{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261031-r07-martingale-0012",
  "pilot_number": "R0066-33737",
  "number": 33737,
  "family_id": "qf-mod31-martdeep-constant-expectation-counterexample",
  "title": {
    "zh-Hant": "第 33737 題：Martingale深度",
    "en": "Question 33737: Constant Total Expectations Can Fail Conditional Martingale Steps"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261031-r07-martingale-0012",
    "batch_id": "batch-20261014-advance-20261031-r07-martingale-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-martingale"
    ],
    "title_zh": "Martingale深度",
    "stem_zh": "同一有限機率空間與filtration的精確模型。操作=martdeep-constant-expectation-counterexample；資料={\"amplitude\":12}。請依 [X0,X1,X2,expectations,martingale,reason] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(0, [12, -12], [-12, 12], [0, 0, 0], 否, constant_expectations_do_not_imply_conditional_martingale_steps)",
      "explanation_zh": "Wave4收束核對=(0, [12, -12], [-12, 12], [0, 0, 0], 否, constant_expectations_do_not_imply_conditional_martingale_steps)；依完整明示資料重算；常期望不足；差分須逐資訊block條件均值為零；不引入optional stopping、收斂、Markov或獨立增量聲明。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "Martingale深度"
      ]
    },
    "variant": {
      "family_id": "qf-mod31-martdeep-constant-expectation-counterexample",
      "seed": 202610317012,
      "parameters": {
        "amplitude": 12,
        "domain": "Exact_finite_martingale_construction",
        "operation": "martdeep-constant-expectation-counterexample",
        "semantic_template": "qf-mod31-martdeep-constant-expectation-counterexample/{\"amplitude\":12}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261031-r07-martingale-0012",
    "batch_id": "batch-20261014-advance-20261031-r07-martingale-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2b5594a60f3a9152b3e8cd508e00d6091ec35306d5e9621ceedc86085a14fb39",
      "family_id": "qf-mod31-martdeep-constant-expectation-counterexample",
      "seed": 202610317012,
      "semantic_template": "qf-mod31-martdeep-constant-expectation-counterexample/{\"amplitude\":12}"
    },
    "fields": {
      "title": "Constant Total Expectations Can Fail Conditional Martingale Steps",
      "stem": "Use two equiprobable outcomes. X0=0, X1=[a,−a], X2=[−a,a] with amplitude a=12>0. F0 is trivial and F1=F2 is the full information partition revealed by X1. Give (X0,X1,X2,expectations,martingale?,constant_expectations_do_not_imply_conditional_martingale_steps).",
      "answer": {
        "kind": "expression",
        "value": "(0, [12, -12], [-12, 12], [0, 0, 0], No, constant_expectations_do_not_imply_conditional_martingale_steps)",
        "explanation": "All three total expectations are zero, and the process is adapted. At time 1, however, the conditional expectation of X2 is X2 itself and differs from X1 on both outcomes. This is the specific counterexample, not a claim that every constant-expectation process fails. The exact result, retaining the requested order, is (0, [12, -12], [-12, 12], [0, 0, 0], No, constant_expectations_do_not_imply_conditional_martingale_steps)."
      },
      "skills": [
        "Constant Total Expectations Can Fail Conditional Martingale Steps"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "同一有限機率空間與filtration的精確模型。操作=martdeep-constant-expectation-counterexample；資料={\"amplitude\":12}。請依 [X0,X1,X2,expectations,martingale,reason] 欄序回答。",
        "en": "Use two equiprobable outcomes. X0=0, X1=[a,−a], X2=[−a,a] with amplitude a=12>0. F0 is trivial and F1=F2 is the full information partition revealed by X1. Give (X0,X1,X2,expectations,martingale?,constant_expectations_do_not_imply_conditional_martingale_steps)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-mod31-martdeep-constant-expectation-counterexample, [amplitude, domain, operation]]",
      "after": "derived = [martdeep-constant-expectation-counterexample, Exact_finite_martingale_construction]",
      "relation": "implies",
      "rule_id": "R66_EXACT_source_bank_keys_local_operation_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原bank的operation是LOCAL hyphen，keys與domain各自精確匹配。欄位順序及數字/Boolean/字串角色依來源，所有有理數文字整數省略/1；不改原題與英文、MKO、AST或狀態。",
        "en": "THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Martingale ORIGINAL X1 total := sum(12, -12)",
      "after": "Martingale ORIGINAL X1 total = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。",
        "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Martingale ORIGINAL X1 total，不以小數近似替代。",
          "en": "Recompute Martingale ORIGINAL X1 total with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "12",
          "-12"
        ],
        "output": "0",
        "label": "Martingale ORIGINAL X1 total"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Martingale ORIGINAL X2 total := sum(-12, 12)",
      "after": "Martingale ORIGINAL X2 total = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。",
        "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Martingale ORIGINAL X2 total，不以小數近似替代。",
          "en": "Recompute Martingale ORIGINAL X2 total with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-12",
          "12"
        ],
        "output": "0",
        "label": "Martingale ORIGINAL X2 total"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [12]",
      "after": "derived = [[omega0, omega1], [F0=trivial, F1=F2=full], 0, [12, -12], [-12, 12], [0, 0, 0]]",
      "relation": "implies",
      "rule_id": "Martingale_EXPLICIT_equally_likely_space_filtration_adapted",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個等機率outcomes，F0平凡、F1=F2完整資訊；三時間值都adapted且可積，總期望皆0。這還不是martingale的條件期望等式。",
        "en": "Use TWO equiprobable outcomes,F0 trivial and F1=F2 full.The three-time process is adapted and integrable;all total expectations are0.That is not yet the conditional martingale equation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Martingale ORIGINAL E X2 given full information minus X1 outcome0 := sub(-12, 12)",
      "after": "Martingale ORIGINAL E X2 given full information minus X1 outcome0 = -24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。",
        "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Martingale ORIGINAL E X2 given full information minus X1 outcome0，不以小數近似替代。",
          "en": "Recompute Martingale ORIGINAL E X2 given full information minus X1 outcome0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-12",
          "12"
        ],
        "output": "-24",
        "label": "Martingale ORIGINAL E X2 given full information minus X1 outcome0"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Martingale ORIGINAL E X2 given full information minus X1 outcome1 := sub(12, -12)",
      "after": "Martingale ORIGINAL E X2 given full information minus X1 outcome1 = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。",
        "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Martingale ORIGINAL E X2 given full information minus X1 outcome1，不以小數近似替代。",
          "en": "Recompute Martingale ORIGINAL E X2 given full information minus X1 outcome1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "12",
          "-12"
        ],
        "output": "24",
        "label": "Martingale ORIGINAL E X2 given full information minus X1 outcome1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "source roles = [[12, -12], [-12, 12]]",
      "after": "derived = [E[X2|F1]=X2, [-24, 24], 否]",
      "relation": "implies",
      "rule_id": "Martingale_COUNTEREXAMPLE_conditional_failure_despite_constant_means",
      "conditions": [],
      "explanation": {
        "zh-Hant": "F1已知實際outcome，所以E[X2|F1]=X2；逐outcome與X1差為非0，第二步失敗。反例否定「常期望充分」，不說每個常期望process都失敗。",
        "en": "F1 reveals the outcome,so E[X2|F1]=X2.The difference from X1 is nonzero on EACH outcome,so the second step fails.This refutes sufficiency of constant expectations,not every constant-expectation process."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    }
  ],
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        -12
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        12
      ],
      [
        0,
        0,
        0
      ],
      false,
      "constant_expectations_do_not_imply_conditional_martingale_steps"
    ]
  },
  "conclusion": {
    "canonical_value": "(0, [12, -12], [-12, 12], [0, 0, 0], 否, constant_expectations_do_not_imply_conditional_martingale_steps)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
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    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
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      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
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