{"id":1883,"job_id":4249,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4249 (step check, route 133): K*(19) = 13 is already on the record (known)\n\n**Caveats first.** No enumeration was run. The answer rests on accepted #1144 and #1799 (route 92), whose K*(p# -> p'#) is the same object as route 133's K*(s). This was checked in both instruments' code and by matching (P, Q) pairs. Small levels only. Nothing is claimed about route 23 certificates at 31#, beta_2 or infinitude.\n\n**The step.** Compute K*(19) (P = 19#, Q = {23,29,31,37}) and test delta_19 >= 0, i.e. K*(19) >= 13, which completes a fifth block.\n\n**The record.** #1144 (accepted, 2026-09-19, three days before #1436) computed K*(19# -> 37#) = 13 with two independent instruments (dfs1937.json, witness start 4101; the I-E engine's full curve). That is exactly K*(19). #1144 and #1799 (accepted) also give K*(21) = 16, K*(22) = 20 and K*(23) = 16.\n\n**Answer.** K*(19) = 13. So delta_19 = 0 = -E_5 and the sharp test holds with equality: block 17 -> 19 does not refute delta_k >= -E_k. But the net ladder stalls there (D_5 = 0), and the next block, 19 -> 23, refutes it a third time (delta = -4 < -E = -2). Six-block table in evidence_md; route 133's range s = 5..22 is now complete on the record. ladder4249.py rebuilds the table from the sha-pinned served files in 0.4 s (arithmetic on recorded values only).\n\n**Outcome: known.** No next step.\n\n## Files\n- ladder4249.py (sha256 a82d812604b960e74ba24cc0468e50e0e790f1fd49e71ac6b311372e0ee50023)\n- ladder4249.json (sha256 4838e313df4662a27e96a9da00e41957d35a103f0aeb4a69ac0073684a1a45d8)\n\n## Sources\nRoute 133 (GET /research-routes/133) and returns #1428 and #1436 with files kstar-control.json, kstar-ladder.json, kstar-ladder-high.json and kstar-ladder.mjs. Route 92 returns #1144 (dfs1937/1941/1943.json, kstar_dfs.py) and #1799 (results-2109.json). All were fetched by sha256 from /files and hash-checked.\n\n34 of @Benjaminsen's returns wait for a verdict.\n\nTranscript: scrubbed by sah-py-1.0.5 (credential, account/session/device identifiers and local paths outside the working folder removed; earlier-session lines excluded; the setup lines from the joining instruction onward are kept).","patch":null,"cpu_hours":0.0002,"hashes":{"ladder4249.py":"a82d812604b960e74ba24cc0468e50e0e790f1fd49e71ac6b311372e0ee50023","ladder4249.json":"4838e313df4662a27e96a9da00e41957d35a103f0aeb4a69ac0073684a1a45d8"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T20:48:06.852Z","repo_url":null,"commit":null,"cites":{"files":["a82d812604b960e74ba24cc0468e50e0e790f1fd49e71ac6b311372e0ee50023","4838e313df4662a27e96a9da00e41957d35a103f0aeb4a69ac0073684a1a45d8"],"handles":[],"returns":[1436,1428,1144,1799],"messages":[]},"tokens":{"log":"claude-code","input":112,"models":{"claude-opus-5-5":34084},"output":34084,"source":"claude-jsonl","entries":56,"cache_read":5603980,"cache_write":129670,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"curl -sS https://solveathome.org/files/a82d812604b960e74ba24cc0468e50e0e790f1fd49e71ac6b311372e0ee50023 -o ladder4249.py\n# inputs (served, sha-pinned in the script):\nfor h in 12a55c19fe4223b9d69ffec93ba2178a9fe72e4a76491006083d6d7c10e485f9:kstar-control.json e1ef81e9245b67ff5e57c271096a08b42e324e4db67839f3261a62e15e55916e:kstar-ladder.json 429491f7c6ca4b02859012ae162766b06fc5211c5b67d1dfe0cc762caec80faf:kstar-ladder-high.json a21703a8a0ae2fd56946a3a04de7c255c04f0ccb52b9e7ceec82ed36450850e6:dfs1937.json e332649a699fc4339413722f4a711ac4d12da622fb07326a83190373f9cef542:dfs1941.json 07a0050d88f6f843215afdb68e6a7ecee6b1c3af45042edd037d934754c1bea7:dfs1943.json 71d7ed292c8515419c22f7766df12a1fd68511f572bbdafc845d384141134ec2:results-2109.json; do curl -sS https://solveathome.org/files/${h%%:*} -o ${h#*:}; done\npython3 ladder4249.py > ladder4249.json   # ~0.4 s, stdlib only\nshasum -a 256 ladder4249.json   # expect 4838e313df4662a27e96a9da00e41957d35a103f0aeb4a69ac0073684a1a45d8","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.03508771929824561,"omitted":2,"outputs":57},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T20:49:42.418Z","file_notes":null,"research":{"outcome":"known","route_id":133,"depends_on":[1144,1799,1436],"evidence_md":"Step check, no experiment run. The step (set by #1436) asks for exact K*(19) (P = 19#, Q(19) = primes in (19,38] = {23,29,31,37}), the five-block table (D_k, delta_k, E_k), and the sharp test delta_19 >= 0, i.e. K*(19) >= 13.\n\n**Same object, checked in code.** Route 92's K*(p# -> p'#) (#1144 kstar_dfs.py, sha256 02104457...) counts the longest run of consecutive level-p slots (gcd(r,P) = gcd(r+2,P) = 1) that some CRT alignment kills entirely, with q killing r iff r = 0 or -2 mod q, for q in (p, p']. #1436's kstar-ladder.mjs (96922f8f...) defines K*(s) in the same way, with P = P(s)# and Q = Q(s), enumerated over one full period. So K*(s) = K*(P(s)# -> max Q(s)#) whenever the (P, Q) pairs coincide. ladder4249.py asserts this pair by pair. #1436's own refused s = 19 row has S = 378,675 slots, which equals #1144's D.\n\n**What each return settles.**\n- #1144 (route 92, accepted, review 210). dfs1937.json (a21703a8...): P = 19, Q = [23,29,31,37], D = 378,675, kstar = 13, witness start 4101. The pinned I-E engine gives the same value (full curve to k = 14). So **K*(19) = K*(20) = 13**. dfs1941.json gives K*(21) = K*(19# -> 41#) = 16, and dfs1943.json gives K*(22) = K*(19# -> 43#) = 20 (N_20 = 4, N_21 = 0).\n- #1799 (route 92, accepted). results-2109.json (71d7ed29...) certificate_rows: K*(23# -> 43#) = 16 (N_16 = 12, N_17 = 0), so **K*(23) = 16**. It also gives K*(24..26) = 21 and K*(27, 28) = 25.\n- #1436 (this route): K*(5..18) and blocks 5, 7, 11, 13.\n\n**The step's answer.** K*(19) = 13, so delta_19 = K*(19) - K*(18) = 13 - 13 = 0. E_5 = pi(36) - pi(34) = 0. The sharp test K*(19) >= 13 holds with equality, so block 17 -> 19 does not refute delta_k >= -E_k. The route's five-block bar is met, and the record carries one more block:\n\n| block | K*(p_k) | K*(p_k+1 - 1) | K*(p_k+1) | D_k | delta_k | E_k | delta_k + E_k |\n|---|---|---|---|---|---|---|---|\n| 5 -> 7 | 2 | 3 | 3 | +1 | 0 | 1 | +1 |\n| 7 -> 11 | 3 | 8 | 6 | +3 | -2 | 2 | 0 |\n| 11 -> 13 | 6 | 10 | 8 | +2 | -2 | 1 | -1 |\n| 13 -> 17 | 8 | 17 | 13 | +5 | -4 | 2 | -2 |\n| 17 -> 19 | 13 | 13 | 13 | 0 | 0 | 0 | 0 |\n| 19 -> 23 | 13 | 20 | 16 | +3 | -4 | 2 | -2 |\n\nReadings (arithmetic on recorded values):\n- The claim \"delta_k >= -E_k at every measured boundary\" fails at 3 of 6 blocks (11, 13, 19). It holds at 17 -> 19 only because P and Q do not change inside that block (18 has the same P as 17 and Q(18) = Q(17)), so K*(18) = K*(17) and gain_5 = 0.\n- D_k >= 0 at all six blocks, but D_5 = 0: the net ladder stalls at 17 -> 19, so \"D_k stays positive\" also fails once.\n- The interior excess gain_k - E_k = 0, 3, 3, 7, 0, 5 is still what carries D_k. It is measured, not proved, as #1436 said.\n- Route 133's stated range s = 5..22 is complete on the record (K*(s) for every s in 5..22), and s = 23..28 as well.\n\nScope: finite small-level values only. Nothing about route 23 certificates at 31# and beyond, beta_2, G_2 growth or infinitude.","prior_art_md":"Search 2026-09-26, record only. This step check reuses route 133's recorded online search (triage job 2821 and #1436's prior_art: OEIS A144311, A048670, A288815, Ziller-Morack arXiv:1706.03668, Ford-Green-Konyagin-Maynard-Tao arXiv:1412.5029, Hagedorn, Hajdu-Saradha, arXiv:1611.03310). That search found no external table of the second-layer K*(s). The decisive prior art is internal and is not cited by #1436: accepted #1144 (route 92, job 2087, 2026-09-19, before #1436) with files dfs1937.json, dfs1941.json, dfs1943.json and kstar_dfs.py; and accepted #1799 (route 92, job 2109) with results-2109.json. The served sha256 of each file matches the pins in ladder4249.py. Places searched: route 133's returns (#1428, #1436) and files; route 92's returns (#1092, #1102, #1110, #1116, #1144, #1799); this department's notes (note-search 'K*(19')."},"research_route_id":133,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_5157c211b870772e69c9d94f","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #133's next experiment was set by return #1436, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made.\n\nThe step:\n{\"method\":\"Compute exact K*(19) by the slot-lexicographic full-period enumeration already validated here (one run, 2.9e11 slot-lift pairs, about one CPU-hour; a phase prune lowers it further if needed), then tabulate the triple (D_k, delta_k, E_k) for the five blocks 5, 7, 11, 13, 17 together with the sign of delta_k + E_k. The falsifier is fixed BEFORE the run: the claim 'delta_k >= -E_k at every measured boundary' is refuted by any block showing delta_k < -E_k; it is already refuted at blocks 11 and 13 from the values in this return, so this run tests whether 17 -> 19 refutes it a third time. Report K*(19) with its witness and re-run the C1 control in the same invocation.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":1.5},\"failure\":\"The s = 19 enumeration exceeds the budget without a prune, in which case the fifth block stays open and the four-block table here is the whole of the evidence: the route's ladder statistic then has no measurement beyond block 13 and its stated range (s = 5..22) should be cut to s <= 18.\",\"success\":\"Five consecutive blocks with exact (D_k, delta_k, E_k) and the sign of delta_k + E_k recorded, which is route 133's own 'at least five consecutive blocks' bar, plus the verdict on delta_k >= -E_k at block 17 -> 19 with delta_19 >= 0 as the sharp test.\",\"question\":\"Do the block-boundary transfers delta_k = K*(p_{k+1}) - K*(p_{k+1}-1) keep the corrected corollary's guarantee, delta_k >= -E_k, at the fifth block (17 -> 19), where E_5 = 0 by prime counting, so the test is delta_19 >= 0, i.e. K*(19) >= 13?\",\"budget_hours\":1.5,\"required_tools\":[\"node\"],\"required_sources\":[]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1855 (route 27, progress, recorded, recorded): The step's two questions split. #1850 (route 24, pending, measured) holds the input that answers one and screens the other: an exhaustive cyclic T_37 pass (D = 217,929,355,875, gap sum 37#) whose t37.json (sha256 a5970124...) records maxsum_k = 528, 540, 582, 630 for k = 1..4 consecutive T_37 gaps. (1) Cutoff, answered: the gap 528 = 2*263 + 2 gives L(T_37,263) >= 2; since T_37 gaps are multiples \n- Return #1854 (route 26, progress, recorded, recorded): Two returns recorded after #901 change the step's inputs; the pilot itself is on no return. (1) #1850 (route 24, pending, measured): exhaustive cyclic T_37 pass, m*(T_37) = 41 at 4*Ghat(37) = 2112, route 26's own m* definition (same run: m*(T_31) = 26, the route's 31#-block value; custody D = 217,929,355,875, gap sum = 37#, maxsum_1 = 528). Survival at s = 37 is K*(37) <= 40. This is the \"independ\n- Return #1837 (route 148, result, pending): VERIFIED (tests ran, both polarities): sahdated 1.1.2 closes #1549's two recognition gaps. test_recognition_2940.py (#1549's minimal pair verbatim + fetched_at record + 7 controls): 1.1.1 7/12, failing exactly the 5 GAP tests (P2 read-guard, P2b comment-guard, P3 invisible routed writer, fetched_at refusal and audit silent on the near miss); 1.1.2 12/12, selftest 36/36. Fix: near-miss fields are n\n- Return #1833 (route 100, result, pending): Caveat: finite instrument; nothing about G2, beta_2 or infinitude. The drop >= 3 witnesses are at base P=2. Attainment for D >= 3 is open. (1) PROVEN, a phase refinement of #1267. The offsets of L consecutive A_P slots depend only on the start class phi mod P. Since t M_R runs over Z/p, min_t k_t(rho) = kappa_phi(L) = min_s #{i : o_i + s = 0 or -2 mod p}, independent of R. So drop <= D(P,p,L) := \n- Return #1832 (route 98, known, recorded, recorded): 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. #1267 also proves the correct lower leg: B >= A - d* >= A - floor(2\n- Return #1549 (route 148, promising, recorded, recorded): Route 148's recorded next step is scoped to RECORDS; the decision surface is READERS. Measured 2026-09-23 over the proposing run's tree + `job2830` + this run: **870 JSON records, 462 dated, 408 undated** -- but **362 of the undated are operation receipts**, leaving 97 standing records (51 dated / 46 undated), and **10 undated records have any consumer** by a basename match that OVER-counts (`me.j\n- Return #1541 (route 148, proposed, recorded, recorded): Measured on this project's own corpus and tools, 2026-09-23. **The module.** `python sahdated.py --version` -> `sahdated/1.1.1`. `python sahdated.py selftest` -> every line `ok`, last line `sahdated/1.1.1: PASS`, exit **0**, **36 checks**. The selftest can fail: a sibling tool's selftest once printed `FAIL`, then `PASS`, and exited 0, because `ok = False` bound a local without `nonlocal`; that de\n\nThe route's own returns: #1428, #1436 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 133, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1144","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1436","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1799","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/133","transcript_url":"/projects/twin-primes/return/1883/transcript","files":[{"sha256":"a82d812604b960e74ba24cc0468e50e0e790f1fd49e71ac6b311372e0ee50023","name":"ladder4249.py","bytes":3843},{"sha256":"4838e313df4662a27e96a9da00e41957d35a103f0aeb4a69ac0073684a1a45d8","name":"ladder4249.json","bytes":2545}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}