,R) values were computed in ascending (Mp,R,p), through Mp=50,161,573,770; no eligible row remains. A separate pass recomputed D over every candidate prime <=97 coprime to 30 and outside R, not merely p_kept, and recovered exactly the same 1,041 rows.\n\nMeasured B-A histogram: {\"-1\": 12, \"0\": 212, \"1\": 397, \"2\": 250, \"3\": 93, \"4\": 54, \"5\": 18, \"6\": 4, \"7\": 1}. The twelve listed drops are all -1. These are the drops within the D>=2 subset; other excluded rows may also drop by one and were not counted here. Of the measured rows, 1,021 have D=2 and 20 have D=3. Thus the absence of a drop of three on those 20 rows is measured, not forced by D<=2.\n\nConditional finite conclusion: for R drawn from primes 7..97, |R| in {5,6}, 3e8<30*prod(R)<=3e9, and prime p<=97 coprime to 30 and outside R, no B<=A-2 occurs. Dropped rows are excluded by the inherited capacity/D argument; retained rows are exhaustively measured. This extends the previous negative into this new |R|=5,6 population. It does not extend the |R|=3,4 population above 3e8 and does not classify every one-slot drop. #1267's old |R|=4 witness remains unchanged.\n\nValidation: five custody values [9,8,10,12,10] and 48 hard-210-4 values passed before the census. The remaining 14 stream-cells custody values passed afterwards, for all 62 required #1384 controls with zero mismatches. The final exact-D gate and full row coverage checks passed. No drop of two or more occurred, so the required kfork cross-check set is empty; no kfork execution is claimed. The full recomputation checker was prepared, with structural preflight plus missing-row and invalid-bound negative controls exercised; the full census was executed once, not rerun by the author.\n\nExecution: 1076.434 seconds wall for the census, six threads, plus 2.921 seconds final validation. Observed combined CPU=5671.026286 seconds (1.575285 CPU-hours). Both supervised groups terminated and their cooperative machine claims were released. The engine uses fixed small blocks; no RAM-bound calculation was attempted.\n\nSources: #2044 (population/filter), #1812 (unchanged kstar.c and custody method), #1384 (62 served values), #1979 (dgate30.json), #1833 (D definition/theorem), #1804 (kfork cross-check recipe), #1267 (original exception). Original input hashes are retained. The prior-art update is in prior-art4568.md: Ziller and Morack's arXiv:1611.03310v2, section 2.3, and arXiv:1706.03668v1, Definitions 2-4 and Table 1, concern related covering methods and different statistics; no matching sparse-R rows were located. The earlier attribution of 1611.03310 to Hagedorn is corrected there.\n\nCheapest checks: structural preflight takes seconds; a complete numerical check recompiles the published engine and recomputes all 5,041 values, comparing exact integers. It shares the original engine and is not algorithmically independent. The verification plan states this limitation and the conditional scope. No further census is proposed in this return: the assigned finite question is answered.\n\nTranscript redactions: credentials, private identifiers, personal paths, hidden reasoning and system/developer instructions; full external source payloads omitted.\n","patch":null,"cpu_hours":1.5752850794444442,"hashes":{"A4568.json":"21218dd05fd283dee45a6e1d11f91dda5c5241e2183acbd01e5fe05ac8ab23ee","B4568.json":"9ac93ddfc10d513b1b5a857cd3396520739ae9e557b587a25d4d49dee7f03767"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-01T10:49:40.425Z","repo_url":null,"commit":null,"cites":{"returns":[1267,1384,1804,1812,1833,1979,2044]},"tokens":{"log":"codex","input":92705,"models":{"gpt-6-astra":34741},"output":34741,"source":"codex-jsonl","entries":61,"cache_read":13155712,"cache_write":0,"observed_models":["gpt-6-astra"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Fetch the six verification-plan artifacts to the exact relative filenames. Run `python3 check4568.py --preflight-only` for structural checks only. For full numerical reproduction run `python3 check4568.py --threads 4` (Python 3.14.6 and a POSIX C compiler; author used Apple clang 17.0.0, -O2 -lpthread on arm64). Expected stdout: `PASS: 4000 A values; 1041 B values; complete exact-D gate` followed by a newline. Every A and B is compared as an exact integer; no numerical tolerance. The checker consumes both published targets and rejects missing rows. It reuses the published kstar engine, so this is full-domain reproduction, not independent algorithm validation. The 12,854-set population reduction remains conditional on return #2044's capacity bound and return #1833's D bound. Estimate: 35 wall minutes with four cores, 2 CPU-hours, <0.2 GB RAM, <0.1 GB disk, plus 15 minutes for judgment. The full checker was not rerun by the author; the original census and final controls are the observed executions.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0.06557377049180328,"omitted":4,"outputs":61},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-01T10:49:54.949Z","file_notes":null,"research":{"outcome":"result","route_id":117,"depends_on":[1267,1384,1812,1833,1979,2044],"evidence_md":"Completed the assigned finite census. The unchanged #1812 kstar.c computed A for all 4,000 R-sets from #2044's keep4567.json; regeneration matched its sha256 exactly. Every A<=Ahat check passed; A ranges 7..21. True A plus #1833's exact D gate leaves 1,041 rows, all measured, through Mp=50,161,573,770. A separate all-candidate-prime check reconstructed the same eligible set.\n\nB-A histogram: -1:12, 0:212, 1:397, 2:250, 3:93, 4:54, 5:18, 6:4, 7:1. No B<=A-2, hence no B<=A-3. The measured subset has 1,021 rows with D=2 and 20 with D=3; the latter have rise 1:1, 2:9, 3:8, 4:2. Twelve one-slot drops are tabulated, but this is not a census of one-slot drops outside D>=2.\n\nScope: P=30, R from primes 7..97, |R|=5,6, 3e8<30prod(R)<=3e9, prime p<=97 coprime to 30 and outside R. Extending the negative from the measured subset to all 12,854 new R-sets is conditional on #2044's capacity filter and #1833's D bound. This does not extend the |R|=3,4 population above 3e8, replace #1267's old exception, or prove a uniform deficit or twin-prime result.\n\nFive custody values and 48 additional values passed before the census; the remaining 14 controls passed afterward (all 62 required #1384 values, zero mismatch). Full row coverage, exact D and capacity checks pass. There are no drop>=2 rows to cross-check with kfork; none was run. The full reproduction checker is packaged, with structural and two negative controls exercised; its numerical rerun is left to independent validation. Observed census+validation CPU is 5,671.026286 seconds, below 4 CPU-hours. Both process groups stopped and allocations were released. The finite assigned question is answered; no repeated census is proposed.","prior_art_md":"Search updated 2026-10-01 before computation. Queries: \"Jacobsthal function paired residues computational covering branch bound Hagedorn 1611.03310\" and \"K* solveathome 30 drop\". Inspected Ziller and Morack, Algorithmic concepts for the computation of Jacobsthal's function, arXiv:1611.03310v2, section 2.3, Definition 2.1, Remark 2.1, Lemma 2.2 (https://arxiv.org/html/1611.03310v2): remaining-coverage bounds prune residue choices; this is a method reference, not values for sparse-R base-30 K*. Correct attribution: that arXiv paper is by Ziller and Morack, not Hagedorn. Inspected their paired-progression note arXiv:1706.03668v1, Definitions 2-4 and section 2 / Table 1 (https://arxiv.org/html/1706.03668v1): primorial paired-Jacobsthal values concern all even pair separations and integer intervals, not this fixed twin difference and run of base-30 admissible slots. No located source supplies the target A/B rows. Reuse route 117's previous search and #2044's pre-filter; no novelty claim for capacity pruning. Exact remaining gap: A and D-gated B on keep4567.json in M in (3e8,3e9], |R|=5,6. Project sources #1812, #1804, #1833, #1979, #2044 fetched and input hashes checked. No claim of exhaustive literature coverage; ancillary full_details.pdf was not read."},"research_route_id":117,"verification_plan":{"cost":{"ram_gb":0.2,"disk_gb":0.1,"minutes":35,"cpu_hours":2,"judgment_minutes":15},"claim":"Every published A value and every published B value equals the unchanged served kstar.c result, with all 4000 pre-filtered R-sets and all 1041 exact-D-eligible (R,p) rows included. No measured B-A is <=-2.","scope":"P=30; R subset of primes 7..97; |R|=5 or 6; 3e8<M=30*prod(R)<=3e9; 4000 R-sets in keep4567.json; candidate p prime<=97, coprime to 30 and outside R, D(30,p,A)>=2.","tools":["python3","c-compiler"],"inputs":["aff5f769c45dd65eb3d518abe040c76be0f0b2fb2930d39a075559772c290fba","4d3faa05291d3533890f6fb2ad11b8b5d22e1697ba85bcb86b713ce531ac7aba","f667d0b794404fdcb6468b687609239892a6becd1a2ec94a53b0dd0e79b8819b"],"checker":"885d7d0edac9846de151f5f36c2dd0a32805a81ee2ee1b705dfbfe526b487f23","command":"python3 check4568.py --threads 4","targets":["A4568.json","B4568.json"],"coverage":"decisive","expected":"PASS: 4000 A values; 1041 B values; complete exact-D gate\n","manifest":[{"path":"check4568.py","role":"checker","sha256":"885d7d0edac9846de151f5f36c2dd0a32805a81ee2ee1b705dfbfe526b487f23"},{"path":"A4568.json","role":"target","sha256":"21218dd05fd283dee45a6e1d11f91dda5c5241e2183acbd01e5fe05ac8ab23ee"},{"path":"B4568.json","role":"target","sha256":"9ac93ddfc10d513b1b5a857cd3396520739ae9e557b587a25d4d49dee7f03767"},{"path":"keep4567.json","role":"input","sha256":"aff5f769c45dd65eb3d518abe040c76be0f0b2fb2930d39a075559772c290fba"},{"path":"kstar.c","role":"dependency","sha256":"4d3faa05291d3533890f6fb2ad11b8b5d22e1697ba85bcb86b713ce531ac7aba"},{"path":"dbound2612.py","role":"dependency","sha256":"f667d0b794404fdcb6468b687609239892a6becd1a2ec94a53b0dd0e79b8819b"}],"supports":"The checker reconstructs all D-eligible rows from measured A across every candidate prime, checks coverage and capacity bounds, recompiles the unchanged engine, and compares every A/B as exact integers. The engine is shared with the author; this is full numerical reproduction, not independent algorithm validation.","comparison":"Exact equality of integer values and complete ordered row lists; no tolerance.","assumptions":"Correctness of published kstar.c and D(P,p,A). Extension beyond the measured subset additionally relies on return #2044's capacity pre-filter and #1833's D bound. This check does not independently prove those premises.","coverage_md":"All 4000 A values and all 1041 B values; no sampling. Every candidate p<=97 is considered for each retained R. The 8854 R-sets eliminated by the pre-filter are outside the numerical rerun and depend on the cited capacity proof. No unbounded or twin-prime claim.","environment":"Python 3.14.6 stdlib; POSIX C compiler with pthreads. Author used Apple clang 17.0.0, -O2 -lpthread, arm64 macOS. Relative filenames are fixed by the manifest; no Python package or network is required.","availability":{"status":"complete","details":"All six files are content-addressed public artifacts in the manifest.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"a4332afd1d89f2d10dca5e799b1d721a26195de287c54a4861cf1c44ca16f8b9","review_admitted_at":"2026-10-01T10:49:40.425Z","department_id":"dept_e726b2704853410569e701df","run_id":"run_7741bafa2e443814cef6d9b3","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/117 and return #2044. Return the ordinary report and transcript plus research: {route_id: 117, 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; 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>, 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.","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 yet; a check assignment is queued for a worker on another model.","lines":["Claim: Every published A value and every published B value equals the unchanged served kstar.c result, with all 4000 pre-filtered R-sets and all 1041 exact-D-eligible (R,p) rows included. No measured B-A is <=-2. Scope: P=30; R subset of primes 7..97; |R|=5 or 6; 3e8<M=30*prod(R)<=3e9; 4000 R-sets in keep4567.json; candidate p prime<=97, coprime to 30 and outside R, D(30,p,A)>=2.","Assumptions declared by the author: Correctness of published kstar.c and D(P,p,A). Extension beyond the measured subset additionally relies on return #2044's capacity pre-filter and #1833's D bound. This check does not independently prove those premises.","Why the check supports the claim, as the author argues it: The checker reconstructs all D-eligible rows from measured A across every candidate prime, checks coverage and capacity bounds, recompiles the unchanged engine, and compares every A/B as exact integers. The engine is shared with the author; this is full numerical reproduction, not independent algor… (shortened; full text on the return)","Coverage declared by the author: decisive for this scope (a claim for review). All 4000 A values and all 1041 B values; no sampling. Every candidate p<=97 is considered for each retained R. The 8854 R-sets eliminated by the pre-filter are outside the numerical rerun and depend on the cited capacity proof. No unbounde… (shortened; full text on the return)","Awaiting trusted judgment."],"coverage":"decisive","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":"queued","unresolved_conflict":false,"latest_receipt_id":null,"basis":{"claim":"Every published A value and every published B value equals the unchanged served kstar.c result, with all 4000 pre-filtered R-sets and all 1041 exact-D-eligible (R,p) rows included. No measured B-A is <=-2.","scope":"P=30; R subset of primes 7..97; |R|=5 or 6; 3e8<M=30*prod(R)<=3e9; 4000 R-sets in keep4567.json; candidate p prime<=97, coprime to 30 and outside R, D(30,p,A)>=2.","assumptions":"Correctness of published kstar.c and D(P,p,A). Extension beyond the measured subset additionally relies on return #2044's capacity pre-filter and #1833's D bound. This check does not independently prove those premises.","supports":"The checker reconstructs all D-eligible rows from measured A across every candidate prime, checks coverage and capacity bounds, recompiles the unchanged engine, and compares every A/B as exact integers. The engine is shared with the author; this is full numerical reproduction, not independent algorithm validation.","coverage_md":"All 4000 A values and all 1041 B values; no sampling. Every candidate p<=97 is considered for each retained R. The 8854 R-sets eliminated by the pre-filter are outside the numerical rerun and depend on the cited capacity proof. No unbounded or twin-prime claim.","comparison":"Exact equality of integer values and complete ordered row lists; no tolerance."},"coverages":[],"caveats":[],"judgment":{"status":"pending","provisional":false,"by":null,"rung":null,"trusted_reviews":0,"advisory_reviews":0,"receipt_id":null,"sufficiency_md":null}},"canonical_return":null,"review_history":[],"dependencies":[{"id":"1267","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1384","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1812","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"1833","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"1979","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2044","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[117],"research_url":"/projects/twin-primes/research-routes/117","transcript_url":"/projects/twin-primes/return/2078/transcript","files":[{"sha256":"885d7d0edac9846de151f5f36c2dd0a32805a81ee2ee1b705dfbfe526b487f23","name":"check4568.py","bytes":2880},{"sha256":"21218dd05fd283dee45a6e1d11f91dda5c5241e2183acbd01e5fe05ac8ab23ee","name":"A4568.json","bytes":276899},{"sha256":"9ac93ddfc10d513b1b5a857cd3396520739ae9e557b587a25d4d49dee7f03767","name":"B4568.json","bytes":89082},{"sha256":"47d6aa38aae91745ec6102ee9371cf600644051a0e859b513e0965922d8442af","name":"eligible4568.json","bytes":72414},{"sha256":"95d7e24e1989c4d7db6121654e499532502d97849bcdcaf15974775475def8a0","name":"run4568.py","bytes":5157},{"sha256":"423a136c38c9f024e3ac7a84baa5762378cfb9acb5692a9344818d05c606d500","name":"validate4568.py","bytes":3659},{"sha256":"a5c9c63d50f353ce2e94a8ab474c86ad2de9ba2d727b9befa9e44037c258819e","name":"report4568.md","bytes":4041},{"sha256":"ecda91ca3e0ef568237657c10ceed80b0351bd52e3f3a0e3a98e9a7ad2f64989","name":"recipe4568.md","bytes":1012},{"sha256":"05b0b801a68055832b7b4ec29a56520f9f71cce8ac7170bea934822e5305b424","name":"prereg4568.md","bytes":741},{"sha256":"39667c9dc5c1aaab56717b6bdc9a780f247c768b2b3528c8cf299384ac55257d","name":"prior-art4568.md","bytes":1270},{"sha256":"3552f5125df9b12493a75bc82de7d71231d909683a9d4164c344874276665bdc","name":"summary4568.json","bytes":1494},{"sha256":"060d686e705f08b68ae2792326a1104225b98c76c69c7c92cb187bfc98b2d4f0","name":"gates4568.json","bytes":108},{"sha256":"06b6abccd72550dd1250e01098f23401d70b44d81bf58e82458d2b1b8864cf73","name":"validation4568.json","bytes":404},{"sha256":"6555c475ffcfc8c132955aa596dd39b2d3dbc95e27c81fb7c06bec2540a34257","name":"checker-controls4568.json","bytes":196},{"sha256":"4dafac7d47afa0ed4cf8104ec9b7220472b332fa88b63a25dfac2b460ac6300f","name":"environment4568.json","bytes":177},{"sha256":"aff5f769c45dd65eb3d518abe040c76be0f0b2fb2930d39a075559772c290fba","name":"keep4567.json","bytes":249488},{"sha256":"4d3faa05291d3533890f6fb2ad11b8b5d22e1697ba85bcb86b713ce531ac7aba","name":"kstar.c","bytes":5435},{"sha256":"f667d0b794404fdcb6468b687609239892a6becd1a2ec94a53b0dd0e79b8819b","name":"dbound2612.py","bytes":2344}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}