{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-markov-inequality-01",
  "mko_id": "mko-markov-inequality",
  "title": {
    "zh-Hant": "由逐點不等式得到Markov界",
    "en": "Deriving a Markov bound pointwise"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-markov-inequality-01",
    "source": {
      "mko_id": "mko-markov-inequality",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "1363d255fe303067c7fcc800bd82bcecddd8331a0e2262159df06012d4c8bdbd",
      "english_sha256": "0c9c249b7d6fc18648349ae126e9311ddbd1af2b9589196a552d85bb6f181567"
    },
    "mode": "theorem_proof",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "證有限law的界並比較exact tail。",
      "en": "Prove the finite-law bound and compare its exact tail."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "X取0、2、6，質量分別1/2、1/3、1/6；閾值a=3。",
          "en": "X takes 0,2,6 with masses 1/2,1/3,1/6; threshold a=3."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "1/2+1/3+1/6=1",
        "after_tex": "X\\ge0,\\quad a=3>0,\\quad E[X]<\\infty",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "law歸一、值非負、閾值正，有限值保證可積。",
          "en": "The law is normalized, X nonnegative, and the threshold positive; finite values ensure integrability."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "三項資格各自必要，不能只核mean為正。",
            "en": "A positive mean alone does not replace these qualifications."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "x<3,\\ x\\in\\{0,2,6\\}",
        "after_tex": "3\\mathbf1_{\\{x\\ge3\\}}=0\\le x",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "在閾值以下，indicator為零且x非負。",
          "en": "Below threshold the indicator is zero and x is nonnegative."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "signed x在此分支可能失敗，正是非負性用途。",
            "en": "A signed x could fail in this branch, locating the nonnegativity requirement."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "x\\ge3",
        "after_tex": "3\\mathbf1_{\\{x\\ge3\\}}=3\\le x",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "閾值以上分支由x≥3直接成立。",
          "en": "Above threshold the inequality follows directly from x≥3."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "兩分支合起來覆蓋全部support。",
            "en": "Together the branches exhaust the support."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "3\\mathbf1_{\\{X\\ge3\\}}\\le X",
        "after_tex": "3P(X\\ge3)\\le E[X]",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "以非負質量逐項相乘再加總，保持不等號方向。",
          "en": "Multiply by nonnegative masses and sum, preserving the inequality."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "使用全部support而非隨機抽幾個值。",
            "en": "Every support value is used, not a random sample."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "E[X]=0/2+2/3+6/6",
        "after_tex": "E[X]=5/3",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "直接完整law計算期望。",
          "en": "Compute the expectation from the complete law."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "6的質量為1/6，不可當均勻三點。",
            "en": "The mass at 6 is 1/6; the three points are not uniform."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "3P(X\\ge3)\\le5/3",
        "after_tex": "P(X\\ge3)\\le5/9",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "除正數3得到Markov上界。",
          "en": "Divide by positive 3 to obtain the Markov upper bound."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "正閾值保證除法合法且不反向。",
            "en": "The positive threshold makes division valid without reversing order."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "\\{X\\ge3\\}=\\{X=6\\}",
        "after_tex": "P(X\\ge3)=1/6<5/9",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "exact tail只有6的質量，因此此界非等號。",
          "en": "Only the mass at 6 is in the tail, so the bound is not equality."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不要把5/9回填為實際事件機率。",
            "en": "Do not substitute5/9 for the actual event probability."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "a\\mathbf1_{\\{X\\ge a\\}}\\le X,\\quad a>0",
        "after_tex": "P(X\\ge a)\\le E[X]/a",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "同一逐點二分支論證適用任何合格有限非負law。",
          "en": "The same pointwise case proof works for any qualifying finite nonnegative law."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "此一般化依逐點證明，不依本例數字看似成立。",
            "en": "This generalization rests on the pointwise proof, not on observed numbers."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "本例tail=1/6，Markov界=5/9；有限law版本由逐點不等式證成。",
      "en": "The exact tail is 1/6 and the Markov bound5/9; the finite-law version follows pointwise."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "tail一定等於E[X]/a。",
          "en": "The tail always equals E[X]/a."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "此例1/6≠5/9。",
          "en": "Here1/6 is not 5/9."
        }
      },
      {
        "claim": {
          "zh-Hant": "a=0仍可除。",
          "en": "Division is valid at a=0."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "正閾值是分母資格。",
          "en": "A positive threshold is a denominator requirement."
        }
      },
      {
        "claim": {
          "zh-Hant": "signed X只要mean正就能直接套。",
          "en": "A signed X with positive mean qualifies directly."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "低閾值分支0≤X可能不成立。",
          "en": "The below-threshold branch0≤X can fail."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "要求X>=0 almost surely、a>0及E[X]<infinity，結論P(X>=a)<=E[X]/a。完整有限有理law可精確算實際tail與bound；bound不等於tail，signed variable不可直接套用，零閾值禁止。",
          "en": "Require X≥0 almost surely, a>0 and finite E[X]; then P(X≥a)≤E[X]/a. Complete finite rational laws allow exact tail and bound calculations. The bound is not the tail probability, signed variables cannot directly use the nonnegative theorem, and threshold0 is excluded."
        },
        "source": {
          "mko_id": "mko-markov-inequality",
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        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "indicator期望等於事件機率；非負加權和保序；除以正數保序。",
          "en": "Indicator expectations equal event probabilities; nonnegative weighted sums and division by positive numbers preserve order."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "0/2+2/3+6/6",
        "rhs": "5/3"
      },
      {
        "id": "N02",
        "lhs": "(5/3)/3",
        "rhs": "5/9"
      },
      {
        "id": "N03",
        "lhs": "1/6",
        "rhs": "5/9",
        "relation": "<"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整有限機率分布上的非負隨機變數；一般測度版本不由有限抽樣證明。",
      "en": "A nonnegative random variable with a fully specified finite probability law; not a sampled proof of the general measure-theoretic theorem."
    },
    "md_symbols": [
      [
        "a",
        "正閾值",
        "positive threshold"
      ],
      [
        "1_A",
        "事件指示函數",
        "event indicator"
      ]
    ],
    "md_error_controls": [
      {
        "title": "非負性 / Nonnegativity",
        "zh": "先核每個正質量值非負，不能只看mean。",
        "en": "Check every positive-mass value, not merely the mean."
      },
      {
        "title": "界非機率值 / Bound versus actual value",
        "zh": "tail獨立由{6}求得1/6，再比較5/9。",
        "en": "Compute tail 1/6 separately from{6}, then compare with 5/9."
      }
    ],
    "md_open_items": []
  },
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}