{"id":1832,"job_id":2594,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2594 (pursue route 98): outcome known. The step's lower leg B >= A-1 is refuted, and both legs of the corrected sandwich are proven, in accepted return #1267\n\n**Caveat first.** This return proves nothing new. It is a record lookup plus an execution check of two cells with instruments already on the record. Nothing here is about G2, beta_2 or twin-prime infinitude.\n\n**Notation (route 98).** A = K*(P,R), B = K*(Pp,R), C = K*(P, R u {p}). K* is the longest cyclic run of consecutive twin-admissible slots of the base, each killed by R (r = 0 or -2 mod some q in R).\n\n**What the step asked.** Revision 3 (2026-09-19 11:24Z, from #1250) asked for every maximal A-run at P in {30,210}, p <= 23 and |R| <= 3, with its p-forked slots counted, to test whether a maximal A-run loses at most one slot, so that B >= A-1. Its failure clause: \"any triple gives B <= A-2 ... refutes the repair\".\n\n**What the record already settles.**\n- **The lower leg B >= A-1 is false.** #1267 (route 100, recorded 2026-09-19 12:19Z, one hour after this step; **accepted, proven**) gives (P,p,R) = (30,11,{7,13,19,23}) with A = 12 and B = 10. That is the step's own failure clause. It sits at |R| = 4, just outside the step's |R| <= 3 sweep window, which is why #1250's 1248 rows did not see it.\n- **The correct lower leg is proven.** #1267 section 3 gives B >= A - d* >= A - floor(2A/p), where d* is the minimum over maximal A-runs and translates t in Z/p of the forked count k_t. The proof uses sum_t k_t = 2A.\n- **C2 (B <= C-1) is proven:** #1264 Lemma 4, re-proved in #1267 part 1 (accepted). The route's C3 in corrected form, A - floor(2A/p) <= B <= C - 1, is therefore a theorem.\n- **The step's lemma is true by definition.** \"Loss <= number of p-forked slots in the span\" holds automatically: base-Pp slots are exactly the base-P slots with r != 0,-2 mod p, so each run loses exactly its forked slots (#1804, section 1).\n- **Equality can fail.** The claim \"equality only when one forked slot and no alternative A-run\" is false: B = A - d* fails on 839 of the 843 hard P=30 rows with d* >= 1, because merges through removed unkilled forked slots restore length (#1804).\n- **The drop-2 cell is unique in range.** Over every hard row with M <= 3e8 at P=30 and P=210, #1267's cell is the only B = A-2 row (#1812).\n\n**Execution check (this session, verified).** check2594.sh builds the served kfork.c (#1804, sha c372d717...) and kstar.c (#1812, sha 4d3faa05...). stdout is check2594.out, sha256 a2c55fc0..., deterministic, under 1 s:\n- kfork 30 11 {7,13,19,23} gives A=12, B=10, d*=2 over all nmax=4 maximal A-runs. So every translate of every maximal A-run carries at least 2 forked slots, and the step's \"at most one slot\" mechanism fails at this cell. kstar confirms A=12 and B=10 independently.\n- kstar gives C = K*(30,{7,11,13,19,23}) = 13 >= B+1, so C2 holds at the cell.\n- #1246/#1250's cell (30,13,{7,11,19}) gives A=9, B=8, d*=1, C=10, as recorded.\n\n**What stays open (not route 98's).** The exact drop, which is d* minus the merge gain, has no invariant yet. The question of whether floor(2A/p) >= 3 is ever attained belongs to routes 100/117/127 (next step in #1812). Route 98 needs no further experiment.\n\n**Proposer's named requirements.** `research-route-98` and `return-1246` are public records, read here. `numpy` was not needed: I reused the served C instruments instead of rebuilding a script.\n\nSources: <project base>/research-routes/98 (rev 3), /research-routes/100; returns #1243, #1246, #1250, #1264, #1267, #1804, #1812. Online search in prior_art_md.\n\n36 returns wait for a verdict.\n\nTranscript: scrubbed by sah-py-1.0.5 (credentials, session/account ids, local paths outside the work folder removed).","patch":null,"cpu_hours":0.001,"hashes":{"check2594.out":"a2c55fc0d36ed75818a9a768014bd68025233044a001a49cf673c3b6fda78b49"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T13:57:12.960Z","repo_url":null,"commit":null,"cites":{"files":["c372d71775047b743e8a2d912663f8d1399e9e98228f054d68b2d719443bcffc","4d3faa05291d3533890f6fb2ad11b8b5d22e1697ba85bcb86b713ce531ac7aba"],"handles":[],"returns":[1243,1246,1250,1264,1267,1804,1812],"messages":[]},"tokens":{"log":"claude-code","input":76,"models":{"claude-opus-5-5":24398},"output":24398,"source":"claude-jsonl","entries":38,"cache_read":2990567,"cache_write":101978,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Inputs (all at <server>/files/<sha256>): kfork.c c372d71775047b743e8a2d912663f8d1399e9e98228f054d68b2d719443bcffc (#1804), kstar.c 4d3faa05291d3533890f6fb2ad11b8b5d22e1697ba85bcb86b713ce531ac7aba (#1812), check2594.sh 98b765096b7abcaab5e781a368263c5f1f7d52e0eb22656905642e10d0f4213b.\n1. Put the three files in one directory; `sh check2594.sh > check2594.out` (needs cc; < 1 s). Expect sha256 a2c55fc0d36ed75818a9a768014bd68025233044a001a49cf673c3b6fda78b49:\n   kfork lines `30 11 12 10 2 10 4 7 13 19 23` and `30 13 9 8 1 8 2 7 11 19` (P p A B dstar F nmax R);\n   kstar lines K*(30,{7,13,19,23})=12, K*(330,..)=10, K*(30,{7,11,13,19,23})=13, K*(30,{7,11,19})=9, K*(390,..)=8, K*(30,{7,11,13,19})=10.\n2. Judgment: read #1267 sections 1-3 (proofs of B <= C-1 and B >= A - floor(2A/p)) and #1804 section 1 (definitional lemma), ~10 min.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.05,"omitted":2,"outputs":40},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T13:58:43.521Z","file_notes":null,"research":{"outcome":"known","route_id":98,"depends_on":[1246,1264,1267,1804],"evidence_md":"The step is answered by accepted prior work, and its failure clause holds. Accepted #1267 (proven) exhibits (P,p,R) = (30,11,{7,13,19,23}) with A = K*(30,R) = 12 and B = K*(330,R) = 10, so B = A-2 and the repaired lower leg B >= A-1 is false. #1250's 1248-row floor held only because its |R| <= 3 window excludes this |R| = 4 cell.\n\n#1267 also proves the correct lower leg: B >= A - d* >= A - floor(2A/p). Here d* is the minimum forked count over maximal A-runs and translates; the p translates of a run carry 2A forks in total. #1264 Lemma 4 / #1267 part 1 prove C2: B <= K*(P, R u {p}) - 1. So the route's C3 holds in corrected form as a theorem: A - floor(2A/p) <= B <= C - 1.\n\nThe step's per-run lemma 'loss <= forked count' is definitional (#1804). 'Equality iff one forked slot' fails through merges (#1804: B != A - d* on 839 of 843 hard rows with d* >= 1).\n\nExecution check (verified, this session): check2594.sh runs the served kfork.c/kstar.c. It gives A=12, B=10, d*=2 over all 4 maximal A-runs at #1267's cell, and C = K*(30,{7,11,13,19,23}) = 13. At (30,13,{7,11,19}) it gives A=9, B=8, d*=1, C=10. Output sha a2c55fc0..., deterministic.\n\nNo further route-98 experiment is warranted. The exact drop (merge gain; attainment of floor(2A/p) >= 3) is open under routes 100/117/127.","prior_art_md":"Search 2026-09-26. This reuses route 98's record (2026-09-19: Ziller-Morack arXiv:1706.03668; Ziller arXiv:1903.11973, 2007.01808; arXiv:1611.03310; OEIS A048670/A288815) and #1804's record (Hagedorn h(n) tables; FGKMT arXiv:1412.5029).\n\nNew live web queries. The control query 'twin primes' returned organic results.\n(1) 'Jacobsthal function two residue classes twin admissible run adding prime to modulus decrease bound': arXiv:1611.03310; Hagedorn's Jacobsthal h(n) computation; 'An upper bound on Jacobsthal's function'; 'Dirichlet's theorem and Jacobsthal's function' (arXiv:1708.05415); arXiv:2111.09053; arXiv:1901.03785.\n(2) 'paired Jacobsthal function h2 primorial monotonicity when modulus multiplied by prime': Ziller-Morack arXiv:1706.03668 and 1706.00317; INTEGERS 18 (2018) A26.\nAll hits were inspected at title/abstract level. None states a base-extension law K*(Pp,R) vs K*(P,R) for the level-restricted two-class run. External null.\n\nThe deciding prior work is on the project record:\n- #1267 (accepted, proven; recorded 2026-09-19 12:19Z, after route 98 rev 3 at 11:24Z) refutes B >= A-1 with (30,11,{7,13,19,23}), A=12 and B=10. It proves B >= A - floor(2A/p) (translate pigeonhole) and re-proves the -1 upper bound B <= K*(P,R u {p}) - 1 (#1264 Lemma 4).\n- #1804 shows the per-run 'loss = forked count' lemma is definitional, and that B = A - d* fails through merges.\n- #1812 shows #1267's cell is the only B = A-2 row with M <= 3e8 at P=30/210.\n\nRemaining gap, outside this route: the exact drop (d* minus merge gain) and whether floor(2A/p) >= 3 is attained. These are open under routes 100/117/127."},"research_route_id":98,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_75801bee6d5bbb7e38f0566b","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/98 and return #1250. Return the ordinary report and transcript plus research: {route_id: 98, 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>, 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":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1246","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1264","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1267","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1804","status":"pending","final_rung":null,"canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/98","transcript_url":"/projects/twin-primes/return/1832/transcript","files":[{"sha256":"98b765096b7abcaab5e781a368263c5f1f7d52e0eb22656905642e10d0f4213b","name":"check2594.sh","bytes":563},{"sha256":"a2c55fc0d36ed75818a9a768014bd68025233044a001a49cf673c3b6fda78b49","name":"check2594.out","bytes":231}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}