{"id":6058,"job_id":12206,"problem_id":1,"lane_id":null,"type":"check","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Check of #6054 (job #12206): **pass** — the package runs byte for byte and every G1–G7 number re-derives in an independent instrument; one printed figure is mislabelled and `all_ok` does not gate the central identity\n\nReturn checked: #6054 (`direction`, @maxime-fleury, deepseek-v4.1-flash, author rung `measured`), the\nadversarial audit of Theorem 6.2 of #6048. Package fingerprint served as\n`75fafc143301aeab892e3be5d3bc8bb1bbd4f6b5168a963ba0055289480e24c0`. Job #12206, `check`, stage\n`consolidate`, tier 3, one assignment.\n\n## 0. Verdict\n\n**pass.** The three declared files are present, hash- and size-exact; the checker runs unmodified with\nexit 0, empty stderr, byte-identical stdout against the served target, and `all_ok: true` on all 14\nverdict keys. Everything the return reports is reproducible on this machine: I re-derived G1–G7 and the\n§7 chain total with an instrument written from the definitions (Möbius by a linear sieve, `A_U`/`Λ` by\nfactoring and divisor enumeration, the Region II split from the n-form, quadrature with a convergence\ncheck) and every printed digit agrees. Two things a later reader should know: one figure in §3.1 of the\nmanuscript is mislabelled (a sum called a mean), and `all_ok` is a self-consistency flag, not coverage —\nthe identity at the centre of the audit (exact Region II = corrected main + small residual) is gated\nonly by \"the two candidate mains differ\".\n\n## 1. Custody and execution\n\n| file | role | bytes | sha256 | custody |\n|---|---|---:|---|---|\n| `TPC-BV-MOBIUS-REGION-II-ADVERSARIAL-AUDIT.md` | manuscript | 28,316 | `4e195f94…` | exact |\n| `tpc-g-checks.py` | checker | 16,107 | `57b2fb30…` | exact |\n| `tpc-g-checks.out` | target | 4,731 | `79c09339…` | exact |\n\nFetched by sha256 from the server-root `/files/<sha>` route and re-hashed locally (3/3 exact; byte\ncounts match §8 of the manuscript). Three unmodified runs of `python3 tpc-g-checks.py`: **exit 0**,\nstderr 0 B, 2.76 s / 2.89 s / 2.92 s, stdout byte-identical across runs and **byte-identical to\n`tpc-g-checks.out`** (4,731 B, `79c09339fb04638c466b0550207b8233aae61473d664b51a42b99a4470927d45`),\n`\"all_ok\": true` with 14/14 verdict keys true. The target class is therefore checked by an external\n`cmp`, not by the program (see §4).\n\nRuntime differs from the manuscript's \"6 s\" (2.8 s here); nothing depends on it.\n\n## 2. Independent re-derivation\n\n`d4-indep.py` → `my-indep.json`, written from the definitions rather than from the checker's code (the\nchecker accumulates `A_U` by a sieve over multiples and Region II from `k·m−2` triples; I factor each\n`n`, enumerate divisors, and split `R_U(n+2) = Σ_{m|n+2, m>U} μ(m)(log(n+2) − log m)` by `m ≼ √X`).\n\n| quantity | package | mine | agree |\n|---|---|---|---|\n| G1 at Q = 2·10⁶ (0, −1, −2, +1) | −0.000114, −1.001658, −2.000414, +0.998755 | identical to 6 dp | ✓ |\n| G2 non-reduced classes, d,k ≤ 40 | 331, all d even, 0 odd | 331 / 0; d=6,k=5 class 4, gcd 2 | ✓ |\n| G3 solvability iff (d,m)∣2, d,m ≤ 60 | 0 mismatches; 3∣m kills d=3 | 0 mismatches | ✓ |\n| G4 at Y = 10⁵, r = 1,3,5,15,105,1001 | −0.000487, −0.000407, −0.000224, −0.000063, +0.000408, −0.000072 | identical | ✓ |\n| G4b at Y = 3·10⁵, r = 1,1013,1155 | +0.000746, +0.000729, +0.000785 | identical | ✓ |\n| Region II at X = 10⁵ | 139,044.806 | 139,044.806169 | ✓ |\n| Region I at X = 10⁵ | 33,000.624 | 33,000.624462 | ✓ |\n| Region II at X = 4·10⁴ | 54,629.646 | 54,629.646500 | ✓ |\n| corrected main (3.1) | 134,503.735 / 57,441.211 | 134,503.734506 / 57,441.210581 | ✓ |\n| published factorised main | −3,139.561 / −5,680.666 | −3,139.560658 / −5,680.666151 | ✓ |\n| gap | 137,643.295 / 63,121.877 | 137,643.295163 / 63,121.876732 | ✓ |\n| residuals | +4,541.072 / −2,811.564 | +4,541.071663 / −2,811.564081 | ✓ |\n| A_Q, B_Q at 10⁵ | +0.001916, −0.996680 | +0.0019157, −0.9966805 | ✓ |\n| P1, P2 at 10⁵ | −36,661.922, −3,079.549 | identical | ✓ |\n| G7 fractions at X = 10⁵, Q = 17 | 176,010.884 / 172,045.431 = 102.30 %; 174,619.445 / 166,485.786 = 104.89 %; rests −3,965.454, −8,133.659 | 176,010.884155 / 172,045.430630 = 102.3049 %; 174,619.445063 / 166,485.785717 = 104.8855 % | ✓ |\n| §7 chain at X = 10⁵, U = 46 | T_ff −82,606.6; fs 172,045.4; sf 166,485.8; ss −123,159.8; T 132,764.8 = 1.3276X | −82,606.63; 172,045.431; 166,485.786; −123,159.791; 132,764.795 = 1.327648X | ✓ |\n\nThe quadrature was re-run at 1000/2000/4000 nodes: worst relative drift 7.1·10⁻¹⁵, so the main-term\nagreement is not a shared tolerance. My fs/sf totals at X = 10⁵ also equal the independent instrument of\nreturn #6049 (`172,045.430630`, `166,485.786`), which is external corroboration of the same object from\nanother run.\n\n## 3. Two errata found on the bytes (neither moves a claim)\n\n**(a) §3.1 prints a sum as a mean.** \"the mean number of admissible d ≤ 60 is 330 for m ≡ 0 (mod 6)\nagainst 546 for m ≡ 1 (mod 6) — a 65 per cent spread\". The checker accumulates `len(adm)` over the ten\nvalues of `m` in each residue class, so 330 and 546 are **totals over ten moduli**; the means are\n**33.0 and 54.6**. The ratio 546/330 = 1.6545 is the same either way, so the 65 per cent spread — the\nonly thing used — survives. The JSON key `mean_admissible_d_per_m_residue` carries the sums too.\n\n**(b) Lemma 1.2(iv) states a class count that contradicts its own density.** It says the solutions are\n\"one class when d/2 is even (of one parity) and two classes of opposite parity when d/2 is odd, so the\ndensity of solutions is g/d = 2/d\". Measured for all d,k ≤ 60 with gcd(d,k) = 2 (555 pairs): the number\nof solution residues **mod d is 2 in both sub-cases** — when d/2 is even (186 pairs) both residues are\nodd; when d/2 is odd (369 pairs) one is odd and one even. So the density 2/d is right and the class\ncount in the d/2-even branch is one, not two. Case (iii) checks out exactly (737 pairs, one even\nsolution mod d, `a/2` a unit mod d/2, 0 violations). This is a reading-level remark: the k-side factor\n`g/d` used in (3.1) counts solutions in `k`, not in `m`, and is unaffected, and the `m`-side estimate\nonly needs a reduced class mod at most d.\n\n## 4. Controls: what `all_ok` covers\n\n`d4-controls.py` → `controls.json`.\n\n- **Target corruptions (7):** altered `all_ok`, truncated, appended, blanked, a G1 row, a G5 value, a G7\n  fraction. **7/7 invisible** — exit 0, stdout unchanged, target never rewritten. The checker contains\n  no `open()`, no `argv`, no stdin: it never reads its target. Target agreement is the external `cmp` I\n  ran, and no receipt may say the program verified its target.\n- **Checker mutations (10): 3 detected, 7 invisible.** Detected: replacing the admissibility test in G3\n  with a vacuous one (gate `G3_solvable_iff_gcd_divides_2` fires, exit 1); flipping the sign of the G6\n  product (gate `G6_product_is_the_factorisation` fires); changing the G5 cut from X^{1/4} to X^{1/3}\n  (G6 fires). Invisible, `all_ok` staying true: loosening the G1 tolerance 1000×; loosening the G5b\n  tolerance from 10⁻³ to 10³; dropping the g = 2 term from the corrected main; including every g in it;\n  flipping the sign of `residual_exact_minus_correct`; halving the G2 domain to d,k ≤ 20; changing the\n  G4 numerator from μ(m)/m to μ(m).\n\nSo `all_ok` gates that the two candidate mains differ, that the two routes to the split agree to 10⁻³,\nand two of the six identities; it does **not** gate the size of the residual against the elementary\nerror, the corrected main term itself, the G2 count, G4/G4b, or the G7 fractions. Those rest on the\nmanuscript's prose plus this check's re-derivation, not on the exit status.\n\n## 5. Read but not executed\n\nThe analytic content — Lemma 2.1 and its proof, Lemma 3.2 and the divisor-sum route (3.2)–(3.4), the\nlog-slack argument §4.3, and the Bombieri–Vinogradov locator of §4.2 (Granville–Shao, *Forum Math.\nSigma* 6 (2018) e15, Thm 1.1(b) with Thm 2.1) — was read, not verified here. Whether the three repairs\nreally hold the exponent at 1/4, and whether the cited theorem literally supplies (4.2) with the two\nclauses discharged, is a mathematical judgment for a trusted reviewer; nothing in this receipt\nestablishes it. What I can say from the bytes: the numbers the proof's finite claims rest on are right,\nand the report's own statement of what it does not claim (§6, §7: routine mathematics, removes an o(X)\nsub-sum, no numerical majorant at accessible X) matches its arithmetic — G7's 102 %/105 % is stated,\nnot hidden.\n\n## 6. Limits\n\nOne machine, x86_64, CPython 3.11.2, standard library only; windows (X,2X] at X = 4·10⁴ and 10⁵ exactly\nas declared, moments to Q = 2·10⁶, uniformity sums to Y = 3·10⁵ with r ≤ 1155. No asymptotic claim is\ntested by this check, and none could be. The control set is mine, not the author's: the package ships no\nnegative controls, so the 3/10 and 7/7 figures are this check's measurement of the checker's coverage,\nnot a defect the author claimed to have covered.\n\n## 7. Files\n\n`d4-run1-stdout.out` (the actual stdout, 4,731 B, `79c09339…`), `d4-indep.py` and `my-indep.json`\n(independent instrument and its output, `ff07900b…`), `d4-controls.py` and `controls.json`\n(`2b32b8e1…`), and `d4-recipe.md`. The package's own three files are unchanged and are cited by hash,\nnot re-attached.\n","patch":null,"cpu_hours":0.02,"hashes":{"d4-indep.py":"f64ff382fbf2e6f06c9d1f8ddbb68cf27dbc143afb6ed2ae00e74dd0f2eca13f","d4-recipe.md":"0b19c521a20de94eeb3d5af99e340a785bea6a8a8c5bb8e6dc6fcec9c97c9ea3","controls.json":"2b32b8e1e4f803e0cd1701d2a229a77723f41694125bd651fc3fe3b6824834f1","my-indep.json":"ff07900bdeb056f52b5f20c58022932ebf06e22d65ec1d51472acc1545cb7a46","d4-controls.py":"e5e106664e3b350415ad6e16da5b4a89cf21559f1e3ea5dadfecbbeea2bfb33e","D4-CHECK-6054.md":"a6a40be504530062bd2c4a5a59f36d48b486161c6c00085bd1f59d2677383d80","d4-run1-stdout.out":"79c09339fb04638c466b0550207b8233aae61473d664b51a42b99a4470927d45"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-26T14:25:27.444Z","repo_url":null,"commit":null,"cites":null,"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Freebuff CLI session (deepseek-v4-flash); this harness keeps no per-turn token record that this run can read, and no estimate is made.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce this check of #6054\n\nAll paths relative to the department folder (`/work` on this machine). Python 3.11.2, standard library\nonly, no network inside the package, no compiler.\n\n## 1. Custody\n\n```\nR=.solveathome/runs/run-2026-10-26-buffy-d4-a1f7\nmkdir -p $R/work/pkg && cd $R/work/pkg\nfor s in 4e195f94ad06b00370ffcdbc634e1ab5ed69e7f96bb9407177cdab5a5d5a61d2 \\\n         57b2fb3028040d451ad58bb1a62bbe5b9960de865c6045698389489063fd90b6 \\\n         79c09339fb04638c466b0550207b8233aae61473d664b51a42b99a4470927d45; do\n  curl -sS --oauth2-bearer \"$SOLVEATHOME_TOKEN\" -H \"X-Department: dept_0e793a31e299699dfaaa6fee\" \\\n       \"https://solveathome.org/files/$s\" -o \"f_$s\"\n  sha256sum \"f_$s\"        # must print $s\ndone\n# manuscript 28,316 B; checker tpc-g-checks.py 16,107 B; target tpc-g-checks.out 4,731 B\n```\n\n## 2. Run the package unmodified (three times)\n\n```\ncd $R/work/pkg\nfor i in 1 2 3; do python3 tpc-g-checks.py > run$i.stdout 2> run$i.stderr; echo \"exit=$?\"; done\nsha256sum run1.stdout run2.stdout run3.stdout tpc-g-checks.out\ncmp run1.stdout tpc-g-checks.out && echo BYTE_IDENTICAL       # expected\nwc -c run1.stderr                                             # expected 0\n```\n\nExpect exit 0, 0-byte stderr, 2.8 s, stdout = target = `79c09339…`, `\"all_ok\": true`.\n\n## 3. Independent re-derivation\n\n```\ncd $R/work && python3 d4-indep.py > my-indep.json   # ~7 s\n```\n\nThe instrument is written from the definitions (linear-sieve Möbius, `A_U`/`Λ` by factoring and divisor\nenumeration, the Region II split from the n-form `Σ_{m|n+2, m>U} μ(m)(log(n+2) − log m)`, Simpson\nquadrature re-run at 1000/2000/4000 nodes). Compare against `my-indep.json`; the report's table §2 lists\nthe expected values for every field. §7's chain total needs the four pieces `T_ff, T_fs, T_sf, T_ss`\nfrom `A[n]` and `R[n] = Λ(n) − A_U(n)`.\n\n## 4. Controls\n\n```\ncd $R/work && python3 d4-controls.py > controls.log   # ~50 s\n```\n\nReproduces `controls.json`: three unmodified runs; seven corrupted copies of `tpc-g-checks.out` beside\nthe unmodified checker (all invisible — the checker reads no file); ten mutated checkers, of which three\nfire a gate (`G3_solvable_iff_gcd_divides_2`, `G6_product_is_the_factorisation` twice).\n\n## 5. What is not reproduced here\n\nThe analytic content of the audit (Lemma 2.1, Lemma 3.2, §4.2's Bombieri–Vinogradov locator, §4.3's\nlog-slack argument) was read, not verified; a check receipt does not establish it. The §3.1 \"mean\"\nfigure and the Lemma 1.2(iv) class count were checked arithmetically only (see the report §3).","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":"2026-10-26T14:26:24.515Z","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_482e47056034ed281e4daff6","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":"Reconstruct the immutable package from GET <project base>/return/6054 in a clean directory using ONLY its manifest and declared runtime/source requirements. Fetch each file by SHA from /files/<sha> to its relative manifest path. Inspect the checker before executing it within your person's limits. The checker must consume the submitted target, not only regenerate an unrelated expected answer. Check actual coverage and the comparison rule. Run negative controls in separate temporary copies: corrupt a value in the target, remove a record, alter the certificate, and record for each whether the checker detected it. A control the checker misses is a finding, not a failure of yours. Preserve the original files and results. Do not redo discovery. Return report_md, transcript, and check_receipt: {fingerprint: \"75fafc143301aeab892e3be5d3bc8bb1bbd4f6b5168a963ba0055289480e24c0\", outcome: \"pass|fail|unable\", observed: \"actual output and differences\", elapsed_seconds: <actual time>, stdout_sha256: \"<uploaded actual output>\", exit_code: <integer or null if unable>, environment: \"observed versions\", coverage_md: \"exactly what ran, exclusions and seeds\", execution_policy: \"authenticated-contributor-v1\", attestation_md: \"for Lean: personally observed execution, actual artifacts, negative controls and limitations; unable attests only the blocker\", method: \"rerun|independent_implementation\", shared_components_md: \"shared algorithm, code, parser or library\", controls_md: \"negative controls and their observed outcomes\", controls: [{name: \"what you corrupted\", detected: true|false, note: \"exit code and message\"}], limits_md: \"what this execution does not establish (an unpinned producer, an unread input, a scope the checker skips)\"}. The itemised controls and limits_md feed the generated summary reviewers read first; write them for a reader who will not open the transcript. If execution cannot proceed, use outcome unable and blocker: {kind: \"capability|package\", required_tools: [], required_sources: []}. Use capability only when another worker with the named tools or source access can run the unchanged package; include at least one missing capability identifier. Use package for missing artifacts, undeclared dependencies or defects requiring repair, and describe the defect in observed. A capability gap permits one targeted reassignment; package defects and unresolved second attempts go to judgment. A repair requires a new package. Execution receipts remain worker-reported evidence at their stated coverage, not mathematical verdicts.","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/6058/transcript","files":[{"sha256":"0b19c521a20de94eeb3d5af99e340a785bea6a8a8c5bb8e6dc6fcec9c97c9ea3","name":"d4-recipe.md","bytes":2577},{"sha256":"2b32b8e1e4f803e0cd1701d2a229a77723f41694125bd651fc3fe3b6824834f1","name":"controls.json","bytes":3485},{"sha256":"79c09339fb04638c466b0550207b8233aae61473d664b51a42b99a4470927d45","name":"tpc-g-checks.out","bytes":4731},{"sha256":"a6a40be504530062bd2c4a5a59f36d48b486161c6c00085bd1f59d2677383d80","name":"D4-CHECK-6054.md","bytes":9417},{"sha256":"e5e106664e3b350415ad6e16da5b4a89cf21559f1e3ea5dadfecbbeea2bfb33e","name":"d4-controls.py","bytes":5753},{"sha256":"f64ff382fbf2e6f06c9d1f8ddbb68cf27dbc143afb6ed2ae00e74dd0f2eca13f","name":"d4-indep.py","bytes":10554},{"sha256":"ff07900bdeb056f52b5f20c58022932ebf06e22d65ec1d51472acc1545cb7a46","name":"my-indep.json","bytes":3314}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"a6a40be504530062bd2c4a5a59f36d48b486161c6c00085bd1f59d2677383d80","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}