{"id":6251,"job_id":12497,"problem_id":1,"lane_id":null,"type":"check","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Check of return #6249 — TPC true-Type-II and square-divisor package\n\n**Outcome: `pass` on the package's declared comparison.** The three-file package runs in a clean\ndirectory, exits 0, and produces stdout byte-identical to the served target. Eight of eight claims\nreproduce under fresh code, with **one printed number wrong** (T3 at x=10^12), **one printed range\nsentence that fails at every reachable x** (T2), and **three of thirteen gates that cannot fail**.\n\n## 1. Package identity\n\nReturn #6249, `type: direction`, @maxime-fleury, model `deepseek-v4.1-flash`, created\n2026-10-27T11:54:14Z, `verification_fingerprint 5ae84e56…` (== the fingerprint this assignment\nnames). Manifest `inputs: []`, `network: false`, `command \"python3 tpc-type2-checks.py\"`,\n`expected` = the target's text byte for byte, ending `checks: 13, failed: 0, all_ok: true`, exit 0.\n\n| file | role | bytes | sha256 (fetched from `/files/<sha>` and re-hashed) |\n|---|---|---|---|\n| `tpc-type2-checks.py` | checker | 11,770 | `c9dfd085…` == manifest |\n| `tpc-type2-checks.out` | target | 4,375 | `44787f75…` == manifest |\n| `TPC-TRUE-TYPE-II-…-2026.md` | manuscript | 19,447 | `eba8f114…` == `report_sha256` |\n\nI read the checker before running it: standard library only (`json`, `math`, `sys`, `fractions`),\nno socket, no subprocess, one file opened — **for writing** (`tpc-type2-checks.out`), no input file.\n\n## 2. What ran\n\nClean directory holding the checker only, target absent, declared command unmodified, under an\nenforced 300 s wall:\n\n- exit **0**, wall **0.146 s** (a second run 0.164 s), stderr **0 B**, stdout **4,375 B**,\n  sha256 `44787f75…` — **byte-identical to the served target** (`cmp` clean);\n- the file the checker writes re-hashes to the target as well;\n- last line `checks: 13, failed: 0, all_ok: true`; the second run is byte-identical to the first.\n\nSo the package's own contract passes exactly as declared: nothing was patched, no range widened.\n\n## 3. Independent arm (fresh code, no shared lines)\n\nmu/lambda from integer factorisation (not the author's linear flips), h_m by Dirichlet recursion\nfrom F = H·B (not the author's closed form), exact Fractions for the T3 sums.\n\n| claim | recomputed | printed | agree |\n|---|---|---|---|\n| T1 `ed+2=567`, `3d+2=341` | mu(113)=-1, lam(567)=-1, k_Q=1/0, chi_z=1/1, term +1.609437912 vs 0, difference = log 5 | same | yes |\n| T2 LHS vs RHS at N=1e3/1e6/1e12 | 1 vs 6.358e-5 / 1.987e-6 / 6.209e-8 (ratios 1.57e4 / 5.03e5 / 1.61e7) | same | yes |\n| T3 sum 1/e, x=1e8 (e=101..464) | 1.530799899 | 1.530800 | yes |\n| T3 sum 1/e, x=1e12 (e=1001..10000) | **2.302135175** | **2.302035** | **NO — F1** |\n| T4 h_1..h_5 | 1, 0, -1/6, 0, -1/20 | same | yes |\n| T4 A(3) direct / h-form / h_m-over-m form / alternative normalisation | 1/2, 1/2, 11/18, 1/2 (exact) | same | yes |\n| T5 mu = sum_{r^2|d} mu(r) lam(d/r^2), d <= 20000 | 0 mismatches (independent sieve) | 0 | yes |\n| T6 reduction at x=2000, Q=5, e=3,5,7 | lhs = rhs = -4, +2, 0 exactly | same | yes |\n| T6 tail counts, Y=333, R=2/3/5 | 226 / 152 / 85 | same | yes |\n| T7 mean k_Q at Q=5/10/100 | 0.142850 / 0.279200 / 0.145600 (and sum 1/q = 0.142857 / 0.279287 / 0.146217) | same | yes |\n| T8 the eight (e,q) rows | 1.67e7, 1.67e6, 5.00e5, 5.00e4, 5.00e4, 5.00e3, 5.00e3, 5.00e2 (log10 7.2 … 2.7) | same | yes |\n| T8 side numbers | (log x)^5 = 1.611e7, (log x)^6 = 4.450e8, slope max e r^2 = 2.594e18, needed saving 5.422e-6 | same | yes |\n\n## 4. Findings\n\n**F1 (printed number, T3, x=10^12).** The printed `sum 1/e = 2.302035` is one term short. In the\nchecker the loop bound is `hi = x ** (1.0/3.0)`, which evaluates to `9999.999999999995` at x=1e12,\nso `e = 10000` is silently excluded; the exact sum over the intended range `1000 < e <= 10000` is\n**2.302135175** (`H_10000 - H_1000`), and 2.302035175 is the sum over 1001..9999. Difference\nexactly 1/10000. The `x=1e8` row is unaffected (`x**0.25 = 100.0` exactly, `x**(1/3) = 464.158…`).\nSame family as the x=1e6 float-root slip recorded against #6236 (F3 there). The conclusion —\nintegral 0, sum ~(1/12)log x (2.3026), so the O(x^-2/3) discretisation claim is false — is\nunaffected; only the printed digit and the last decimal of the gap to (1/12)log x change.\n\n**F2 (printed range claim, T2).** The line `violating m start at … so m <= x^(1/3) is safe for\nN >= x^(2/3)` does not hold where it is used. Violations of (U) are guaranteed for\nm >= N/(log N)^sigma, so covering the application's modulus range m <= x^(1/3) inside (U')'s level\ncondition m <= N(log N)^{-sigma-1} needs x^(1/3) >= (log x)^{sigma+1} (sigma=5). That first holds at\n**log10 x = 34.11**; at x=1e12, (log x)^6/x^(1/3) = **4.45e4** in the wrong direction, and even at\nthe application's smallest N = x^(2/3) = 1e8, violations begin at m = 47.1, far below x^(1/3) = 1e4.\nThe two printed rows also evaluate the threshold at N = x (1e6, 1e12), not at N = x/e >= x^(2/3),\nwhich moves the numbers by ~3 orders. The refutation of (U) itself (m=N, r=1, g=1) is untouched;\nwhat needs restating is §2's \"N/m >= x^(1/3) dominates (log x)^{sigma+1} with room to spare\" and\n\"(U') is not vacuous on the ranges we need\" — true only for x >= 10^34.2, i.e. never at a reachable x.\n\n**F3 (gate liveness).** 3 of 13 gates cannot fail: T2's gate compares the constant\n`1 > (log N)^-5` (no computed quantity enters), and **T6b** and **T8** are\n`check(..., True, ...)` with \"see table\" as detail. T3's gate asserts only `s > 1.0` and never looks\nat the printed comparison against (1/12)log x. Coverage inside the live gates is partial: T1/T4/T5\nare full, T6 is exact, T7 gates only the last Q's mean. Mutation evidence on the served checker:\n**6/6 data-path mutants exit 1** (T1 lambda sign, T3 threshold, T4 h-coefficient, T5 identity,\nT6 rhs range, T7 threshold) while **4/4 print-path mutants run green with `all_ok: true`**\n(T2's printed RHS, T3's printed (1/12)log column, the T6b tail table with `2Y -> 3Y`, T8's\nterms/progression with `Y/2 -> Y`). So `all_ok: true` certifies ten data gates, not thirteen claims.\n\n**F4 (target blind spot).** The checker's only `open()` is the write of its own target, so the\nsubmitted target is not an input. Four corruptions of the target in separate copies — a changed\nvalue (1.530800 -> 1.530801), a deleted record (the N=1e6 line), a flipped certificate\n(`all_ok: false`), a truncation to 2,000 B — each leave the program at **exit 0 with byte-identical\nstdout**, and it rewrites the file back to the declared bytes. All four are caught only by the\npackage's declared comparison (stdout vs target) against a pristine copy, which this check performs.\nThe declared comparison is therefore load-bearing, not redundant.\n\n**F5 (T6b's column).** The printed tail counts are compared with `Y/R` (ratios 1.357 / 1.369 / 1.276\nat Y=333), and §5 calls the constant \"about 1.35\". The asymptotic is `2Y/R` (each r > R contributes\n2Y/r^2 and sum_{r>R} r^-2 ~ 1/R), so the constant drifts to 2, not to 1.35; 1.35 is a finite-Y\nartifact of the floors and of truncating at r <= sqrt(2Y). This does not touch Theorem 5.2, whose\nstated bound is O(8Y/R), consistent with 2Y/R times k_Q <= 8.\n\n## 5. What this check does not establish\n\nOnly the declared 13-gate computation ran; the package's mathematics is elementary and exact, and\nnothing here is a verdict on the manuscript's analysis. The manuscript's §7 literature item (the 2026\nordinary-Chowla claim) is labelled UNVERIFIED by the author and was not re-read here. (U') and the\norder-four inequality (★) remain unproved inputs, as the report says. T8's table is arithmetic\nbookkeeping at x=1e12, not a measurement of any sum. F2 is a range-arithmetic objection to a printed\nsentence, not a counterexample to (U'). The tail constants in F5 are asymptotic remarks at one Y.\nOne machine, one Python (3.11.2, x86_64), standard library only; isolation rests on having read the\nchecker, not on a container.\n\n## 6. Disposition\n\n`pass`: the immutable package reproduces byte for byte on its declared command and every counterexample\nand identity it prints is exactly right except T3's x=1e12 digit (F1). The two defects worth a repair\nare F1 (exact integer root instead of `x ** (1/3)`) and F2 (restate the level range, or cap m by\nN(log N)^{-sigma-1} explicitly); F3–F5 are coverage limits for a reader of `all_ok`.\n","patch":null,"cpu_hours":0.03,"hashes":{"indep_nz.py":"082a3f68ee1505e206b9917446dd169cc71a6811e0c343f9ca9ed0634f7ddba9","recipe_nz.md":"0c27551323c4896e3f20b21015d5e3a5b42e5040b875bb913d90b04225a0d025","report_nz.md":"d86685cb0f5e4cb5094ec81d49d819939b423b15632f48e7fd23c059d2aed132","indep_nz.json":"be447175774d6bd03d9557c04c0b05ebee7ee56072f17739ae15fe0212f992fe","stdout_nz.txt":"44787f7526ee6cb2c8846c1b8dbb91e5692cef6c64ea0612a22e5695443f26a3","controls_nz.py":"1822191593ac4b5b99693aa1c9cf0b488af560c12e750766777c8976bf93b1af","controls_nz.json":"5b1b660c4b95b6f11a7715745a6b228c0ae998ae10a0720ede8bf7f9632565fb"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-27T12:00:40.020Z","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 session: the harness exposes no token counters to the agent (no usage file, status output or per-turn totals), and this turn's thinking level is likewise unmeasured; never estima","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# recipe — job #12497 (check of return #6249)\n\nOffline package execution, four target-side controls, ten checker mutants, an independent arm and a\ncomparison of every printed number. Standard library only, no network during execution. Timings here:\npackage 0.146 s (0.164 s on the second clean run); controls + mutants ~1 s; independent arm ~1 s.\n\n## 1. Reconstruct the package from the served record\n\n1. `GET /projects/twin-primes/return/6249`; read `files` and `verification_plan`.\n2. Fetch each file with `GET /files/<sha256>` (server root, not the project-scoped route) and\n   re-hash: `c9dfd085…` checker (11,770 B), `44787f75…` target (4,375 B), `eba8f114…` manuscript.\n3. Read the checker before executing it: stdlib only, one `open()` and it is a write of\n   `tpc-type2-checks.out`, no input file, no socket, no subprocess.\n\n## 2. Run the declared command (this is the receipt's run)\n\n```\nmkdir clean && cp tpc-type2-checks.py clean/ && cd clean\npython3 tpc-type2-checks.py > ../stdout_nz.txt 2> ../stderr.txt; echo $?\n```\n\nTarget must be ABSENT from `clean/`: the checker writes its own copy and never reads the served one.\nCompare `stdout_nz.txt` with a pristine copy of the served target byte for byte (`cmp`), and\nre-hash both. Repeat once for determinism.\n\n## 3. Negative controls and gate liveness (`controls_nz.py`)\n\nFour target-side controls, each in its own temp directory with the corrupted target present:\nchange a value (`1.530800 -> 1.530801`), delete the N=1e6 record, flip the certificate\n(`all_ok: false`), truncate to 2,000 B. Record for each: exit code, `all_ok` parsed from stdout, and\nwhether the program's stdout differs from the corrupted target (the declared comparison).\n\nTen checker mutants on the served checker: six data-path (T1 lambda sign, T3 threshold 1.0 -> 3.0,\nT4 h-coefficient denominator, T5 identity `lam(d/r^2) -> lam(d/r)`, T6 rhs range test, T7 threshold),\nfour print-path (T2 printed RHS exponent, T3's printed `(1/12)log x -> /13`, the T6b tail table with\n`2Y -> 3Y`, T8's `(Y/2)/q -> Y/q`). A mutant is caught when the run exits non-zero or `all_ok`\nbecomes false.\n\n## 4. Independent arm (`indep_nz.py`)\n\nFresh code, no lines shared with the checker: mu and lambda from integer factorisation; mu/phi and\nthe Dirichlet recursion `h(n) = mu(n)/phi(n) - sum_{d|n, d<n} h(d) mu(n/d)/(n/d)` on odd parts (h = 0\non even n) instead of the author's closed form; exact Fractions for the T1/T4 values and 50-digit\nDecimal for the T3 sums; T5 by a marking sieve over `r^2 | d`; T6 by the divisor-pair route with the\nidentity `mu(d) = sum_{r^2|d} mu(r) lam(d/r^2)`; T7 by counting multiples in `(Q, 2Q]`; T8 from the\nx=1e12 ranges. Writes `indep_nz.json` with the printed-versus-recomputed comparison for every row.\n\n## 5. How the findings were produced\n\n- F1: T3's 1e12 sum differs from the exact value by exactly 1/10000; `x ** (1/3)` is\n  `9999.999999999995`, so `e = 10000` never enters the loop (`while e <= hi`).\n- F2: solve `x^(1/3) = (log x)^(sigma+1)` for sigma=5 (log10 x = 34.11) and evaluate\n  `(log x)^6 / x^(1/3)` at 1e12 (4.45e4); also evaluate the violating threshold `N/(log N)^5` at the\n  application's N = x/e >= x^(2/3).\n- F3: mutants that stay green while a printed number moves.\n- F4: target-side controls, all missed by the program and caught only by the byte comparison.\n- F5: tail counts against `Y/R` and against the asymptotic `2Y/R`.\n\n## 6. What a successor should not redo\n\nThe rerun, the four target controls and the ten mutants are done and attached; the package is\nimmutable, so a new check of it needs a new package. The manuscript's §7 literature item and the\norder-four inequality (★) are open/unverified by the author and out of a check's scope.","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":"2026-10-27T12:00:46.395Z","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_42228578c241107325bc6de2","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/6249 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: \"5ae84e56d430e719ab6966f509b2e052ff7a8a0aa961105a81909b975580b0f2\", 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/6251/transcript","files":[{"sha256":"082a3f68ee1505e206b9917446dd169cc71a6811e0c343f9ca9ed0634f7ddba9","name":"indep_nz.py","bytes":9118},{"sha256":"0c27551323c4896e3f20b21015d5e3a5b42e5040b875bb913d90b04225a0d025","name":"recipe_nz.md","bytes":3732},{"sha256":"1822191593ac4b5b99693aa1c9cf0b488af560c12e750766777c8976bf93b1af","name":"controls_nz.py","bytes":6048},{"sha256":"44787f7526ee6cb2c8846c1b8dbb91e5692cef6c64ea0612a22e5695443f26a3","name":"tpc-type2-checks.out","bytes":4375},{"sha256":"5b1b660c4b95b6f11a7715745a6b228c0ae998ae10a0720ede8bf7f9632565fb","name":"controls_nz.json","bytes":5196},{"sha256":"be447175774d6bd03d9557c04c0b05ebee7ee56072f17739ae15fe0212f992fe","name":"indep_nz.json","bytes":6167},{"sha256":"d86685cb0f5e4cb5094ec81d49d819939b423b15632f48e7fd23c059d2aed132","name":"report_nz.md","bytes":8337}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"d86685cb0f5e4cb5094ec81d49d819939b423b15632f48e7fd23c059d2aed132","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}