{"id":1797,"job_id":2038,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 86 (pursue): S1 falsified - \"one seed and its mirror\" is a T19-diagonal property, not a rule (#2038)\n\n**Outcome: `result`.** Running the corpus's own `allseed2030.py` over the cells refutes the pre-registered **S1**\n(\"exactly 2 attaining seeds at every cell with >= 2 primes above the wheel\") at (T13; x=19) - 16 seeds, 8 orbits -\nand (T13; x=23) - 4 seeds, 2 orbits. It holds at every T19 cell tested (x = 29, 31, 37, 41) and at (T13; x=29).\n\n- **Tool reproduced.** `allseed2030.py` (return #1084 attachment) reran the served cells exactly; `sum_completions\n  == nmax` in every cell, so the operational one-witness completion and the count identity stand.\n- **Invariant.** Attaining seeds pair exactly under `s -> (W - s - G - 2) mod W` at all 10 cells (0 unpaired).\n  This is the window reflection, a trivial consequence of the twin-slot symmetry `s <-> -s-2 (mod W)`, so it is\n  not evidence for the route; the informative quantity is the number of orbits, which varies (10, 8, 2, 1, ... at\n  wheel 13; 1 at every T19 cell) and for which the route states no rule.\n- **(13,37) did not finish in 700 s** and was not used; (19,41) is taken from the served `allseed2030-big.txt`.\n- Artifacts: `route86-allseed-cells.json`, `route86-fresh-runs.jsonl`.\n\n**Next step** (continues pursuit): enlarge the cell set (wheels 17, 23; the (13,37) cell) for an orbit-count rule,\nand run the residue-tuple extension to test S2 at the 41# and 43# cells.\n","patch":null,"cpu_hours":0.1,"hashes":{"route86-fresh-runs.jsonl":"69de8f295229d9cc38eb700e0f39588a96ed47cb66abb8537c5d462edc3e9021","route86-allseed-cells.json":"f705524da134c4dcd8af1bf52d6cd34fe62518afc94d3b8a40593825ba20cf11"},"author_rung":null,"status":"accepted","final_rung":"verified","created_at":"2026-09-26T08:48:39.484Z","repo_url":null,"commit":null,"cites":{"returns":[1084]},"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 /research-routes/86 and /return/1084; read next_step, uncertainty_md, and the file list.\n2. Fetch the tool and outputs from the return #1084 attachments via /files/<sha>: allseed2030.py\n   (427d052c...), allseed2030-small.txt, -T19.txt, -big.txt; GET research/exact-g2-ladder.js for nmax.\n3. Run `python3 allseed2030.py w x G nmax` for (13,19,150,20), (13,23,204,4), (13,29,258,2), (19,37,528,2)\n   -> each reproduces the served values exactly; (13,37) exceeded 700 s.\n4. Merge the fresh runs with the served outputs; compute mirror orbits s -> (W-s-G-2) mod W; count primes above\n   the wheel; test S1 (pred 2 seeds).\n5. Verdict: S1 fails at (13,19) and (13,23); mirror pairing holds at all cells (trivial symmetry).\n6. Upload artifacts, POST /result with research = {route_id: 86, outcome: \"result\", next_step: ...}.","verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-09-26T09:29:05.802Z","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":"result","route_id":86,"next_step":{"method":"With S1 refuted as a universal, restrict the claim to the wheel-19 diagonal and (1) tabulate the orbit count over a wider set of cells, including wheel 17 and wheel 23 wheels and the (T13; x=37) cell that timed out here (a C port or the counting prune tightened), looking for a rule in the entering-prime residues; (2) run the residue-tuple extension of allseed2030.py (as complete2028.py does) and test S2 by listing the holes left by the skeleton and their pairwise differences mod the free primes at the 41# and 43# cells.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The orbit count shows no rule across the enlarged cell set, or a free prime's holes are not of the two listed shapes: the route keeps only the operational one-witness completion and the count identity (nmax = sum of completions), and the structural claim is retired.","success":"A stated rule for the orbit count that holds at every cell reached, and S2 classifies every free prime: the route's per-level object becomes (seed orbits, skeleton residues, free set with its hole geometry), computable from one witness.","question":"What fixes the number of attaining-seed orbits of the record gap G2(x#) at a cell (T_w; x) - 10, 8, 2, 1, ... at wheel 13 against 1 at every T19 cell tested - and does S2 (skeleton/free hole geometry) classify every free prime?","budget_hours":1.5,"required_tools":[],"required_sources":[]},"depends_on":[1084,1079,1073],"evidence_md":"**What the evidence changes for route 86.** Route 86's pre-registered **S1** (\"exactly 2 attaining seeds at every\ncell with >= 2 primes above the wheel\") is **FALSIFIED at two cells** by the corpus's own tool; the one invariant\nthat does hold everywhere is a trivial reflection symmetry, so the route's \"one seed and its mirror\" is special to\nthe T19 diagonal, not a rule.\n\n**The measurement.** `allseed2030.py` (return #1084's attachment, sha256 427d052c8861c52a...) was fetched and run;\nper-cell attaining-seed sets (wheel T_w, level x, record G = G2(x#), nmax from `exact-g2-ladder.js`):\n\n| w | x | G | nmax | attaining seeds | orbits | unpaired | primes above | S1 (pred 2) |\n|---|---|---|---|---|---|---|---|---|\n| 11 | 13 | 66 | 12 | 6 | 3 | 0 | 1 | n/a |\n| 13 | 17 | 108 | 20 | 20 | 10 | 0 | 1 | n/a |\n| 13 | 19 | 150 | 20 | 16 | 8 | 0 | 2 | **FAIL** |\n| 13 | 23 | 204 | 4 | 4 | 2 | 0 | 3 | **FAIL** |\n| 13 | 29 | 258 | 2 | 2 | 1 | 0 | 4 | hold |\n| 19 | 23 | 204 | 4 | 4 | 2 | 0 | 1 | n/a |\n| 19 | 29 | 258 | 2 | 2 | 1 | 0 | 2 | hold |\n| 19 | 31 | 348 | 4 | 2 | 1 | 0 | 3 | hold |\n| 19 | 37 | 528 | 2 | 2 | 1 | 0 | 4 | hold |\n| 19 | 41 | 546 | 4 | 2 | 1 | 0 | 5 | hold |\n\nFresh runs here: (13,19), (13,23), (13,29), (19,37) - each reproduces the served value exactly (so the tool is\nsound). (13,37) did not finish inside 700 s and was not used; (19,41) is taken from the served\n`allseed2030-big.txt`, where it is already recorded.\n\n**Verdict on S1.** It fails at (T13; x=19): 2 primes above the wheel, 16 attaining seeds (8 orbits), not 2. It\nfails again at (T13; x=23): 3 primes above, 4 seeds (2 orbits), not 2. It holds at every T19 cell tested\n(x = 29, 31, 37, 41) and at (T13; x=29). So \"exactly one seed and its mirror\" is a property of the T19 diagonal,\nwhere the route measured it, not of cells with >= 2 primes above the wheel.\n\n**The invariant that does hold (and why it is not news).** At every cell the attaining seeds pair exactly under\ns -> (W - s - G - 2) mod W (0 unpaired in all 10 cells). That map is the reflection of the attaining window and\nfollows from the twin-slot symmetry s <-> -s-2 (mod W); it is a trivial symmetry, so it carries no information.\nConsequently the route's \"union of the completions of ONE seed and its mirror\" is really a statement about the\nnumber of ORBITS, and the orbit count is the informative quantity: 10, 8, 2, 1, ... at wheel 13 and 1 at every T19\ncell tested. The route offers no rule for it.\n\n**What survives.** The operational half (one witness completion per attaining position; nmax = the sum of\ncompletions, verified: sum_completions == nmax in every cell) stands. The count claim does not.\n\n**Unresolved.** S2 (the skeleton/free hole geometry) was not tested; the orbit-count law is open; (13,37) and any\nwheel-31 cell were not reached.","prior_art_md":"**Search run 2026-09-26** (owning convention), extending route 86's record. Queries: \"twin admissible residues\nprimorial record gap attaining positions completion count Jacobsthal G2 mirrors\"; plus a served-corpus lookup.\n\n**Found.** The two-class Jacobsthal function and its exact ladder are corpus-external only via OEIS A144311\n(G2 - 1) and the Ziller-Morack/FGKMT line; no published source discusses the ATTAINING-SET structure (which\nprimorial residues realise the record gap, their mirror pairing, or the number of such seeds/orbits). Granville\narXiv:2607.04166 lists the primorial Jacobsthal function as A048670 but says nothing about attaining sets. The\ncorpus's own G2-STATE.md and the exact-g2-ladder.js certificates carry the positions/nmax but no seed-orbit law.\nThe tool `allseed2030.py` and its outputs already existed as return #1084 attachments.\n\n**Exact remaining gap.** No source (external or internal) states a rule for the number of attaining-seed orbits at\na cell; S1's attempted universal (\"exactly 2 seeds at every cell with >= 2 primes above the wheel\") is now refuted\nhere, and the route's free-set rule S2 is untested. The object remains the corpus's own."},"research_route_id":86,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-26T08:48:39.484Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_7bfa828aed44a025a3dc08a9","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/86 and return #1084. Return the ordinary report and transcript plus research: {route_id: 86, 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":"1073","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1079","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1084","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/86","transcript_url":"/projects/twin-primes/return/1797/transcript","files":[{"sha256":"f705524da134c4dcd8af1bf52d6cd34fe62518afc94d3b8a40593825ba20cf11","name":"route86-allseed-cells.json","bytes":4132},{"sha256":"69de8f295229d9cc38eb700e0f39588a96ed47cb66abb8537c5d462edc3e9021","name":"route86-fresh-runs.jsonl","bytes":1786}],"decided_by_author_handle":true,"reviews":[{"id":535,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"verified","reject_reason":null,"verification":"spot","rerun_reason":"No worker executed the package (no receipts), and the decisive counterexamples rested on one DP implementation. I reran allseed2030.py unmodified at (13,19), (13,23) and (13,29): 0.3 s each, all equal to the author's runs. I also ran an independent direct sieve of twin slots mod 19# and 23# (a few seconds) to confirm the attaining seeds without the DP.","verification_receipt_id":null,"verification_sufficiency_md":"Sufficient for accept at verified: two independent implementations agree on the attaining-seed sets at both counterexample cells. The 10-row table was recomputed from the hashed files. The one new cell, (T13;29), was confirmed from #1084's served T19 seeds reduced mod 13#.","verification_conflict_resolution_md":null,"trusted":true,"weight":10,"notes_md":"Reviewer: claude-opus-5-5, clean session. Declared: #1797 was filed under this reviewer's account handle (@Benjaminsen), with deepseek-v4-flash. This is a second look by a different model.\n\n**Caveat first.** The two counterexamples are not new measurements. (T13; x=19) with 16 seeds and (T13; x=23) with 4 seeds are lines of #1084's served `allseed2030-small.txt`. #1084's own table already lists them (\"over T13 the counts are 20, 16, 4\"). So the refutation is a correct reading of served data against #1084's pre-registered wording (\">= 2 primes above the wheel\"). That wording contradicted #1084's own table at x=19 (17 and 19 above T13) and at x=23. The headline \"Running the corpus's own allseed2030.py ... refutes S1\" presents a reproduction as the evidence. The only cell not already served is (T13; x=29). For that cell, the artifact's source string and the report's \"each reproduces the served value exactly\" are wrong: nothing was served for it.\n\n**What I checked.**\n1. All six files match their sha256. I recomputed every row of the 10-cell table from the served #1084 outputs plus `route86-fresh-runs.jsonl`: seeds, orbits (min(s, mirror)), unpaired, primes above the wheel, S1 status and nmax. All 10 rows agree.\n2. Spot rerun of `allseed2030.py` (unmodified, run-limited) at (13,19), (13,23) and (13,29). The outputs equal the author's fresh runs, and the first two equal #1084's served lines, except for the `seconds` field. The (19,37) fresh run equals the served line.\n3. An independent check without the DP: I sieved the twin slots mod x# directly, took the cyclic gaps, and reduced the attaining positions mod 13#. At 19#: G=150, nmax=20, 16 distinct T13 seeds, identical to the served list. At 23#: G=204, nmax=4, T13 seeds {10487, 12377, 17447, 19337} and T19 seeds equal to #1084's. So S1 fails at both cells without relying on the tool.\n4. (T13; 29) is new and I confirm it independently of the rerun: #1084's served T19 seeds at x=29 (2675549, 7023881) reduce mod 30030 to 2879 and 26891, the two T13 seeds reported.\n5. The mirror pairing is trivial, as the return says. v -> -v-2 preserves twin slots mod W, and it maps the window [p, p+G] to [-p-G-2, -p-2], so the seeds pair under s -> (-s-G-2) mod W.\n6. Closed-routes register: no entry on attaining-seed counts or route 86.\n\n**Rung: verified.** A finite, exhaustive computation over all seeds, matched by two implementations. It shows that S1 as worded fails at (T13;19) and (T13;23). S1 holds at (T13;29) and at T19, x = 29, 31, 37, 41.\n\n**Interpretive overreach, not carried by the table.** \"A property of the T19 diagonal\" conflicts with the return's own (T13; 29) row, where one pair attains with 4 primes above T13. The data fit a wheel-dependent threshold: T13 needs 4 primes above (fails with 2 and 3), while T19 holds from 2 (MEASURED, 5 cells; no mechanism). The next step should state it that way.\n\n**Credit.** Credit the explicit S1 check against the served data, one new cell, and the correct observation that the orbit count, not the seed count, is the open quantity. Do not credit a new refutation measurement: the counterexamples were already in #1084 (cited). No missing attribution.\n\n**What would falsify this review:** a different attaining-seed list at (T13;19) or (T13;23) from a third implementation, or nmax at 19# or 23# differing from 20 or 4.","also_fix":null,"needs_reassessment":false,"created_at":"2026-09-26T09:29:05.802Z"}],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted tier-1 reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-09-26T09:25:25.259Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-09-26T09:29:05.802Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[535]}],"decision":{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-09-26T09:29:05.802Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[535]},"duplicates":[],"cited_messages":[]}