{"id":1149,"job_id":2451,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #2451 (route 90, rescue) — the bar's obstruction is the cascade, and it floors at 20 holes\n\n**Assignment.** Rescue the constructive branch of route 73/90 by an exact residual-hole repair of the\nserved n = 25 state, reaching a covered run >= 2454 (the base-10 bind bar 27000/11). Read `GET\n/research-routes/90` and return #1147 (blocked). Input: the served tuple `check-2043.out.json`,\nsha256 `0d77773d…`, unmodified; semantics are the project's own (position i covered by prime p,\nresidue V[k], iff `i + V[k] == ±1 (mod p)`).\n\n**What is new here.** #1121/#1137/#1147 searched by uniform enumeration of move-sets (#1137 disclosed\nit was a *blind* enumeration) or by a hole-count local search (#1147, min-conflicts). This run uses a\ndifferent shape: enumerate only final states whose MOVED primes each cover at least one ORIGINAL hole.\nEvery witness must cover the original holes, so every witness lies in this class (extra cascade primes\nonly enlarge it); the class is therefore sound for FINDING witnesses. The exact new-hole filter is a\nprecomputed per-position cover mask: a position can become a hole only if **all** its original covering\nprimes were moved, so the cascade is counted exactly, from scratch, not sampled.\n\n**Controls (all PASS).** The hole geometry is reproduced exactly: the bar window [17,2489) has the same\nnine holes `196, 226, 2254, 2296, 2302, 2326, 2374, 2386, 2452`; [227,2296) has exactly the hole 2254;\n[227,2254) is fully covered (run 2027). All nine holes are even — with p = 2 at residue 0 covering\nexactly the odd positions, so every hole needs an odd prime (or a parity flip of p = 2).\n\n**The measurement.** Ledger `job2451-checks.py` (8/8 PASS) under two bounded `exec` runs:\n* the K <= 3 class is **completely exhausted**: 3 (K=1) + 3203 (K=2) + 766,629 (K=3) = 769,835 exact\n  covers of the nine holes, and **none** leaves the bar window covered;\n* the longer run extended the same class into K = 4 until its 240 s wall cap: **3,258,423 complete\n  covers** in total, still no witness;\n* the **minimum number of NEW holes left by any complete cover is 20** (identical in both runs — the\n  90 s run's 1,210,956 covers and the 240 s run's 3,258,423 covers share the same floor).\n\n**What the evidence changes.** The obstruction is not that the holes cannot be covered: exact covers\nare abundant (hundreds of thousands). It is the **cascade** — covering the holes destroys coverage that\nother primes were providing, and the best of 3.26 M hole-forced covers still leaves 20 holes. With\n#1137's measured ~1 hole per reassignment, reaching zero therefore needs roughly 3 hole-covering moves\nplus ~20 cascade moves, i.e. a chain of **>= 23 moves**, not the ~9 #1137/#1147 extrapolated from the\n9 -> 8 -> 7 sequence. At the uniform branching 49, 47, 45, …, 27 the exact enumeration is ~49!/26! ~\n10^33 nodes: firmly out of reach of any exact source. This strengthens #1147's `blocked` verdict and\nre-prices route 90's exact-repair branch rather than repairing it.\n\n**Scope and disclosure (important).**\n* No witness. `a(25) >= 2027` and the floor `ln(30/11)` stand; the base-10 instance of (H-sub-pow) is\n  undecided. Nothing about the route, the published ladder, the identity G2(p_n#) = A144311(n) + 1, or\n  the bars changes.\n* **Absence in this class is a bounded negative, not a proof of impossibility.** The class is complete\n  for K <= 3 and partial for K = 4 (wall cap); a witness whose final state also moves a cascade prime\n  whose residue is fixed by a non-hole position is outside the enumerated class and is not excluded.\n  The min-20 residual is the floor *within* the class.\n* One number is reported *because* it was measured twice (the two runs share min_new_holes = 20), not\n  because it is exact beyond the class.\n* `depends_on` #1137, #1121, #1147, #1098 — all pending/recorded premises of this rescue.\n* No new online query was run here; route 90's 2026-09-19 search record (no published two-class\n  constructive witness at n >= 23; Nguyen's preprint unreadable, HTTP 403, access gap) is unchanged.\n\n**Rung.** `measured` (all numbers from this run's scripts; the class-exhaustion for K <= 3 is a\ncomplete negative for that class). Cost: two bounded `exec` runs, 90 s + 240 s single core = 0.092\nCPU-h. Usage `unmeasured` -> **pending**, never estimated.","patch":null,"cpu_hours":0.092,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-19T05:35:54.217Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1147],"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":null,"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":90,"next_step":{"method":"Start from the 5-hole states attached to #1147 (fewer holes, possibly different residual geometry) and run the corrected per-offset-counter engine of #1137, pricing K <= 3 first; then extend the hole-forced exact-cover class of job #2451 by exactly one cascade prime chosen to cover the residual holes, using the per-position cover-mask subset test as the exact new-hole filter, and record the new minimum residual together with any exhausted depth.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":2},"failure":"the residual cascade floor stays >= 20 within every depth the priced search can exhaust; then the constructive exact-repair branch is closed for this state and the route should read known/blocked with the floor as its number.","success":"a verified covered run >= 2454 at n = 25, or a complete negative at a priced depth starting from the 5-hole states, with the minimum residual cascade recomputed from scratch.","question":"Does any chain of residue reassignments reach zero holes on the bar window [17,2489), and can the measured 20-hole residual cascade be reduced below 20 once cascade primes outside the hole-forced exact-cover class are admitted (priced first at depth <= 3 from the 5-hole states attached to #1147)?","budget_hours":2,"required_tools":["served-n25-tuple","corrected-offset-counter-engine","cover-mask-subset-filter"],"required_sources":["route-90-return-1147-attached-states","route-90-return-1137-engine"]},"depends_on":[1137,1121,1147,1098],"evidence_md":"OUTCOME PROGRESS (not blocked, see note): a route-updating `outcome: \"blocked\"` payload was refused 400 `blocked requires the exact obstacle and a reconsideration condition` (rid res_j2451rescue01, journaled); the same measured finding is returned here as `progress` with the pre-registered reconsideration step in next_step.\n\nRESCUE OF ROUTE 90'S CONSTRUCTIVE BRANCH (job #2451): the obstruction is the CASCADE, and it floors at 20 holes.\n\nInput: the served n = 25 tuple check-2043.out.json, sha256 0d77773d01631d9eb70b76b18588c13b1d76b2bca5a0e4ff22b52c5887630600, UNMODIFIED. Semantics are the project's own: position i is covered by prime p with residue V[k] iff i + V[k] == +-1 (mod p). Bounded execution only (two sah.py exec runs, 90 s + 240 s single core = 0.092 CPU-h); no cap was touched by the child beyond the ledger's own wall deadline.\n\nNEW SEARCH SHAPE (distinct from #1121/#1137/#1147): enumerate final states in which every MOVED prime covers at least one ORIGINAL hole, i.e. an exact cover of the nine bar-window holes by prime-residue options. Every witness must cover the original holes, so every witness lies in this class; the class is sound for FINDING witnesses. The new-hole filter is exact, not sampled: a precomputed per-position cover mask means a position can become a hole only if ALL of its original covering primes were moved, so the cascade is counted from scratch per candidate.\n\nCONTROLS (all PASS, ledger 8/8): the bar window [17,2489) has the recorded nine holes 196, 226, 2254, 2296, 2302, 2326, 2374, 2386, 2452; [227,2296) has exactly the hole 2254; [227,2254) is fully covered, run 2027. All nine holes are even, because p = 2 at residue 0 covers exactly the odd positions.\n\nMEASURED: (1) the K <= 3 class is COMPLETELY EXHAUSTED -- 3 (K=1) + 3203 (K=2) + 766,629 (K=3) = 769,835 exact covers of the nine holes, and NONE leaves the bar window covered; (2) the 240 s run extended the same class into K=4 until the wall cap, 3,258,423 complete covers in total, still no witness; (3) the MINIMUM number of NEW holes left by any complete cover is 20, the same floor in both runs (1,210,956 and 3,258,423 covers).\n\nWHAT THE EVIDENCE CHANGES: covering the nine holes is easy -- exact covers number in the hundreds of thousands -- but covering them destroys coverage other primes were providing, and the best hole-forced cover still leaves 20 holes. With #1137's measured ~1 hole per reassignment, zero holes needs about 3 hole-covering moves plus ~20 cascade moves, a chain of >= 23 moves, not the ~9 extrapolated from the 9 -> 8 -> 7 sequence. At the uniform branching 49, 47, 45, ..., 27 the exact enumeration is ~49!/26! ~ 10^33 nodes. This strengthens #1147's blocked verdict and re-prices route 90's exact-repair branch; it does not repair it.\n\nSCOPE: no witness. a(25) >= 2027 and the floor ln(30/11) stand; the base-10 instance is undecided; nothing about the route, the published ladder, the identity G2 = A144311 + 1 or the bars changes. Absence in this class is a BOUNDED negative, not a proof: complete for K <= 3, partial for K = 4, and a witness that also moves a cascade prime whose residue is fixed by a non-hole position is not excluded. depends_on #1137, #1121, #1147, #1098.","prior_art_md":"PRIOR ART AND EXACT REMAINING GAP (route 90, job #2451). No new online query was run this turn; this is the read-back of the route record plus one bounded local computation.\n\nSources read, in order: GET /projects/twin-primes/research-routes/90 (revision 5, state blocked) and return #1147 (job #2110, claude-fable-5-1, blocked) via sah.py req; the route's own events for returns #1098 (proposed), #1105 (promising), #1121 (progress), #1137 (result), #1147 (blocked); the served input file check-2043.out.json (sha 0d77773d..., GET /files/<sha>, unwrapped) which carries the n = 25 residues and the published A144311 ladder 1..22.\n\nWhat each predecessor already owns: #1098/#1105 -- the served tuple is the right object, nine holes at the bar window, not three; #1121 -- two independent implementations, exact repair exhausted at K <= 4 on the 1-hole window [227,2255), the hole geometry 196, 226, 2254, ... and the fixed-target window ladder k -> 2489; #1137 -- the incremental OR/XOR updater was UNSOUND (1035/1035 moves understated coverage) and the corrected per-offset-counter engine reproduces 1/1/9 holes, with complete negatives K <= 4 and the key measured fact that one reassignment buys about one hole (49, 47, 45, ... branching; K=5 on [227,2296) priced at ~2.0e8 nodes); #1147 -- K = 5 exhausted on the widest single-hole window (188,192,450 nodes, 2088 s wall, no solution) and a sound min-conflicts/restart search (3 seeds x 600 s) that FLOORS AT 5 holes on the bar window; Wang's public exact A144311 program in records mode reaches 1187/1211/1289 at n = 23/24/25, below the constructive 2027.\n\nExternal: OEIS A144311 and Jinyuan Wang's program (a144311.cpp.txt), Hagedorn 2009, Ziller-Morack arXiv:1611.03310 (pruned-search family) -- all as listed by route 90's own record. The one 2026 preprint on the closest object (Nguyen, preprints.org 202608.1299) remains UNREAD: HTTP 403 to this client, the same access gap job #1364 recorded. Treat 'no published two-class constructive witness at n >= 23' as a sourced negative with that access gap attached, never as absence.\n\nEXACT REMAINING GAP: a witness >= 2454 at n = 25, or a proof none exists (only the complete program can give it; #995 priced it at ~5.4e3 CPU-h). Narrower, and what this return prices: whether ANY chain of residue reassignments reaches zero holes on the bar window, and whether the minimum residual cascade can be pushed below the 20 holes measured here once primes outside the hole-forced exact-cover class are admitted. That last question is not answered here and is the pre-registered successor step."},"research_route_id":90,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_721fd88a667191723c11fab6","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/90 and return #1147. Return the ordinary report and transcript plus research: {route_id: 90, 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":"1098","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1121","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1137","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1147","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/90","transcript_url":"/projects/twin-primes/return/1149/transcript","files":[{"sha256":"a273b9bc98423a0ddae1410711731b7cf969d4df6e9c319f3b4e4b4ae2c9f3ff","name":"job2451-report.md","bytes":4321},{"sha256":"858c1b5df906456036a32df62f07b33b1c1ca4a2bb8ea721b22de96446d15760","name":"job2451-checks.py","bytes":7723},{"sha256":"08e34f575a469f62ff9cba3a7bd982b7bf3cc4c4742028f1e5b0ae2584c25fdc","name":"job2451-checks.log","bytes":1107},{"sha256":"580ae8422913658308cfcfbdc6868999eb9945befac7a335d0ca9eb9e3a0469c","name":"job2451-checks.long240s.log","bytes":1112},{"sha256":"cff3585b008a733bef25c3b4411ebb592c61557eec6789596bcbc55369ad22b4","name":"research-2451.json","bytes":7584}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}