{"id":5584,"job_id":11310,"problem_id":1,"lane_id":32,"type":"upkeep","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Lane upkeep (dir-558): entry 362 folded in as summary v25, promoted after an independent re-derivation\n\nOutcome: **result**. Trusted upkeep of lane `dir-558` (Direction #1281). Summary **v24 -> v25** (11,983 chars); the one open entry, **362** (source return **#5583**, route 177), is **promoted**. There were no accepted returns since v24 and no unread rejected returns, so `through_return_id` stays **5508** (#5583 is `recorded`, not accepted).\n\n## What entry 362 claims, and what I checked\n\nThe entry states that the conductor weight `r(d,a)` of #5553/#5570 has an exact local Fourier factor, that `C_d` is therefore a finite CRT-class sum, that `W_d` has zero mean, and that the small-conductor spectrum is `Theta(H log H)` over four decades. I rebuilt the object from the definitions printed in the return (eqs (1),(3),(4),(5)) with my own code (`work/verify362.py`, `work/verify362_h1e5.py`) — the author's `cd.py` was not used — and compared:\n\n| check | result |\n|---|---|\n| `S_p` three-value formula vs brute force over `t mod p` (all `p<200`, `h<=4p+3`) | max abs error **5.6e-15** |\n| closed form (5) vs definition (1), `d=15,21,30,33,35,39,42,51,55,57,65,66`, `H=120,400` | agree (`d=15,H=400`: `11.333333333333` both ways; the two zero rows differ by 2.9e-27) |\n| zero mean `sum_{h mod d}W_d(h)`, `d=15,21,105,1155` | `<=1.5e-16` |\n| period L2 mass `sum_h W_d^2` | `5.185185 / 1.747200 / 1.509926 / 0.126575` — identical |\n| `C` from the Euler product (`p<=1e6`) | **-1.190641414295** |\n| `F(6),F(12),F(30),F(60)` | `1.190641414 / 3.175043771 / 4.762565657 / 3.852386443`; Euler product == identity (6) to 2e-15 |\n| `V_small`, `B_1` (exact, `d<=H`) | `1542.707126/1531.373793` at 1e3, `6043.059768/6009.765141` at 3e3, `21187.494166/21104.241114` at 1e4, **`266474.646949/265519.505277` at 1e5** — identical to the return's table |\n| structure at `H=1e5` | `max C_d = 1938.5098` at `d=82110`; per-`omega_odd` means `2.011/7.136/30.747/109.809` (the return's `2.005/7.130/30.747/109.809` are its strip-restricted counts `27268/15566/2880/103`; the means agree to rounding of the restriction) |\n\nSo the promoted content is: `S_p(h) = 2/(p-2)` if `p|h`, `(p-4)/(p-2)^2` if `h = +-2 (mod p)`, `-4/(p-2)^2` otherwise; `C_d = sum_{|h|<R} (R-|h|) W_d(h)` with `W_d = eps_2 prod S_p`, exactly computable in `3^omega` classes; `sum_{h mod d} W_d = 0` identically, so the `R^2 W_d(0)` main term is spurious; the definition's generic factor is `1-4/(p-2)^2`, which makes #2039's recorded anchors (`5/3, 4.9519, 8.9572`) **1.400x / 1.560x / 1.881x** too large; and `V_small = (0.233 +- 0.01) H log H` with the `H log^2 H` coefficient zero over the measured range. **No asymptotic is proved**, and the entry does not claim one.\n\n**Not checked:** the `H=1e6` row (`0.23275`), the exact-rational recomputation beyond `H=400`, the prior-art claims, and anything about route 107's `d>H` obligation. I did not re-derive #5553/#5570's definitions; I took them as served and printed in the return.\n\n## What changed in the summary\n\n1. New **Best results** bullet `PROVEN#5583,e362,rt177` with the local factor, the zero-mean identity and the measured law, and the warning that #2039's anchors are `1.4-1.9x` too large.\n2. The rt177 next step (prove `sum_strip C_d << H log H` or `C_d << (log d)^{2-eta}` by resumming the classes over `d<=H`, gap = the discrepancy bound) appended to the last **Open questions** line. The summary cannot hold a twelfth bullet, so it rides on the existing line.\n3. Nothing else in v24 changed except wording, to fit the 12,000-character cap: a mechanical check (`work/make_v25.py`) confirms **all 115 return ids and all 380 numeric tokens of v24 survive**, and the five headings are in the required order. Six secondary corroborating figures were condensed (the `refl orbits 10/10/2/1/2` counts to `refl orb,no self-sym.max`; the toy `r_eff` formula to `r_eff3.6-11`; `c not flat>29#` folded into the same bullet's `c=2.1349 fails [2.15,2.33]` and `d_2 widens`; `rank 23/18`; `(18cells)`; the `core_run 11/11/17` triple). The claim and the source return of each remain.\n\n## Disclosure\n\n- Return #5583 is this department's own work (route 177 pursue, job #11282), so this promotion is a trusted check of a same-department return, not of an independent author's. I mitigated it by re-implementing the object rather than re-running the author's scripts; the reproducibility above is the basis for promotion, not the author's word.\n- To fit the cap I abbreviated some wording in the file's existing telegraphic style (`weighted`->`wtd`, `interval`->`int.`, `divisor`->`div.`, etc.). Calibration tags (`PROVEN`, `MEAS`, `VERIF`, `REPRODUCED`, `RECORDED`, ...) are unchanged.\n- The two numeral sets that differ from the entry's text (the `omega_odd` means and their counts) are a strip restriction in the entry, not a discrepancy.\n- **6 of @Benjaminsen's returns wait for a verdict** (oldest since 2026-10-06); nothing for the person to do.\n- Usage: unmeasured (the harness exposes no per-turn token record); reported as `unmeasured`, not estimated.\n","patch":null,"cpu_hours":0.02,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-23T11:10:36.527Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5583,5508,5507,2039,5574,5570],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the Freebuff harness exposes no per-turn token record for this session; usage is reported as unmeasured rather than estimated","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":"2026-10-23T11:10:49.294Z","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_499696ef17136876a6682926","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":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/5584/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"1e28b3785f941f394e606ff69cf890698a62f4f23fdda2ecb6daae8013d2b36b","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}