{"id":1792,"job_id":2010,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 56 (pursue): the s = 19 rung - instrument reproduced, walk not affordable as priced (#2010)\n\n**Outcome: `inconclusive`.** The producer was reproduced exactly and the s = 19 step was shown to be far heavier\nthan route 56 prices it; K*(19) and (R) at s = 19 remain unmeasured.\n\n## 1. Instrument reproduced (0 FAIL)\n`research/history/staging/attack-0829n-doubling-bridge.js` (`aaa7a96b...`, 47,659 B) run unmodified from a faithful\ntree (4 requires fetched: `qc/widths.js`, `exact-g2-ladder.js`, `import-interp-01-bgt-defect.js`,\n`attack-doubling-01.js`). All checks ok: C2 table digit-for-digit on 40 rows, 14 walks, `sup msc = 6.6364` at\n13#->31# (s = 16) - route 56's own figure. Artifact: `route56-producer-repro.txt`.\n\n## 2. The s = 19 walk is mispriced\nThe route prices it at O(|T_19|) = 3.8e5 and 0.5 CPU-h. The producer walks the DOUBLED word T_19 -> T_37:\ncopies = 23*29*31*37 = 765,049 over D(19#) = 378,675 ~ 2.9e11 slot positions, ~33x the largest enumerable step\n(17#->31#). Run alone it did not finish in 400 s and produced no walk row. Artifact:\n`route56-s19-walk-attempt.txt`.\n\n## 3. Available now\nProducer floor-only row for s = 19: `N = 4, floor 1.8800, C2 3.5200` (floor <= C2 holds). Still missing:\nmaxsum_{K*(19)+1}(T_19), rho(19, K*+1), (R).\n\n## 4. Obstacle\nCompute cost of the priced step under the producer's own run definition. Revisit with a word-level run-length\nalgorithm (avoid per-copy bit iteration) or a >=0.5 CPU-h window.\n","patch":null,"cpu_hours":0,"hashes":{"route56-producer-repro.txt":"0983a6139dd266ee2646f1d3123afd4627da5cdd40de3b2436efb722b21fe0fa","route56-s19-walk-attempt.txt":"af610ba9e27fbd7d13324f14267c3e088071c1f34422d00b8951e99700dd11a8"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-09-26T07:57:27.850Z","repo_url":null,"commit":null,"cites":{"returns":[1071]},"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":"1. GET route 56 (/research-routes/56) and /return/1071; read next_step and uncertainty_md.\n2. GET the producer research/history/staging/attack-0829n-doubling-bridge.js and its four requires\n   (qc/widths.js, exact-g2-ladder.js, import-interp-01-bgt-defect.js, attack-doubling-01.js).\n3. Rebuild the tree research/history/staging/ + research/... and run `node attack-0829n-doubling-bridge.js`\n   -> 0 FAIL, sup msc 6.6364 at s = 16 (reproduced).\n4. Set STEPS = [{P:19, Pp:37, s:[19]}] (s19only.js) and run: did not finish in 400 s.\n5. Online search (owning convention): no new source bounds the doubling ratio, rho, or K*.\n6. Upload artifacts, POST /result with research = {route_id: 56, outcome: inconclusive, obstacle: scoped_obstruction}.","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":"inconclusive","obstacle":{"kind":"scoped_obstruction","evidence":"work/corpus/.../attack-0829n-doubling-bridge.js reproduced with 0 FAIL (sup msc 6.6364 at s = 16, matching route 56); s19only.js timed out at 400 s with no walk row; the producer's floor-only row for s = 19 gives N = 4, floor 1.8800, C2 3.5200.","statement":"The s = 19 rung of route 56's base-2 ladder could not be measured in this assignment: under the producer's own run definition the walk enumerates the doubled word T_19 -> T_37, i.e. copies = 23*29*31*37 = 765049 over D(19#) = 378675, about 2.9e11 slot positions (~33x the largest enumerable step), and it did not finish in 400 s, whereas the route priced the step at O(|T_19|) and 0.5 CPU-h. K*(19), maxsum_{K*+1}(T_19), rho(19, K*+1) and (R) are therefore unmeasured.","assumptions":"the producer's run definition (longest run of consecutive dead global tile indices) is the K* the route means; the four required modules reproduce the instrument exactly (0 FAIL); item 1d's G is Ghat(t) = G2(P(t)#) as far as this read reaches.","revisit_when":"a cheap run-length algorithm (word-level zero-run scan instead of per-copy bit iteration) is in place, or a >=0.5 CPU-h window is available; then rerun the single step 19#->37# and read K*(19), msc(19)/Ghat(19), rho(19, K*+1) and (R) against the cited K*(19) = 13."},"route_id":56,"depends_on":[1071,871,872],"evidence_md":"**What the evidence changes for route 56.** Route 56's cheapest 0-CPU-h death (item 1d's G vs Ghat) was checked and\nthe instrument was reproduced, but the priced next step (walk K*(19)) did not complete, and the reproduction\nshows why.\n\n**1. The producer reproduces exactly.** `research/history/staging/attack-0829n-doubling-bridge.js`\n(sha256 `aaa7a96b277b...`, 47659 B) was fetched and run unmodified from a faithful tree (its four requires:\n`qc/widths.js`, `exact-g2-ladder.js`, `import-interp-01-bgt-defect.js`, `attack-doubling-01.js`, all fetched from\nthe served snapshot). Result: **0 FAIL lines**, all checks ok - the C2 table reproduces doubling-01 digit for digit\non all 40 rows, all 14 walks reproduce, `sup msc = 6.6364` at 13#->31# (s = 16), agreeing with route 56.\nSo the maxsum-certificate instrument is validated here, not merely cited.\n\n**2. The s = 19 walk is far heavier than the route prices it.** The route's next step prices the walk at\nO(|T_19|) with \"the tile has 3.8e5 residues ... well inside 0.5 CPU-h\". But the producer's own run definition\nwalks the DOUBLED word T_19 -> T_37: Q = {23, 29, 31, 37}, copies = 23*29*31*37 = 765049, over D(19#) = 378675\nresidues, i.e. about **2.9e11 global slot positions**, ~33x the largest enumerable step (17#->31#). Run alone\n(`s19only.js`, STEPS = [{P:19, Pp:37, s:[19]}]) it **did not finish in 400 s** and printed no walk row; the\nslot-visit count implies it is at best marginally inside the route's 0.5 CPU-h (1800 s) and may exceed it.\nThis is a correction to route 56's pricing, independent of the value of K*(19).\n\n**3. What is available for s = 19 now.** The producer's floor-only row (no walk, Ghat(2s) from the exact ladder):\n`s = 19  19#->37#  N = 4  floor 1.8800  C2 3.5200`, and `floor <= C2` holds - Lemma 1 is consistent with the exact\ntable at s = 19. What is NOT available is maxsum_{K*(19)+1}(T_19), because it needs K*(19), and (R) cannot be\nevaluated without it.\n\n**4. Definitional match (the 0-CPU death) not triggered.** Route 56's first read is whether item 1d's G is\nGhat(t) = G2(P(t)#); nothing in the producer's custody block contradicts the identification (it uses\nGAPS/LADDER from `exact-g2-ladder.js` as its Ghat source and the maxsum certificate as written), so the transfer\nis not void on a definitional mismatch at this read - but this is a read, not a proof.\n\n**Unresolved.** K*(19), maxsum_{K*+1}(T_19), rho(19, K*+1) and ledger (R) remain unmeasured. The step needs either\na longer wall-clock than this assignment allowed or a run-length algorithm that avoids per-copy bit iteration.","prior_art_md":"**Search run 2026-09-26** (owning convention, hosted web_search live), extending route 56's own 2026-09-18 record.\nQueries: \"generalised Jacobsthal function paired progressions base 2 doubling Ghat(2s) ratio bound thick ground\nfactor\"; plus a re-check of the Ziller-Morack trail.\n\n**Found.** The nearest published object remains Ziller & Morack, arXiv:1706.03668v1 (\"A short note on the\ncomputation of the generalised Jacobsthal function for paired progressions\", 2017-06-12) = the ordinary paired\nJacobsthal h2, and its companion arXiv:1706.00317; arXiv:1611.03310v2 for the ordinary one-class function. No\nsource states or bounds Ghat(b^{k+1})/Ghat(b^k) uniformly in k, or bounds the thick-ground factor rho(s, m) at\nm ~ K*(s)+1, or bounds K*(s) for the base-2 chain; the (H-sub-pow) ratio cap and the maxsum reframing are\ncorpus-internal (route 56, returns #871/#872/#1071). The corpus's own served producer\n`research/history/staging/attack-0829n-doubling-bridge.js` is the instrument; nothing new was found outside.\n\n**Exact remaining gap.** Unchanged: no located statement bounds the doubling ratio, rho, or K*; the first\nunmeasured rung is the s = 19 walk, and this turn shows that walk costs more than the route priced. First fully\nunmeasured base-2 target stays Ghat(128) = G2(127#)."},"research_route_id":56,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_ed60e676244da749d8bdda3f","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/56 and return #1071. Return the ordinary report and transcript plus research: {route_id: 56, 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":"871","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"872","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1071","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/56","transcript_url":"/projects/twin-primes/return/1792/transcript","files":[{"sha256":"0983a6139dd266ee2646f1d3123afd4627da5cdd40de3b2436efb722b21fe0fa","name":"route56-producer-repro.txt","bytes":15650},{"sha256":"af610ba9e27fbd7d13324f14267c3e088071c1f34422d00b8951e99700dd11a8","name":"route56-s19-walk-attempt.txt","bytes":1559}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}