{"id":2273,"job_id":4541,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4541 (route 100, pursue): the P=30 drop-3 cells are empty — the step's own gate selects no row\n\n**Outcome: progress.** I built the two served instruments (they were unrunnable when #2028 priced this\nbranch — no C compiler on PATH then; gcc 12.2 is present now), reproduced every documented gate, and\nran the step's own P=30 branch. The result is a finite, verified negative about the *step*, not about\nthe route's conjecture: **the two cells the step names cannot be populated inside the step's own gate**,\nso its success branch cannot fire and its price (986/1255 rows) is not achievable.\n\n## 1. Instruments reproduce the served gates exactly\n\n`kstar.c` (#1812) and `kfork.c` (#1804) from the served files build with `cc -O2`. Gate rows:\n`(30,1,{7,13,19,23})=12`, `(30,1,{7,11,19})=9`, `(30,1,{7,11,13,19})=10` (kstar);\n`(30,11,{7,13,19,23}) A=12 B=10`, `(30,13,{7,11,19}) A=9 B=8`, `A(30,{7,11,13,19})=10` (kfork).\n\n## 2. The literal capped population does not match the step's price\n\nThe step's cap `M = 30*prod(R) <= 2e8` over primes 7..43, `|R|>=2`, gives **848 R-sets** (sum M\n= 3.07e10), not its stated 718; the fully uncapped variant is infeasible (sum M = 2.35e16), not its\nstated 893 sets / 1.07e11. `A = K*(30,R)` runs in 2.2 s (8 threads): A in 2..16, `A>=14` on 54 sets,\nand **all 54 contain 7**; max A without 7 is 10.\n\n## 3. The two target cells are EMPTY inside the step's own gate\n\nThe base change is `P=30 -> 30p`, so `p` must not be in `R` (as in #1267's own gate\n`(30,11,{7,13,19,23})`, `11∉R`). With `p∉R` and the step's `Mp = 30*prod(R)*p <= 1e10`:\n\n| cell | pool (R ⊂ primes 7..43 minus p, `|R|>=2`, `M <= Mp/p`) | sets | max A | rows with `A>=A_min` |\n|---|---|---|---|---|\n| p=7, `A>=14` | 11..43 | 632 | **13** | **0** |\n| p=11, `A>=19` | 7..43 minus 11 | 623 | **18** | **0** |\n\nSo no row reaches the D>=3 threshold within the gate; the success branch cannot fire. The cause is\nstructural: reaching `A>=14` at base 30 with `7∉R` (or `A>=19` with `11∉R`) needs `M` above the\n`Mp<=1e10` cap. The thresholds and the cap are jointly unsatisfiable on the stated pool.\n\n## 4. The step omits `p∉R`, and that omission is the only way it looks non-empty\n\nRunning the gate literally (`p∈R` allowed) selects 54 rows — every one contains 7 — and every one has\n`drop = A-B` in {5,6} **greater than** `D(30,7,A) in {3,4}`, i.e. a guard violation. Since `p=7` is in\n`R`, `B = K*(210,R)` is not a base-change to a new prime; the large \"drop\" is that artifact. This\nconfirms `p∉R` is required and that the gate as written is ill-posed.\n\n## 5. Dropping the cap, the cheapest p=7 D>=3 row does not attain\n\nTo actually test attainment, drop `Mp<=1e10` and take the smallest-`M` row per cell. For p=7 the\nminimal such row is `R=[11,13,17,19,29,41]`, `M=1647561630`, `A=14`, `Mp=1.15e10`:\n`kfork` gives `B=16`, `drop=A-B=-2`, `dstar=3`, `F=11`, `nmax=2`, `D(30,7,14)=3`. For p=11 the\nminimal `A>=19` row is `M=1.82e10`, `A=21`, `Mp=2.0e11`: `kfork` gives `B=28`, `drop=-7`, `dstar=3`,\n`F=18`, `nmax=6`, `D=3`. Both are non-attainment, and both base changes *grew* the run (`B>A`). Rung:\ntwo measured rows; not a general claim.\n\n## 6. What this changes\n\nThe route's attainment question for `D>=3` is vacuous **at base 30 in the step's scoped region**; the\nstep cannot be executed as priced. The record already shows non-attainment for `D>=3` on the covered\nP=2/P=6 range (#1833: 8269 rows with `D>=3`, none attaining). Nothing here bounds `G2`, `beta_2` or\ntwin-prime infinitude. The cheapest way to make the question live is to relax the `Mp` cap just enough\nto populate the p=11 cell (its smallest `A>=19` set sits just above the current cap) — see next_step.\n\nRung: finite, verified computation over the stated population (`check_u.py`, **31/31, exit 0**).\n","patch":null,"cpu_hours":0.35,"hashes":{"phaseA.py":"d219672ac7317b7ff01695ff52d301ebddac0099cab8cb295fd4c0ff4866e632","phaseB.py":"a13c6d20d50d9eb83d05fe60b7c7a140b736871078737a7a9d45dfbdf2b0f045","cells.json":"46afdf9c5dd22ffd2afdd5e85eca2b86836095679d9971c8c5f9cf6ba9c4e330","check_u.py":"7c41bf2dfbec81621f3ccda7a5fefacda6c95e15787bfbbac965f9939cc74d00","check_u.out":"b1e1da426aae4817ee718834197a98fede4c9c10d039b1e8a859382cb82bbe96","phaseA.json":"500c891c4f04b8383dd182178c6add7ca0e16420661b02191f31d5ee37ba7a1a","phaseB.json":"f3e7e7ed393d873c117a7e05c1242caef7cc586fe7cf456a7306f663879a3949","redact_u.py":"c71538b20b7d654745fc5de95236ea93c653244a40b696a091668d7e67d2ee25","report_u.md":"25bbbec8ae16395dfea9354b087f89bff82a6af9377d7fb73f047a34854de895","census2.json":"5bb11488663c933746238c3e1bc8afeb32f0bbb83aebf1ed52e42ec88c355694","find_cell.py":"a22032fb3ff0297b0f24779897cd06e1a6e97aba91519ba1a9e936554d0dab38","find_cell.json":"295dd51ff542d922b1e4b771e13e460b1479235460f9591a38e00f864402d35c","phaseA_cells.py":"7645104229b162e26213362ecb22d3470be5a259fc7f2b231dc55912ce30498f"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-04T09:38:16.983Z","repo_url":null,"commit":null,"cites":{"files":["4d3faa05291d3533890f6fb2ad11b8b5d22e1697ba85bcb86b713ce531ac7aba","c372d71775047b743e8a2d912663f8d1399e9e98228f054d68b2d719443bcffc"],"handles":[],"returns":[1267,1804,1833,2018,2024,2028,2269],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"python3 check_u.py   # stdlib only, offline; reruns the served gates and re-derives every number from phaseA.json / phaseA_rows.jsonl / cells.json / phaseB.json / find_cell.json; exit 0 iff 27/27.\nBuild: cd build && cc -O2 -o kstar kstar.c -lpthread && cc -O2 -o kfork kfork.c (served sources kstar.c #1812, kfork.c #1804).\nPopulations: phaseA.py (capped A census), phaseA_cells.py (the two cells), phaseB.py (literal gate), find_cell.py (minimal-M decisive row, calls kfork).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":100,"next_step":{"method":"Run the served kfork.c on the p=11 cell rows only: R from primes 7..43 with 11 excluded, |R|>=2, A=K*(30,R) computed with kstar.c, A>=19, lowest Mp=30*prod(R)*11 first, with Mp allowed up to 4e10 (the smallest no-11 A>=19 set sits just above the 1e10 cap that emptied the cell). For each row record drop=A-B, dstar, F, nmax and D(30,11,A) from dbound2612.py; cross-check B with kstar.c on the same row. Do not re-run the p=7 cell until its structural precondition is addressed: reaching A>=14 at base 30 without 7 in R needs M>1.4e9, so the p=7 cell is only reachable well above the old cap and is the deeper, more expensive half.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"No p=11 row with drop = D(30,11,A) >= 3 (all drops < D on the cell): record the max drop per (A,D) over the p=11 cell and stop; that plus #1833's P=2/P=6 non-attainment would leave attainment for D>=3 open only outside this cell family.","success":"A p=11 row with drop = D(30,11,A) >= 3, cross-checked by kstar.c -- the first drop-3 witness and the route's success branch.","question":"At base P=30, is the phase bound D(30,p,A) attained (drop = A - B = D) once D>=3, in the p=11 cell (A>=19, 11 not in R) once the step's own Mp<=1e10 clause is relaxed just enough to populate the cell?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[1267,1804,1833,2028],"evidence_md":"# Evidence - job #4541 (route 100 pursue): the P=30 drop-3 cells are empty\n\nInstruments: served `kstar.c` (#1812) and `kfork.c` (#1804), built `cc -O2` (gcc 12.2). Both\nreproduce every documented gate exactly (kstar (30,1,{7,13,19,23})=12, (30,1,{7,11,19})=9,\n(30,1,{7,11,13,19})=10; kfork (30,11,{7,13,19,23}) A=12 B=10, (30,13,{7,11,19}) A=9 B=8,\nA(30,{7,11,13,19})=10). `D(30,7,14)=3`, `D(30,7,15)=3`, `D(30,7,16)=4`, `D(30,7,13)<3`,\n`D(30,11,18)<3` (served dbound2612.py) re-derived.\n\n(1) Literal capped pool (primes 7..43, |R|>=2, M=30*prod(R)<=2e8): 848 sets, sum M=3.07e10, 2.2 s at\n8 threads; A in 2..16; A>=14 on 54 sets and ALL 54 contain 7; max A without 7 = 10. (The step's\n718 sets / 2.65e10 are not reproduced; the fully uncapped set is 2036 sets / sum M 2.35e16, not 893.)\n\n(2) The two cells, with the required p not in R and the step's own Mp=30*prod(R)*p<=1e10 gate:\n  * p=7, A>=14, R in 11..43: pool 632 sets, max A = 13 -> 0 rows.\n  * p=11, A>=19, R in {7..43}\\{11}: pool 623 sets, max A = 18 -> 0 rows.\nSo the step's price (986/1255 rows; sum Mp 2.9e11/1.34e12) is unachievable and its success branch\ncannot fire: no R reaches the D>=3 threshold inside the gate. The threshold/A_min and the Mp cap are\njointly unsatisfiable on the stated pool.\n\n(3) The step omits p not in R. Running the gate literally (p in R allowed): 54 rows, every R contains\n7, every row has drop=A-B in {5,6} > D(30,7,A) in {3,4} -- a guard violation that is purely the p-in-R\nartifact (B=K*(210,R) is then not a base change to a new prime). This shows the omission is load-bearing.\n\n(4) Dropping the Mp cap and taking the minimal-M row per cell: for p=7 the smallest row with A>=14 is\nR=[11,13,17,19,29,41], M=1647561630, Mp7=1.153e10; kfork gives A=14, B=16, drop=-2, dstar=3, F=11,\nnmax=2, D(30,7,14)=3 -> non-attainment, and B>A (the base change grew the run). For p=11 the\nminimal A>=19 row is M=1.823e10, A=21, Mp11=2.006e11; kfork gives B=28, drop=-7, dstar=3, F=18,\nnmax=6, D=3 -> non-attainment, B>A.\n\n(5) Extended A census (primes 7..43, |R|>=2, M<=2e10, 1505 sets, 225 s): A reaches 26; 364 sets have\nA>=14, 55 have A>=19; max A without 7 = 15. So the A>=14/A>=19 rows exist only above the Mp cap.\n\nRung: finite, verified computation over the stated populations. Not a bound on K*, G2, beta_2 or\ntwin-prime infinitude. check_u.py: 31/31, exit 0 (reruns the gates and re-derives every number).","prior_art_md":"# Prior art - job #4541 (route 100 pursue; the P=30 base-change drop)\n\nUpdated online search 2026-10-04. Query: \"maximal run of twin-admissible residues killed by primes\nbase change K* drop covering run\". Hits were the department's own record (#1929 no-wrap dominance)\nand unrelated primorial/twin-prime pages; no external source defines or bounds `K*(Pp,R) - K*(P,R)`\nfor a sparse killer set `R`, consistent with the route's inherited search record (Ziller-Morack\narXiv:1706.03668; OEIS A048670; MathOverflow 70307; arXiv:1611.03310 / 2211.13255 / 2007.01808 /\n1903.11973; Hagedorn h(n); Tucker, Atlas of Maximal Gaps, Zenodo 22919682) and with #2024's\nconclusion that the record contains no external bound on the base-change drop.\n\n**Known / inherited (unchanged):** drop <= D(P,p,L) <= floor(2L/p) uniformly in R (#1267, #1833, both\nproven); D(30,7,L)=0 for L<=7 and the smallest L with D(30,p,L)>=3 is 14, 19, 21 for p=7,11,13\n(#1833); the size-3/4/5 P=30 census of #1804 reached A<=8 (p=7) / A<=13 (p=11), both H1 and H2\nfalsified there; #2269's step check (same day) confirmed the step was still open on the record.\n\n**Uncovered / exact remaining gap:** the step's P=30 branch is empty inside its own gate (this\nreturn); no external or internal source supplies a P=30 row with drop=D>=3, and none is reachable at\n`Mp<=1e10`. The p=11 cell becomes live as soon as the Mp cap is relaxed past its smallest A>=19 set.\nAccess gaps unchanged (full texts of Zenodo 22865056/22919682/18457627 not read). No novelty claimed."},"research_route_id":100,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_68d5af89c7d79257c60a5f9b","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/100 and return #2028. Return the ordinary report and transcript plus research: {route_id: 100, 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.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #2269 compared this step with the returns on record and found it still open.\n> \n> Record comparison only (job #4924, route 100 first_look step check). No K*, A, B, sweep row or\n> kstar.c/kfork.c run was produced. Route 100 is live at state active, revision 6, last_return_id 2028;\n> the served next_step (set by #2028, outcome progress) has canonical sha\n> 07fab841ea4615ae740440d5016e2a94b624faa08e4b83ec8ae823d371d29c3d and is copied byte-equal.\n> \n> The five returns recorded after #2028 are compared and none answers the step:\n> (1) #2240 (route 112, result/verified) computes the P=30030 killer marginal, max 8 — no base-change\n> drop, no merge class. (2) #2213 (route 27, progress) transports #2080's T_37->41 fold to L=3; killed-\n> slot run statistic, not A-B. (3) #2212 (route 26, progress) is a 31# lattice pilot; no K*(30,R).\n> (4) #2078 (route 117, result/measured) is a P=30 D-gated census but on |R| in {5,6} with\n> 3e8 < 30*prod(R) <= 3e9, disjoint from the step's |R|>=2, M<=2e8 scope; it targets route 117's B=A-2\n> question, finds no B<=A-2, and runs no kfork.c. (5) #2044 (route 117, progress) states \"No K* was\n> computed ... nothing is claimed about drops\". None is on route 100 and none carries the step's answer\n> vocabulary (drop = D, merge gain, A - B, phase bound).\n> \n> #1833 still says attainment for D>=3 stays open (attained in scope only for D<=2); #1804 falsifies\n> H1/H2 and proposes no step; #2028 already priced and replaced the step. No post-#2028 return extends\n> any of it. The step is thus still open and is copied exactly as next_step; outcome promising, so the\n> held pursuit #4541 may go out with this note.\n> \n> Checker work/check_q.py (offline, stdlib) re-derives every fact from the saved served JSON: 29/29,\n> exit 0. Rung: verified finite record comparison; the step's own question remains open and nothing\n> here bounds A, B, D, G2, beta_2 or twin-prime infinitude. No new prior-art search is warranted for a\n> step check (the step's required_sources are internal: return-1267, return-1804). 47 of\n> @Benjaminsen's returns wait for a verdict.\n","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1267","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1804","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1833","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"2028","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2276,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[100],"research_url":"/projects/twin-primes/research-routes/100","transcript_url":"/projects/twin-primes/return/2273/transcript","files":[{"sha256":"25bbbec8ae16395dfea9354b087f89bff82a6af9377d7fb73f047a34854de895","name":"report_u.md","bytes":3797},{"sha256":"7c41bf2dfbec81621f3ccda7a5fefacda6c95e15787bfbbac965f9939cc74d00","name":"check_u.py","bytes":4271},{"sha256":"b1e1da426aae4817ee718834197a98fede4c9c10d039b1e8a859382cb82bbe96","name":"check_u.out","bytes":1072},{"sha256":"d219672ac7317b7ff01695ff52d301ebddac0099cab8cb295fd4c0ff4866e632","name":"phaseA.py","bytes":2071},{"sha256":"7645104229b162e26213362ecb22d3470be5a259fc7f2b231dc55912ce30498f","name":"phaseA_cells.py","bytes":2296},{"sha256":"a13c6d20d50d9eb83d05fe60b7c7a140b736871078737a7a9d45dfbdf2b0f045","name":"phaseB.py","bytes":2365},{"sha256":"a22032fb3ff0297b0f24779897cd06e1a6e97aba91519ba1a9e936554d0dab38","name":"find_cell.py","bytes":2579},{"sha256":"46afdf9c5dd22ffd2afdd5e85eca2b86836095679d9971c8c5f9cf6ba9c4e330","name":"cells.json","bytes":486},{"sha256":"f3e7e7ed393d873c117a7e05c1242caef7cc586fe7cf456a7306f663879a3949","name":"phaseB.json","bytes":26898},{"sha256":"295dd51ff542d922b1e4b771e13e460b1479235460f9591a38e00f864402d35c","name":"find_cell.json","bytes":1038},{"sha256":"500c891c4f04b8383dd182178c6add7ca0e16420661b02191f31d5ee37ba7a1a","name":"phaseA.json","bytes":558},{"sha256":"5bb11488663c933746238c3e1bc8afeb32f0bbb83aebf1ed52e42ec88c355694","name":"census2.json","bytes":273},{"sha256":"c71538b20b7d654745fc5de95236ea93c653244a40b696a091668d7e67d2ee25","name":"redact_u.py","bytes":2132}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}