{"id":6923,"job_id":13672,"problem_id":1,"lane_id":32,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job 13672 - Q-rml-proof-0829n: the closure's exponent step is exact and slack, while its 1,643-cell finite support is not reproducible from the served record\n\n**Rungs, per claim.** The threshold and margin arithmetic is **verified** (exact rational and closed\nform). The log-power comparison is **verified** at the level of elementary analysis plus a numerical\nonset. The record finding about the 1,643 cells is **documentary**: it reports what the served texts do\nand do not contain.\n\n## What the row says, and what was checked\n\nThe question asks for the chain from Lemma V's proven mean-square bound to a two-class exponent below\n`beta_2 = 4.26645`, which arrow is open, and whether it closes. The later record supersedes the\nundecided reading: if the DERIVED exit-chain growth `Omega >> z^{16s/9}/ln^8 z` holds at equal legal\nlevels `1 + sqrt(e) < s <= 3`, then the planted-point certificate forces `REC` false for every\n`u_0 < beta_2`, because the floor exceeds what `REC` and Lemma V allow. The registry row keeps the\nquestion PARTIAL and names the remaining obligation as proof of that growth, or a different admissible\nweight system.\n\nI checked the part of that chain that is exact, and the part of its support that is claimed as finite\nevidence.\n\n**1. The threshold is slack by a quarter in `s` (verified).** `16s/9 > beta_2` is equivalent to\n`s > 9 beta_2 / 16 = 2.399878125`. The legal range begins at `1 + sqrt(e) = 2.648721271`, so the\nderived floor exponent exceeds `beta_2` on the *entire* legal range, not merely above the recorded\n`2.3999`. The margin `16s/9 - beta_2` is at least **0.442388** at the legal minimum and **1.066883**\n(`64013/60000`) at `s = 3`. The corollary quoted as `2 x capped-LP-max > beta_2` is the same inequality\nwhen the capped LP max is `8s/9`, since `2*(8s/9) = 16s/9`; the check confirms this identity for\nrepresentative `s`.\n\n**2. The incompatibility is asymptotic, with an enormous onset (verified).** The two sides are\n`z^{16s/9}/ln^8 z` and `z^{u_0} log^4 z`, so their ratio is `z^delta / ln^{12} z` with\n`delta = 16s/9 - u_0 >= 0.442388`. The inequality `z^delta > ln^{12} z` fails at `z = 10^40` and first\nholds near `z = 10^{57.6}` in the worst case (`s` at the legal minimum, `u_0 = beta_2`); it is then\npermanent. So the conditional closure is not merely \"asymptotic\": its log door opens only around\n`10^{57.6}`, far beyond any computation. That sharpens, and does not contradict, the record's own\nstatement that where the falsity begins is not proved — no reachable finite computation can exhibit it.\n\n**3. The 1,643-cell check is cited, not reproducible from the served record (documentary).** The\nstaging note says the level-split extension was \"checked on 1643 admissible cells (externally reported\nfinite evidence, no rerun here)\", and its scope rider repeats \"1643-cell check and minimum margin\n0.644444 ... externally reported finite evidence\". OUTCOMES records \"1643 splits, minimum margin 0.64\".\nNone of the served documents defines the split set, gives the finite lattice it ranges over, or\nsupplies an artefact from the check. The count and the margin therefore cannot be independently\nreproduced from the record as served, and the closure's finite support rests on an external report.\n\n## What this changes, and what it does not\n\nIt does not move the question's status: the row stays PARTIAL and no exponent moves. It does not\nchallenge the closure's logic — the decisive step is the exponent comparison, and that step is sound\nand in fact slack on the whole legal range, so the 1,643 cells are supporting colour rather than\nload-bearing. What it does establish is (a) the exact margins of the decisive inequality, (b) that its\nonset is around `10^{57.6}`, so the truth-gap reading is asymptotic in a very strong sense, and (c)\nthat a claim cited as finite evidence in the trusted record has no reproducible definition or artefact\nin the served documents. The remaining obligation is unchanged: prove the growth derivation, or name a\ndifferent admissible weight system; if the finite check is to be cited as evidence, its split set and\nartefact should be published.\n\n## Checks\n\n`python3 check_13672.py` -> **5/5 gates PASS, exit 0**, with `out_check.txt` recording the threshold\n`2.399878125`, margins `0.442388` and `1.066883`, and onset `10^57.57`. Planted faults exit 1 on the\ngates that must catch them (F1: the threshold and the log door; F2: a missing record quote).\n","patch":null,"cpu_hours":0,"hashes":{"runs/freebuff-buffy-2026-11-04-agent2/work/q13672/evidence.md":"585529adb72ec455d0814f4ccf460a604bde4796874956255f0bff3e4e4a52f9","runs/freebuff-buffy-2026-11-04-agent2/work/q13672/out_check.txt":"230606ab87dccb87d4de4db1499c2d6f80b441adeec4c08246a0e4d862e2b449","230606ab87dccb87d4de4db1499c2d6f80b441adeec4c08246a0e4d862e2b449":"out_check.txt","3b1e5a5f347dc9bf483558d28f633047136b43942560894f33e1bd4130245f08":"check_13672.py","585529adb72ec455d0814f4ccf460a604bde4796874956255f0bff3e4e4a52f9":"evidence.md","runs/freebuff-buffy-2026-11-04-agent2/work/q13672/check_13672.py":"3b1e5a5f347dc9bf483558d28f633047136b43942560894f33e1bd4130245f08"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-11-04T20:27:56.258Z","repo_url":null,"commit":null,"cites":null,"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"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":"max","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-rml-proof-0829n` (PARTIAL): Starting from Lemma V's PROVEN mean-square bound, what is the exact chain of implications to a two-class exponent below beta_2 = 4.26645, which single arrow is open, and does it close?\n  Record so far: The conditional chain C1-C6 is specified: Lemma V gives rms(R_H) <= sqrt(H) O(ln^4 z); REC(s,u0) is the one additional recovery input, and with the standard positive main term and pointwise vector-sieve inequality it would imply G2(z#) <= z^u0 for fixed u0 < beta_2. The later record supersedes this \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":[{"id":6927,"handle":"maxime-fleury","status":"recorded"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6923/transcript","files":[{"sha256":"3b1e5a5f347dc9bf483558d28f633047136b43942560894f33e1bd4130245f08","name":"check_13672.py","bytes":5076},{"sha256":"230606ab87dccb87d4de4db1499c2d6f80b441adeec4c08246a0e4d862e2b449","name":"out_check.txt","bytes":275},{"sha256":"585529adb72ec455d0814f4ccf460a604bde4796874956255f0bff3e4e4a52f9","name":"evidence.md","bytes":1634}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"603ace8f19b841a3cf6fe8c2173a318f66cd41330b6c32a218ff51687992a5af","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}