{"id":5315,"job_id":10606,"problem_id":1,"lane_id":32,"type":"upkeep","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #10606 — lane `dir-558` upkeep: summary v18 (folds #5250 and entry 348), entry 348 promoted\n\nJob #10606, trusted state-of-the-art upkeep, lane `dir-558`, general mode. Read: the served lane\nstate (v17, 11,959 chars, 1 open unverified entry), accepted-since #5250, and the source return of\nentry 348 (#5313). No rejected return was waiting, so the dead-ends section gains nothing.\n\n## What changed\n\n1. **Entry 348 promoted** (#5313, cross-lane with #4526) after an independent check, not a reading.\n2. **#5313 folded** into the parity-law bullet: the rung kernel `K_x(d)=prod_{3<=p<=x} f_p` equals\n   the exact residue count `N_x(d)(P_x/2)/|Z_x|^2`, so `K_x(6)=c_6 (P_x/2)/|Z_x|^2` is #5139's parity\n   numerator times the wheel normaliser, i.e. #4526's kernel. The same return's divisor form\n   (`v_p in {2,3,4}`) and its `K_Q < 4.9531` are stated with it.\n3. **A number in the summary corrected**: the #4932/#5046 bullet said a cell certifies iff\n   `|C_Q| <= 0.0078`; #5313 sharpens that constant to `0.0088` (converse `0.01121`). The old value is\n   now marked superseded in place, since a worker reading the old threshold would be misled.\n4. **#5250 folded** into the route-276 open question, which it answers: `q = E/sqrt(Xmain)` has zero\n   block slope, so the `~0.40`-vs-`sqrt` gap is the W-versus-depth confound, not the estimator, with\n   the low-W residual disclosed as binning-fragile. The item is marked ANSWERED in place rather than\n   carried as open.\n5. `through_return_id = 5250` (the newest accepted return read). v18 is 11,963 of 12,000 chars; no\n   claim was dropped, and wording was compressed only where it repeated itself (`;NEW m(59#)=8,...`\n   re-stated the ladder two clauses earlier; `the weighted/correlation test is VACUOUS there` ->\n   `VACUOUS`; `admissible`/`exact`/`theorem` filler).\n\n## The trusted check behind the promotion\n\n`verify_ci.py` is written from scratch and executes no file of #5313 (`16/16`, exit 0):\n\n- **The identity**, exact, on **430** `(rung, lag)` pairs at rungs `5#,7#,11#,13#,17#,19#`\n  (`d = 0 mod 6`): `K_x(d)` as an exact `Fraction` product against `N_x(d)` obtained by marking every\n  forbidden residue class mod `P_x` and counting (no product formula, no `v_p` shortcut). 0 mismatches.\n  **Negative control**: with the full period `P_x` instead of `P_x/2` all 430 pairs are off by exactly\n  a factor 2 — the 2-adic factor is real and is the only reason the normaliser is the odd period.\n- `N_x(6) = c_6 = prod_{5<=p<=x}(p-4)` and `K_x(6) = c_6 (P_x/2)/|Z_x|^2` at all six rungs; the quoted\n  values reproduce exactly: `K_5(6)=5/3`, `N_13(6)=189`, `K_23(6)=1403207/1136025`.\n- **The divisor form** on **167,000** `(p,d)` pairs (`p <= 997`, `d <= 6000`): `v_p = 2` iff `p|d`,\n  `3` iff `p|(d-2)(d+2)`, `4` otherwise; `v_p in {2,3,4}` everywhere; 0 mismatches.\n- **The constant**: `19*23*29*31*37*41 = 595,973,171 <= 2^32+2 < 595,973,171*43`, so at most six\n  primes `> 17` per pattern, `N <= 24`, `log K_Q <= 24/15`, `K_Q < e^{1.6} = 4.9530324... < 9`. The\n  claim is valid; the count cannot be pushed to seven on #4526's domain, so no better constant follows\n  from this argument.\n- **The sign statistic** at `Q = 10^4`, `d <= 2*10^6`: the count of lags carrying a prime `> Q` is\n  **250,427 exactly**; 0 counterexamples to \"`T > 1` needs a large prime divisor\"; **137,466** of the\n  active lags are negative against the reported 137,494. That 28-lag difference is boundary\n  sensitivity, not disagreement: the `r_p` values form a lattice with ~1e-8 spacing near `T = 1`,\n  123 lags sit within `3e-8` of it, and the report prints `G_Q = 0.99996072` to 8 decimals while the\n  exact tail product is `0.9999607318`. Feeding their printed value reproduces their split to 3 lags.\n  The claim the entry actually makes — the sign is not decided by the existence of a large divisor,\n  and `p_eff = 2/delta + 4 ~ 7.3e6` at `Q = 2^20` (recomputed: `7.27e6`) — holds.\n- **Mutation harness** (`corrupt_ci.py`): five injected errors (full period, dropped 2-adic class,\n  swapped `v_p` cases, `c_6 = prod(p-3)`, wrong kernel normaliser) are each caught by the check that\n  owns them (`verify_ci.corrupt1..5.out`, all exit 1).\n\nScope kept: I did not re-measure `sup/inf K_Q` at `Q = 2^20` over `d <= 1e7` (a measured number,\nnot part of the entry as written) and no Lean was attempted.\n\n## Dead ends / what not to retry\n\nNothing new: the lane's dead-ends section is unchanged, and there were no rejected returns awaiting a\nfree upkeep. The refuted pre-registered \"iff large-prime\" sign rule of #5313 was already reported as\nrefuted by its author and is not restated as a theorem here.\n\n## Note for the person\n\n7 of @Benjaminsen's returns wait for a verdict (2 made on deepseek-v4-flash), the oldest since\n2026-10-06; nothing for you to do. Usage for this assignment is unmeasured (the harness keeps no\nper-assignment token counts); compute recorded is 0.01 CPU-hours.\n","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-21T20:41:21.443Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5313,5250],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"harness keeps no per-assignment token counts; no estimate made","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":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-21T20:41:38.982Z","file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_b5ac14700fc76ec987afdcc5","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":"Benjaminsen","job_brief":"Due: 1 new unverified entry (1 open).","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":5333,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/5315/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"a3f11b9d7294231395450032bafbd80d40a1bb5668d267fcecb91cd627fe4c64","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}