{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261108-r04-chebyshev-inequality-0031",
  "pilot_number": "R0040-20781",
  "number": 20781,
  "family_id": "qf-mod39-cheb-tail-probability",
  "title": {
    "zh-Hant": "第 20781 題：Chebyshev不等式",
    "en": "Question 20781: The Absolute-Deviation Event Includes Equality at Epsilon"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261108-r04-chebyshev-inequality-0031",
    "batch_id": "batch-20261014-advance-20261108-r04-chebyshev-inequality-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-chebyshev-inequality"
    ],
    "title_zh": "Chebyshev不等式",
    "stem_zh": "完整有限law、finite mean與variance及positive epsilon。操作=cheb-tail-probability；資料={\"masses\":[[1,11],[2,11],[8,11]],\"values\":[0,6,9],\"epsilon\":[3,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(84/11, 3, 1/11, exact_absolute_deviation_tail)",
      "explanation_zh": "Wave6第二層核對=(84/11, 3, 1/11, exact_absolute_deviation_tail)；依完整明示資料獨立重算；P(abs(X-mu)>=epsilon)小於等於Var(X)/epsilon squared且不需要independence；對nonnegative squared deviation套Markov；bound不等於tail並保持zero-mass與zero-variance a.s.語義。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "Chebyshev不等式"
      ]
    },
    "variant": {
      "family_id": "qf-mod39-cheb-tail-probability",
      "seed": 202611084031,
      "parameters": {
        "masses": [
          [
            1,
            11
          ],
          [
            2,
            11
          ],
          [
            8,
            11
          ]
        ],
        "values": [
          0,
          6,
          9
        ],
        "epsilon": [
          3,
          1
        ],
        "domain": "Finite_rational_Chebyshev_inequality",
        "operation": "cheb-tail-probability",
        "semantic_template": "qf-mod39-cheb-tail-probability/{\"epsilon\":[3,1],\"masses\":[[1,11],[2,11],[8,11]],\"values\":[0,6,9]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261108-r04-chebyshev-inequality-0031",
    "batch_id": "batch-20261014-advance-20261108-r04-chebyshev-inequality-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "e087ad252f740934fd9e97bcea0bdfdd6927f5cc159c53af2a7a94173dfd53e5",
      "family_id": "qf-mod39-cheb-tail-probability",
      "seed": 202611084031,
      "semantic_template": "qf-mod39-cheb-tail-probability/{\"epsilon\":[3,1],\"masses\":[[1,11],[2,11],[8,11]],\"values\":[0,6,9]}"
    },
    "fields": {
      "title": "The Absolute-Deviation Event Includes Equality at Epsilon",
      "stem": "Use complete normalized finite law masses=[1/11, 2/11, 8/11] and aligned values X=[0, 6, 9]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Give (mean, epsilon, tail_probability, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(84/11, 3, 1/11, exact_absolute_deviation_tail)",
        "explanation": "Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (84/11, 3, 1/11, exact_absolute_deviation_tail)."
      },
      "skills": [
        "The Absolute-Deviation Event Includes Equality at Epsilon"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "完整有限law、finite mean與variance及positive epsilon。操作=cheb-tail-probability；資料={\"masses\":[[1,11],[2,11],[8,11]],\"values\":[0,6,9],\"epsilon\":[3,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use complete normalized finite law masses=[1/11, 2/11, 8/11] and aligned values X=[0, 6, 9]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Give (mean, epsilon, tail_probability, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=cheb-tail-probability;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[mean, epsilon, tail_probability, certificate]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。標籤不自動是數學證據，No保持來源的聲稱範圍。",
        "en": "Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source claim scope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Chebyshev complete probability law mass := sum(1/11, 2/11, 8/11)",
      "after": "Chebyshev complete probability law mass = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev complete probability law mass，不以小數近似替代。",
          "en": "Recompute Chebyshev complete probability law mass with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/11",
          "2/11",
          "8/11"
        ],
        "output": "1",
        "label": "Chebyshev complete probability law mass"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Chebyshev original atom0 weighted SIGNED X := mul(1/11, 0)",
      "after": "Chebyshev original atom0 weighted SIGNED X = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev original atom0 weighted SIGNED X，不以小數近似替代。",
          "en": "Recompute Chebyshev original atom0 weighted SIGNED X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/11",
          "0"
        ],
        "output": "0",
        "label": "Chebyshev original atom0 weighted SIGNED X"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Chebyshev original atom1 weighted SIGNED X := mul(2/11, 6)",
      "after": "Chebyshev original atom1 weighted SIGNED X = 12/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev original atom1 weighted SIGNED X，不以小數近似替代。",
          "en": "Recompute Chebyshev original atom1 weighted SIGNED X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2/11",
          "6"
        ],
        "output": "12/11",
        "label": "Chebyshev original atom1 weighted SIGNED X"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Chebyshev original atom2 weighted SIGNED X := mul(8/11, 9)",
      "after": "Chebyshev original atom2 weighted SIGNED X = 72/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev original atom2 weighted SIGNED X，不以小數近似替代。",
          "en": "Recompute Chebyshev original atom2 weighted SIGNED X with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8/11",
          "9"
        ],
        "output": "72/11",
        "label": "Chebyshev original atom2 weighted SIGNED X"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Chebyshev complete SAME-law mean := sum(0, 12/11, 72/11)",
      "after": "Chebyshev complete SAME-law mean = 84/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev complete SAME-law mean，不以小數近似替代。",
          "en": "Recompute Chebyshev complete SAME-law mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "12/11",
          "72/11"
        ],
        "output": "84/11",
        "label": "Chebyshev complete SAME-law mean"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Chebyshev original atom0 X minus SAME mean := sub(0, 84/11)",
      "after": "Chebyshev original atom0 X minus SAME mean = -84/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev original atom0 X minus SAME mean，不以小數近似替代。",
          "en": "Recompute Chebyshev original atom0 X minus SAME mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "84/11"
        ],
        "output": "-84/11",
        "label": "Chebyshev original atom0 X minus SAME mean"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Chebyshev original atom0 centered SQUARE := mul(-84/11, -84/11)",
      "after": "Chebyshev original atom0 centered SQUARE = 7056/121",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev original atom0 centered SQUARE，不以小數近似替代。",
          "en": "Recompute Chebyshev original atom0 centered SQUARE with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-84/11",
          "-84/11"
        ],
        "output": "7056/121",
        "label": "Chebyshev original atom0 centered SQUARE"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Chebyshev original atom0 weighted centered square := mul(1/11, 7056/121)",
      "after": "Chebyshev original atom0 weighted centered square = 7056/1331",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev original atom0 weighted centered square，不以小數近似替代。",
          "en": "Recompute Chebyshev original atom0 weighted centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/11",
          "7056/121"
        ],
        "output": "7056/1331",
        "label": "Chebyshev original atom0 weighted centered square"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Chebyshev original atom1 X minus SAME mean := sub(6, 84/11)",
      "after": "Chebyshev original atom1 X minus SAME mean = -18/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev original atom1 X minus SAME mean，不以小數近似替代。",
          "en": "Recompute Chebyshev original atom1 X minus SAME mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "6",
          "84/11"
        ],
        "output": "-18/11",
        "label": "Chebyshev original atom1 X minus SAME mean"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Chebyshev original atom1 centered SQUARE := mul(-18/11, -18/11)",
      "after": "Chebyshev original atom1 centered SQUARE = 324/121",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev original atom1 centered SQUARE，不以小數近似替代。",
          "en": "Recompute Chebyshev original atom1 centered SQUARE with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-18/11",
          "-18/11"
        ],
        "output": "324/121",
        "label": "Chebyshev original atom1 centered SQUARE"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Chebyshev original atom1 weighted centered square := mul(2/11, 324/121)",
      "after": "Chebyshev original atom1 weighted centered square = 648/1331",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev original atom1 weighted centered square，不以小數近似替代。",
          "en": "Recompute Chebyshev original atom1 weighted centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2/11",
          "324/121"
        ],
        "output": "648/1331",
        "label": "Chebyshev original atom1 weighted centered square"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Chebyshev original atom2 X minus SAME mean := sub(9, 84/11)",
      "after": "Chebyshev original atom2 X minus SAME mean = 15/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev original atom2 X minus SAME mean，不以小數近似替代。",
          "en": "Recompute Chebyshev original atom2 X minus SAME mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "9",
          "84/11"
        ],
        "output": "15/11",
        "label": "Chebyshev original atom2 X minus SAME mean"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Chebyshev original atom2 centered SQUARE := mul(15/11, 15/11)",
      "after": "Chebyshev original atom2 centered SQUARE = 225/121",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev original atom2 centered SQUARE，不以小數近似替代。",
          "en": "Recompute Chebyshev original atom2 centered SQUARE with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "15/11",
          "15/11"
        ],
        "output": "225/121",
        "label": "Chebyshev original atom2 centered SQUARE"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Chebyshev original atom2 weighted centered square := mul(8/11, 225/121)",
      "after": "Chebyshev original atom2 weighted centered square = 1800/1331",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev original atom2 weighted centered square，不以小數近似替代。",
          "en": "Recompute Chebyshev original atom2 weighted centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8/11",
          "225/121"
        ],
        "output": "1800/1331",
        "label": "Chebyshev original atom2 weighted centered square"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Chebyshev full variance E centered square := sum(7056/1331, 648/1331, 1800/1331)",
      "after": "Chebyshev full variance E centered square = 864/121",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev full variance E centered square，不以小數近似替代。",
          "en": "Recompute Chebyshev full variance E centered square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "7056/1331",
          "648/1331",
          "1800/1331"
        ],
        "output": "864/121",
        "label": "Chebyshev full variance E centered square"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "ORIGINAL complete normalized masses=[1/11, 2/11, 8/11];ALL aligned SIGNED values X=[0, 6, 9];SAME-law mean=84/11;all deviations=[-84/11, -18/11, 15/11]",
      "after": "ALL nonnegative squared deviations Y=[7056/121, 324/121, 225/121];variance=E[Y]=864/121;finite moments;variance0 iff X=mean on EVERY POSITIVE-mass atom,not necessarily null atoms",
      "relation": "implies",
      "rule_id": "cheb_same_law_finite_signed_variable_no_independence_premise",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此題是完整law的一個隨機變數，X可有正質量負值，不要偷加X≥0或independence。平方Y在每個原子都非負；零變異只迫正質量處偏差0，null-outlier仍保原資料。",
        "en": "This is one random variable under a complete law.X may be negative at positive mass;do not add X≥0 or independence.Each squared Y is nonnegative,and zero variance forces zero deviations only at positive-mass atoms;retain null outliers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Chebyshev original atom0 absolute-event probability mass := mul(1/11, 1)",
      "after": "Chebyshev original atom0 absolute-event probability mass = 1/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev original atom0 absolute-event probability mass，不以小數近似替代。",
          "en": "Recompute Chebyshev original atom0 absolute-event probability mass with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/11",
          "1"
        ],
        "output": "1/11",
        "label": "Chebyshev original atom0 absolute-event probability mass"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "ORIGINAL atom=0;SIGNED centered deviation=-84/11;absolute deviation=84/11;positive epsilon=3",
      "after": "event abs(deviation)>=epsilon=是;EQUALITY INCLUDED;probability contribution=1/11",
      "relation": "implies",
      "rule_id": "cheb_every_absolute_event_atom_including_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先對同一law的mean中心化再取絕對值，事件用≥而非>；正epsilon本身的嚴格資格不會把尾事件改成嚴格。",
        "en": "Center at this same law's mean before taking the absolute value.Use ≥ for the event,not >;strict positivity of epsilon does not make the event inequality strict."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Chebyshev original atom1 absolute-event probability mass := mul(2/11, 0)",
      "after": "Chebyshev original atom1 absolute-event probability mass = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev original atom1 absolute-event probability mass，不以小數近似替代。",
          "en": "Recompute Chebyshev original atom1 absolute-event probability mass with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2/11",
          "0"
        ],
        "output": "0",
        "label": "Chebyshev original atom1 absolute-event probability mass"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "ORIGINAL atom=1;SIGNED centered deviation=-18/11;absolute deviation=18/11;positive epsilon=3",
      "after": "event abs(deviation)>=epsilon=否;EQUALITY INCLUDED;probability contribution=0",
      "relation": "implies",
      "rule_id": "cheb_every_absolute_event_atom_including_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先對同一law的mean中心化再取絕對值，事件用≥而非>；正epsilon本身的嚴格資格不會把尾事件改成嚴格。",
        "en": "Center at this same law's mean before taking the absolute value.Use ≥ for the event,not >;strict positivity of epsilon does not make the event inequality strict."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Chebyshev original atom2 absolute-event probability mass := mul(8/11, 0)",
      "after": "Chebyshev original atom2 absolute-event probability mass = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev original atom2 absolute-event probability mass，不以小數近似替代。",
          "en": "Recompute Chebyshev original atom2 absolute-event probability mass with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8/11",
          "0"
        ],
        "output": "0",
        "label": "Chebyshev original atom2 absolute-event probability mass"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "ORIGINAL atom=2;SIGNED centered deviation=15/11;absolute deviation=15/11;positive epsilon=3",
      "after": "event abs(deviation)>=epsilon=否;EQUALITY INCLUDED;probability contribution=0",
      "relation": "implies",
      "rule_id": "cheb_every_absolute_event_atom_including_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先對同一law的mean中心化再取絕對值，事件用≥而非>；正epsilon本身的嚴格資格不會把尾事件改成嚴格。",
        "en": "Center at this same law's mean before taking the absolute value.Use ≥ for the event,not >;strict positivity of epsilon does not make the event inequality strict."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Chebyshev COMPLETE absolute-deviation tail probability := sum(1/11, 0, 0)",
      "after": "Chebyshev COMPLETE absolute-deviation tail probability = 1/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Chebyshev COMPLETE absolute-deviation tail probability，不以小數近似替代。",
          "en": "Recompute Chebyshev COMPLETE absolute-deviation tail probability with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/11",
          "0",
          "0"
        ],
        "output": "1/11",
        "label": "Chebyshev COMPLETE absolute-deviation tail probability"
      }
    }
  ],
  "math_model": {
    "result": [
      "84/11",
      "3",
      "1/11",
      "exact_absolute_deviation_tail"
    ]
  },
  "conclusion": {
    "canonical_value": "(84/11, 3, 1/11, exact_absolute_deviation_tail)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261014-advance-20261108-r04-chebyshev-inequality-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261108-r04-chebyshev-inequality-foundations/questions.jsonl",
      "batch_sha256": "c90bc449291772f89479d9f3a367a2ffcec971aa6328ff0535e2cd63991a9827",
      "original_line": 31,
      "original_line_sha256": "b777af0eed72302b2a0b295eef724c9e91799b03a08936946d5f90b0f3f0c1e8",
      "question_compact_sha256": "e087ad252f740934fd9e97bcea0bdfdd6927f5cc159c53af2a7a94173dfd53e5",
      "english_row_compact_sha256": "875486a9e65d1d7730fd9edfa5d60565361a61d399fea9b83b4d3ef57991f6e8",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261108-r04-chebyshev-inequality-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261108-r04-chebyshev-inequality-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-chebyshev-inequality": {
        "sha256": "53f883dd0d24e24dd2c178a77c932cdf6caf9b65e2b443bee96a237be01b73e3",
        "english_sha256": "af0da60b30b2be710d729655f44307db4c48210cbe727eded9d899eefcd264a5",
        "original_path": "public/data/mko/chebyshev-inequality.json",
        "english_path": "public/data/i18n/editions/mko-chebyshev-inequality.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "f1ed3239c9185524aa4f0839ab4cfbe8f7d74200b283ea09e850a792f0178a3b",
    "original_candidate_sha256": "d363916bf85d407228fd63301313258b00c3ce2bdb9bb41e3eb87f577e20ad40",
    "original_candidate_line_sha256": "d363916bf85d407228fd63301313258b00c3ce2bdb9bb41e3eb87f577e20ad40"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-008-p41",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d",
      "question_first_witness_sha256": "e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 24,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "3880bad8ba01d1ac366fcccb5618270074793b24cadfad7009eabc82fbede972",
      "arithmetic_steps": 19
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}