{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-estimator-variance-01",
  "mko_id": "mko-estimator-variance",
  "title": {
    "zh-Hant": "同law二次矩與中心矩交叉核對",
    "en": "Same-law second and centered moment cross-checks"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-estimator-variance-01",
    "source": {
      "mko_id": "mko-estimator-variance",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "8e31f372ae9c9485e8d279c1850f034c30f4a1d657c84cd603be31da9a55e53f",
      "english_sha256": "10812015f30a09fe7768421828193eb4aff15eab016edddf0f99b5f3601d9953"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "對全部theta算variance並核對中心定義。",
      "en": "Compute variance at all theta values and verify centered definitions."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "Θ={1/4,3/4}，Ω={00,01,10,11}；P_(1/4)=(9,3,3,1)/16，P_(3/4)=(1,3,3,9)/16。",
          "en": "Theta={1/4,3/4}, Omega={00,01,10,11}; P_(1/4)=(9,3,3,1)/16 and P_(3/4)=(1,3,3,9)/16."
        }
      },
      {
        "id": "G2",
        "statement": {
          "zh-Hant": "U(x)=(x1+x2)/4，依Ω順序的total table為(0,1/4,1/4,1/2)，目標參數為g(theta)=theta。",
          "en": "U(x)=(x1+x2)/4 has total table(0,1/4,1/4,1/2) in Omega order, with target g(theta)=theta."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "G2",
          "work"
        ],
        "before_tex": "P_{1/4}^U=(9,6,1)/16,\\quad P_{3/4}^U=(1,6,9)/16",
        "after_tex": "E_{1/4}U=1/8,\\quad E_{3/4}U=3/8",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整抽樣law先固定各自中心。",
          "en": "Fix each center from its complete sampling law."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "work"
          ],
          "explanation": {
            "zh-Hant": "同一U可有不同中心。",
            "en": "One U may have different centers under different laws."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "G2",
          "S01",
          "work"
        ],
        "before_tex": "E_{1/4}U^2=(1/16)(6/16)+(1/4)(1/16)",
        "after_tex": "E_{1/4}U^2=5/128",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "平方的是U值，不是機率。",
          "en": "Square U values, not probabilities."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "中間值1/4平方為1/16。",
            "en": "The squared middle value is 1/16."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "G2",
          "S02",
          "work"
        ],
        "before_tex": "\\operatorname{Var}_{1/4}U=E_{1/4}U^2-(E_{1/4}U)^2",
        "after_tex": "5/128-(1/8)^2=3/128",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "用同theta的mean平方扣除。",
          "en": "Subtract the square of the mean from the same theta law."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不可扣(3/8)平方。",
            "en": "Do not subtract(3/8) squared here."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "G2",
          "S03",
          "work"
        ],
        "before_tex": "E_{1/4}(U-1/8)^2",
        "after_tex": "(9/16)(1/8)^2+(6/16)(1/8)^2+(1/16)(3/8)^2=3/128",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "以中心平方差完整三項獨立核第一variance。",
          "en": "Independently verify the first variance using all three centered-square terms."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "零support值的中心偏差不是零。",
            "en": "The zero support value does not have zero centered deviation."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "G2",
          "S04",
          "work"
        ],
        "before_tex": "E_{3/4}U^2=(1/16)(6/16)+(1/4)(9/16)",
        "after_tex": "E_{3/4}U^2=21/128",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二law重新使用末項9/16。",
          "en": "Use the last mass 9/16 for the second law."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不混入第一law權重。",
            "en": "Do not import first-law weights."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "G2",
          "S05",
          "work"
        ],
        "before_tex": "\\operatorname{Var}_{3/4}U=21/128-(3/8)^2",
        "after_tex": "\\operatorname{Var}_{3/4}U=3/128",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二variance恰與第一相同，但mean不同。",
          "en": "The second variance happens to equal the first despite a different mean."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "相同variance不代表相同sampling law。",
            "en": "Equal variance does not imply equal sampling laws."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "G2",
          "S06",
          "work"
        ],
        "before_tex": "E_{3/4}(U-3/8)^2",
        "after_tex": "(1/16)(3/8)^2+(6/16)(1/8)^2+(9/16)(1/8)^2=3/128",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "用第二中心平方和再次核對。",
          "en": "Cross-check again with second-law centered squares."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "每項均非負，所以負variance必是計算錯誤。",
            "en": "Every summand is nonnegative, so a negative variance signals error."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "G2",
          "S07",
          "work"
        ],
        "before_tex": "V(x)\\equiv0,\\quad\\theta\\in\\{1/4,3/4\\}",
        "after_tex": "\\operatorname{Var}_\\theta(V)=0,\\quad E_\\theta[V]\\ne\\theta",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "常數estimator提供零variance但有偏的明示邊界。",
          "en": "A constant estimator explicitly witnesses zero variance with nonzero bias."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "零variance只保證該law下a.s.常數。",
            "en": "Zero variance only gives a.s. constancy under that law."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "兩個theta的variance皆3/128，已由中心矩與二次矩各自核對。",
      "en": "Both theta variances are 3/128, verified by centered and raw second moments."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "第一variance可用第二mean3/8中心化。",
          "en": "Use the second mean 3/8 for the first variance."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "第一law mean為1/8，必須同law。",
          "en": "The first law has mean 1/8; centers must match laws."
        }
      },
      {
        "claim": {
          "zh-Hant": "零variance一定無偏。",
          "en": "Zero variance implies unbiasedness."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "V≡0有零variance但對theta>0有偏。",
          "en": "V identically0 has zero variance but is biased for theta>0."
        }
      },
      {
        "claim": {
          "zh-Hant": "兩variance相同則兩laws相同。",
          "en": "Equal variances imply equal laws."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "U=0質量9/16與1/16不同。",
          "en": "The U=0 masses differ:9/16 and 1/16."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "sampling law、support與mean全部綁同一theta，精確核Var_theta(T)=E_theta[(T-E_thetaT)^2]=E_theta[T^2]-(E_thetaT)^2。零變異只代表該theta下幾乎必然常數，不代表等於theta或無偏。",
          "en": "Keep sampling law, positive-mass support and mean at the same theta. Check Var_theta(T)=E_theta[(T−E_theta[T])^2]=E_theta[T^2]−(E_theta[T])^2 exactly. Zero variance means almost-sure constancy under that law, not equality with theta or unbiasedness."
        },
        "source": {
          "mko_id": "mko-estimator-variance",
          "sha256": "8e31f372ae9c9485e8d279c1850f034c30f4a1d657c84cd603be31da9a55e53f",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "有限law機率非負且總和一；statistic只依樣本，期望為所有outcomes的加權和。 Var(U)=E[(U-EU)^2]=E[U²]-(EU)²。",
          "en": "Finite laws have nonnegative probabilities summing to one; a statistic depends only on data, and expectation sums all weighted outcomes. Var(U)=E[(U-EU)^2]=E[U²]-(EU)²."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "5/128-(1/8)**2",
        "rhs": "3/128"
      },
      {
        "id": "N02",
        "lhs": "21/128-(3/8)**2",
        "rhs": "3/128"
      },
      {
        "id": "N03",
        "lhs": "(9/16)*(1/8)**2+(6/16)*(1/8)**2+(1/16)*(3/8)**2",
        "rhs": "3/128"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "variance只量隨機散布，不代表估計正確。",
      "en": "Variance measures random spread, not correctness of estimation."
    },
    "md_symbols": [
      [
        "θ",
        "固定但未知參數",
        "fixed unknown parameter"
      ],
      [
        "U",
        "同一total statistic",
        "one total statistic"
      ]
    ],
    "md_error_controls": [
      {
        "title": "同law中心 / Same-law center",
        "zh": "theta=1/4配1/8，theta=3/4配3/8。",
        "en": "Pair theta=1/4 with 1/8 and theta=3/4 with 3/8."
      },
      {
        "title": "雙公式 / Two formulas",
        "zh": "中心平方和與二次矩扣mean平方獨立比對3/128。",
        "en": "Independently compare centered sums and raw-moment subtraction at 3/128."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
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    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
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      "BR142-S115",
      "BR142-S111",
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    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}