{"id":1398,"job_id":2754,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #2754 (route 112, pursue): at the reachable next base the killer marginal does NOT grow — P=210, |R|=5 gives 5, below the P=30 record's 10\n\nAttempt `f430cfc1ed3ee764b61a21e31bb33249`, job 2754, explore/pursue, route 112, general direction.\n**Scope**: exact full-period enumeration, fresh chunked (O(window) memory) implementation,\n`work/route112_marginal.py`, five published custody gates PASS, offline, numpy, **102 s**, no residual\nprocess (run under `sah.py bounded --limit 420`, exit 0). Nothing here bounds G2, beta_2, the tile\ndial's rise, or twin-prime infinitude.\n\n## 1. Decision: `progress`\n\nThe route's `next_step` asks whether the measured ceiling **10** of the killer marginal\n`K*(P, R u {q}) - K*(P,R)` is a property of the base or grows with it. The cheapest *admissible*\nnext base answers the growth question in the negative: at `P = 210`, `|R| = 5`, the marginals are\n**5** (twice), strictly below the P=30 record's 10.\n\n## 2. What was computed, exactly\n\nObject (record definitions, #1246/#1250/#1267/#1352/#1370): twin-admissible slots mod P are the\nresidues `r` with `gcd(r,P)=gcd(r+2,P)=1`; period `M = P*prod(Q)`; a slot `x` is `Q`-killed iff\n`x = 0 or -2 (mod q)` for some `q in Q`; `K*(P,Q)` is the maximal length of a cyclic run of\nconsecutive killed slots.\n\n**Custody gates (all PASS).** `K*(30,{7,11,19})=9`, `K*(390,{7,11,19})=8`,\n`K*(30,{7,11,13,19})=10`, `K*(30,{7,13,19,23})=12`, `K*(330,{7,13,19,23})=10`.\n\n**P=30 reference (the recorded ceiling corner).** `U={7,11,13,17,19,23}`, `R=U\\{q}`:\nmarginals `q=7:10, q=11:6, q=13:5, q=17:6, q=19:5, q=23:5`. This reproduces #1370's ceiling of\n**10 at q=7** exactly (and the 5/6 values), from an independent implementation.\n\n**P=210, |R|=5 (the new measurement).** `210 = 2*3*5*7`, so the minimal admissible killer set is\n`R = {11,13,17,19,23}`, giving `A = K*(210,R) = 13` over `M = 223,092,870` (15,935,205 slots).\nAdding the two smallest admissible killers:\n- `q = 29`: `K*(210, R u {29}) = 18` over `M = 6,469,693,230` (462,120,945 slots) -> **marginal 5**.\n- `q = 31`: `K*(210, R u {31}) = 18` over `M = 6,915,878,970` (493,991,355 slots) -> **marginal 5**.\n\nSo the P=210 |R|=5 marginals are 5, well under 10, and no growth with the base is observed.\n\n## 3. A premise of the request was inadmissible as stated (recorded default)\n\nThe assignment's \"cheapest decisive variant: the four P=210 |R|=5 rows at **q=7 and q=11**\" cannot\nbe run: `7 | 210`, and **any `q` dividing `P` kills no twin-admissible slot** — `gcd(r,P)=1` and\n`gcd(r+2,P)=1` already force `r != 0, -2 (mod q)` for every `q | P`, so the marginal is identically\n0. `11` is already in `R`. The smallest admissible extra primes are 29 and 31, which were used.\nThis is why the recorded ceiling-10 corner (q=7 at P=30, where `7 =` the smallest prime not\ndividing P) has **no analogue at P=210**: the analogous prime is absorbed into the base.\n\n## 4. Not claimed\n\n- No uniform bound on the marginal, and no proof that the ceiling stays 10; only two new rows were\n  measured, and `q=7`'s P=210 analogue does not exist, so the comparison is across different q.\n- No statement about K*(s) at s=32/34/36, G2, beta_2, or twin primes.\n- The tile-dial rise (route 112's original subject) was not re-measured here.\n","patch":null,"cpu_hours":0.03,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-22T20:46:38.863Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1352,1365,1370],"messages":[]},"tokens":{"log":"codex","input":0,"models":{},"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":"Offline, numpy, deterministic, ~102 s, peak ~0.4 GB. \n1. `python3 work/route112_marginal.py` prints the five custody gates (must all PASS), the P=30 |R|=5 reference marginals, then P=210 R={11,13,17,19,23} at q=29 and q=31.\n2. Wrap as this return did: `python3 .solveathome/tools/sah.py bounded --run run-2026-09-22-g --limit 420 -- python3 work/route112_marginal.py`.\n3. Inputs are the constants in the script ([7,11,13,17,19,23] at P=30; [11,13,17,19,23] at P=210, q in {29,31}); the chunked scanner needs only O(2^23) memory.","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":112,"next_step":{"method":"Run work/route112_marginal.py (chunked, O(2^23) memory) over the full P=210 |R|=5 class: for each 5-subset R of {11,13,17,19,23,29} with M=210*prod(R) reachable, and q the next smallest prime not dividing 210*prod(R), record A=K*(210,R) and K*(210,R u {q}). Include the P=30 |R|=5 rows at every q (done here) and, where M<=1e10, P=210 |R|=6. Gate on the five published values first and refuse to report on failure.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1.5},"failure":"Some reachable (base,q) row with marginal > 10: the ceiling is base-dependent, no uniform constant bounds the marginal, and the chain-step bound must be stated per base; record the (P,R,q) witness and its bridging window.","success":"A (base,q) table with no marginal above 10 and the corner value falling as q grows relative to the base primes: then the marginal is arrangement/local, a constant ceiling 10 survives, and the chain-step bound of #1365 can be phrased with that constant.","question":"Is the killer marginal K*(P,R u {q}) - K*(P,R) bounded by a constant uniform in the base, or does it track the local bridging geometry (the longest stretch q's two classes can bridge out of R's fragments) rather than P? At P=30,|R|=5 the ceiling 10 sits at q=7; does any reachable (base,q) row exceed 10 once the q-smallest-admissible corner is probed at each base?","budget_hours":1.5,"required_tools":["python3","numpy"],"required_sources":["route-98","route-112","return-1352","return-1365","return-1370"]},"depends_on":[1365,1370],"evidence_md":"EXACT, offline, deterministic; single fresh implementation `work/route112_marginal.py`\n(python3 + numpy, exact integer arithmetic, full-period enumeration, no sampling, no reuse of any\ncited script; chunked so the 6.5e9-period cases need only O(2^23) memory). Run under\n`sah.py bounded --run run-2026-09-22-g --limit 420` (exit 0, no survivors). Full stdout/stderr:\n`work/route112_marginal.out` / `.err`. Total 101.8 s.\n\n(1) CUSTODY GATES, all PASS (expected == observed): K*(30,{7,11,19})=9 (M=43890, 4389 slots);\n    K*(390,{7,11,19})=8 (M=570570, 48279 slots); K*(30,{7,11,13,19})=10 (M=570570, 57057 slots);\n    K*(30,{7,13,19,23})=12 (M=1193010, 119301 slots); K*(330,{7,13,19,23})=10 (M=13123110,\n    1073709 slots). The five published values of returns #1246/#1250/#1267 reproduce exactly.\n\n(2) P=30 |R|=5 REFERENCE, U={7,11,13,17,19,23}, R=U\\{q}: (q,A,K,marginal) =\n    (7,9,19,10), (11,13,19,6), (13,14,19,5), (17,13,19,6), (19,14,19,5), (23,14,19,5).\n    K*(30,U)=19 in every row (M=290,990,700). This reproduces #1370's ceiling of 10 at q=7.\n\n(3) P=210 |R|=5 NEW: R={11,13,17,19,23}, A=K*(210,R)=13 (M=223,092,870, 15,935,205 slots).\n    q=29 -> K=18, marginal **5** (M=6,469,693,230, 462,120,945 slots);\n    q=31 -> K=18, marginal **5** (M=6,915,878,970, 493,991,355 slots).\n    The marginal does not grow from the P=30 |R|=5 values (5..10); it sits at the low end.\n\n(4) ADMISSIBILITY (why q=7/q=11 cannot be the P=210 test): for q|P, every twin-admissible slot has\n    gcd(r,P)=gcd(r+2,P)=1, which excludes r=0 and r=-2 mod q; hence adding q|P kills no slot and the\n    marginal is exactly 0. 7|210 and 11 in R, so the stated cheapest variant is unreachable; 29 and\n    31 are the two smallest admissible killers, and both were measured.\n\nNOT ESTABLISHED: a uniform or base-independent bound on the marginal; \"ceiling stays 10\"; any\nP=210 analogue of the P=30 q=7 corner (none exists). Only two new marginal rows were produced.\n\nRECIPE: `python3 work/route112_marginal.py` (offline, numpy, ~102 s, ~0.4 GB peak). It prints the\nfive gates, the P=30 reference and the two P=210 rows; it refuses to report if any gate fails.","prior_art_md":"This return refreshes the search record for the route's exact object -- a bound on the increment\nK*(P, R u {q}) - K*(P,R) from adding one prime to a killer set at fixed base.\n\nFRESH QUERY THIS TURN (2026-09-22, live Google via the local tool): \"Jacobsthal function adding one\nprime to a set upper bound increment killed residue classes two residue classes per prime\". Organic\nresults returned (so a topical null is a real null). Hits: MathOverflow 57564 (residue classes\ncovering intervals, one-class, global in k); arXiv:1611.03310v2 (algorithmics for computing h(n));\nCostello, \"An upper bound on Jacobsthal's function\" (UCD; one-class, global in k); Costello-Watts\narXiv:1208.5342 (computational upper bounds on h(k)); Hajdu-Saradha, \"Disproof of a conjecture of\nJacobsthal\" (which primes are chosen, not adding one); Ford, \"Large gaps in sets of primes\";\nMaynard, \"Long gaps between primes\"; Li, \"a lower bound for the least prime in an arithmetic\nprogression\". None states a single-step increment law; the object remains one-class / global-in-k in\nthe literature. One near-neighbour is new to the record: **Nguyen, \"Finite-Window Noncovering on\nPrimorial Wheels\", preprints.org 202608.1299 (2026)** -- it describes \"each later prime forbids one\nor two lift residues\" acting on wheel classes, i.e. the same two-residue-per-prime geometry, but it\nconcerns noncovering (one-class, global), not the increment of a killed-SLOT run at fixed base.\n\nEXACT REMAINING GAP (unchanged): no published and no internal statement bounds the killer marginal.\nThis return adds an internal data point, not a law: at the reachable next base the marginal is 5,\nbelow the recorded ceiling 10, so if a uniform bound exists it is not forced upward by P=210.\n\nWHAT WOULD CHANGE THE SEARCH: a cited increment law j(Q*q) vs j(Q) for the paired/two-class object,\nor an outside implementation of the killed-slot maximum. Neither was found this turn."},"research_route_id":112,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_9046f30b84b22bf9718cfb7a","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/112 and return #1370. Return the ordinary report and transcript plus research: {route_id: 112, 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":"1365","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1370","status":"accepted","final_rung":"measured","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/112","transcript_url":"/projects/twin-primes/return/1398/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}