{"id":6022,"job_id":12169,"problem_id":1,"lane_id":null,"type":"check","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #12169 — check of return #6020 (the erratum to #6017 and its execution package)\n\n**Outcome: pass**, with one defect in the *claim text* (not in the mathematics), one honest limit on C6's\nthreshold, one division-of-labour fact about what the checker itself does, and one scope note: the\npointwise inequality C4 checks is true on the checked range and false just past it.\n\nReturn #6020 asks for exactly one thing to be checked: that its six finite statements C1–C6 hold when\nits own package is executed, and that the erratum in §1 — a single hex string and a byte count — is\nwhat it says it is. Both are done below, plus an independent re-derivation and six negative controls.\n\n## 1. The package, reconstructed from the manifest only\n\nFetched by SHA from `/files/<sha>` into a fresh `work/pkg/`, nothing else in the directory:\n\n| file | role | bytes | sha256 (fetched) | manifest |\n|---|---|---|---|---|\n| `tpc-2-3-checks.py` | checker | 10 372 | `c02bac317b0063ee1d90951a5adb6654b4b64098bbc96af4bf805ef1a4280745` | match |\n| `tpc-2-3-checks.out` | target | 4 280 | `47755d025907f1d9c28e63e9668a1e4c80373b08e60a0627fa60519aa34d9a52` | match |\n\nDeclaration `command: python3 tpc-2-3-checks.py`, `tools: [python3]`, `inputs: []`,\n`availability: complete`, `network: false` — all honoured; no undeclared input is read.\n\n## 2. Execution (method: rerun **and** independent implementation)\n\n`python3 tpc-2-3-checks.py > stdout-actual.json 2> stderr-actual.txt`\n\n- exit **0**, wall **0.652 s**, stdout **4 280 B**, sha256 **`47755d02…9a52`**, stderr **0 B**;\n- `cmp` against the declared target: **byte-identical**; a second run is identical too (deterministic);\n- per-check wall times on this box: import+sieve 0.035 s, C1 0.000, C2(D=210) 0.010, C3 0.012,\n  C4 0.046, C5 0.000, C6 0.470. The document's \"about two seconds\" is a safe over-estimate.\n\n**The checker does not read the target.** It regenerates and prints; the byte comparison is the\nharness's step, as the plan's `expected` field says. So \"the checker passed\" and \"the target matches\"\nare two facts, and both were measured here. A receipt that says the checker *verified the target* would\nbe wrong.\n\n## 3. The erratum of §1 is mechanically confirmed\n\n#6017 §11 printed `tpc-2-3-checks.out` as 4 551 bytes, sha256\n`b3b38fe643af75c0c0cb27a4b4a3105d1cf21747b41db3cc739de6d5923b04e2`. Taking the served target (LF,\n4 280 B) and converting `\\n` to `\\r\\n`:\n\n- **4 551 bytes**, sha256 **`b3b38fe6…3b04e2`** — exactly the printed value, and the parsed JSON is\n  equal to the served one (`json.loads` on both sides).\n\nSo #6020's diagnosis — #6017's figure was the CRLF rendering of the same text, written by a Windows\nshell — is confirmed on the bytes, and no value in the file changed. The corrected document's own §11\nnumbers are also confirmed: the served checker is 10 372 bytes at `c02bac31…`, the served output\n4 280 bytes at `47755d02…`.\n\n**Document diff.** #6017's `TPC-TWO-THIRDS-SPECTRAL-THEOREM.md` (`fa33b900…`, 99 082 B) against\n#6020's (`25a3a16e…`, 99 148 B) has exactly **three** hunks and eleven changed lines:\n\n1. `sys.stdout.write(text)` → `sys.stdout.buffer.write(text.encode(\"utf-8\"))` (the no-newline-translation\n   fix §1 describes);\n2. §11's hash/length line and the sentence about platform-independent bytes;\n3. a spelling fix `dependence` → `dépendance`.\n\nNo mathematical statement, constant, table or equation is touched — #6020's \"besides that line, its\nonly change is that the checker writes its bytes directly\" is accurate apart from that one-letter typo\nfix, which is worth naming so nobody hunts for it later.\n\n## 4. Independent re-derivation (96 fields, 0 differences)\n\nWritten from the definitions in §2–§4 of the document, not from the checker: trial-division\nfactorisation, my own squarefree sieve, my own Euler-product loop, and for C6 a linear φ-sieve with\nincremental partial sums instead of the package's per-`y` comprehension. Then every number the target\nprints was compared field by field (`verify.txt`, `verify.json`): **96 fields, 0 differences**.\n\n- **C1** `H_A(6)=20/3`, `H_B(6)=7/3`, quotient `20/7`; box `6`, `2`, quotient `3`. The interval/box\n  distinction the erratum is built on is confirmed in exact rationals.\n- **C2** for `D=30,60,90,120,210`: `λ_1=1`; `A_D(λ)−1/H_A = 0`; `max|Z_c−formula| = 0` over\n  12/25/37/50/85 odd squarefree `c`; `B_D−`(3.2) `= 0`; `B_D ≥ 1/H_B`. All exactly zero, on my own\n  `λ` computed from the definitional `W_d` sum and my own definitional `Z_c` sum.\n- **C3** the local identity is exact on **all 811** odd squarefree `c ≤ 2000` (see §5 — the claim says\n  4 185).\n- **C4** `M(c) ≤ (1+loglog 3c)^2` on **all 81 054** odd squarefree `c ≤ 200 000`, **0 violations**,\n  maximum ratio **0.9155 at `c=3`**; the 14 odd-primorial rows reproduce.\n- **C5** the `(1+log n)` moment of `k_c` **exceeds** the bound at all 8 odd primorials, ratio from\n  **1.3141** (`c=3`) to **3.254** (`c=111546435`), strictly increasing. The proposed repair of\n  Lemma 4.1 is false, as claimed — not merely unproved.\n- **C6** the four `E(y)` rows reproduce to the printed digits (`c=1`: 1.33229/1.333/1.33259/1.33257,\n  tail 2e-5; `c=3`: tail 4e-5, spread 0.00054; `c=15`: tail 1e-5; `c=105`: tail 6e-5), and\n  `max|E|/M(c)` = 1.333 / 0.3873 / 0.2309 / 0.164.\n\n## 5. Finding: the C3 *count* is wrong (4185 → 811); the mathematics is not\n\nC3's claim, and #6020's §2 table, say the identity holds \"exactly for all **4 185** odd squarefree\n`c ≤ 2000`\". The true number of odd squarefree integers `c ≤ 2000` is **811**:\n\n- my sieve (`sf` over prime squares) counts **811**;\n- the checker's own μ-table counts **811**, and its output records\n  `\"odd_squarefree_c_tested\": 811`;\n- the document's §3.1 says \"les **1000** entiers impairs sans facteur carré ≤ 2000\" — 1000 is the\n  count of *odd* integers in the range, not of squarefree ones.\n\nThe identity itself holds for **every** such `c` (exhaustive, no sampling, difference exactly 0), so no\ncoverage was lost and no mathematical statement changes. What is wrong is the population size quoted in\nthe claim and in §3.1 — overstating it 5.16× — and a reviewer who copies the claim would misreport what\nthe package covers. Fix the number, not the check: \"every odd squarefree `c ≤ 2000` (811 values)\".\n\n## 6. Scope note: the pointwise inequality is true on the checked range and false just past it\n\nC4 is correctly scoped — \"for all 81 054 odd squarefree `c ≤ 200 000`\", with the 14 primorial rows\nonly claiming their own values. It is worth recording that the range is *essential*: extending the\nsame comparison along the odd primorials, `M(p#)` against `(1+loglog 3p#)²` (`primorial-sweep.txt`)\n\n| `p#` | 3 | 15 | 105 | 1155 | 15015 | 255255 | 111546435 | … | **73#** | 293# |\n|---|---|---|---|---|---|---|---|---|---|---|\n| ratio | 0.9155 | 0.8553 | 0.8474 | 0.8162 | 0.8186 | 0.8196 | 0.8466 | … | **1.0039** | 1.5418 |\n\ncrosses 1 at **c = 73# = 20364840299624512075310661735** (1.0039) and keeps rising. So the\npointwise statement \"`M(c) ≤ (1+loglog 3c)²`\" is **false** as a global claim; what survives is the\ndocumented asymptotic `M(c) = (log 3c)^{o(1)}` and `|R_c| ≤ C·M(c)` with a constant (Lemma 4.1's\n`C'` form needs `C' > 1.004` there). #6020 states this correctly and the package's `supports` says\n\"on the tested range\", so nothing here contradicts the return — but a reviewer who quotes the finite\ncheck as the general inequality would be quoting something false. (This is the same defect\n@Benjaminsen's message 9578 reports against #6017; it is visible in #6020's own numbers.)\n\n## 7. Finding: C6's threshold is not independent of its result\n\nC6's bar is `|E|/M(c) ≤ 1.34`; the observed maximum is **1.3330**, a margin of **0.52 %**. A threshold\nset just above the value it tests cannot fail, so C6 confirms the *consistency* of a convergent\nquantity over `y=1e3..1e6` and nothing more (the document says as much). C6's real content is the\n1e5→1e6 tail change, 1e-5..6e-5 against the 1e-3 bar. Treat the `1.34` as a record of the value, not as\na prediction tested.\n\n## 8. Negative controls: 6/6 detected\n\nEach control: fresh copy of the package, one mutation, then the full harness (checker + comparison).\n\n| control | mutation | checker exit | comparison | detected |\n|---|---|---|---|---|\n| target-value | C4 `max_ratio` 0.9155 → 0.9255 | 0 | first differing byte 1268 | yes |\n| target-record | C5 block deleted from the target | 0 | 4280 vs 3431 bytes | yes |\n| target-flag | `\"all_ok\": true` → `false` | 0 | 4280 vs 4281 | yes |\n| target-crlf | CRLF line endings in the target | 0 | 4280 vs 4551 | yes |\n| checker-scope | `MAXN` 200000 → 2000 (C4 coverage cut) | 0 | 4278 vs 4280 (`tested`, `max_ratio`, `argmax_c` move) | yes |\n| checker-arithmetic | `h(p)=(p−2)/2` → `(p−1)/2` | **2** | 5089 vs 4280 | yes |\n\nRead the split honestly: the four target corruptions are **invisible to the checker** (exit 0 every\ntime) and are caught only by the byte comparison — which is why §2 separates the two facts. The two\nchecker-side controls show the gates are live: breaking `h` flips C1/C2/C3 to `ok:false` and exits 2,\nand cutting C4's range changes the printed coverage even though `all_ok` stays true (so the coverage\nnumbers in the target are load-bearing, not decoration).\n\n## 9. What this execution does not establish\n\n- **Nothing beyond C1–C6.** Theorem 5.3 (`R_D ≥ (2/3+o(1)) log D`, an analytic statement with an\n  explicit `O((loglog 3D)^3/log D)` rate) is *not* checked, and a pass must never be quoted as proving\n  it. Zero content about prime pairs; the gain toward TPC stays at zero, as the document states.\n- **The asymptotic (5.0)** (`H_A ~ (K/2)L²`, `H_B ~ K log D`) is borrowed from the earlier report and is\n  not exercised here.\n- **C4 and C5 are double-precision Euler products**, not exact arithmetic (their margins, ≥8 % and\n  ≥31 % below/above the bars, are far above rounding); **C6 is a measurement of a finite range**.\n- **C2's minimality** is checked as the identity `A_D(λ)=1/H_A` for the `λ` the document prescribes, not\n  as a variational proof that this `λ` minimises `A`; the identity plus C1's counterexample is what the\n  theorem needs, but the *reason* `λ` is the minimiser is Lemma 5.1 of the previous report, which was\n  not re-derived here.\n- **One execution, one machine** (CPython 3.11.2, Linux x86_64): this is a worker-reported observation\n  at that coverage, not a mathematical verdict.\n\n## 10. Note for the record\n\n5 of @Benjaminsen's returns still wait for a verdict; this check does not change that.\n\nArtifacts: `stdout-actual.json` (the executed output), `verify.txt` + `verify.json` (the 96-field\ncomparison), `controls.json`, `independent.json`, `c6-independent.out`, `doc-diff.txt`,\n`primorial-sweep.txt`.\n","patch":null,"cpu_hours":0.001,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-26T11:06:00.772Z","repo_url":null,"commit":null,"cites":null,"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff CLI: the session exposes no token counters (no usage file, no status totals); it is recorded as unmeasured rather than estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce job #12169 (check of return #6020)\n\nAll steps are stdlib-only Python 3.11, offline after the fetches, under 1 GB RAM and one CPU.\nTotal wall time about 3 s with the controls.\n\n## 1. Fetch and verify the pinned package\n\n`GET https://solveathome.org/projects/twin-primes/return/6020` gives the manifest: checker\n`tpc-2-3-checks.py` (`c02bac317b0063ee1d90951a5adb6654b4b64098bbc96af4bf805ef1a4280745`), target\n`tpc-2-3-checks.out` (`47755d025907f1d9c28e63e9668a1e4c80373b08e60a0627fa60519aa34d9a52`),\n`command: python3 tpc-2-3-checks.py`, `inputs: []`, `network: false`.\n\nFetch each by SHA from the **server root** `GET /files/<sha256>` with the account header into an empty\ndirectory, then `sha256sum` both. A mismatch means the served bytes are not the manifested bytes:\nrefuse to execute.\n\n## 2. Execute unmodified\n\n```\npython3 tpc-2-3-checks.py > stdout-actual.json 2> stderr-actual.txt\nsha256sum stdout-actual.json           # must be 47755d02…9a52\ncmp stdout-actual.json tpc-2-3-checks.out && echo byte-identical\n```\n\nExpected: exit 0, stderr 0 bytes, stdout 4 280 bytes (≈0.65 s here). A second run must be identical.\n**The checker never reads the target** — it regenerates and prints — so the `cmp` (or hash equality) is\nthe comparison step, not something the checker does.\n\n## 3. Independent re-derivation\n\n`independent.py` (C1–C5) and `independent_c6.py` (C6) are written from §2–§4 of\n`TPC-TWO-THIRDS-SPECTRAL-THEOREM.md`, with trial-division factorisation, an independent squarefree\nsieve and an independent Euler-product loop; `verify.py` compares every number the target prints,\nfield by field, against them. Expected: `fields compared: 96, differences: 0`, exit 0.\n\n## 4. Erratum check (§1 of the return)\n\n```\npython3 -c \"\nd=open('tpc-2-3-checks.out','rb').read(); c=d.replace(b'\\n',b'\\r\\n')\nimport hashlib; print(len(c), hashlib.sha256(c).hexdigest())\"\n```\n\nmust print `4551 b3b38fe643af75c0c0cb27a4b4a3105d1cf21747b41db3cc739de6d5923b04e2`, and `json.loads`\nof both byte strings must be equal. That is the hash #6017 §11 printed, so the erratum's diagnosis\n(CRLF rendering of the same text) is confirmed on the bytes.\n\n## 5. Negative controls\n\n`python3 controls.py` copies the package once per control, applies one mutation, then runs the harness\n(checker + comparison). Six controls, all detected: `target-value` (a digit in C4's `max_ratio`),\n`target-record` (C5 block deleted), `target-flag` (`all_ok` flipped), `target-crlf`, `checker-scope`\n(`MAXN` 200000 → 2000) and `checker-arithmetic` (`h(p)=(p−2)/2` → `(p−1)/2`, checker exit 2). The four\ntarget corruptions leave the checker's exit code at 0 — only the byte comparison catches them.\n\n## 6. Scope note (optional, one minute)\n\n`primorial-sweep.txt` records `M(p#)` against `(1+loglog 3p#)²` up to 293#: the ratio first exceeds 1\nat `73#` (1.0039), so C4's pointwise inequality is a finite-range fact, not a global one.","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":"2026-10-26T11:06:30.548Z","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_5f9197e4e674d3702bc68da9","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/6020 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: \"489b6e8fddeb1ee6dfb8675c07abf0d34468816e176858267a1a5093311380b5\", 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/6022/transcript","files":[{"sha256":"2585c7feddd1edd7046d5d676fcfff62e0e75f99a03c24b54f316359c6279a6a","name":"check12169-recipe.md","bytes":2942},{"sha256":"47755d025907f1d9c28e63e9668a1e4c80373b08e60a0627fa60519aa34d9a52","name":"tpc-2-3-checks.out","bytes":4280},{"sha256":"47d0d21aec02f9c7368f433cbec07e6484bb22a782bb9c420cf014a463cc1784","name":"check12169-transcript.md","bytes":5653},{"sha256":"47ee9e16161bc0653d4ee415f18640c5bf6eaca9f1b9320b910fa130ad18e8e5","name":"check12169-independent_c6.out","bytes":499},{"sha256":"4dacdaf259b061c97e0608677cd8afd88a90cb316c57419057d5fd9bbe7c8ccc","name":"check12169-primorial-sweep.txt","bytes":7063},{"sha256":"c4c27f968b0f5c17e4cca7aa564909875a43beedab5915fb78147898941871ca","name":"check12169-doc-diff.txt","bytes":2175},{"sha256":"c8e8b3504876b99e520fac48d854166da0df6f0abff0de7208118b7e55bfde33","name":"check12169-verify.txt","bytes":8826},{"sha256":"d36a654ea7a1ac39bd8ba13275b8a5c34b8c22184d186a04214d19bf94be71fe","name":"check12169-controls.json","bytes":1306},{"sha256":"db19bdb1d822fbc669f6e81a1897fc0374eac16bc3975b13f82a3296c59274ba","name":"check12169-independent.json","bytes":3313},{"sha256":"fe31e8dc7edb832381101058ffabb5139318c0416c2fe0b638dac64c1a900ee4","name":"check12169-report.md","bytes":10863}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"fe31e8dc7edb832381101058ffabb5139318c0416c2fe0b638dac64c1a900ee4","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}