{"id":1116,"job_id":2081,"problem_id":1,"lane_id":null,"type":"explore","user_id":34,"model":"claude-fable-5.1","provider":"anthropic","report_md":"# Triage of route 92 (19#->43#: certificate unreachable in the inclusive engine): promising, with the cheaper engine already in the literature\n\nRung per claim: the prior-art statements below are **verified** against the cited sources (read this turn); the investment judgement is **heuristic**. No new computation was run (cpu_hours 0); published numbers are cited, not reproduced.\n\n## What the evidence changes\n\n1. **The route's \"new algorithmic fact\" is prior art, and it is exactly the cheaper engine the route asks for.** Route 92 says pruning an alternating sum over subsets (terms with a factor `q - nu_q(J) = 0` vanish) is \"standard\" and that \"the next investment must be a cheaper engine (pruned subset DFS, or bit-parallel)\". Both halves exist in print for the one-class Jacobsthal object on primorials: Hagedorn, *Computation of Jacobsthal's function h(n) for n<50*, Math. Comp. 78 (2009) -- pruned backtracking over residue classes, exact to h(49)=742; Ziller-Morack, arXiv:1611.03310 -- sequential and greedy-permutation algorithms with structural symmetries, exact primorial values far beyond 43#; Ziller, arXiv:1903.11973 -- exact maxima for k-prime products to k=43, counterexamples to Jacobsthal's conjecture; Costello-Watts, arXiv:1208.5342 -- an inclusion-exclusion recursion over prime-pair divisors with vanishing terms dropped, giving strict upper bounds to k=10000. A dedicated 64-bit \"word sieve\" for covering runs was **not found** as a named primitive (bitsets are implementation detail in the above).\n2. **The exact remaining gap is narrower than the route states, and it is the project's object, not the method.** None of these sources computes the project's two-classes-per-prime `K*` or the pre-registered alternating-sum identity of attack-kstar-01; Ziller's exact ladder is the ONE-class `h(p_n#)`. So `[ABSENT]` stands for the object, while the route's recorded claim that the pruning idea is \"what is NEW here\" should be read as \"new to this lane\", not new.\n3. **Mapping the borrowed method onto K*.** Hagedorn/Ziller backtracking assumes a feasibility predicate that is monotone under extension (a partial arrangement that already fails cannot be rescued). K* is defined by the first k with `N_k = 0`, and return #1110 measured `N_k` **non-increasing** in k and proved `K* >= 16` by monotonicity in the entering set -- exactly the monotonicity a pruned search needs. The assumption that does NOT transfer: the one-class algorithms kill one residue per prime; the two-class convention kills two, so the branching factor and the symmetry reductions of Ziller-Morack (which rely on the one-class residue structure) must be re-derived, not copied.\n4. **Weakest assumption inspected.** The obstruction \"~932 h\" rests on a log-quadratic extrapolation calibrated on three steps with worst bias +1.87; the PROVEN part is only `K* >= 16` (10.6 h at k=16 in the current engine). The route's own honest range [16,24] means the cost obstruction is between \"one long run\" and \"infeasible\" -- which is why the next step must be a validation at a KNOWN answer, not another extrapolation.\n\n## Decision\n\n**Promising.** One bounded experiment is justified and it is the route's own: run the 64-bit engine `kstar-1943.py` (return #1110, sha dc1a15e4...) at 19#->37# where the served period-free `K* = 13` is on record, check the k=1 closed form first, ~1.6 h single-process at the measured 1.18 ms/shape. Success cross-validates the only instrument the lane has at base 19; failure localises a representation defect BEFORE anyone spends 10-900 h at 19#->43#. Recommended in the same step: a pruned two-class backtracking in the Hagedorn/Ziller style, validated on the same 25 committed cells, as the \"cheaper engine\" the route's `revisit_when` names -- its assumptions are mapped in (3).\n\nDepends on #1110 (the engine, the 25-cell regression and the K*>=16 argument), which is pending review; this triage does not elevate it.\n\nTranscript: harness messageHistory export from the Freebuff CLI (agent-written format); removed: credential values and fragments, non-project absolute paths, session identifiers. Usage: this application version records no per-turn token counts, so none is claimed. 96 of this handle's returns wait for a verdict.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-19T00:25:45.089Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1110,1100],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"claude-fable-5.1":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["claude-fable-5.1"]},"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":"promising","route_id":92,"next_step":{"method":"Run kstar-1943.py (return #1110, sha dc1a15e4...) at P=19, P'=37, Q=23,29,31,37, D=378675, kmax=14 in resumable --lo/--hi shape chunks under a job-object time/memory limit; check N_1 against the closed form first; compare the first zero with K*=13. In parallel implement a two-class pruned DFS with the monotone feasibility predicate N_k non-increasing, validate it on the 25 committed cells of #1110 before any new step.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"N_1 misses the closed form at P'=37 (representation defect in the 64-bit code), or the first zero differs from 13 (a finding about the served record, localised to one step), or the DFS disagrees on a committed cell.","success":"K*=13 reproduced with N_1 exact; the pruned DFS matches all 25 cells -- the lane has two instruments at base 19 and a price for 19#->43# measured, not extrapolated.","question":"Does the independent 64-bit engine reproduce the served period-free certificate K* = 13 at 19#->37#, with the k=1 closed form exact, and can a two-class pruned backtracking (Hagedorn/Ziller style) reproduce the same 25 committed cells?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[1110],"evidence_md":"Prior-art check this turn: the pruned-subset / backtracking engine that route 92 names as the next investment already exists for the one-class Jacobsthal object on primorials (Hagedorn 2009 Math. Comp. 78; Ziller-Morack arXiv:1611.03310; Ziller arXiv:1903.11973 to 43 primes; Costello-Watts arXiv:1208.5342 inclusion-exclusion with vanishing terms dropped). Its feasibility-monotonicity assumption transfers to K* because #1110 measured N_k non-increasing and proved K*>=16 by monotonicity in the entering set; its one-class symmetry reductions do NOT transfer (two residues killed per prime). The obstruction (~932 h) is an extrapolation with worst bias +1.87; only K*>=16 (10.6 h) is proven. Hence the cheapest justified step is the route's own validation at a known answer (19#->37#, K*=13), not a rerun at 19#->43#.","prior_art_md":"Search re-run 2026-09-19 (web, 1 query set: Jacobsthal computation pruned search / inclusion-exclusion / bit-parallel). Sources read: https://www.semanticscholar.org/paper/7edfd307c62f7784ffe572d861697537ac344079 (Hagedorn 2009: pruned backtracking, exact h(n) for n<50, h(49)=742); https://arxiv.org/abs/1611.03310 (Ziller-Morack: sequential + greedy-permutation algorithms, structural symmetries of maximal covering sequences, exact primorial values); https://arxiv.org/abs/1903.11973 (Ziller: exact maxima for k-prime products to k=43); https://arxiv.org/abs/2007.01808 (Ziller: gaps between consecutive coprimes to primorials, exhaustive to k=44); https://arxiv.org/abs/1208.5342 (Costello-Watts: alternating subset sums over prime-pair divisors, vanishing terms pruned, bounds to k=10000). NOT FOUND: a named 64-bit word-sieve primitive for covering runs; any computation of the project's two-classes-per-prime K* or of the attack-kstar-01 alternating-sum identity. Exact remaining gap: the two-class K* at 19#->37# and beyond has a single instrument (kstar-1943.py) validated on 25/43 committed cells and no published value to compare against except the served period-free K*(19#->37#)=13 and K*(19#->41#)=16."},"research_route_id":92,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_eea4d9d2f9904e6b441503c2","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/92 and return #1110. Return the ordinary report and transcript plus research: {route_id: 92, 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":"1110","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/92","transcript_url":"/projects/twin-primes/return/1116/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}