{
  "schema_version": 1,
  "release_id": "goldbach-exception-069697-certificate-v4",
  "status": "computer-assisted proof; finite certificates independently replayed, analytic source correspondence pending independent verification",
  "claim": "E(X) <<_epsilon X^(23/33+epsilon), with the strict corollary E(X) << X^(69697/100000)",
  "exact_constants": {
    "base_exponent": "23/33",
    "decimal_corollary_exponent": "69697/100000",
    "decimal_exponent_gap": "1/3300000",
    "historical_exponent_gap": "1/330",
    "packet_coefficient": "33/10",
    "strict_bridge_gap": "1/303030",
    "secondary_packet_ceiling": "198479/200000",
    "secondary_ceiling_slack": "1521/200000",
    "active_scalar_packet_ceiling": "983/1000",
    "near_siegel_C_public_bound": "99729/1000",
    "near_siegel_epsilon_z_max": "1/1000",
    "near_siegel_fixed_tail_coefficient": "100",
    "near_siegel_total_T_public_bound": "1/50",
    "near_siegel_packet_tail_coefficient": "202",
    "near_siegel_tail_rate": "164/75",
    "near_siegel_tail_power": "4469/1875",
    "near_siegel_small_lambda_slope": "54811/10000",
    "near_siegel_mid_gap_lower": "6329235347/200000000000"
  },
  "runtime": {
    "python_minimum": [
      3,
      9
    ],
    "numpy": "2.0.2",
    "validated_environment": {
      "python": "3.9.6",
      "numpy": "2.0.2",
      "platform": "macOS-26.5-arm64-arm-64bit"
    },
    "optimization": "disabled; python -O is forbidden"
  },
  "source_files": [
    {
      "path": "experiments/zhao_069697_full_active_audit.py",
      "sha256": "87e1068035c36d04fa15ef6fb95a3d5b1ab4d83d80901b4099fb2235deabd255",
      "role": "active-packet entry point"
    },
    {
      "path": "experiments/zhao_secondary_aligned_trial.py",
      "sha256": "8eac9e17d8a04051d2fd0e4818d432a6a44ceab97aaaf4a8f6a0ad7ef762c540",
      "role": "aligned secondary-packet entry point"
    },
    {
      "path": "experiments/zhao_secondary_scalar_audit.py",
      "sha256": "4fa2a701fe7b2365fd44498f742e241f97ad12419e184703a701bb5c28ebe129",
      "role": "scalar secondary-packet entry point"
    },
    {
      "path": "experiments/zhao_high_class_positional_trial.py",
      "sha256": "6675c7bb67d045a898789793e11b0d1fb68dc59def0e173f27558efa9a737fd0",
      "role": "high-class positional cap"
    },
    {
      "path": "experiments/zhao_aligned_rt_staircase.py",
      "sha256": "8eb9afc6020a180cbc29cd37f735f8b868b9ece1dc198bc22bfc57a89f089be3",
      "role": "aligned cap-and-mass optimizer"
    },
    {
      "path": "experiments/zhao_positional_fixed_active.py",
      "sha256": "54c798799b25cac3586fef4360bdb8fe2196069532be500fbe039253db061fd5",
      "role": "fixed-position active bounds"
    },
    {
      "path": "experiments/zhao_positional_fixed_density.py",
      "sha256": "9a4ac5267743ffa1a188689028a8d8eac1f72e70e076fda671f3b7a9e3702a20",
      "role": "fixed-position density cap"
    },
    {
      "path": "experiments/zhao_positional_staircase_audit.py",
      "sha256": "bbd2ec0440551ff0f0e300c554408f073f8ebf7241152a03adf77771120a4de4",
      "role": "positional staircase bounds"
    },
    {
      "path": "experiments/zhao_adaptive_density.py",
      "sha256": "21f380bc2524ed29935e268aa8e849c896bf01d07d97f50b50e8fded70090a38",
      "role": "adaptive exact density enclosure"
    },
    {
      "path": "experiments/zhao_packet_certificate.py",
      "sha256": "64fb7f6cf563ad7c6ae0ac9e9c7f9e7bfffc141b8cd476907d2e8f3e7e8e865e",
      "role": "exact packet arithmetic"
    },
    {
      "path": "experiments/zhao_packet_numeric.py",
      "sha256": "1f1e4477185e1ace41461527e7d2262a261acbdb9f1395982412f2bafb51622b",
      "role": "floating-point parameter selection"
    },
    {
      "path": "experiments/zhao_l2_thresholds.py",
      "sha256": "23469bfa2be08bf06dbeaaf195d023dfcb10ee0646161ae8f96915f755e2c4ef",
      "role": "L2 threshold bounds"
    },
    {
      "path": "experiments/zhao_ordered_staircase_audit.py",
      "sha256": "7d6090bd659ef936452e77bc74069d03a89fe409e3fe7e881fbb4766faf26f40",
      "role": "ordered staircase bounds"
    },
    {
      "path": "experiments/zhao_near_siegel_exact.py",
      "sha256": "e37dff33a587c8bed30c3baa7426badba6372979047fe6070866645d0393279c",
      "role": "standard-library exact near-Siegel verifier"
    },
    {
      "path": "preprints/goldbach-exception-069697/near-siegel-witness.json",
      "sha256": "6ceee077c77eff8ede3ddc6cf4722e49ef273d5185c5d3d7c72bfbcb79bc671d",
      "role": "machine-readable near-Siegel branch witness"
    },
    {
      "path": "experiments/export_secondary_aligned_witness.py",
      "sha256": "e5a4a5b95667bb8cfce0711406b0842b050b314ebe8b0d121bfbc1b4ba87f513",
      "role": "exact aligned-secondary witness producer"
    },
    {
      "path": "experiments/check_secondary_aligned_witness.py",
      "sha256": "8909489a48dc24bf0d19ea74f68cb43ffe9e6435caa19449615603f69d8b9711",
      "role": "independent standard-library aligned-secondary checker"
    },
    {
      "path": "preprints/goldbach-exception-069697/secondary-aligned-witness.json",
      "sha256": "94cff613c74ac36eac9129e5f61ea80dbeb45ef7ca828d8744af42eb6a489d62",
      "role": "machine-readable exact witnesses for sixteen aligned secondary rows"
    },
    {
      "path": "experiments/export_active_scalar_witness.py",
      "sha256": "ce110b4323114ca270f5a60be5ee6ed4e6fc8e4f4342a4d0d67c31ba9f2902d6",
      "role": "exact active-and-scalar witness producer"
    },
    {
      "path": "experiments/check_active_scalar_witness.py",
      "sha256": "9efd9aaec630e947e6d88a6e2e684b17896da86f909c7773415e480143c9accc",
      "role": "independent standard-library active-and-scalar checker"
    },
    {
      "path": "preprints/goldbach-exception-069697/active-scalar-witness.json",
      "sha256": "4c5a5d59a074253933d1efb7161bd69ae4656eed1dc70b9e21c5cedfb8cccc84",
      "role": "machine-readable exact witnesses for sixty-three active and four scalar rows"
    }
  ],
  "commands": [
    {
      "id": "active",
      "script": "experiments/zhao_069697_full_active_audit.py",
      "args": [
        "--A",
        "33/10",
        "--cutoff",
        "1.466",
        "--tolerance",
        "1/10000",
        "--step",
        "1/200",
        "--end",
        "1.24",
        "--positional-boundaries",
        ".92,1,1.05,1.10,1.15,1.20,1.25,1.30,1.40,1.466"
      ],
      "success_markers": [
        "strict 0.69697 bridge gap=1/303030",
        "A=33/10 FULL ACTIVE PACKET CLOSES"
      ],
      "expected_output_sha256": "b39bb14f470cfcac846f769189d8419ba9895e959d92d1b9cec82781088a7723",
      "regex_counts": [
        {
          "label": "active packet rows",
          "pattern": "^\\[[^\\n]+\\]; (?:two classes/two zeros|one class/one zero|one class/two zeros); packet<=",
          "count": 63
        }
      ]
    },
    {
      "id": "secondary-aligned",
      "script": "experiments/zhao_secondary_aligned_trial.py",
      "args": [
        "--scope",
        "all",
        "--cap-bins",
        "0",
        "--tolerance",
        "1/10000"
      ],
      "success_markers": [
        "SECONDARY ALIGNED TRIAL CLOSES"
      ],
      "expected_output_sha256": "8c544e756d93c235888f6fad0b0f511233ee4e8257c60328daa808b6e4a6082c",
      "regex_counts": [
        {
          "label": "aligned secondary packet rows",
          "pattern": "^\\[[^\\n]+; allocation=\\([12], [12]\\); packet<=",
          "count": 16
        }
      ]
    },
    {
      "id": "secondary-scalar",
      "script": "experiments/zhao_secondary_scalar_audit.py",
      "args": [],
      "success_markers": [
        "SECONDARY SCALAR PACKET CLOSES"
      ],
      "expected_output_sha256": "7f85ad8f8afe317af27807dac0feb56c436ed460cd4f9b41a76ceb7f15a5c0c9",
      "regex_counts": [
        {
          "label": "scalar secondary packet rows",
          "pattern": "^\\[[^\\n]+\\](?: old scalar<=|; packet<=)",
          "count": 4
        }
      ]
    },
    {
      "id": "near-siegel",
      "script": "experiments/zhao_near_siegel_exact.py",
      "args": [
        "--witness",
        "preprints/goldbach-exception-069697/near-siegel-witness.json"
      ],
      "success_markers": [
        "C<99.729",
        "100*(2+1/50)=202",
        "NEAR-SIEGEL EXACT CERTIFICATE CLOSES"
      ],
      "expected_output_sha256": "17c18319123e79c3aded878f5aeb1adfe4305150d078e477159af3940ad6929b"
    },
    {
      "id": "secondary-aligned-independent",
      "script": "experiments/check_secondary_aligned_witness.py",
      "args": [
        "preprints/goldbach-exception-069697/secondary-aligned-witness.json"
      ],
      "success_markers": [
        "INDEPENDENT EXACT CHECK: ALL 16 SECONDARY ROWS CLOSE"
      ],
      "expected_output_sha256": "e37b115216edb2077baabb268912bdf7e648bbc1b88d0e43967d6cb326ff089d",
      "regex_counts": [
        {
          "label": "independently checked aligned secondary rows",
          "pattern": "^s[^:]+: objective=",
          "count": 16
        }
      ]
    },
    {
      "id": "active-scalar-independent",
      "script": "experiments/check_active_scalar_witness.py",
      "args": [
        "preprints/goldbach-exception-069697/active-scalar-witness.json"
      ],
      "success_markers": [
        "INDEPENDENT EXACT CHECK: ALL 63 ACTIVE AND 4 SCALAR ROWS CLOSE"
      ],
      "expected_output_sha256": "124cceea0347177d5a1ccc3c17767f2f376cba349379f013fd727f444bb4032e",
      "regex_counts": [
        {
          "label": "independently checked active rows",
          "pattern": "^a-[^:]+: packet=",
          "count": 63
        },
        {
          "label": "independently checked scalar rows",
          "pattern": "^scalar-[^:]+: objective=",
          "count": 4
        }
      ]
    }
  ],
  "scope_note": "The master verifier checks release integrity and orchestrates the exact audits. The near-Siegel command is a standard-library exact reimplementation of that branch. The independent finite-row commands separately verify the exact row covers, class-count recurrences, common cap coordinates, suffix majorants, greedy fills, and objectives for all sixteen aligned-secondary rows and all sixty-three active plus four scalar rows. They take the exported analytic caps, integer count inequalities, and mass budgets as their explicit trust boundary; they do not independently derive those inputs from the Zhao-Pintz analytic estimates."
}