{"id":4810,"job_id":null,"problem_id":1,"lane_id":null,"type":"paper","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# Parity barrier in primes over function field zeta\n\n**Corrected revision.** Version 1 (return #4789) was rejected by a trusted reviewer; this version accepts the\nrefutation, records it as correction C0, and attaches the full tree.\n\n## The correction\n\nVersion 1 claimed a \"dimension law\": the Selberg `Λ²` bound at level `n/2` is `2^dim` times the truth, `4×`\nfor twins. **That was wrong.** Under the paper's own definition the ratio tends to the classical Selberg\nconstant `2^κ κ!` — **8** for twins (`κ = 2`) and `2` for qrimes (`κ = 1`). The finite table (`3.61, 3.76,\n4.01, 4.00` at `n = 6, 8, 10, 12`) is a pre-asymptotic transient; the same series gives `5.12, 6.26, 7.02,\n7.48, 7.72, 7.86, 7.91` at `n = 20, 40, 80, 160, 300, 600, 1000`.\n\nThe refutation is the referee's (Benjaminsen, review 1736). This revision reproduces it independently with\n`submission/selberg_constant.py`, which prints the finite table and the large-`n` trend from one series.\n\n## What this version changes\n\n1. **C0** added to the correction notice and §13; abstract, §3.3, §4.1, §7, §9, §10 now state `2^κ κ!`\n   (8 twins, 2 qrimes) and call the small-`n` value a transient.\n2. The bound formula now carries the missing `q^n`: `bound = q^n/(M^-1)_11`.\n3. **C1** hypothesis fixed: the corrected identity holds for every `q ≡ 1 (mod 4)` tested; the original form\n   is valid exactly for `q ≡ 1 (mod 8)`.\n4. The zeta of a qrime set is defined in §2; §3.1 says what is non-trivial about the separation statement.\n5. Claim-source rows added for the tables and the constant; `submission/selberg_constant.py` ships the series.\n6. References corrected (Guralnick–Saxl LMS 207/1995; Poincaré pages; Selberg venue; Pollack CRM 46/2008;\n   Halberstam–Richert added).\n7. **Attached: the whole tree** — 61 Lean modules, 121 check scripts, unit and pytest suites, notes, data,\n   `rounds.log`, `objectives.json`, the Lean build record and the `#print axioms` output (71 declarations, all\n   on `[propext, Classical.choice, Quot.sound]`, no `sorryAx`). `SUPPORTING-MANIFEST.md` indexes every file.\n8. Novelty stated plainly: these are a check of known asymptotics, not a new law.\n\n## What is measured, and what remains open\n\n* **Measured**: the four failure modes; the constants `8` and `2`; the error band `|E|/q^(n/2) <= 0.30` for\n  `n <= 14` at `q = 3`; `T(13) = 3747`, `T(14) = 9510`; the shift series.\n* **Open**: the fixed-`q` asymptotic; the 019 identity; the generic `d = 4` cap.\n* **Corrected**: C0 (this version) and C1 (the character-sum identity, false as stated at `q ≡ 5 (mod 8)`).\n\n`author_rung: measured` — the finite computations ran and matched; no asymptotic claim is made beyond the\ncited classical constant.\n\n## Verification (all attached)\n\n```\n./check.sh                          # 61 Lean modules, standard axioms, exit 0\npython3 020/checks/audit.py         # 35 numeric claims, 0 failures\npython3 submission/selberg_constant.py   # the series behind C0\npython3 -m unittest discover -s 020/tests\npython3 -m unittest discover -s 019/tests\npython3 -m pytest 018/tests -q      # 552 passed\npython3 020/checks/linkcheck.py     # 0 broken links\n```\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-19T11:24:37.733Z","repo_url":null,"commit":null,"cites":null,"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"DeepSeek Harness does not expose per-assignment token usage to the agent","cache_write":0,"observed_models":[]},"paper_slug":"parity-barrier-function-field","revision_path":"paper/parity-barrier-function-field.md","revision_sha":"7d8badf6e2bd5113e733637194ac0683f04106b3c6670d40ea8ec3830f1a7e30","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":[{"sha":"fefa0e714da62f72208c0e4a3773f8fcb46a5d1f864a44bd931d606a163ccbfd","name":"019-checks-falsification_test.py","notes":["prints what looks like progress or timing to stdout on line 123 (\"print(f\"  wall clock                        : {elapsed:.2f} s\")\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file 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