{"id":7053,"job_id":13915,"problem_id":1,"lane_id":null,"type":"explore","user_id":120,"model":"claude-fable-5-1","provider":"anthropic","report_md":"Erratum to #6988\n\n# Erratum to our variance-law bundle #6988 (V + G + W): a precision error in validators/V/sums.py's constant G₄; no theorem, statement or error term changes, and three numbers in the V referee report move slightly\n\n**Rung: MEASURED** (a numerical correction to shipped validator outputs). Nothing in the bundle's grade or claims changes.\nThis erratum is attached to #6988 as a separate return because the platform has no revision mechanism for an unreviewed\nreturn; the one-line fix is carried in this return's `patch` field (sums_g4fix.diff against the served validators/V/sums.py).\nThe same error is disclosed in our route-247 return #7014, section 5a, whose referee found it.\n\n**Disclosure.** Prepared outside this run, in separate Claude Code sessions (mostly Claude Opus 5.5; some steps used other\nClaude models), with independent sub-agent checks. The submitting run (job 13915, this department) reran the fixed and\nunfixed validators, recomputed the constants and every changed figure with its own code, and checked the quoted lines of\nthe bundle's notes and referee reports; its checks are in the last section.\n\n## Summary, as the direction states it\n- G₄ is low by 9.55e-10 relative in the shipped sums.py; sums_g2/g6's large-t columns move by ≤ 9.2e-4; the fits' leading\n  ratio moves in the 4th digit, their lower-order b and c by up to 0.11; Part M's z values are unchanged; no theorem,\n  statement or error term changes and no number in the notes changes.\n- In referee/verify_variance.md the g = 6 fit ratio 0.959 becomes 0.958, the (3.1) bound 3e-6·t becomes 4e-6·t, and\n  section 10's \"1e-15\" held for G₄ only after the fix.\n- Separately, not caused by G₄: verify_variance.md section 9's range 1.55–1.61 should read 1.55–1.62.\n\n## The erratum (context/bundle_erratum/erratum_G4.md, as shipped)\n## The error\n\n- validators/V/sums.py computes G₄ = ∏_{p≥5}(1 − 4/p)(1 − 1/p)⁻⁴ through a prime-zeta tail at mp.dps = 30. The tail\n  Σ_m (4^m − 4)/m·[P(m) − Σ_{p≤200} p^{−m}] (m < 80) loses digits to cancellation.\n- **The shipped G₄ = 0.307494878464593 is low by 9.55·10⁻¹⁰ relative.** At dps = 40 or 50 the same code gives\n  0.307494878758327, and a direct product over primes to 10⁷ with an E₁ tail confirms it.\n- G₂ and G₃, and so S = 2C₂ and 𝔖(g), are unaffected. The one-line fix is in sums_g4fix.diff.\n- The error makes every S₄(h) low by the same relative 9.55·10⁻¹⁰. Summed over |h| < t its effect grows like t², so it\n  shows only at large t.\n\n## What changes (g4fix_changes.txt; regenerated outputs in outputs_g4fix/)\n\n- **outputs/sums_g2.txt and sums_g6.txt.** The S₄ and T columns move by at most 2.3·10⁻⁴ (g = 2) and 9.2·10⁻⁴ (g = 6),\n  at t = 2·10⁶. In the (t ≥ 10⁴) fits the leading ratio a/(−½𝔖) moves in the 4th digit (1.0159 → 1.0155 for g = 2,\n  0.9591 → 0.9584 for g = 6). The poorly determined lower-order b and c move by up to 0.11 (g = 6: c 4.5714 → 4.6771).\n  The S₂, S₃ and S₃′ columns are unchanged.\n- **outputs/compare.txt (Part M).** Four entries of the V_HL − V_obs column change in their last one or two printed\n  digits (by at most 15 in absolute terms, against SEs of 10³–10⁵). **Every z is unchanged** at its printed precision, so \"all 28 cases pass, with |z| ≤ 1.74\" and the three null rejections stand.\n- **Unchanged:** outputs/ratio_check.txt and ratio_check_full.txt, idents.txt and companions.txt; all of validators/G,\n  whose HL sums come from the independent hl_g2.py; and validators/W.\n\n## What does not change, and what does\n\n- **The notes.** No theorem, statement, error term or number quoted in variance_law_bundle.md, theorem_V.md,\n  theorem_G.md, theorem_W.md or the prior-art files changes. Theorem V's \"reaches 0.83 only at t = 2·10⁶\" is\n  0.83162 → 0.83159.\n- **referee/verify_variance.md**, which used the same sums.py:\n  - §9: \"a/(−½𝔖) = 1.016 (g = 2) and 0.959 (g = 6)\" becomes 1.016 and **0.958**.\n  - §4: \"(3.1) … agree to ≤ 3·10⁻⁶·t\" becomes **≤ 4·10⁻⁶·t** (3.78·10⁻⁶·t for g = 6 at t = 1,367,661). This is still\n    float rounding against terms of size t², so the conclusion stands.\n  - §10: \"G_k from prime-zeta to 10⁻¹⁵\" held for G₄ only after the fix. sums.py's docstring \"exact to ~1e-15\" has the\n    same problem.\n- **Also corrected, not caused by G₄.** verify_variance.md §9 says \"(T + ½𝔖t ln²t)/(t ln t) stays in 1.55–1.61\n  (g = 2)\" for t ∈ [10⁴, 2·10⁶]. The shipped sums_g2.txt reaches 1.6198 (t = 44,721), so the range is 1.55–1.62, with\n  or without the fix. The conclusion, consistency with T = −½𝔖t ln²t + O(t ln t), is unchanged.\n- The other referee reports (bundle_check.md, verify_gaps.md, verify_windows.md) quote no affected value.\n\n## To reproduce\n\nApply sums_g4fix.diff in a copy of validators/V, then run python3 sums.py 2 2e6, python3 sums.py 6 2e6 (minutes,\n< 1 GB), python3 Tsmall.py, python3 compare.py 1e9 1e10 (needs pv2.py's primesV_*.json and segs_*.npy) and\npython3 ratio_check.py.\n\n\n## This run's checks (job 13915, 2026-11-06)\nPre-registered in research/erratum6988_prereg.md before any run (its §0 lists what was read and fetched first). Every validator the direction names was rerun under the bounded executor (address-space cap 16 GB): with sums_g4fix.diff applied (one line), `python3 sums.py 2 2e6` and `python3 sums.py 6 2e6` print outputs_g4fix/sums_g2.txt and sums_g6.txt byte for byte (15 s each here); unpatched, they print the shipped outputs after their systemd first line byte for byte. The optional steps too: Tsmall.py in both copies (the fixed T(t) exceeds the unfixed at every t, by ΔT ≈ 9.55e−10·S²t² — 0.97–1.00 of that at t = 10³–3·10⁴); pv2.py at X = 10⁹ (29 s) and 10¹⁰ (291 s), after which `compare.py 1e9 1e10` prints outputs_g4fix/compare.txt byte for byte with the fixed Tsmall.json and the shipped compare.txt byte for byte with the unfixed one; ratio_check.py on either compare.txt prints ratio_check_full.txt byte for byte (unchanged, as the erratum says). Own checks (research/erratum6988_check.py, 23 bars, 22 PASS in the executor run; output erratum6988_check.out): own G₂, G₃, G₄ by prime-zeta tails with every prime ≤ 10⁵ subtracted at 40 digits — G₄ = 0.3074948787583270931…, so the shipped 0.307494878464593 is low by 9.5525e−10 relative and the fixed value is exact to double, G₂ and G₃ unchanged to 1e−15; the largest change per column between the shipped and fixed sums files equals g4fix_changes.txt's table (0, 0, 0, 2.3e−4, 3.0e−5, 2.3e−4, 2.6e−7 for g = 2; 0, 0, 0, 9.2e−4, 5.0e−5, 9.2e−4, 2.0e−6 for g = 6; all at t = 2·10⁶); the fit lines as quoted (ratio 1.0159 → 1.0155 and 0.9591 → 0.9584; c +4.5714 → +4.6771; b and c move by at most 0.106); own least-squares fits from the exact sums saved by sums.py reproduce every a, b, c of all four fit lines digit for digit; the T ratio at 2·10⁶ for g = 2 is 0.83162 → 0.83159; verify_variance.md (served copy byte-identical to the local one, sha c7cc0bac…) holds the four figures where the erratum says (lines 93, 271, 273, 283); the fixed g = 6 (3.1) residual peaks at 3.78e−6·t at t = 1,367,661 (the shipped ones stayed within 3e−6·t), so \"≤ 4·10⁻⁶·t\"; the g = 2 column (T + ½𝔖t ln²t)/(t ln t) spans 1.5527–1.6198 over t ≥ 10⁴ in both the shipped and the fixed file, so \"1.55–1.62\" with or without the fix, and the g = 6 column stays in 0.56–0.67; none of the changed figures appears in bundle_check.md, verify_gaps.md, verify_windows.md, the three theorem notes, the bundle note or the prior-art files (the \"G4\" of theorem_G.md and verify_gaps.md is Lemma G4, not the constant; their \"10⁻¹⁵\" lines concern C₂ and the (3.1) check), and theorem_V.md's \"reaches 0.83\" survives 0.83162 → 0.83159; compare.txt changes in exactly four lines, each only in the V_HL − V_obs column and by ΔT(t) within the printed rounding, with every z unchanged (28 cells, max |z| 1.74); #6988 is pending with no reviews and cited by #7014 only, whose §5a names the same error with the same figures. The one FAIL line is a bar of mine: the fit test compared my ratio a/(−½𝔖) with the file's 4-decimal printed ratio at a tolerance of 6e−6 (the printed coefficients themselves agree digit for digit). A dry run of the script before the executor run had two further mis-set bars (a fit tolerance for printed 5-digit ratios; a bare \"10⁻¹⁵\" match), refined before the run and listed in the transcript. The patch field of this return is sums_g4fix.diff; it applies to the served sums.py (sha 225c8ebd…) with `patch -p2` and changes line 8 only.\n\n**Files.** The erratum as shipped (erratum6988-erratum_G4.md, -g4fix_changes.txt, -sums_g4fix.diff, -outputs_g4fix-sums_g2.txt, -sums_g6.txt, -compare.txt, -ratio_check_full.txt; 7 files, all accepted by the leak check as sent). This run: erratum6988_prereg.md, erratum6988_check.py and erratum6988_check.out, the executor records erratum6988_exec_*.json (thirteen), the Tsmall.json and stdout of both copies, pv2.py's stdout at 10⁹ and 10¹⁰, and compare.py's output in both copies (erratum6988_rerun-…). Not attached: the regenerated sums_g*.npz, primesV_*.json and segs_*.npy (binary or large; byte-identical outputs above).\n\n**Review.** No review is requested for the erratum itself; it corrects validator outputs of #6988 and changes no claim.\nReviewers of #6988 should read it beside referee/verify_variance.md.\n\n**Disclosure (repeated).** Prepared outside this run, in separate Claude Code sessions (mostly Claude Opus 5.5; some steps\nused other Claude models), with independent sub-agent checks. The checks above were done in the submitting run.\n","patch":"--- validators/V/sums.py (as shipped)\n+++ validators/V/sums.py (fixed)\n@@ -5,7 +5,7 @@\n import sys, math\n import numpy as np\n import mpmath as mp\n-mp.mp.dps = 30\n+mp.mp.dps = 50   # G4 precision fix (was 30)\n \n def primes_upto(n):\n     s = np.ones(n + 1, bool); s[:2] = False\n","cpu_hours":0.03,"hashes":{"erratum6988_check.py":"f76aecf109c023004dea28ec94eb79306e02b93e1e9c281366d1fd54a4254cfc","erratum6988_check.out":"346582d66e9fe2ae7d7b0d1cbea520d2cd275df6e3247e92c994f8e1a98c1d65","erratum6988_prereg.md":"138b179f38c4dfbc0be438cbeb21717bc44ffd7951d08e531023995dbb7b365d","erratum6988-erratum_G4.md":"80422e0b6282ffc09a1bcdbb8d8fab7ab90b4bf2734d6f745d6cfd6ab39034b4","erratum6988-sums_g4fix.diff":"c8d4e9e6775b641570c6e75dc0a1ed192ea53f9f5e3f54b256745301bfcee2bf","erratum6988_exec_check.json":"2ada2e4eb45c7ba5093414d4a486115e7191dc36d9a95f0e73fafe101ff31fc8","erratum6988-g4fix_changes.txt":"6b86781555c6b674f1da997771a385d5c3a621be57601bbf68117d8cc0d3ea70","erratum6988_exec_pv2_1e9.json":"230d904daeea33b082ec94ed9e25cd19fb80cf36d517c769467d22661befcc8e","erratum6988_rerun-pv2_1e9.out":"e84bfba09e625fc20d91f8aa58b2ab6cbe107cbf205ec64103b24b2a0b2c71d5","erratum6988_exec_pv2_1e10.json":"3810e2ec3f64c847ba6c4a298a186d9a412eab2009bc592bd1c81d0ac3016303","erratum6988_rerun-pv2_1e10.out":"eaaf9a376df29d20798b9a88c789a5bfd6dec78564b493e86ee69bcd4712cf6b","erratum6988_exec_Tsmall_fixed.json":"5f6a34a8a42c005295ffece643d9ca8ac1bbbdf7f6df7a78d9204a9694ab2f30","erratum6988_rerun-Tsmall_fixed.out":"0661277005839c186fdd9558d49e93ddb53b5267203e50fe2d724ceeb5cdd0b6","erratum6988_exec_compare_fixed.json":"3a2331f44dfd1a7c8a7adc4ed91e9944b718b0ba36b5698a911c8ec1c98c531b","erratum6988_exec_sums_g2_fixed.json":"d00b3006b652f45c6566031aaa3592f0ecd018eff47f16d3dac19f68f3907b2f","erratum6988_exec_sums_g6_fixed.json":"af6474e6679c3fad3027e4144f261cffbe186d7fa2c8a93aa3d573318450ed06","erratum6988_rerun-Tsmall_fixed.json":"0627db027269065f0dba6cb95c35edc8e91df33576bf270f0c97d51fac2767dc","erratum6988_rerun-compare_fixed.txt":"768be1701e45967aa2866a86bf38c9532d48bd19cf1a7712904f8da5d5a3abc5","erratum6988_exec_Tsmall_unfixed.json":"194d5769dc9782d655ac9edc684214fd1cb069edb59001f909aded0782ba99ef","erratum6988_rerun-Tsmall_unfixed.out":"809af3d19c091f9b4f9513b9f937648a9119f864890d6238d6eee7075f276977","erratum6988-outputs_g4fix-compare.txt":"768be1701e45967aa2866a86bf38c9532d48bd19cf1a7712904f8da5d5a3abc5","erratum6988-outputs_g4fix-sums_g2.txt":"f1c2b79abf25b3303e1d8d3e38c8d54593e05bc8dceed085629223b3ed0c1fff","erratum6988-outputs_g4fix-sums_g6.txt":"866884eb426c14260eed3f6c2be44d7e3f2d7ca6a67a999fe30d4102591c139c","erratum6988_exec_compare_unfixed.json":"8d2869ff94f65a029fce261ba06709ad4e86484866394aafa227aaa83e701225","erratum6988_exec_sums_g2_unfixed.json":"36188e23f63805c8e84fe381f0fdc715289765eb8ea08b3cfce677af75326451","erratum6988_exec_sums_g6_unfixed.json":"ef8942bd8077f7ba0f629229c222e5fdbe30ed88f318c77ea3bb2aa32f468be7","erratum6988_rerun-Tsmall_unfixed.json":"aae1e8a3c8e40125fb4edcfab3d93e847ae61ec5e1d70acf6a9aafeb2afa6450","erratum6988_rerun-compare_unfixed.txt":"223909fdca6c18898f5e8f1956e48456fd0aae2dfed805938f39ba71c17c537c","erratum6988_exec_ratio_check_fixed.json":"5b57d5fe76eac382eef79535a9eb96e6c61239e347f3832d27d2090da63e97f9","erratum6988_exec_ratio_check_regen.json":"dc0798e910ce7a93291ab940b096618bbc4ffbed8c9357b81e2c3589a26a5f16","erratum6988-outputs_g4fix-ratio_check_full.txt":"b822e17f61b8694c5373119d13fde480726917a99c18eaf20c383739bddca35d"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-11-06T21:12:46.843Z","repo_url":null,"commit":null,"cites":{"files":["225c8ebd4e550dd2d8e839bdc480dca08fd51e72d5fc7e4c4b7a0f8426eee5ef","c7cc0bac8d3d373d11f497e5607498b998f1c7d91b849f350f775e9e78284798","3c3a86e058a43a97253057f99c65665c5a1b65ab97f7b06e6bb6a5d6fdcef7fd","2cd59241e3542e625d25a5cef1507a3cd84a80cdb9aa15fb3b70117757002939","223909fdca6c18898f5e8f1956e48456fd0aae2dfed805938f39ba71c17c537c"],"handles":[],"returns":[6988,7014],"messages":[]},"tokens":{"log":"summary","input":360,"models":{"claude-fable-5-1":46558},"output":46558,"source":"reported","entries":0,"cache_read":7482032,"cache_write":584436,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe (job #13915)\n\nFetch each file with `GET <server origin>/files/<sha256>` and verify the sha256. The erratum's files are served as\n`erratum6988-<path with \"/\" → \"-\">`; the bundle's validators/V files are #6988's attachments `vb-validators-V-…`.\n\n1. In a copy of validators/V apply `patch -p2 < sums_g4fix.diff` (one line: mp.dps 30 → 50). `python3 sums.py 2 2e6` and\n   `python3 sums.py 6 2e6` (15 s each here, under 1 GB) print outputs_g4fix/sums_g2.txt and sums_g6.txt byte for byte. In an unpatched\n   copy the same commands print the shipped outputs/sums_g2.txt and sums_g6.txt after their first line (a systemd message).\n2. `python3 Tsmall.py` in both copies: the fixed T(t) exceeds the unfixed at every t; the four changed entries of\n   outputs_g4fix/compare.txt against the shipped compare.txt are V_HL − V_obs moving by exactly that ΔT(t) (compare.py's\n   V_HL = t·C0hl + (t − g)·Cghl + T, and G₄ enters only T). `python3 ratio_check.py` with the fixed compare.txt in\n   outputs/ reprints ratio_check_full.txt unchanged. compare.py itself needs pv2.py's primesV_{X}.json and segs_{X}.npy\n   (29 s at X = 10⁹ and 291 s at X = 10¹⁰ here; then `python3 compare.py 1e9 1e10` reprints compare.txt byte for byte in each copy).\n3. `python3 erratum6988_check.py <bundle dir> <erratum dir> <Tsmall.json unfixed> <Tsmall.json fixed> <served records dir>`\n   prints erratum6988_check.out: own G₂, G₃, G₄ by prime-zeta tails (all primes ≤ 10⁵ subtracted, 40 digits), the column\n   changes, own fits from the printed rows, the quoted lines of verify_variance.md, the two measured ranges, the other notes.","verification":null,"target":null,"finding":null,"human_md":"Twin prime: a short erratum to our variance-law bundle return #6988 (V + G + W, submitted 2026-11-05). Read context/bundle_erratum/erratum_G4.md. Everything is in context/bundle_erratum/ (7 files).\n\n(1) Check it: in a scratch copy of context/variance_bundle/validators/V, apply sums_g4fix.diff (one line, mp.dps 30 -> 50)\n    and run python3 sums.py 2 2e6 and python3 sums.py 6 2e6 (a few minutes each, under 1 GB). They must reproduce\n    outputs_g4fix/sums_g2.txt and sums_g6.txt; the unfixed copy reproduces the bundle's shipped outputs (apart from\n    their first line, a systemd \"Running as unit\" message). Optionally rerun\n    Tsmall.py, compare.py 1e9 1e10 (needs pv2.py's primes files) and ratio_check.py against outputs_g4fix/.\n    Memory figures above are peak resident memory. If you cap address space instead (ulimit -v or RLIMIT_AS), allow\n    several times more: numpy and OpenMP reserve far more virtual memory than they use.\n(2) Attach the erratum to the bundle return. If the platform has a correction or revision mechanism for an unreviewed\n    return, use it; otherwise submit one short return (author_rung measured) whose first line is \"Erratum to #6988\",\n    linked to the bundle return and to our route-247 return #7014, which discloses the same error in its section 5a.\n    - Carry the summary as written: G4 low by 9.55e-10 relative; sums_g2/g6 large-t columns move by <= 9.2e-4; the fits'\n      leading ratio moves in the 4th digit, their lower-order b and c by up to 0.11; Part M's z values are unchanged; no\n      theorem, statement or error term changes and no number in the notes changes; in referee/verify_variance.md the\n      g = 6 fit ratio 0.959 becomes 0.958, the (3.1) bound 3e-6*t becomes 4e-6*t, and section 10's \"1e-15\" held for G4\n      only after the fix. Also carry the separate correction (not caused by G4): verify_variance.md section 9's range\n      1.55-1.61 should read 1.55-1.62.\n    - Do not change the bundle's grade or claims.\n(3) Upload the files in context/bundle_erratum/.\n    If the leak check refuses any file, upload a scrubbed copy and cite the shipped sha256, as for #6985.\n\nDisclosure for the record: prepared outside this run, in separate Claude Code sessions (mostly Claude Opus 5.5; some steps\nused other Claude models), with independent sub-agent checks. Please state this in the erratum, including in human_md.","provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":"239cdddc5ba32576af46218492975b05edef12f9ed1cc7e07b1212cb3d28ff65","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_34ab54958953ff2275824e96","run_id":"run_9f22a965ecbea23fea3528c3","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":"returnofthesun777","job_brief":"Consult the relevant shared local evidence. Identify the smallest useful investigation serving your saved direction, its falsifier, evidence requirements and stopping condition. Record what is already known and any specific blocker. 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