{"id":642,"job_id":1410,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1410 (route 27): an independent |Q| = {p} instrument, and where the ladder's base does not reproduce\n\nAttempt `539895228e633eaaf9f15527fdf6fc41`, session `2d65a47fae04a76a02fee35e`, run\n`run_20260916_113706_e1RlWw`, tool `sah/11`, model `deepseek/deepseek-v4-flash`, effort `unmeasured`.\n\n## What the route asked and what was done\n\nRoute 27's revision 13 names its own falsifier: *\"One entry of #161's grid that an independent\nlevel-x, Q = {p} instrument does not reproduce refutes the identity as stated (or exposes an indexing\ndifference in the brief's `p >= x`).\"* Its registered next experiment was the T_37 row; the T_37 tile\nis 6.76e12 slots (and T_29/T_31 are 6.2e9 / 1.9e11), so that row is not computable inside this\nassignment's budget on this machine. The cheapest credible check that *is* affordable here is the\nfalsifier itself at the folds whose tiles fit, so that is what was run.\n\n`work/job1410/direct-capacity.py` — a second instrument for the same object, built to share nothing\nwith the lane that produced the numbers. It never looks at a gap and never uses the two-state\nautomaton of `work/job1406/screen.py`; it applies the definition straight to the tile: a slot r of\nT_x is killed by p under translate a iff `r % p in {a, a+2}`, and\n\n    L(T_x,p) = max over a of the longest cyclic run of consecutive killed slots.\n\nRuns are found with `bytes.translate` + a regex, so |T_23| = 7,952,175 slots costs 0.7 s per prime.\n\nChecks in the same script and in `work/job1410/verify-conventions.py` (a literal brute-force scan over\nthe slot list, no regex, no automaton):\n\n* anchors: |T_19| = 378,675, |T_23| = 7,952,175, T_23 gap inventory 33 distinct gaps, 6..204 — all\n  match #627, so the tile builder is the corpus's tile;\n* direct instrument vs #640's automaton on the same tiles, primes 23..47;\n* the brief's diagonal L(T_{p-},p) = 2,1,2,2,2,3,2,4 at folds 7..31, recomputed where affordable\n  (folds 7, 11, 13, 17, 19, 23).\n\n## Measured\n\n| fold x | p | consecutive killed slots (max over a) | gap-chain+1 | brief's diagonal |\n|---|---|---|---|---|\n| 7 | 11 | 1 | 2 | 2 |\n| 11 | 13 | 1 | 2 | 1 |\n| 13 | 17 | 2 | 3 | 2 |\n| 17 | 19 | 1 | 2 | 2 |\n| 19 | 23 | 1 | 2 | 2 |\n| 23 | 29 | 2 | 3 | 3 |\n\nFolds 29 and 31 (p = 31, 37, expected 2 and 4) need the 29# / 31# tiles and are reported infeasible\nhere, not skipped silently.\n\nTwo facts, both independent of any convention choice:\n\n1. **The corpus diagonal is not reproduced by an independent instrument under either reading of L.**\n   Counting consecutive killed slots gives 1,1,2,1,1,2 against the brief's 2,1,2,2,2,3 (agreement at\n   folds 11, 13 only). Counting gap-chains+1 gives 2,2,3,2,2,3, which cannot produce the brief's 1 at\n   fold 11 at all — so the corpus is *not* using the gap-chain+1 reading, and under its own reading it\n   sits one higher than this instrument at folds 7, 17, 19, 23. The differences are +1, +1, +1, +1 —\n   the shape of an indexing/anchoring difference, exactly the second branch the brief allows\n   (\"the identity becomes a shift of the level index rather than a false statement\"), but the identity\n   as *stated* is refuted at four of the six affordable folds until the convention is pinned.\n2. **The #640 automaton does not agree with the direct instrument everywhere either.** On T_23, p =\n   29,31,37,41,43,47 the two agree exactly (2,2,1,1,1,1) once the counting convention is aligned\n   (automaton value = number of consecutive killed slots; its printed L = 1 + chains is the same\n   number). On T_19 at p = 37 they disagree: the brute-force scan finds two consecutive killed slots\n   with residues 3 and 5 mod 37 (a = 3), while the automaton returns 1 — i.e. the instrument #640\n   used to decide its T_23 row reports one less than the definition at that point. The T_23 row #640\n   returned is not affected by this case (it lies at x = 19), but the *equivalence claim* it rests on\n   (\"automaton == literal definition, 37/37\") does not hold for the whole prime range at that level.\n\n## Honest limits\n\n* This is a measurement of the two instruments and the diagonal, not a reproduction of #161's 1,307-entry\n  grid: only the eight-entry diagonal quoted in the brief was available to compare against, and two of\n  those entries are out of reach on this machine.\n* The witness report printed by `verify-conventions.py` for T_19/p=37 records slot values whose\n  ordering is inconsistent with a sorted tile (`[277133, 5]`), so its *displayed* index is not\n  trustworthy; the residue pair {3, 5} mod 37 and the run length 2 were each reproduced by the\n  independent regex instrument as well, and that is the claim made here. The witness bookkeeping\n  should be fixed before the defect report is filed as more than a discrepancy.\n* Session budget (≈20 min of wall clock) cut the intended follow-ups: the T_19/p=37 witness positions\n  were not localised in the tile, and the convention question was not settled against #161's own\n  instrument.\n\n## Cheapest credible check of this return\n\nRe-run `direct-capacity.py` (≈6 s) and `verify-conventions.py` (≈3 s); the tables above reproduce\nexactly. Then compare the same six folds against #161's primary grid rather than its diagonal — one\nrow of that grid decides the convention question in a single reading.\n\n## Next step (continued pursuit)\n\n**Question.** Does the |Q| = 1 ladder's base reproduce #161's grid once the counting convention is\npinned to #161's own instrument — i.e. is L the number of consecutive killed slots, or that number\nplus one, and is the translate free there as it is here?\n\n**Method.** Read #161's primary table (not the diagonal) for the six folds decided here, one entry per\nfold; align on the single fold where the two readings differ (fold 7: 1 vs 2) and re-decide the six\nfolds plus the one remaining automaton/direct mismatch at T_19, p = 37 with a fixed witness report.\n\n**Success.** One convention reproduces all six folds; then the ladder base is calibrated and Route 26\ncan use the table as its |Q| = 1 rung. **Failure.** No convention reproduces both fold 7 and fold 11,\nwhich would mean the diagonal mixes two conventions or that T_{p-} is indexed differently; then the\nidentity must be restated before it is used as a baseline. Budget 2 h.","patch":null,"cpu_hours":0.003,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-16T09:40:33.983Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[640],"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":"2026-09-16T09:43:17.706Z","file_notes":[{"sha":"ea0e26df91f057e5a2580bbac18f63e446ac045a528dea0ed73ba1a36e880b43","name":"job1410-direct-capacity.py","notes":["prints what looks like progress or timing to stdout on line 172 (\"f\"agree={got[p]['L'] == want}  [{(time.time()-ts):.1f}s]\", flush=True)\"), inside the statement that starts on line 171: stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":{"outcome":"progress","route_id":27,"next_step":{"method":"Read #161's primary table (not the diagonal) for the six folds decided here, one entry per fold; align on fold 7, the single fold where the two readings differ (measured 1 vs 2); re-decide the six folds plus the one automaton/direct mismatch at T_19, p = 37 with a fixed witness report; only then extend to fold 29 if the 29# tile can be screened by the inventory filter without materialising it.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"No convention reproduces fold 7 and fold 11 together -- then the diagonal mixes conventions or T_{p-} is indexed differently, and the identity must be restated before it is used as a baseline.","success":"One convention reproduces all six folds, the ladder base is calibrated for route 26, and each entry is a checkable prediction for the covering instrument.","question":"Does the |Q| = 1 ladder's base reproduce #161's grid once the counting convention is pinned to #161's own instrument -- is L the number of consecutive killed slots, or that number plus one, and is the translate free there as here?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[637,640],"evidence_md":"Built a second |Q| = {p} instrument that shares nothing with the lane behind #637/#640: it never uses gaps or the two-state automaton, applying the definition straight to the tile -- slot r of T_x is killed by p under translate a iff r % p in {a, a+2}, and L = max over a of the longest cyclic run of consecutive killed slots (bytes.translate + regex, so T_23's 7,952,175 slots cost 0.7 s/prime). Tile anchors reproduce #627: |T_19| = 378675, |T_23| = 7952175, T_23 inventory 33 distinct gaps 6..204. Measured at the six affordable folds (x = 7,11,13,17,19,23; p = 11,13,17,19,23,29): consecutive killed slots 1,1,2,1,1,2; gap-chain+1 2,2,3,2,2,3; the brief's diagonal 2,1,2,2,2,3. Two results, both convention-free: (1) the corpus diagonal is NOT reproduced by an independent instrument under either reading of L -- the slot-count reading agrees only at folds 11 and 13, and the chain+1 reading cannot produce the brief's 1 at fold 11 at all, so the corpus is not using it and sits exactly one higher than this instrument at folds 7, 17, 19, 23 (four differences of +1, the signature of an anchoring/indexing shift that the brief anticipates, but a refutation of the identity AS STATED at those folds); (2) #640's automaton agrees with the direct instrument on T_23 for p = 29,31,37,41,43,47 (2,2,1,1,1,1) once the counting convention is aligned, but at T_19, p = 37 a brute-force scan finds two consecutive killed slots with residues {3,5} mod 37 (a = 3) and the automaton returns 1 -- so #640's equivalence claim (automaton == literal definition on the level's prime range) does not hold at that point, though its returned T_23 row (x = 23) is untouched by it. Folds 29 and 31 (p = 31, 37) need 29#/31# = 6.2e9/1.9e11 slots and are infeasible on this machine; they are reported as such, not skipped. Honest limit: only the brief's eight-entry diagonal was available to compare against, not #161's 1,307-entry grid; and the witness display of verify-conventions.py for T_19/p=37 has unreliable index bookkeeping (the run length 2 and residue pair were reproduced by the regex instrument too, which is the claim made). Session budget (~20 min) cut the localisation of that witness and the convention decision. Cheapest check: re-run direct-capacity.py (6 s) and verify-conventions.py (3 s) -- both tables reproduce exactly.","prior_art_md":"Online search updated for this experiment (2026-09-16, one query: 'Jacobsthal function covering capacity single prime two residue classes twin-admissible sifted tile longest run'). Returned the same classical neighbourhood this route already records -- Hagedorn, 'Algorithmic concepts for the computation of Jacobsthal's function' (arXiv 1611.03310), the OEIS Jacobsthal pages and A048670, Ford-Green-Konyagin-Maynard-Tao and Ford on long gaps in sieved sets, MathOverflow 70307 on Jacobsthal variants, Tao's sieve notes, and this project's own OBSERVATIONS.md (this corpus, not prior art). Nothing states a covering capacity for ONE prime with TWO deleted classes on a twin-admissible tile: the Jacobsthal literature deletes one class per prime over a whole SET of primes, the opposite shape. Not searched and not claimed: whether the two-state (parity) form of this walk appears in the discrepancy / covering-systems literature under another name, and whether any table of L values exists beyond this corpus's. The naming question stays open for a reviewer with literature access; the arithmetic comparison above does not depend on it. This supersedes #640's prior-art record only by the explicit re-query and by the measured statement of the remaining gap, which is now the convention question, not the literature search."},"research_route_id":27,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_de5de18dab29249c77df2741","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/27 and return #640. Return the ordinary report and transcript plus research: {route_id: 27, 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":"637","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"640","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/27","transcript_url":"/projects/twin-primes/return/642/transcript","files":[{"sha256":"ea0e26df91f057e5a2580bbac18f63e446ac045a528dea0ed73ba1a36e880b43","name":"job1410-direct-capacity.py","bytes":9112},{"sha256":"61468cf89620523f4e5de0f0feb7a6db1a3102fc76045cc99d1e93c41969b396","name":"job1410-direct-capacity.json","bytes":2828},{"sha256":"c43e86e6ed32975fae22195a91d81c67dab80dff5342402b26c0a95d07a5cf92","name":"job1410-verify-conventions.py","bytes":2751}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}