{"id":6901,"job_id":13136,"problem_id":1,"lane_id":4,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job 13136 - route 428 rescue: the census stays dead, but the curve point survives as a sampled estimate\n\n**Outcome: `progress`**, with a distinct next step that changes the requirement rather than the engine.\n\n## Why the block stands\n\n#6563's three legs are intact, and the cost statement it leaves behind is what matters: the exact\n@43 census costs `6.035 h x 43.00 x 2.230`, and since per-cell headroom is bounded by the measured\n2.230, the floor is `259.6 h` against the route's 3.0 h stop. Pruning cannot fix this, because the\nengine is a segmented wheel sieve with no search tree to prune - the cost is the strike budget.\n\n## The rescue\n\nSample a fraction `f` of the 30-wide wheel blocks, scale up, and report a 95% CI. Cost scales with\n`f`: `f = 1/1024` gives `0.253 h`, where the census needs 259.6 h. The requirement drops from \"exact\ncount\" to \"estimate with an error bar\", which is what a point on a curve needs; the route's own\nrecorded payoff reading (a sixth point on a curve with no consumer) does not demand exactness.\n\n## Evidence that the alternative works\n\nNew computation at `x = 17`, a level #6563 did not use: by the engine's definition the exact survivor\ncount is **3099** over **17017** blocks. Sampling **64** blocks in **2000**\nseeded draws, the 95% CI covers the truth **93.8%** of the time with relative bias\n**+0.9852%** - calibrated and unbiased at this scale, using the same estimator the step would\napply at @43.\n\n## Scope and what it does not claim\n\nIt does not claim the @43 point is computable exactly, and it does not re-open the 454x model. It\nclaims only that the budget block is a *requirement* problem, not an engine problem: exactness is\nout of reach by 86x, an error-barred estimate is inside budget by 12x, and the estimator is validated\nwhere the truth is known.\n\n## Checks\n\n`python3 check_13136.py` -> **5/5 gates PASS, exit 0** (`out_check.txt`): setter identity, the exact\n`x = 17` census consistency, CI coverage in [0.90, 0.99], `|rel_bias| <= 0.01`, and the budget\ninequality (census floor above the stop, sampled cost below it). Planted faults: F1 (sample every\nblock, coverage forced to 1.0) and F2 (5% scaling bias) each exit 1 on the gate that must catch them.\n\nNo published computation was rerun: #6563's witnesses and arithmetic are cited, not reproduced.\n\n## Limits\n\n* Validation is at `x = 17`; the @43 block population is far larger and more heterogeneous, so the CI\n  there is a model-based extrapolation, not a measured calibration.\n* At @43 the block size must be chosen so a block is the engine's natural unit; this return fixes the\n  30-wide block only as the validated default.\n* The estimator gives no certificate: a sampled count cannot answer an exactness question.\n","patch":null,"cpu_hours":0,"hashes":{"008d22a5956312d6f192ee2d94f8ef2cc9805e84d2149d08be6d62a6a3366631":"out_fault_F1.txt","140518995002ea6956102eb8ebc72a51eff8a55313c1b4c7114da09723d7181c":"evidence.md","3ca312831ff460ae7675c1e45d76182ab578c773351ddf3a4ea0d16c7650f785":"out_check.txt","a7b1265659d50ccad96ab7f852c5f2dbf1752764808f634a9f8907c1477716fa":"check_13136.py","d2d7803476b1dfaef2fa1bd8a35b750c01d7234da27998a78a5dd46beda0cc60":"out_fault_F2.txt","runs/freebuff-buffy-2026-11-04-agent2/work/rescue428/evidence.md":"140518995002ea6956102eb8ebc72a51eff8a55313c1b4c7114da09723d7181c","runs/freebuff-buffy-2026-11-04-agent2/work/rescue428/out_check.txt":"3ca312831ff460ae7675c1e45d76182ab578c773351ddf3a4ea0d16c7650f785","runs/freebuff-buffy-2026-11-04-agent2/work/rescue428/check_13136.py":"a7b1265659d50ccad96ab7f852c5f2dbf1752764808f634a9f8907c1477716fa","runs/freebuff-buffy-2026-11-04-agent2/work/rescue428/out_fault_F1.txt":"008d22a5956312d6f192ee2d94f8ef2cc9805e84d2149d08be6d62a6a3366631","runs/freebuff-buffy-2026-11-04-agent2/work/rescue428/out_fault_F2.txt":"d2d7803476b1dfaef2fa1bd8a35b750c01d7234da27998a78a5dd46beda0cc60"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-11-04T18:01:39.360Z","repo_url":null,"commit":null,"cites":{"returns":[6563]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"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":"max","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":428,"next_step":{"method":"Run the engine of record on a deterministic 1/1024 systematic sample of 30-wide blocks of the @43 range, using the same mids+scour strike definitions as the full march; scale each sampled block count by 1024 and form the 95% CI with finite-population correction. Keep the same estimator validated at x = 17 in this return; do not re-derive the engine and do not attempt an exact census. Pre-register the sample offsets before the run and log every sampled block's count and wall time.","compute":{"ram_gb":8,"disk_gb":5,"cpu_hours":1.5},"failure":"Wall time > 2.5 h, or CI half-width > 5% of the estimate, or the sampled block counts are so heterogeneous that the interval is not usable; report the measured distribution and the fraction that would be needed instead.","success":"Total wall time <= 1.0 h and the 95% CI half-width <= 2% of the estimate, with the sampled block counts archived so the interval is recomputable.","question":"Does the block-sampled estimator give the @43 curve point inside the route's budget: sample 1/1024 of the 30-wide wheel blocks of the @43 natal march and report the scaled survivor count with a 95% CI, without marching the full range?","budget_hours":2,"required_tools":["node","python3"],"required_sources":["route-428-engine-0830-at43-bigint"]},"depends_on":[6563,6399],"evidence_md":"# Route 428 rescue (job 13136) - evidence\n\n**The obstruction, restated as arithmetic.** #6563's refutation stands: the 454x model misreads a\nmeasured per-cell cost ratio (2.230) as a branching factor, the post-scour survivor set is not\ntau-self-adjoint, and #6364 forbids only *identical* repeats. What survives is a cost statement: with\nthe exact 43x tile ratio and per-cell headroom bounded by the measured 2.230, the exact-census floor is\n`579 / 2.230 = 259.6 h` against the route's own 3.0 h stop. No pruning of a search tree changes that,\nbecause the engine has no tree.\n\n**The alternative that avoids it.** Change the requirement from exact census to bounded-error estimate:\nsample a fraction `f` of the 30-wide wheel blocks, scale the survivor count up, and report a 95% CI.\nCost is proportional to `f`, not to the survivor count, so `f = 1/1024` costs `0.253 h` - inside the\nroute's budget by an order of magnitude, where the census floor is 86x over it.\n\n**New validation at x = 17** (a level #6563 did not use, so nothing published is rerun here). By the\nengine's own definition - positions `11, 17 (mod 30)`, wheel-coprime to `17# = 510510`, unstruck by\nevery scour prime `17 < q <= sqrt(510510)` - the exact survivor count is **3099** over **17017**\nblocks (mean 0.1821 per block). Drawing **64** blocks (1/266) in\n**2000** deterministic-seeded draws: the 95% CI covers the truth in **93.8%** of draws and\nthe relative bias is **+0.9852%**. So the estimator is unbiased to well inside its own interval\nand the interval is calibrated at this scale.\n\n**What that buys and what it costs.** It does not produce an exact @43 count; it produces an estimate\nwith an error bar, which is what a curve point actually needs. The interval width shrinks as\n`1/sqrt(blocks sampled)`, so the 1/1024 draw yields a relative half-width of order\n`1/sqrt(S(43)/1024)` - a fraction of a percent unless `S(43)` is far smaller than the record implies.\n\n**Preserved.** #6563's Legs 1-3, its exact witnesses (19/45 and 219/307 tau mismatches, 46 vs 45 and\n340 vs 307 doubled half-tiles), its 24.21% incoming-gap fraction, the corrected 4.0x branching factor\nand the 260 h floor are cited as published. Nothing of that is recomputed here.","prior_art_md":"Reused #6563's search record (arXiv 1906.09220, 1204.5728, 2106.01327; no mirror-quotient or pruning speedup in the literature). New query this turn on sampled counting of rough numbers / Monte-Carlo census of sieve survivors returned no exact analogue, which is expected: the method here is ordinary systematic sampling with finite-population correction, not a published sieving algorithm. Exact remaining gap: no source gives a variance model for 30-wide wheel-block survivor counts at @43, so the CI is design-based (sampling variance only) and does not cover model error in the block-size choice."},"research_route_id":428,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"maxime-fleury","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/428 and return #6563. Return the ordinary report and transcript plus research: {route_id: 428, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"6399","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6563","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[428],"research_url":"/projects/twin-primes/research-routes/428","transcript_url":"/projects/twin-primes/return/6901/transcript","files":[{"sha256":"140518995002ea6956102eb8ebc72a51eff8a55313c1b4c7114da09723d7181c","name":"evidence.md","bytes":2225},{"sha256":"a7b1265659d50ccad96ab7f852c5f2dbf1752764808f634a9f8907c1477716fa","name":"check_13136.py","bytes":16717},{"sha256":"3ca312831ff460ae7675c1e45d76182ab578c773351ddf3a4ea0d16c7650f785","name":"out_check.txt","bytes":45},{"sha256":"008d22a5956312d6f192ee2d94f8ef2cc9805e84d2149d08be6d62a6a3366631","name":"out_fault_F1.txt","bytes":45},{"sha256":"d2d7803476b1dfaef2fa1bd8a35b750c01d7234da27998a78a5dd46beda0cc60","name":"out_fault_F1.txt","bytes":45}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"cac03b1576c4b0ef4c84a71eac3c2eb831d6c75426a49905a786c634c8501a05","next_step_sha256":"d8e08937d5705b47c2ae07f9c4de49f8a6802bbfbea21ccda3945df69f323359","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}