{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-sampling-distribution-01",
  "mko_id": "mko-sampling-distribution",
  "title": {
    "zh-Hant": "依fiber推送的完整抽樣分布",
    "en": "A complete fibre-pushforward sampling distribution"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-sampling-distribution-01",
    "source": {
      "mko_id": "mko-sampling-distribution",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "0878462afb0517679eb337ad1d5f12bd6025d7039d27a541544de8493ab9cdc9",
      "english_sha256": "1872de8113252797ce3bf87aa73156e94042d971e9ca7591f474298d6905770c"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "對兩個theta完整合併U的相同值outcomes。",
      "en": "Aggregate equal-valued outcomes of U under both theta laws."
    },
    "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": "U=(0,1/4,1/4,1/2)",
        "after_tex": "U(\\Omega)=\\{0,1/4,1/2\\}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "列出不同statistic值，避免重複支撐。",
          "en": "List distinct statistic values without duplicate support entries."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "work"
          ],
          "explanation": {
            "zh-Hant": "01與10將屬同fiber。",
            "en": "01 and 10 belong to one fibre."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "G2",
          "S01",
          "work"
        ],
        "before_tex": "U^{-1}(0),U^{-1}(1/4),U^{-1}(1/2)",
        "after_tex": "\\{00\\},\\{01,10\\},\\{11\\}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三個fibers互斥且覆蓋四outcomes。",
          "en": "The three fibres are disjoint and cover all four outcomes."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "每個樣本恰落一fiber，不能雙算。",
            "en": "Each sample belongs to exactly one fibre."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "G2",
          "S02",
          "work"
        ],
        "before_tex": "P_{1/4}(U=0)",
        "after_tex": "9/16",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "零值fiber僅00。",
          "en": "The zero-value fibre contains only 00."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "此entry是statistic機率，不是參數機率。",
            "en": "This is a statistic probability, not a parameter probability."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "G2",
          "S03",
          "work"
        ],
        "before_tex": "P_{1/4}(U=1/4)",
        "after_tex": "3/16+3/16=3/8",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "中間fiber包含兩個觀察，必須相加。",
          "en": "The middle fibre contains two observations whose masses must be added."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不能只用01質量3/16。",
            "en": "Using only the01 mass 3/16 would omit10."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "G2",
          "S04",
          "work"
        ],
        "before_tex": "P_{1/4}(U=1/2)=1/16",
        "after_tex": "P_{1/4}^U=(9/16,3/8,1/16)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "補完第三entry形成第一完整pushforward。",
          "en": "Complete the third entry to form the first full pushforward."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "三項依0、1/4、1/2順序。",
            "en": "Entries are ordered by 0,1/4,1/2."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "G2",
          "S05",
          "work"
        ],
        "before_tex": "P_{3/4}\\text{ on the same fibres}",
        "after_tex": "P_{3/4}^U=(1/16,3/8,9/16)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二law用相同fibers重新加總，不能變更U。",
          "en": "Reaggregate the second law on the same fibres without changing U."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "中間entry相同不代表整個laws相同。",
            "en": "Equal middle entries do not make the full laws equal."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "G2",
          "S06",
          "work"
        ],
        "before_tex": "9/16+3/8+1/16,\\quad1/16+3/8+9/16",
        "after_tex": "1,\\quad1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩個抽樣分布皆非負且總質量一。",
          "en": "Both sampling distributions are nonnegative and normalized."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "pushforward歸一由完整fiber partition支撐。",
            "en": "Normalization relies on the complete fibre partition."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "G2",
          "S07",
          "work"
        ],
        "before_tex": "x_{\\rm obs}=01",
        "after_tex": "U(x_{\\rm obs})=1/4\\ne\\text{the entire law }P_\\theta^U",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "一次estimate只是一個值，與重複抽樣的完整law分開。",
          "en": "One estimate is a value, distinct from the full repeated-sampling law."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "這個不等是物件區別，未聲稱任意數值不會偶合。",
            "en": "This distinguishes objects, not an assertion that numerical values never coincide."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "U的兩個抽樣law為(9/16,3/8,1/16)與(1/16,3/8,9/16)。",
      "en": "The sampling laws of U are(9/16,3/8,1/16) and(1/16,3/8,9/16)."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "P_(1/4)(U=1/4)=3/16。",
          "en": "P_(1/4)(U=1/4)=3/16."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "01、10兩者質量都須加，得3/8。",
          "en": "Both01 and 10 contribute, giving 3/8."
        }
      },
      {
        "claim": {
          "zh-Hant": "兩law完全相同。",
          "en": "The two laws are identical."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "U=0的質量9/16與1/16不同。",
          "en": "Their U=0 masses are 9/16 and 1/16."
        }
      },
      {
        "claim": {
          "zh-Hant": "抽樣分布只需觀察到的01。",
          "en": "Only observed 01 is needed for the sampling distribution."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "需對全部carrier加權，否則總質量不足。",
          "en": "All carrier masses are required for normalization."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "在共同有限樣本空間，固定同一個不依未知theta的total statistic T，逐theta把各fiber機率加總並重驗非負與總質量一。抽樣分布不是觀察後estimate、likelihood、posterior或參數本身的分布。",
          "en": "On one common finite sample space, fix a total statistic T independent of the unknown parameter. For each parameter, sum probabilities over statistic fibres and recheck nonnegativity and total mass 1. A sampling distribution is not an observed estimate, likelihood, posterior or distribution of the parameter itself."
        },
        "source": {
          "mko_id": "mko-sampling-distribution",
          "sha256": "0878462afb0517679eb337ad1d5f12bd6025d7039d27a541544de8493ab9cdc9",
          "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。",
          "en": "Finite laws have nonnegative probabilities summing to one; a statistic depends only on data, and expectation sums all weighted outcomes."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "3/16+3/16",
        "rhs": "3/8"
      },
      {
        "id": "N02",
        "lhs": "9/16+3/8+1/16",
        "rhs": "1"
      },
      {
        "id": "N03",
        "lhs": "1/16+3/8+9/16",
        "rhs": "1"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "抽樣分布不是一次觀察後的參數分布。",
      "en": "A sampling distribution is not a parameter distribution after one observation."
    },
    "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": "fiber完整 / Complete fibres",
        "zh": "列{00}、{01,10}、{11}後再求機率。",
        "en": "List{00},{01,10},{11} before summing masses."
      },
      {
        "title": "逐theta / Parameterwise",
        "zh": "分開兩個pushforward，不平均兩套laws。",
        "en": "Keep pushforwards separate rather than averaging laws."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "34dc1cd85c3efadbcfd415dbe36f253fdad355a17c78ddeef279fa0cbb2bf808",
    "original_source_sha256": "e67f4ec52577c1384b1c87cc1e0aa76a24df5697a005983130df8ae74bd5704c",
    "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": [
      "mko_definition"
    ],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
      "BR142-S093",
      "BR142-S094",
      "BR142-S095",
      "BR142-S088",
      "BR142-S089",
      "BR142-S090",
      "BR142-S091",
      "BR142-S092"
    ],
    "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."
  }
}