{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-finite-sufficient-statistic-01",
  "mko_id": "mko-finite-sufficient-statistic",
  "title": {
    "zh-Hant": "逐fiber條件law的充分性證書",
    "en": "A sufficiency certificate from every fibre conditional law"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-finite-sufficient-statistic-01",
    "source": {
      "mko_id": "mko-finite-sufficient-statistic",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "5b80cc27ea20e17864cb5068dedde0919001d6163cabe652473150401c131231",
      "english_sha256": "cd5defd0a6ff5e44e92db46ae4d8f1c71413883dfb875e6539fbcd7cac62fef6"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "完整核正質量fiber跨theta不變，保留null未定義。",
      "en": "Audit invariance across theta on every positive-mass fibre, retaining undefined null conditions."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "Θ={0,1}，Ω={a,b,c,d,z}；P0=(1/8,3/8,1/4,1/4,0)，P1=(1/16,3/16,3/8,3/8,0)。S(a)=S(b)=0，S(c)=S(d)=1，S(z)=2。",
          "en": "Theta={0,1}, Omega={a,b,c,d,z}; P0=(1/8,3/8,1/4,1/4,0), P1=(1/16,3/16,3/8,3/8,0). S(a)=S(b)=0, S(c)=S(d)=1, S(z)=2."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "P_0,P_1",
        "after_tex": "1/8+3/8+1/4+1/4=1,\\quad1/16+3/16+3/8+3/8=1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩個完整law歸一且非負。",
          "en": "Both complete laws are normalized and nonnegative."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "z在兩law質量零但仍在carrier。",
            "en": "z has zero mass in both laws but remains in the carrier."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "S^{-1}(0),S^{-1}(1),S^{-1}(2)",
        "after_tex": "\\{a,b\\},\\{c,d\\},\\{z\\}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "S total，三個非空fibers形成完整partition。",
          "en": "S is total and its three nonempty fibres form a complete partition."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不漏掉null fiber{z}。",
            "en": "Do not omit null fibre{z}."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "P_\\theta(S=t),\\quad t=(0,1,2)",
        "after_tex": "P_0^S=(1/2,1/2,0),\\quad P_1^S=(1/4,3/4,0)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先算全部fiber質量才決定哪些可conditioning。",
          "en": "Compute every fibre mass before deciding where conditioning is defined."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "sufficiency不要求S的邊際law不變。",
            "en": "Sufficiency does not require the marginal law of S to be invariant."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "t=0:\\quad P_0(a\\mid S=0)=\\frac{1/8}{1/2},\\quad P_1(a\\mid S=0)=\\frac{1/16}{1/4}",
        "after_tex": "P_0(\\cdot\\mid S=0)=P_1(\\cdot\\mid S=0)=(1/4,3/4,0,0,0)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "a、b的條件質量跨theta相同，其他三outcomes為零。",
          "en": "The conditional masses of a,b agree across theta; the other three outcomes have mass zero."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "列完整五點條件law，不能只核a。",
            "en": "List the full five-point conditional law, not only a."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "t=1:\\quad\\frac{1/4}{1/2}=\\frac{3/8}{3/4}=1/2",
        "after_tex": "P_0(\\cdot\\mid S=1)=P_1(\\cdot\\mid S=1)=(0,0,1/2,1/2,0)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二個正質量fiber也完整核跨theta不變。",
          "en": "The second positive-mass fibre also has a fully invariant conditional law."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "兩個theta此cell分母都正。",
            "en": "Both theta denominators on this cell are positive."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "P_0(S=2)=P_1(S=2)=0",
        "after_tex": "P_\\theta(\\cdot\\mid S=2)\\text{ undefined for both }\\theta",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "不造0/0分布；此fiber沒有有效theta的比較義務。",
          "en": "Do not invent a0/0 distribution; this fibre has no valid theta conditions to compare."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "undefined不是零向量機率law。",
            "en": "Undefined is not a zero-vector probability law."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "\\text{all positive fibre conditionals invariant across valid }\\theta",
        "after_tex": "S\\text{ is sufficient for this finite model}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "已窮盡兩theta與三fibers，依有限條件law定義得充分性。",
          "en": "Both parameters and all three fibres are exhausted, giving sufficiency by the finite conditional-law definition."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "沒有以factorization theorem替代逐表檢查。",
            "en": "No factorization theorem replaced the table audit."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "R(x)\\equiv0",
        "after_tex": "P_0(a\\mid R=0)=1/8\\ne1/16=P_1(a\\mid R=0)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "常數統計量 R 的唯一 fiber 為 Ω，所有 theta 下機率均為 1；給定 R=0 後，樣本結果的條件分布仍是 P_theta 且隨 theta 改變，所以 R 對此有限模型不充分。",
          "en": "The constant statistic R has the single fibre Omega, with probability 1 under each theta. The conditional distribution of the sample outcome given R=0 is P_theta itself and varies with theta, so R is not sufficient for this finite model."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "identity statistic則每有效fiber為點質量，總是充分；兩邊界不同。",
            "en": "The identity statistic instead has point-mass conditionals on valid fibres and is sufficient; the boundaries differ."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "S充分，常數R不充分；null fiber的條件law保持undefined。",
      "en": "S is sufficient and constant R is not; null-fibre conditional laws remain undefined."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "充分要求S的law跨theta相同。",
          "en": "Sufficiency requires an invariant law of S."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "本例S充分但P0(S=0)=1/2、P1(S=0)=1/4。",
          "en": "S is sufficient here although P0(S=0)=1/2 and P1(S=0)=1/4."
        }
      },
      {
        "claim": {
          "zh-Hant": "對S=2填(0,0,0,0,0)為law。",
          "en": "Use(0,0,0,0,0) as the law at S=2."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "總和為0且conditioning分母零，應undefined。",
          "en": "It sums to 0 and conditioning has zero denominator, so it is undefined."
        }
      },
      {
        "claim": {
          "zh-Hant": "常數R在此model充分。",
          "en": "Constant R is sufficient here."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "其條件a機率隨theta為1/8或1/16。",
          "en": "Its conditional probability of a varies as1/8 or1/16."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "statistic須是sample carrier上的total map，完整fibers形成partition。對每個(theta,t)先算P_theta(T=t)；只在正質量cell定義並exact計算fiber conditional law。跨所有有效theta逐t核對conditional-law invariance；zero-mass cell保持undefined。nonsufficient witness、identity statistic與constant statistic邊界分開；不提前使用factorization theorem或宣稱估計品質。",
          "en": "A statistic is a total map on the full sample carrier and its fibers form a partition. For every (theta,t) first compute P_theta(T=t). Define and calculate the exact fiber conditional law only on a positive-mass cell, then compare all valid theta laws at each t. Leave zero-mass conditions undefined, encoded as null in these outputs. Separate explicit nonsufficiency witnesses, the identity statistic and the constant-statistic boundary. No factorization theorem or estimator-quality conclusion is introduced."
        },
        "source": {
          "mko_id": "mko-finite-sufficient-statistic",
          "sha256": "5b80cc27ea20e17864cb5068dedde0919001d6163cabe652473150401c131231",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "有限充分性定義：對每個fiber，只比較正質量conditioning下的完整conditional laws是否與theta無關；零質量商保持undefined。",
          "en": "Finite sufficiency: on each fibre compare full conditional laws only where conditioning has positive mass, requiring theta invariance; null quotients remain undefined."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "(1/8)/(1/2)",
        "rhs": "1/4"
      },
      {
        "id": "N02",
        "lhs": "(1/16)/(1/4)",
        "rhs": "1/4"
      },
      {
        "id": "N03",
        "lhs": "(3/8)/(3/4)",
        "rhs": "1/2"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "不使用factorization theorem或估計品質推論。",
      "en": "No factorization theorem or estimator-quality inference."
    },
    "md_symbols": [
      [
        "S",
        "total statistic",
        "total statistic"
      ],
      [
        "t",
        "fiber標籤",
        "fibre label"
      ]
    ],
    "md_error_controls": [
      {
        "title": "先質量 / Mass first",
        "zh": "每(theta,t)先求fiber質量再除。",
        "en": "Compute each(theta,t) fibre mass before division."
      },
      {
        "title": "完整條件law / Complete conditionals",
        "zh": "每個有效cell列全部五樣本的機率。",
        "en": "List all five sample probabilities on each valid cell."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
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    "original_source_sha256": "99f4f978f27d4111a01876f24ceb88ab0e95d184de6346874988ea73556fac8f",
    "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": [
      "BR142-S222",
      "BR142-P080",
      "BR142-P080-PAIRED-ZH"
    ],
    "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."
  }
}