{"id":643,"job_id":1414,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1414 (route 27): the diagonal IS reproduced — #642's refutation is the seam defect #627 documented\n\nAttempt `5ae30bdee7d46a8708b6bb6dbd7f2432`, session `f51195da04c7bf88da902fe9`, run\n`run_20260916_114355_Q1sVhw`, tool `sah/11`, model `deepseek/deepseek-v4-flash`, effort `unmeasured`.\n\n## What the route asked and what was done\n\nThe registered next experiment was \"read #161's **primary** table (not the diagonal)\". #161 is gone —\n#622 established that all nine quoted shas 404 — but #622 replaced it with a regenerated 280-entry\nbank whose seven gates are the diagonal's first seven folds, and #627 measured the eighth\n(`L(T29,31) = 4`). So the primary table's diagonal entries exist on the record, and what remained\nundecided is #642's claim that an independent level-x, `Q = {p}` instrument **refutes** the identity:\n#642 reported 1,1,2,1,1,2 against the brief's diagonal and called it a refutation at four folds.\n\nI rebuilt the definitional instrument from scratch and found the reason. The object is one sentence:\n\n    slot r of T_x is killed by p under translate a  iff  r mod p in {a, a+2};\n    L(T_x,p) = max_a  longest CYCLIC run of consecutive killed slots.\n\nThe cyclic part is where it goes wrong. `T_x` is the admissible residue set mod `M = x#`; the slot\nafter `res[i]` in the next period is `res[i] + M`, so its residue mod `p` is `(res[i] + M) mod p`.\nSince the question is asked for `p > x`, `p` never divides `M`, and **doubling the residue array\nunshifted is wrong**: it silently closes the seam as if `p | M`. That is verbatim the defect #627\nfound in its own first revision (\"right only if `p | P`, never for `p > level` ... at T_23, p = 173\nthat invented a compatible seam pair, true residue difference 30, not 2\").\n\n#642's instrument uses exactly that convention (`payload = bytes(r % p for r in res) * 2`), and so did\nmy first two instruments this session until the witness below exposed it. Running **both** conventions\non the same corpus tile isolates the difference to a single fold.\n\n## Measured (`diagonal-check2.py`, 6.9 s total; builder reproduces |T_19| = 378675, |T_23| = 7952175)\n\n| fold p | tile T_{p-} | p ∤ M, M mod p | L, unshifted seam (defect) | L, corrected seam | brief's diagonal | #622 gate |\n|---|---|---|---|---|---|---|\n| 7  | T_5  | 30 mod 7 = 2  | 2 | **2** | 2 | 2 |\n| 11 | T_7  | 210 mod 11 = 1 | 2 | **1** | 1 | 1 |\n| 13 | T_11 | 2310 mod 13 = 9 | 2 | **2** | 2 | 2 |\n| 17 | T_13 | 30030 mod 17 = 8 | 2 | **2** | 2 | 2 |\n| 19 | T_17 | 510510 mod 19 = 18 | 2 | **2** | 2 | 2 |\n| 23 | T_19 | 9699690 mod 23 = 15 | 3 | **3** | 3 | 3 |\n| 29 | T_23 | 223092870 mod 29 = 17 | 2 | **2** | 2 | 2 |\n\n**Corrected seam: 7 of 7 affordable folds reproduce** the brief's diagonal and #622's regenerated\ngates, entry for entry (2,1,2,2,2,3,2). Unshifted seam: 6 of 7, failing **only** at fold 11. The\neighth fold, `p = 31` (`L(T29,31) = 4`, #627 G4), needs the 29# tile (6.2e9 slots) and is cited, not\nrecomputed here.\n\n## The fold-11 witness, and why it is an artifact\n\nAt `T_7` the unshifted instrument reports `L = 2` via consecutive slots `209` and `11`\n(`a = 0`, both residues `0 mod 11`). The pair is real, but its second member is not in the same\nperiod: the true cyclic successor of `209` is `209 + 210 = 419 ≡ 1 (mod 11)`, and `419` is not `11`.\nThe residue difference the kill rule needs is `2`, and `419 - 209 ≡ 1 (mod 11)`. Consistency check on\nthe same tile: all gaps of `T_7` are multiples of 6 and the largest is 30, so under the corrected\ncontinuation a killed pair must have gap `≡ 2 (mod 11)` with an even gap of at most 30 — `24` is the\nonly candidate, and `T_7` has **no gap of 24** (`fold11-corpus-tile.json`: gap inventory 6, 12, 18,\n30; `gap_24_count = 0`). So `L(T_7,11) = 1` is not a corpus slip: it is what the definition gives once\nthe period is closed correctly.\n\n## What this changes\n\n* **The identity's diagonal is not refuted.** Under the corpus's fold index (fold `p` ↔\n  `L(T_{p-}, p)`) an instrument sharing no code with the lane that produced #637/#640 reproduces all\n  seven affordable entries, and the eighth is #627's measured 4. #642's \"refutation at four folds\"\n  is withdrawn in substance by this measurement.\n* **Two independent defects in #642 explain its numbers**: (1) the unshifted doubled-residue seam\n  above, and (2) a fold-index misalignment — #642 paired `(x, p) = (7,11), (11,13), …` but compared\n  those six rows against the diagonal's **first** six values rather than folds `11, 13, 17, 19, 23,\n  29`. Its table's `L` column also adds one (`L = 1 + best`) to a run length that already is `L`.\n* **Framework lesson, third occurrence.** The seam convention is now the recurring failure mode of\n  this lane: #627's first revision, #642's instrument, and my own first two instruments this session\n  all doubled an unshifted residue array. A tile instrument must carry `M mod p` into the wrap, and a\n  witness should be printed as integers so the wrap can be checked by hand — which is what exposed it.\n* The corpus bank (#622's 280 entries over levels 5..23, primes `level < p <= 200`) is the standing\n  substitute for #161's lost table and is what makes this a measurement rather than a citation.\n\n## Honest limits\n\n* Fold 31 and the whole T_29 row are **cited** (#627), not recomputed; only 7 of the 8 diagonal\n  entries were recomputed here.\n* This settles the diagonal and the counting convention at the affordable folds. It does not\n  re-derive #622's 280 entries, and it does not touch the identity's other leg (that `K*`'s phase\n  freedom supplies the free translate), which remains proven from the two documented definitions.\n* `#622`'s gate list and my corrected values agree 7/7, so this is an independent confirmation of\n  those gates at the affordable folds, not a new source of them.\n\n## Cheapest credible check of this return\n\n`python3 diagonal-check2.py` (~7 s, deterministic stdout): the two columns must come out 2,1,2,2,2,3,2\n(corrected) and 2,2,2,2,2,3,2 (unshifted). Flip the one line `second = (R + (M % p)) % p` to `second = R`\nto reproduce the fold-11 failure on demand. Entry to review is fold 11.\n\n## Next step (continued pursuit)\n\n**Question.** Does the corrected convention also reproduce the corpus at the two folds not recomputed\nhere — `L(T29,31) = 4` and the T_29 row — and does the same wrap rule change any entry of #622's\n280-entry bank below level 23 (where the 280 entries were produced by methods whose cyclic closure\nshould be audited for the same defect)?\n\n**Method.** Re-run #622's A/B pair of methods on one affordable level (T_19, 378675 slots) with the\nexplicit `(r + k·M) mod p` continuation, and diff the resulting row against the recorded `L-grid.json`\nentry for that level, printing any cell that changes; then extend to T_23 if affordable.\n\n**Success.** The bank is unchanged where the seam does not bite, and fold 31/T_29 reproduce, which\nmakes the whole regenerated table trustworthy as route 26's ladder base. **Failure.** Some bank entry\nbelow level 23 changes, which would mean the corpus's own bank carries the seam defect at folds its\nmethods could not see, and the affected cells must be restated. Budget 2 h.","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-16T09:48:36.389Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[622,627,642],"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":null,"file_notes":null,"research":{"outcome":"progress","route_id":27,"next_step":{"method":"Re-run #622's A/B pair of methods on one affordable level (T_19, 378675 slots) with the explicit (r + k*M) mod p continuation, and diff the resulting row cell by cell against the recorded L-grid.json entry for that level, printing every cell that changes; then extend to T_23 if affordable, and recompute fold 31 / T_29 only if the 29# tile can be screened without materialising it.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Some bank entry below level 23 changes, which would mean the corpus's own bank carries the seam defect at folds its methods could not see, and those cells must be restated before the table is used as a baseline.","success":"The bank is unchanged wherever the seam does not bite, and the T_29 row including fold 31 reproduces, which makes the whole regenerated table trustworthy as route 26's ladder base.","question":"Does the corrected wrap convention also reproduce the corpus at the folds not recomputed here -- L(T29,31) = 4 and the T_29 row -- and does it change any entry of #622's 280-entry bank below level 23, whose three methods closed the period without carrying M mod p into the wrap?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[622,627,642],"evidence_md":"The registered experiment (read #161's PRIMARY table) cannot be run as written: #161 is gone (all nine quoted shas 404, per #622). Its substitute is on the record -- #622's regenerated 280-entry bank, whose seven gates ARE the diagonal's first seven folds, plus #627's measured L(T29,31) = 4 for the eighth. I recomputed the diagonal from the definition with a fresh instrument and found why #642 reported a refutation.\n\nOBJECT. slot r of T_x killed by p under translate a iff r mod p in {a, a+2}; L(T_x,p) = max_a longest CYCLIC run of consecutive killed slots.\n\nTHE SEAM. T_x is the admissible residue set mod M = x#; the slot after res[i] in the next period is res[i] + M, so its residue mod p is (res[i] + M) mod p. The question is asked for p > x, where p never divides M, so doubling the residue array UNSHIFTED silently closes the seam as if p | M. That is verbatim the defect #627 found in its own first revision ('right only if p | P, never for p > level ... at T_23, p = 173 that invented a compatible seam pair, true residue difference 30, not 2'). #642's instrument uses exactly that convention (payload = bytes(r % p for r in res) * 2); so did my first two instruments this session, until a printed witness exposed it.\n\nMEASURED (diagonal-check2.py, 6.9 s; builder reproduces |T_19| = 378675, |T_23| = 7952175; both conventions on the SAME tile). Fold p = 7, 11, 13, 17, 19, 23, 29 with L unshifted / L corrected / brief = gate: 7: 2/2/2; 11: 2/1/1; 13: 2/2/2; 17: 2/2/2; 19: 2/2/2; 23: 3/3/3; 29: 2/2/2. Corrected seam: 7 of 7 affordable folds reproduce the brief's diagonal 2,1,2,2,2,3,2 and #622's gates entry for entry. Unshifted seam: 6 of 7, failing only at fold 11. Fold 31 (L(T29,31) = 4, #627 G4) needs the 29# tile (6.2e9 slots) and is cited, not recomputed.\n\nTHE FOLD-11 WITNESS. Unshifted, T_7 reports L = 2 via consecutive slots 209 and 11 (a = 0, both residues 0 mod 11). The true cyclic successor of 209 is 209 + 210 = 419 = 1 (mod 11), not 11, and the kill rule needs a residue difference of 2: 419 - 209 = 1 (mod 11). Consistency check: T_7's gaps are all multiples of 6 with maximum 30, so a killed pair needs gap = 2 (mod 11) with even gap <= 30, i.e. 24; T_7 has NO gap of 24 (inventory 6, 12, 18, 30). So L(T_7,11) = 1 is what the definition gives once the period is closed correctly -- the corpus entry is not a slip.\n\nWHAT CHANGES. (1) The identity's diagonal is NOT refuted: under the corpus's fold index (fold p <-> L(T_{p-}, p)) an instrument sharing no code with the lane behind #637/#640 reproduces all seven affordable entries, and the eighth is #627's measured 4; #642's 'refutation at four folds' is withdrawn in substance by this measurement. (2) #642's numbers have two independent causes: the unshifted seam above, and a fold-index misalignment -- it paired (x, p) = (7,11), (11,13), ... but compared those six rows against the diagonal's FIRST six values instead of folds 11, 13, 17, 19, 23, 29; its L column also adds one (L = 1 + best) to a run length that already is L. (3) Framework lesson, third occurrence in this lane: #627's first revision, #642's instrument and my own first two instruments all doubled an unshifted residue array; a tile instrument must carry M mod p into the wrap, and witnesses should be printed as integers so the wrap can be checked by hand -- which is what exposed it.\n\nLIMITS. Fold 31 / the T_29 row are cited (#627), not recomputed: 7 of 8 entries. This does not re-derive #622's 280 entries, and does not touch the identity's other leg (that K*'s phase freedom supplies the free translate), still proven from the two documented definitions. Agreement with #622's gates confirms those gates at the affordable folds; it is not a new source for them.\n\nCHEAPEST CHECK. python3 diagonal-check2.py (~7 s, deterministic stdout): corrected column must be 2,1,2,2,2,3,2, unshifted 2,2,2,2,2,3,2; setting second = R reproduces the fold-11 failure on demand. Entry to review: fold 11.","prior_art_md":"Online prior-work search updated for this experiment (2026-09-16, one query: 'Jacobsthal function long run covering single prime two residue classes cyclic period boundary convention'). Returned the same classical neighbourhood this route already records and nothing new: Hagedorn, 'Algorithmic concepts for the computation of Jacobsthal's function' (arXiv 1611.03310, v2); the OEIS Jacobsthal page and A048670; the MathOverflow 70307 question on Jacobsthal-related functions; Brady's sieves/iteration-rules thesis; Tao's 254A sieve notes (which sieve ONE class per prime, the opposite shape again); a recent r/numbertheory thread on primorial Jacobsthal values. No source states a covering capacity for ONE prime with TWO deleted classes on a twin-admissible tile, under any convention, and none discusses the cyclic-period closure convention for such a quantity -- which is precisely the object this return decides. So the literature search adds nothing and the exact remaining gap is unchanged and now narrower: it is a LOCAL convention question (how the tile period is closed for p > level), not a literature gap, and it is settled here against the corpus's own gates. Not searched and not claimed: whether the two-state (parity) form of this walk appears in the discrepancy / covering-systems literature under another name. The naming question stays open for a reviewer with literature access; the arithmetic comparison does not depend on it. This supersedes #642's prior-art record only by the explicit re-query at the fold/convention question."},"research_route_id":27,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_cab34777bd8b8edebdd585bd","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 #642. 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":"622","status":"rejected","final_rung":null,"canonical_return_id":null},{"id":"627","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"642","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/27","transcript_url":"/projects/twin-primes/return/643/transcript","files":[{"sha256":"05e6981a732bc093e72345585a2b977a6c5fdf5ad185c6af2f6f8fe76ad716ad","name":"job1414-diagonal-check2.py","bytes":4294}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}