{
  "table": "T441",
  "baseline_sha256": "c5598a58a753ef07ad376efe5ed3707743e2bb062a23fcfecbd278d5a094ca29",
  "candidate_sha256": "ea44451ccead08ad02f2bb67f3532c5881f43f4785341ef1a3790c244630c9fc",
  "proof": {
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    "all_entries_checked": true,
    "independent_checks": {
      "method": "Direct Bornemann Fredholm determinant identities with Gauss-Legendre Nystrom discretisation; no import of generate.py.",
      "environment": {
        "python": "3.12.3 (main, Mar 23 2026, 19:04:32) [GCC 13.3.0]",
        "platform": "Linux-7.0.0-1012-aws-x86_64-with-glibc2.39",
        "python-flint": "0.6.0"
      },
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          "reason": "left endpoint, small F1"
        },
        {
          "entry": "1,0",
          "reason": "ordinary central F1"
        },
        {
          "entry": "1,3",
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        {
          "entry": "2,-6",
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        },
        {
          "entry": "2,-11/2",
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        },
        {
          "entry": "2,-13/3",
          "reason": "disputed old interval F2"
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        {
          "entry": "2,0",
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        {
          "entry": "4,-11/2",
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        {
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        {
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        {
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        {
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        {
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          "finite": true
        },
        {
          "configuration": {
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          "entry": "4,-6",
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          "relative_difference_to_entries_json": "4.86666052836e-101",
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          "finite": true
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        {
          "configuration": {
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            "length": 56,
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          "entry": "4,-11/2",
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          "scaling": "sqrt2",
          "fredholm_value": "2.481261946380455144752892360090282881334052465448006349525686285220907624164087149488269067699780603619641557040555e-7",
          "relative_difference_to_entries_json": "1.53292337197e-100",
          "relative_difference_to_baseline": "8.84811277452e-11",
          "absolute_radius": "5.45319501020e-228",
          "finite": true
        },
        {
          "configuration": {
            "name": "B",
            "nodes": 260,
            "length": 56,
            "dps": 230
          },
          "entry": "4,0",
          "reason": "ordinary central F4 with sqrt2 scaling",
          "scaling": "sqrt2",
          "fredholm_value": "0.9985741973581685402167048089588001920045534684113751159403404896516268942355433412683067653814745766639137479186539",
          "relative_difference_to_entries_json": "3.61377774199e-101",
          "relative_difference_to_baseline": "1.68780864908e-13",
          "absolute_radius": "1.81222826133e-227",
          "finite": true
        },
        {
          "configuration": {
            "name": "B",
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            "length": 56,
            "dps": 230
          },
          "entry": "4,3",
          "reason": "right endpoint, near-one F4 with sqrt2 scaling",
          "scaling": "sqrt2",
          "fredholm_value": "0.9999999998328804780003568407875292745376769407884345820823704710984784225072746219733942991292294488389921250517125",
          "relative_difference_to_entries_json": "3.89921250582e-101",
          "relative_difference_to_baseline": "1.19521999663e-13",
          "absolute_radius": "5.57722702737e-228",
          "finite": true
        }
      ],
      "quadrature_convergence": [
        {
          "entry": "1,-6",
          "relative_difference_between_quadratures": "0e-15"
        },
        {
          "entry": "1,0",
          "relative_difference_between_quadratures": "0e-15"
        },
        {
          "entry": "1,3",
          "relative_difference_between_quadratures": "0e-15"
        },
        {
          "entry": "2,-11/2",
          "relative_difference_between_quadratures": "0e-15"
        },
        {
          "entry": "2,-13/3",
          "relative_difference_between_quadratures": "0e-15"
        },
        {
          "entry": "2,-6",
          "relative_difference_between_quadratures": "0e-15"
        },
        {
          "entry": "2,0",
          "relative_difference_between_quadratures": "0e-15"
        },
        {
          "entry": "2,3",
          "relative_difference_between_quadratures": "0e-15"
        },
        {
          "entry": "4,-11/2",
          "relative_difference_between_quadratures": "0e-15"
        },
        {
          "entry": "4,-6",
          "relative_difference_between_quadratures": "0e-15"
        },
        {
          "entry": "4,0",
          "relative_difference_between_quadratures": "0e-15"
        },
        {
          "entry": "4,3",
          "relative_difference_between_quadratures": "0e-15"
        }
      ],
      "max_relative_difference_to_entries_json": "4.09545589338e-100",
      "max_relative_difference_between_quadratures": "0e-15"
    },
    "limitations": "Arb is used internally, but Taylor step midpoints are retained. Taylor truncation and nonlinear boundary errors are tested by convergence and independent determinants, not rigorously enclosed.",
    "convergence": {
      "configurations": [
        [
          230,
          180,
          8,
          40
        ],
        [
          270,
          230,
          10,
          46
        ]
      ],
      "agreement_tolerance": "1e-110",
      "identity_tolerance": "1e-112",
      "worst_relative_difference": "1.36026781873e-144",
      "worst_relative_difference_entry": "4,-6",
      "first_run": {
        "worst_hamiltonian_error": "0e-228"
      },
      "second_run": {
        "worst_hamiltonian_error": "0e-268"
      }
    },
    "fresh_recomputation": {
      "configurations": [
        [
          230,
          180,
          8,
          40
        ],
        [
          270,
          230,
          10,
          46
        ]
      ],
      "agreement_tolerance": "1e-110",
      "identity_tolerance": "1e-112",
      "worst_relative_difference": "1.36026781873e-144",
      "worst_relative_difference_entry": "4,-6",
      "first_run": {
        "worst_hamiltonian_error": "0e-228"
      },
      "second_run": {
        "worst_hamiltonian_error": "0e-268"
      }
    },
    "cdf_controls": "Every value lies in (0,1); each beta family is strictly increasing in s.",
    "authorization": {
      "date": "2026-09-26",
      "decision": "User explicitly approved these three corrections and the 100-digit update.",
      "corrections": {
        "2,-11/2": {
          "before": "6.7138387900e-7",
          "after": "6.713838789865410167674493916412417817737075455005658174976848233127922634513016497128519800710851266e-7"
        },
        "2,-13/3": {
          "before": "0.000825047383733",
          "after": "8.250473837316331135362759508688492256831557302400394180558870408082542082115456260086533170251676255e-4"
        },
        "4,-11/2": {
          "before": "2.4812619466e-7",
          "after": "2.481261946380455144752892360090282881334052465448006349525686285220907624164087149488269067699780604e-7"
        }
      },
      "comparison_counts": {
        "refines": 162,
        "contradicts": 3
      }
    },
    "original_critique": {
      "decision": "approve",
      "independent_check_performed": true,
      "summary": "Approved the repair's blocked disposition, not a publication candidate. There is no candidate.json or verification.json to approve. The saved 100-digit entries preserve all 165 entry identities and the documented beta/s coverage, and the repair correctly stopped because a values-only candidate would refine 162 entries but contradict the old claimed intervals for 2,-11/2, 2,-13/3, and 4,-11/2. An independent Fredholm determinant check, written separately from generate.py and run through the required Sage wrapper, checked ordinary, endpoint and disputed values for beta=1,2,4, including the beta=4 sqrt(2) convention; its worst relative difference to entries.json was 4.09545589338e-100. The numerical support is heuristic rather than proven, and the repair states that limitation. Human review is genuinely needed before correcting the three old intervals or choosing a partial candidate that leaves them unchanged.",
      "findings": [
        {
          "severity": "info",
          "summary": "No candidate artifact is present; approval applies to repair-result.json status=blocked.",
          "detail": "candidate.json and verification.json are absent. Since the repair did not submit a candidate, candidate_sha256, generator_sha256 and verification_sha256 are intentionally not included."
        },
        {
          "severity": "info",
          "summary": "Definitions, formulas, conventions, parameter coverage and entry identities were preserved in the saved values.",
          "detail": "The structural audit found 165 baseline entries and 165 saved entries, with identical identity order and set. Coverage remains beta=1,2,4 at the 55 reduced rationals with denominator at most 4 in [-6,3]. The beta=4 sqrt(2) convention is present in baseline.json and references/T451.json and is used by the proposed generator and by the critique Fredholm check."
        },
        {
          "severity": "info",
          "summary": "Full-table convergence evidence supports the attempted 100-digit values heuristically, not rigorously.",
          "detail": "diagnostics.json records two Painleve/Taylor configurations, (230 digits, 180 terms, step 1/8, boundary 40) and (270 digits, 230 terms, step 1/10, boundary 46), with worst relative difference 1.36026781873e-144 at 4,-6. All entries in entries.json have digits='100' and 100-significant-digit scientific decimal strings. The generator describes its rigour as heuristic (agreement-checked), which is appropriate because midpoint Taylor steps do not retain rigorous enclosures."
        },
        {
          "severity": "info",
          "summary": "Independent Fredholm controls corroborate representative values and the three disputed corrections.",
          "detail": "critique_artifacts/fredholm_independent.py directly evaluates the Bornemann determinant identities with Gauss-Legendre Nystrom matrices and does not import generate.py. It checked 12 entries under two quadrature settings. The worst relative difference to entries.json was 4.09545589338e-100 at 2,-6; the disputed entries agreed with entries.json to 5.38625984382e-101, 4.70189548139e-101 and 1.53292337197e-100 relative."
        },
        {
          "severity": "info",
          "summary": "The reported block is genuine.",
          "detail": "The structural audit reproduced the coordinator comparison counts {'refines': 162, 'contradicts': 3}. The three contradictions are outside the old decimal intervals by about 1.35, 1.37 and 2.20 old last-place units respectively, so silently publishing them as refinements would violate the repair rules."
        }
      ],
      "evidence": [
        {
          "path": "critique_artifacts/structural_audit.py",
          "command": "python3 critique_artifacts/structural_audit.py | tee critique_artifacts/structural-audit-output.txt",
          "outputs": [
            "critique_artifacts/structural-audit.json",
            "critique_artifacts/structural-audit-output.txt"
          ]
        },
        {
          "path": "critique_artifacts/fredholm_independent.py",
          "command": "bash /home/ubuntu/numberdb-precision-20260925/repo/agents/precision-repair/sage.sh critique_artifacts/fredholm_independent.py | tee critique_artifacts/fredholm-independent-output.txt",
          "outputs": [
            "critique_artifacts/fredholm-independent.json",
            "critique_artifacts/fredholm-independent-output.txt"
          ]
        },
        {
          "path": "critique_artifacts/citations.md",
          "command": "web search/open for arXiv:0904.1581 and local inspection of baseline.json/references/T451.json",
          "outputs": [
            "critique_artifacts/citations.md"
          ]
        },
        {
          "path": "progress.md",
          "command": "updated with critique commands and limitations",
          "outputs": [
            "progress.md"
          ]
        }
      ],
      "artifact_hashes": {
        "baseline_json_sha256": "d607cbd9ead2180e98d53cd6c31646cbbd743800f97b3d10604384f3c6639b9f",
        "baseline_generate_py_sha256": "95378719afc431e7711e9b9645d82e80614a10a382e1b283457e2be2b86a8ed8",
        "proposed_generate_py_sha256": "31b06e0db21c4791ff1b77b176c81f2dc31d097e3f42f7fc7c8fe93042e9270f",
        "repair_result_json_sha256": "1bb471851d50154a67a0d07e572d850c2684d13338a895ab8205592dd41e791c",
        "entries_json_sha256": "b52df6b32b4e43f67e7b330fc1748a49d9f2e6bbfebc663bf1083839dc9c2986",
        "diagnostics_json_sha256": "3ce4a9a4dea5870f3ec34f374f73ad770a6b071e087da514d04ac2a7c140b135",
        "structural_audit_json_sha256": "b25a420bdb5b697a0b1996bd3718dd6253d1cfedd4ba9210be46189d01182743",
        "fredholm_independent_json_sha256": "6976bad6baafa391d275135c4559a80b6d1940a62a2f041ff18ac8027356ad8e"
      }
    },
    "application_script_sha256": "e42e3e03bab7dd1ee2e0c1821a26e9a2d0aac88aa77cbb214e7ea11f2a5530fc"
  },
  "provenance": {
    "repair": {
      "stage": "repair",
      "engine": "codex",
      "session": "01a0da43-5c0b-7e73-be8e-91c585974c66",
      "model": "gpt-5.5",
      "log": "/home/ubuntu/numberdb-precision-20260925/queue/jobs/T441/20260925T203042Z-repair-codex-5415eeee.jsonl",
      "started": 1790368242.1826644,
      "exit_code": 0,
      "finished": 1790369224.2563584,
      "native_session": "/home/ubuntu/numberdb-precision-20260925/queue/jobs/T441/native-repair-01a0da43-5c0b-7e73-be8e-91c585974c66.jsonl",
      "native_session_sha256": "c538d36494ecc20cec8959cabf2c48118670758549b5ddb702b79f7ae46c212f"
    },
    "critique": {
      "stage": "critique",
      "engine": "codex",
      "session": "01a0da52-5830-7a31-b11a-2601561d90f0",
      "model": "gpt-5.5",
      "log": "/home/ubuntu/numberdb-precision-20260925/queue/jobs/T441/20260925T204704Z-critique-codex-bf56841e.jsonl",
      "started": 1790369224.3183372,
      "exit_code": 0,
      "finished": 1790369870.0505667,
      "native_session": "/home/ubuntu/numberdb-precision-20260925/queue/jobs/T441/native-critique-01a0da52-5830-7a31-b11a-2601561d90f0.jsonl",
      "native_session_sha256": "28c55f5fe66e081693322f4cb986f6df1431cdf3b6cd8482f439eea7cec56511"
    },
    "approval": {
      "date": "2026-09-26",
      "decision": "User explicitly approved these three corrections and the 100-digit update.",
      "corrections": {
        "2,-11/2": {
          "before": "6.7138387900e-7",
          "after": "6.713838789865410167674493916412417817737075455005658174976848233127922634513016497128519800710851266e-7"
        },
        "2,-13/3": {
          "before": "0.000825047383733",
          "after": "8.250473837316331135362759508688492256831557302400394180558870408082542082115456260086533170251676255e-4"
        },
        "4,-11/2": {
          "before": "2.4812619466e-7",
          "after": "2.481261946380455144752892360090282881334052465448006349525686285220907624164087149488269067699780604e-7"
        }
      },
      "comparison_counts": {
        "refines": 162,
        "contradicts": 3
      }
    },
    "application": {
      "pipeline": "interactive",
      "engine": "codex-cli",
      "model": "gpt-5.2",
      "stage": "approved-correction"
    }
  },
  "batch": "approved-20260926-T441",
  "authority": "Website database and revisions; cloud files are supporting evidence."
}