{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-martingale-01",
  "mko_id": "mko-martingale",
  "title": {
    "zh-Hant": "兩次公平擲幣的完整鞅檢查",
    "en": "A complete two-coin martingale audit"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-martingale-01",
    "source": {
      "mko_id": "mko-martingale",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "25744aeb70a10daa6dcabb09f5b71bc0ee3e9f2c266412b727873f5e2abddf7b",
      "english_sha256": "baf1b76a7fdf33a34d203015ec0ab002e6398566cc74b3d6decc7de93b52132e"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "在同一空間逐時核鞅的全部責任。",
      "en": "Audit every martingale obligation on one space."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "Ω={HH,HT,TH,TT}，每原子質量1/4，σ代數為冪集。",
          "en": "Omega={HH,HT,TH,TT}, every atom has mass 1/4, with the power-set sigma-algebra."
        }
      },
      {
        "id": "G2",
        "statement": {
          "zh-Hant": "P0={Ω}；P1={{HH,HT},{TH,TT}}；P2為四個singleton。F_t=σ(P_t)。",
          "en": "P0={Omega}; P1={{HH,HT},{TH,TT}}; P2 consists of four singletons. F_t=sigma(P_t)."
        }
      },
      {
        "id": "G3",
        "statement": {
          "zh-Hant": "依上述原子順序，X0=(0,0,0,0)，X1=(1,1,-1,-1)，X2=(2,0,0,-2)。",
          "en": "In that atom order X0=(0,0,0,0), X1=(1,1,-1,-1), X2=(2,0,0,-2)."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "G2",
          "G3",
          "work"
        ],
        "before_tex": "p_\\omega=1/4",
        "after_tex": "\\sum_{\\omega\\in\\Omega}p_\\omega=1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "四個非負質量總和一，三個變數共享這個law。",
          "en": "The four nonnegative masses sum to one; all variables use this law."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "G3",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不得把不同時刻的邊際law拼成共同空間。",
            "en": "Separate marginal laws would not supply this common space."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "G2",
          "G3",
          "S01",
          "work"
        ],
        "before_tex": "\\mathcal P_0,\\mathcal P_1,\\mathcal P_2",
        "after_tex": "\\mathcal F_0\\subseteq\\mathcal F_1\\subseteq\\mathcal F_2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "P1每塊包含於Ω，P2每singleton包含於一個P1塊，故資訊嵌套。",
          "en": "Every P1 block lies in Omega and each singleton in one P1 block, proving nesting."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "G3",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "反查每個細塊都有唯一母塊，沒有交叉分割。",
            "en": "Every finer block has one parent; no crossing partition is hidden."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "G2",
          "G3",
          "S02",
          "work"
        ],
        "before_tex": "X_t\\big|_B\\quad(B\\in\\mathcal P_t)",
        "after_tex": "X_t\\text{ is }\\mathcal F_t\\text{-measurable}\\quad(t=0,1,2)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "X0全域常數，X1在兩塊各為1與-1，X2在singleton可測。",
          "en": "X0 is constant, X1 is constant on each of its two blocks, and X2 is singleton-measurable."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "G3",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "若X1在HH與HT不同，這一步就失敗。",
            "en": "Different X1 values at HH and HT would fail adaptedness."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "G2",
          "G3",
          "S03",
          "work"
        ],
        "before_tex": "E|X_0|,E|X_1|,E|X_2|",
        "after_tex": "(E|X_0|,E|X_1|,E|X_2|)=(0,1,1)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "直接四原子加權，三個絕對期望有限。",
          "en": "Direct four-atom sums give finite absolute expectations."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "G3",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "負值先取絕對值，不能用期望相消冒充可積。",
            "en": "Take absolute values before summing; cancellation is not integrability."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "G2",
          "G3",
          "S04",
          "work"
        ],
        "before_tex": "E[X_1\\mid\\Omega]",
        "after_tex": "(1+1-1-1)/4=0=X_0",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "t=0只有一個正質量資訊塊，下一值條件平均等於目前值。",
          "en": "At t=0 the sole positive block has next-value mean equal to X0."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "G3",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "分母是P(Ω)=1，且已使用全部四原子。",
            "en": "The denominator is P(Omega)=1 and all four atoms are included."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "G2",
          "G3",
          "S05",
          "work"
        ],
        "before_tex": "E[X_2\\mid\\{HH,HT\\}]",
        "after_tex": "\\frac{2/4+0/4}{1/2}=1=X_1\\big|_{\\{HH,HT\\}}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "在第一個P1塊，以該塊質量1/2重新正規化。",
          "en": "On the first P1 block renormalize by its mass 1/2."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "G3",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "直接用未除1/2的分子會錯得1/2。",
            "en": "The unnormalized numerator would incorrectly give 1/2."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "G2",
          "G3",
          "S06",
          "work"
        ],
        "before_tex": "E[X_2\\mid\\{TH,TT\\}]",
        "after_tex": "\\frac{0/4-2/4}{1/2}=-1=X_1\\big|_{\\{TH,TT\\}}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二塊也核對，不能只檢查正分支。",
          "en": "Audit the second block too, rather than only the positive branch."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "G3",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "兩個P1塊互斥且覆蓋Ω，因此沒有遺漏條件情況。",
            "en": "The two disjoint P1 blocks cover Omega, exhausting conditional cases."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "G2",
          "G3",
          "S07",
          "mko_definition"
        ],
        "before_tex": "\\text{nesting, adaptedness, integrability, both time equations}",
        "after_tex": "(X_t,\\mathcal F_t)_{t=0}^{2}\\text{ is a martingale}",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "所有t=0、1的條件期望等式及前置資格均已完成。",
          "en": "Every conditional equation for t=0,1 and all prior qualifications are established."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "G3",
            "S07",
            "mko_definition"
          ],
          "explanation": {
            "zh-Hant": "結論只綁這三時刻；零質量塊的一般值僅a.s.決定。",
            "en": "The conclusion concerns these three times; null-block values generally are only a.s. determined."
          }
        },
        "leaf_basis": "mko_definition"
      }
    ],
    "result": {
      "zh-Hant": "此有限過程是相對F0⊆F1⊆F2的鞅。",
      "en": "This finite process is a martingale relative to F0⊆F1⊆F2."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "只要三時刻期望都零就足夠。",
          "en": "Zero means at all three times suffice."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "仍須核適應與每個資訊塊的條件平均。",
          "en": "Adaptation and every block conditional mean remain required."
        }
      },
      {
        "claim": {
          "zh-Hant": "第一塊條件平均為1/2。",
          "en": "The first-block conditional mean is 1/2."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "那是未正規化的分子，還須除塊質量1/2。",
          "en": "That is the numerator, which must be divided by block mass 1/2."
        }
      },
      {
        "claim": {
          "zh-Hant": "任意零質量塊必有唯一條件值。",
          "en": "Every null block has a unique conditional value."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "零分母不定義商；版本只在a.s.意義唯一。",
          "en": "A zero denominator defines no quotient; versions are only a.s. unique."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "在共同有限機率空間上完整核filtration nesting、adaptedness、可積性及每個t的E[X_{t+1}|F_t]=X_t。零質量資訊block只a.s.決定；不推出獨立增量、Markov、optional stopping或收斂。",
          "en": "On one common finite probability space, completely check filtration nesting, adaptedness, integrability and E[X_(t+1)|F_t]=X_t for every t. On zero-mass information blocks conditional values are determined only almost surely. No independent-increment, Markov, optional-stopping or convergence conclusion follows."
        },
        "source": {
          "mko_id": "mko-martingale",
          "sha256": "25744aeb70a10daa6dcabb09f5b71bc0ee3e9f2c266412b727873f5e2abddf7b",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "有限partition生成σ代數；適應表示在每時刻資訊塊上常數；正質量塊條件平均為加權和除塊質量。",
          "en": "Finite partitions generate sigma-algebras; adaptedness means constancy on the information blocks at each time; on a positive-mass block, a conditional mean is the probability-weighted sum divided by that block's probability mass."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "(1+1-1-1)/4",
        "rhs": "0"
      },
      {
        "id": "N02",
        "lhs": "(2/4)/(1/2)",
        "rhs": "1"
      },
      {
        "id": "N03",
        "lhs": "(-2/4)/(1/2)",
        "rhs": "-1"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "只處理時間0、1、2；不使用停止或收斂定理。",
      "en": "Only times 0,1,2; no stopping or convergence theorem."
    },
    "md_symbols": [
      [
        "Ω",
        "完整樣本空間",
        "complete sample space"
      ],
      [
        "P_t",
        "資訊partition",
        "information partition"
      ],
      [
        "X_t",
        "時刻t值",
        "value at time t"
      ]
    ],
    "md_error_controls": [
      {
        "title": "共同空間 / Common space",
        "zh": "每一表均使用HH、HT、TH、TT同順序。",
        "en": "Keep HH,HT,TH,TT in the same order in every table."
      },
      {
        "title": "逐塊檢查 / Blockwise check",
        "zh": "t=1兩塊都除以1/2；不以總期望代替。",
        "en": "Divide by 1/2 on both t=1 blocks; total expectation is not a substitute."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "3e4962dc9cb35d34a2042159a6a6d4be004e5b0186957eb56a98adc6741ea543",
    "original_source_sha256": "641e4e51f6819b6df72016d7daa4b23f5fb3f6e935cefe065795b2f2fedfc3c4",
    "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": [
      "work"
    ],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
      "BR142-S001",
      "BR142-S006",
      "BR142-S004",
      "BR142-S005",
      "BR142-P003",
      "BR142-S002",
      "BR142-S003"
    ],
    "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."
  }
}