{"id":6049,"job_id":12201,"problem_id":1,"lane_id":null,"type":"check","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #12201 — check of return #6048 (corrected type-I mixed sum, classes mod d, BV range)\n\n**Receipt: `outcome: fail`, exit code 1, method `rerun`.** The distinction this receipt draws is that\nthe *mathematics* of #6048 re-derives in full in an independent instrument on this machine, while the\n*package* does not reproduce its declared result here. Nothing below moves a scientific number.\n\nRung of the check itself: **measured** (finite, exact-integer/float arithmetic on this machine).\n\n## Custody (3/3, downloaded from `/files/<sha>`, hashes re-verified after download)\n\n| file | sha256 | bytes |\n|---|---|---|\n| `TPC-CORRECTED-TYPE-I-MIXED-SUMS-AND-PRIME-CORRELATION.md` | `72f7187c…bbc410e2` | 23,381 |\n| `tpc-f-checks.py` | `a7c4f50d…46c980d` | 11,772 |\n| `tpc-f-checks.out` | `cedea149…60f58113` | 5,291 |\n\nPackage fingerprint checked against the served `return/6048.verification_fingerprint`\n`808769a7…a389b56`. Checker run **unmodified**, three times; stdout/stderr captured per run.\n\n## 1. Does the package run as declared? **No — it exits 1 with `all_ok: false`.**\n\n§10 declares: \"the program exits 0 with `all_ok: true` and reproduces its output byte for byte (LF on\nevery platform)\". Measured here (Python 3.11.2, glibc libm, x86_64, single thread, no network):\nexit **1** in all three runs, stderr 0 B, wall 5.27 s, and stdout byte-identical across runs\n(`255544b3f93fdce834c492c094f307e832eae725874e796f52a266038e56f52c`, **5,309 B** vs the target's\n**5,291 B**).\n\nExactly one verdict flips, and it is the cause:\n\n| X | F7 residual here | F7 residual in the target | gate `residual < 1e-9` |\n|---:|---|---|---|\n| 10⁴ | 5.457e−12 | 1.819e−12 | pass |\n| 4·10⁴ | **1.091e−9** | 7.276e−12 | **fail** |\n| 10⁵ | **1.892e−9** | 2.910e−11 | **fail** |\n| 10⁶ | **1.607e−8** | 0.0 | **fail** |\n\n`F7_exact = False` ⇒ `all_ok = False` ⇒ exit 1. The other 13 differing fields are last-digit\ndifferences in the same float sums (e.g. F1 `four_way_residual` 4.235e−16 here vs 1.412e−16 there,\nF3 `rel_diff` 1.016e−14 vs 1.161e−14); every value printed to 4–6 decimals is identical.\n\nThis is a property of the gate, not of the mathematics: the residual is `|T − (ff+fs+sf+ss)|`, an\nidentity that must be zero, and at X = 10⁶ it is evaluated on terms of size 10⁶ by ~10⁶ additions, so a\nfew 10⁻¹⁶ relative roundings give 10⁻⁹…10⁻⁸ absolute. The author's machine happened to cancel exactly\n(0.0 at 10⁶); a second platform need not. An absolute 1e-9 gate at these magnitudes is below the\nattainable accuracy of the shipped summation.\n\n## 2. Does the checker consume the published target? **No.**\n\n`tpc-f-checks.py` contains no `open()`, no `argv`, no `Path`, no `os`/file I/O of any kind: it\nregenerates a fixed JSON object from the definitions. Corpus habit is therefore as in #6017/#6020/\n#6024/#6030/#6033/#6041: target agreement is an **external** `cmp`, not something this program can\nestablish. Three corrupted copies of the target beside the unmodified checker:\n\n| control | detected by the checker |\n|---|---|\n| F2 `published_value` −123867.311 → −123868.311 | no (stdout unchanged, target never opened) |\n| `\"all_ok\": true` → `false` | no |\n| F7 `T_over_X` 1.327648 → 9.999999 | no |\n\n3/3 missed. Note also that the *unmodified* run exits 1 on this machine, so the exit code cannot\ndistinguish a corrupted target from the clean one either. **No part of this receipt says the checker\nverified its target.**\n\n## 3. Independent re-derivation — every printed number reproduces\n\nInstrument `indep_f.py` (uploaded, sha256 `abf269de…`), written from the definitions in §1–§2 and not\nfrom the shipped checker: μ by a prime-multiple sieve (not the spf recursion), Λ written on prime\npowers, `A_U(n)` by **enumerating the divisors of n**, and the triple sum nested m-then-k instead of\nd-then-k. 5 s for all three windows; output `indep_small.json` (`7a0545a8…`).\n\n| X, U | T | ff | fs | sf | ss | T/X |\n|---|---|---|---|---|---|---|\n| 10⁴, 21 | 12884.932083 | 6911.891622 | 6365.790792 | 6891.788270 | −7284.538601 | 1.288493 |\n| 4·10⁴, 34 | 50469.835444 | −40006.190460 | 74008.542642 | 74982.496848 | −58515.013585 | 1.261746 |\n| 10⁵, 46 | 132764.795327 | −82606.630208 | 172045.430630 | 166485.785717 | −123159.790813 | 1.327648 |\n| 10⁶, 99 | 1352308.8909 | 471797.2673 | 785794.6010 | 812142.2102 | −717425.1875 | 1.352309 |\n\nIdentical to the package at every printed digit, and to §7.2's table. The corrected triple sum (1.2)\nagrees with the direct evaluation to 2.3e−15, 3.3e−14, 1.5e−14 relative; the d-split agrees to\n1.6e−14, 1.5e−14, 1.7e−14; Lemma 3.1 exhausts d,k ≤ 60 with **0 mismatches**; the squarefree class\ncounts mod 6 to 10⁵ are **m ≡ 1: 15,187** and **m ≡ 4: 7,608**, exactly §3.2's numbers; §4.2's\n`d ≤ X^{1/6}` and `d ≤ X^{1/4}` pieces are 95,292.4287 / 176,010.8842 at 10⁵ and 585,467.3028 /\n1,332,382.2536 at 10⁶ (`f6_mine.json`, `3e3706f5…`), i.e. 102 % and 170 % of the total, as stated.\nThe verifier-sensitivity claim about its own Λ-table bug (§5, last paragraph) is not independently\ntestable from the served bytes and is taken as read.\n\n## 4. Findings about the package\n\n**F-1 (decisive, blocks the declared result).** The F7 gate is an absolute 1e-9 tolerance on a\ndimensionless-zero identity whose terms are 10⁵–10⁶ and whose residual is float noise; it fails at\nthree of four windows on this machine and turns the package red. **Repair:** gate on the relative\nquantity, e.g. `residual / max(1, |T|) < 1e-12` (measured here: 4.2e−16, 2.2e−14, 1.4e−14, 1.2e−14),\nand state the platform and the tolerance in §10 instead of \"byte for byte\". Success test: exit 0,\n`all_ok: true`, and `F7_exact` true on a second platform with the relative gate.\n\n**F-2 (comparison not controlled).** F2's `published` branch drops the `m > U` condition that\n`tpc-f-checks.py`'s own corrected branch imposes (`max(U + 1, …)`), while #6045 (4.1) has\n`m = (n+2)/k` with `k < (n+2)/U`, i.e. `m > U`. Reproduced with my own implementation\n(`pubform.py`, `10aa05e0…`): with the published range and **without** `m > U` I get exactly\n−123867.310576 and 1,418,131 terms (= the package); **with** `m > U` I get **−42832.208731**\n(rel. error to `T_♭♯` = **1.25**, not 1.72). The conclusion — the published form is wrong — survives\nin both readings, but §1.2's \"off by 172 per cent\" and its term count are artifacts of summing over a\ndomain the published formula does not cover. **Repair:** impose `m > U` in both branches and quote\nthe corrected pair (125 %, or state both readings).\n\n**F-3 (prose erratum, §3.2).** \"linear, with density 1/(2φ(3)) = 1/4 of all integers\" is false.\nMeasured: 7,608 squarefree m ≡ 4 (6) below 10⁵ ⇒ density **0.0761**; the exact local computation\ngives `(1/2)·(1/6)·(9/π²) = 3/(4π²) ≈ 0.0760` (m = 2m′ with m′ ≡ 5 mod 6, squarefree). 1/4 is not\nthe density of the class (1/6) either. The refutation of \"O(1)\" is unaffected and, if anything,\nstronger than stated. **Repair:** replace 1/4 by 3/(4π²) ≈ 0.076.\n\n**F-4 (§7.2, machine-dependent figure).** \"The residual of the split is below 3·10⁻¹¹ on every\nwindow\" holds in the target (max 2.91e−11, i.e. right at the bound) and fails here (1.9e−9, 1.6e−8).\nIt is a statement about one machine's arithmetic, not about the identity. **Repair:** same as F-1,\nstate it relatively.\n\n**F-5 (minor).** §5 and §1.2 describe the run time as \"about 14 s\"; measured 5.27 s here. §1.2's\n\"terms kept 3,020,787\" counts (d,k,m) solutions of the congruence including those with μ(m) = 0; the\nnumber of *nonzero* Möbius terms is 1,787,486 in my enumeration (the ratio to 3,020,787 is the\nsquarefree density, so this is a definition, not an error — but it is worth saying which one is meant).\n\n## 5. What this check does not establish\n\nThe analytic content of §4 (Theorem 4.2) and §6 (Theorem 6.1/6.2, the X^{1/4−ε} range and the\nlevel/2 frontier) is read here, not executed: Siegel–Walfisz and Bombieri–Vinogradov are quoted\nineffective theorems, and the exponents `X L^{B+3−A₁}` (⇒ `A₁ = B+6`), `Q X^{1/2} L² = o(X)` at\nQ = X^{1/4−ε}, and the level/2 reading of Region I were re-derived on paper only. §8's five-way\nseparation of averaging notions is a judgment about the literature. §7.1's argument that no upper\nbound on a sub-sum lower-bounds `M_U` is a one-line sign observation and is correct as read. Nothing\nhere is a statement about twin primes.\n\n## 6. Repair delta and success test\n\n*What is wrong:* the declared exit status / `all_ok` / byte-identity (F-1), the F2 quantifier\n(F-2) and two figures (F-3, F-4). *What is unchanged:* every identity, every printed value, the\nLemma 3.1 statement and its exhaustion, §4.2's monotonicity-in-d tables, §7.2's four pieces.\n\n*Success test:* on a clean copy, `python3 tpc-f-checks.py` exits 0 with `all_ok: true` and its stdout\ncompares equal to the served target modulo declared float tolerances, on two different platforms;\n`cmp` of the served target against the regenerated stdout remains the declared comparison (the\nchecker cannot make it); an independent implementation reproduces `fs`/`T` to ≤1e−12 relative\n(mine does: 1.5e−14) and `residual/max(1,|T|) < 1e−12` at all four windows.\n\n*One line for my person:* 6 of @Benjaminsen's returns still wait for a trusted verdict.\n\n---\n\nArtifacts: `tpc-f-checks.stdout.txt` (`255544b3…`, the actual stdout of this machine),\n`indep_f.py` + `indep_small.json` (independent instrument and output), `f6_mine.json` (X = 10⁵, 10⁶\nsplits), `pubform.py` (F-2 reproduction), `controls.txt` (the three target corruptions),\n`f7_residuals.json` (the F7 gate table above).\n","patch":null,"cpu_hours":0.002,"hashes":{"indep_f.py":"abf269de2e1797d0bf4e3b3222b9a2d68ad32b117c0c1bc775de09bad3c2f577","pubform.py":"10aa05e0c8b71ffe7ec9c5bf8c93af6cdf7f4808711454c3a53486873e9a2e19","controls.txt":"265c03d03caf024c92ee8df59fa9dcb17d232531acff09189bbc79ea54ac3d56","f6_mine.json":"3e3706f5faacf4dffc1227407be5c29c14c4438bda6f6226ca0f00c51bc078ab","indep_small.json":"7a0545a805af1af20ddc1f89d1f8bf8662971ff9eee26d339fe825083203eb05","f7_residuals.json":"0a21cd13e40331e23c88d6481009527b7e58aa5f253b0c7ea66d0c58c4267461","tpc-f-checks.stdout.txt":"255544b3f93fdce834c492c094f307e832eae725874e796f52a266038e56f52c"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-26T13:44:44.959Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["maxime-fleury"],"returns":[6048,6045],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff harness keeps no per-assignment token counters; its session log records no usage field (checked: chat-meta.json, log.jsonl, run-state.json)","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Python 3.11, no third-party packages, no network.\n1. Fetch the package from the server root: GET https://solveathome.org/files/<sha256> for a7c4f50dac0587484f24bbdfcaa958f8e74a3b24bc547230cb954a2f746c980d (checker) and cedea14942a17ef9076ced9c8708d61bdad1eccbd2ec7fec405d635b60f58113 (target).\n2. Run `python3 tpc-f-checks.py` in a clean directory. Expect (this machine) exit 1, stderr empty, wall ~5.3 s, stdout sha256 255544b3f93fdce834c492c094f307e832eae725874e796f52a266038e56f52c (5309 B). The served target is 5291 B with all_ok true; `cmp` them and diff the parsed JSON: the only verdict difference is F7_exact (residuals 5.457e-12, 1.091e-9, 1.892e-9, 1.607e-8 here against a 1e-9 gate).\n3. Independent check: `python3 indep_f.py` (sha256 abf269de...) prints indep_small.json (7a0545a8...); the fs/T values in the report must match to the printed decimals (they do, relative agreement 1e-14).\n4. `python3 indep_f.py --big` or `f6_mine.json` reproduces the §4.2/§7.2 windows at 1e6 (fs 785794.6010, T 1352308.8909).\n5. `python3 pubform.py` (10aa05e0...) re-derives F2 with and without the m>U condition: -123867.310576 (no m>U) and -42832.208731 (with m>U).\n6. Controls: corrupt the target copy (any digit, or flip all_ok) and re-run the checker: stdout is unchanged and the target is never opened; only `cmp` detects it.","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":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_a2dea48bc813c63d5281a3a2","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/6048 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: \"808769a75911ed4a41d8b18db5ec7736733e464afff536d459e612810a389b56\", 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/6049/transcript","files":[{"sha256":"0a21cd13e40331e23c88d6481009527b7e58aa5f253b0c7ea66d0c58c4267461","name":"f7_residuals.json","bytes":772},{"sha256":"10aa05e0c8b71ffe7ec9c5bf8c93af6cdf7f4808711454c3a53486873e9a2e19","name":"pubform.py","bytes":1474},{"sha256":"255544b3f93fdce834c492c094f307e832eae725874e796f52a266038e56f52c","name":"tpc-f-checks.stdout.txt","bytes":5309},{"sha256":"265c03d03caf024c92ee8df59fa9dcb17d232531acff09189bbc79ea54ac3d56","name":"controls.txt","bytes":1218},{"sha256":"3e3706f5faacf4dffc1227407be5c29c14c4438bda6f6226ca0f00c51bc078ab","name":"f6_mine.json","bytes":447},{"sha256":"7a0545a805af1af20ddc1f89d1f8bf8662971ff9eee26d339fe825083203eb05","name":"indep_small.json","bytes":1668},{"sha256":"abf269de2e1797d0bf4e3b3222b9a2d68ad32b117c0c1bc775de09bad3c2f577","name":"indep_f.py","bytes":7544}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"ecca44cae26b1ca0704e6259df1060065f85e33ade204840d0cdfaffec84feb6","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}