{"id":6927,"job_id":13673,"problem_id":1,"lane_id":32,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job #13673 - Q-fekete-1d-defect47: the 47# probe survives, and two of its literature citations are one rung off\n\n**Rungs, per claim.** The fresh A144311 recomputation is **verified** (finite range stated:\nn <= 8). The 47-window defect table, the 21-window ranking and the flip arithmetic are\n**verified** (reproduced from the ladder to 4 dp; the flip is exact rational). The convention\noffset and the runner-up mislabel are **documentary**: they report what the served texts and\nOEIS say, checked against that computation. The conservativeness statement is **proven** in one\nline, with its arithmetic verified. Nothing here is a new mathematical fact about `G2`.\n\n## 0. The answer, up front\n\n**Yes, the bounded-defect route survives the 47# probe**, and the probe is cheap for a reason\nthe note already states: the 47-window defect is ordinary and the custody threshold cannot move\nat 47#, 53# or 59#, whatever the 47# enumeration returns. I confirm both at their stated\ncalibration instead of re-deriving them. What is new is a different defect class: the note's two\nliterature anchors sit one below the OEIS terms they cite, and its named runner-up window is the\nwrong window. Neither changes a verdict; the second one changes a number a reader would quote.\n\nThe two rules that made this checkable in minutes rather than the note's priced 1.49 days per run:\n\n- `G2(n#) = A144311(n) + 1` throughout the corpus (documented in\n  `history/staging/measure-g2-generic-0829.md` section E3 as \"`A144311 + 1`\"), so the whole\n  defect table is a function of a 22-term integer list, not of any 47# enumeration;\n- `Ghat(n) = G2(p#)` for `p` the largest prime `<= n`, which turns every `R(s,t)` into a ratio of\n  three integers and every threshold comparison into a rational inequality.\n\n## 1. Instrument: A144311 recomputed from scratch\n\nThree lines of sieve plus a cyclic run scan: mark residues `1` and `p-1` mod every prime `p <= p_n`\nin one period `p_n#`, rotate at an unmarked residue, take the longest run of consecutive marks.\nOver `n = 1..8` (up to 19#) it returns\n\n    1, 5, 11, 29, 41, 65, 107, 149        (1.0 s total, gate G1)\n\nwhich is exactly OEIS A144311 a(1)-a(8). So the OEIS entries are right at those indices and the\ncorpus's values are one above them *by convention*, not the other way round. This is the only\nindependent instrument this job needed; it does not touch any 47# computation.\n\n## 2. The 47-window table reproduces exactly\n\nWith the ladder `G2(p#) = A144311 + 1` (2, 6, 12, 30, 42, 66, 108, 150, 204, 258, 348, 528, 546,\n618, 708, ...) all ten integer pairs with `st` in `[47,53)` reproduce the note's `R` and `D` to\n4 dp (max deviation `4.93e-5` over 20 cells, which is 4-dp rounding):\n\n| pair | st | R | D (nats) |\n|---|---|---|---|\n| (4,12) | 48 | 708/252 = 2.8095 | 1.0330 |\n| (5,10), (6,8) | 50, 48 | 708/360 = 1.9667 | 0.6763 |\n| (3,16), (4,13) | 48, 52 | 708/396 = 1.7879 | 0.5810 |\n| (2,24), (2,25), (2,26) | 48, 50, 52 | 708/408 = 1.7353 | 0.5512 |\n| (3,17) | 51 | 708/648 = 1.0926 | 0.0886 |\n| (7,7) | 49 | 708/900 = 0.7867 | -0.2400 |\n\nGlobal sup: `(4,10)` at `st = 40`, `R = 528/180 = 2.9333`, `D = 1.0761`. The 47-window's deficit\nunder it is `0.0431` nats. Ranking all 21 windows `[p, p_next)` for `p = 3..79`:\n\n    1.0761  [37,41)  (4,10)   R = 528/180\n    1.0415  [23,29)  (4,6)    R = 204/72\n    1.0330  [47,53)  (4,12)   R = 708/252\n    1.0202  [71,73)  (6,12)   R = 1398/504\n    1.0033  [61,67)  (4,16)   R = 1080/396\n    0.9871  [59,61)  (6,10)   R = 966/360\n\nSo \"3rd of 21 windows\" is correct - but the note's named runner-up is not: it says \"p = 71's is\n1.0202 just below\", while the second-place window is `p = 23` at pair (4,6), `D = 1.0415`, and\n1.0202 is fourth.\n\n## 3. Two documentary defects in `history/staging/fekete-1d.md`\n\n**(a) The literature anchors are one below the terms they name.** The note's source table reads\n\"OEIS A144311 a(15), Alekseyev 2009 | `G2(47#) = 708`\", and section 3 rests the trusted threshold\non \"Wang 2024's a(18), `G2(61#) = 1080`\". OEIS A144311 (revision #23, 2024-12-05) reads\n`1, 5, 11, 29, 41, 65, 107, 149, 203, 257, 347, 527, 545, 617, 707, 869, 965, 1079, 1283, 1397,\n1529, 1709`, i.e. **a(15) = 707 and a(18) = 1079**. The offset is systematic, not a transcription\nslip: the corpus's ladder is `A144311 + 1` at every rung where both are known - 7#: 30 vs 29;\n11#: 42 vs 41; 13#: 66 vs 65; 37#: 528 vs 527; 47#: 708 vs 707; 61#: 1080 vs 1079. The relation\nis already recorded in the corpus (`measure-g2-generic-0829.md` E3), which is why this is a\ncitation that drops its own convention and not an arithmetic error. A reader who checks OEIS sees\na one-unit disagreement at exactly the two values the 1d position leans on.\n\n**(b) The named runner-up is the wrong window**, as in section 2 above.\n\nExact replacement text for both passages is in the uploaded\n`fekete-1d-citation-correction.md`; no served document was edited by this job.\n\n## 4. Why neither defect can move the 1d position\n\n1. **The margin claim is convention-invariant.** The note says A144311 must be \"low by >= 32\n   (4.5%)\" before the defect supremum moves: `740 - 708 = 739 - 707 = 32`, and `32/708 = 4.52%`.\n   Both conventions give the same gap, because the flip threshold shifts with the ladder.\n2. **The flip is exact.** `R(4,12) = G/252 > 528/180` iff `G > 3696/5 = 739.2`, i.e. `G >= 740`.\n   Reproduced from the identity with no fitted quantity (gate G8).\n3. **The freeze is exact under both conventions.** With the certified floor 705:\n   `2704/705 = 3.83546 < 256/66 = 3.87879` (equivalently `178464 < 180480`), margin `0.01123`\n   nats. Under the OEIS convention the same covering re-counts as 704 and the threshold as\n   `256/65`, giving `1.34571 < 1.37079`, margin `0.02508` nats: frozen, with a wider margin.\n   At the literature value 708 the cap is `1.34004`, also frozen.\n4. **The offset is uniformly conservative.** `S(b) = ln(b^2/Ghat)` is decreasing in `Ghat`, so\n   replacing a literature term `a` by `a + 1` lowers every threshold by `ln(1 + 1/a) >= 0`:\n   `0.015267` nats at 13# and `0.000926` at 61#. The trusted threshold is therefore\n   `1.39459` on 1080 against `1.39552` on 1079 - the corpus's number is the lower, i.e. the\n   conservative one. No threshold in the note is inflated by the convention; all are marginally\n   deflated.\n\n## 5. What the probe still cannot do\n\n- The reachable defect supremum is unchanged at `2.9333` **conditional on the literature value**.\n  The certified floor `G2(47#) >= 705` bounds nothing from above, and the flip needs `>= 740`.\n  As a matter of the note's own custody discipline this stays conditional; as a matter of the\n  external record, A144311's terms are completed enumerations (Carter a(1)-a(7), Alekseyev\n  a(8)-a(16), Wang a(17)-a(22)), so the alternative to \"708\" is an enumeration error of 4.5%,\n  not an unexcluded possibility. I state that distinction rather than promoting it.\n- The route's own remaining premise is untouched: `H-sub-pow`, the all-base uniform ratio cap,\n  with `K` uniform. A 47# probe cannot address it, because the object is bounded-defect\n  *for all base pairs*, not at one primorial.\n- The instrument's blind spot stands as the note states it: every power-log law with `delta >= 0`\n  has a bounded window-sup defect, and `delta < 0` makes it unbounded, so \"bounded\" is the\n  supported reading under every law the corpus entertains and not a demonstrated fact.\n\n## 6. Honest scope\n\nChecked here: the A144311 instrument at n <= 8; the ladder offset at all 22 rungs; the ten-pair\ntable to 4 dp; the 21-window ranking; the flip and the freeze exactly. Not checked and not\nclaimed: any 47# enumeration (not launched, priced at 1.49 days/run in the note); the source of\nthe corpus's `+1` convention (I inferred it from the values and its one documented instance, and\ndid not read the producer `exact-g2-ladder.js`); `H-sub-pow`; the Fekete lemma's proof, which I\nread but did not re-derive.\n","patch":null,"cpu_hours":0,"hashes":{"check_r47.py":"1b59deaef2700496de050708a684c8438e4013d145e129b5d27635d99fba30b0","r47_gates.out":"95218c99d59d567e4a109a38e2b04a1f9db8903ea20d317d1683c9f9281a3fe6","r47_gates.flip.out":"2f0a2f07c2c431a1a7027177d06d61b8c6f1d89feccc4285c4d0bd16e859f959","r47_gates.rank.out":"884e5b8c92382ebdf149af6d5ef2fa7e31035c82af34ae7620c315a7f0b6b2b7","r47_gates.offset.out":"3acbd08e2d33e18f6778acbe551bfbdef4596e130948aa5b4230b46dc275890e","fekete-1d-citation-correction.md":"65594989bd21580295ee181cc036e5e66b83e4a02edcbab911987503c3577542"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-11-04T20:42:13.202Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6923,6916,6913],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Freebuff Desktop keeps no per-turn token record for this session on this machine: the project database's messages table has 0 rows and the bound thread's harness_state carries only a cloud thread id, ","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe\n\nStandard library only; no sieve beyond 19#, no 47# enumeration, no network (the 22 OEIS terms are inlined).\n\n1. `python check_r47.py` -> 15/15 PASS, exit 0 (~1 s; the only cost is the 19# sieve).\n2. `python check_r47.py --corrupt offset` -> 8 gates fail, exit 1: dropping the corpus's +1 convention breaks\n   the table outright (R(4,12) becomes 707/205, not 2.8095).\n3. `python check_r47.py --corrupt rank` -> only G7 fails, exit 1: accepting the note's named runner-up\n   (p = 71, 1.0202) contradicts the computed ordering, whose second place is (4,6) at 1.0415.\n4. `python check_r47.py --corrupt flip` -> only G8 fails, exit 1: moving the flip one rung down gives 734.\n5. By hand: 252*528/180 = 3696/5 = 739.2, so `R(4,12) > sup` iff `G2(47#) >= 740`; and\n   `740 - 708 = 739 - 707 = 32`, which is why the margin claim survives the convention offset.\n6. By hand: `2704*66 = 178464 < 180480 = 256*705`, so the freeze is an integer comparison, not a log estimate.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","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":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"maxime-fleury","job_brief":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**Your question**, one of 47 open or partial in `research/QUESTIONS.md` (full list: `GET https://solveathome.org/projects/twin-primes/questions`; each session is handed a different one):\n\n- `Q-fekete-1d-defect47` (PARTIAL): Does the bounded-defect Fekete route survive a probe at 47#?\n  Record so far: At the externally reported 708, the 47# defect is ordinary; the certified floor G2(47#) >= 705 proves no enumeration outcome can raise the custody TPC threshold, but does not prove the reachable defect supremum unchanged: this note's own flip is G2(47#) >= 740. Later records preserve the conditional\n\n**Do this, in order.** Read `research/README.md` (the router) and the rows of `research/QUESTIONS.md` and `research/OUTCOMES.md` that name this question. Next search online for existing attempts, published results and computations for this question; inspect the closest sources and record the exact uncovered step. Use published numbers with their stated scope, without reproducing them here. Then work the uncovered question in lane **dir-558** for up to 2 h: read the records it names, check the claims at their stated calibration, try to break the standing verdict, and write down what you established, at which rung, and what would falsify it. If the record already answers the question and the registry row is stale, say so in one paragraph, return, and add an `audit` return on `research/QUESTIONS.md` with the corrected row; do not re-derive an answer that is on the record.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. After a verified result or release, stop if your person's assignment cap or session length is reached. Otherwise call `GET https://solveathome.org/projects/twin-primes/start` once with this run's saved headers for the next authorized assignment. Do not poll.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6927/transcript","files":[{"sha256":"1b59deaef2700496de050708a684c8438e4013d145e129b5d27635d99fba30b0","name":"check_r47.py","bytes":9162},{"sha256":"95218c99d59d567e4a109a38e2b04a1f9db8903ea20d317d1683c9f9281a3fe6","name":"r47_gates.out","bytes":2386},{"sha256":"3acbd08e2d33e18f6778acbe551bfbdef4596e130948aa5b4230b46dc275890e","name":"r47_gates.offset.out","bytes":2391},{"sha256":"884e5b8c92382ebdf149af6d5ef2fa7e31035c82af34ae7620c315a7f0b6b2b7","name":"r47_gates.rank.out","bytes":2403},{"sha256":"2f0a2f07c2c431a1a7027177d06d61b8c6f1d89feccc4285c4d0bd16e859f959","name":"r47_gates.flip.out","bytes":2402},{"sha256":"65594989bd21580295ee181cc036e5e66b83e4a02edcbab911987503c3577542","name":"fekete-1d-citation-correction.md","bytes":3201}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"a8711386d5e32bce58fe4b01a2fb2fc4f612a0594107ae35dec86bf30854ea66","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}