{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261023-r06-discrete-stochastic-process-0026",
  "pilot_number": "R0020-10751",
  "number": 10751,
  "family_id": "qf-mod23-process-time-marginal",
  "title": {
    "zh-Hant": "第 10751 題：有限時域joint process",
    "en": "Question 10751: Aggregate a Marginal from the Complete Joint Path Law"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261023-r06-discrete-stochastic-process-0026",
    "batch_id": "batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-discrete-stochastic-process"
    ],
    "title_zh": "有限時域joint process",
    "stem_zh": "共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[0,0,1],[0,1,1],[1,1,0]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0}。請依 [time,marginal_rows] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(0, [[0, 1/2], [1, 1/2]])",
      "explanation_zh": "現代橋梁核對=(0, [[0, 1/2], [1, 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "有限時域joint process"
      ]
    },
    "variant": {
      "family_id": "qf-mod23-process-time-marginal",
      "seed": 202610236026,
      "parameters": {
        "paths": [
          [
            0,
            0,
            1
          ],
          [
            0,
            1,
            1
          ],
          [
            1,
            1,
            0
          ]
        ],
        "masses": [
          [
            1,
            4
          ],
          [
            1,
            4
          ],
          [
            1,
            2
          ]
        ],
        "time": 0,
        "domain": "Finite_horizon_joint_path_law",
        "operation": "process-time-marginal",
        "semantic_template": "qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[0,0,1],[0,1,1],[1,1,0]],\"time\":0}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261023-r06-discrete-stochastic-process-0026",
    "batch_id": "batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "32716da528d08a29b2dd2f6375d64ff8c4521ea5b253f8ed8b100e23c9a1c540",
      "family_id": "qf-mod23-process-time-marginal",
      "seed": 202610236026,
      "semantic_template": "qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[0,0,1],[0,1,1],[1,1,0]],\"time\":0}"
    },
    "fields": {
      "title": "Aggregate a Marginal from the Complete Joint Path Law",
      "stem": "Use one common finite probability space specified by the complete distinct equal-horizon path list=[[0, 0, 1], [0, 1, 1], [1, 1, 0]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.",
      "answer": {
        "kind": "expression",
        "value": "(0, [[0, 1/2], [1, 1/2]])",
        "explanation": "At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[0, 1/2], [1, 1/2]])."
      },
      "skills": [
        "Aggregate a Marginal from the Complete Joint Path Law"
      ],
      "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": "共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[0,0,1],[0,1,1],[1,1,0]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0}。請依 [time,marginal_rows] 欄序回答。",
        "en": "Use one common finite probability space specified by the complete distinct equal-horizon path list=[[0, 0, 1], [0, 1, 1], [1, 1, 0]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "original support index=0;path=[0, 0, 1];raw mass=[1, 4]",
      "after": "aligned reduced mass=1/4",
      "relation": "implies",
      "rule_id": "path_mass_original_support_pairing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整路徑須相異且同一正時域長度，質量綁原支持索引；只約分，不重排路徑、不除總量把非法候選洗成機率。",
        "en": "Complete paths are distinct and share one positive horizon.Masses remain paired to original support indices.Reduce fractions only:do not sort paths or divide by a total to repair an unqualified candidate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "original support index=1;path=[0, 1, 1];raw mass=[1, 4]",
      "after": "aligned reduced mass=1/4",
      "relation": "implies",
      "rule_id": "path_mass_original_support_pairing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整路徑須相異且同一正時域長度，質量綁原支持索引；只約分，不重排路徑、不除總量把非法候選洗成機率。",
        "en": "Complete paths are distinct and share one positive horizon.Masses remain paired to original support indices.Reduce fractions only:do not sort paths or divide by a total to repair an unqualified candidate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "original support index=2;path=[1, 1, 0];raw mass=[1, 2]",
      "after": "aligned reduced mass=1/2",
      "relation": "implies",
      "rule_id": "path_mass_original_support_pairing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整路徑須相異且同一正時域長度，質量綁原支持索引；只約分，不重排路徑、不除總量把非法候選洗成機率。",
        "en": "Complete paths are distinct and share one positive horizon.Masses remain paired to original support indices.Reduce fractions only:do not sort paths or divide by a total to repair an unqualified candidate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "whole path-law mass sum := sum(1/4, 1/4, 1/2)",
      "after": "whole path-law mass sum = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 whole path-law mass sum，不以小數近似替代。",
          "en": "Recompute whole path-law mass sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/4",
          "1/4",
          "1/2"
        ],
        "output": "1",
        "label": "whole path-law mass sum"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "horizon=3;normalized aligned masses=[1/4, 1/4, 1/2]",
      "after": "all nonnegative=是;total=1;qualified joint law=是",
      "relation": "implies",
      "rule_id": "whole_finite_joint_law_qualification",
      "conditions": [],
      "explanation": {
        "zh-Hant": "候選資格題可保留負或未歸一質量來報失格；取邊際／事件／聯合分布前則必須整份非負且總量恰1。有限時域不推出Markov或stationarity。",
        "en": "The qualification task may retain negative or unnormalized masses to report failure.Marginal/event/joint queries require the whole law nonnegative with total exactly1.A finite horizon establishes neither Markov behavior nor stationarity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "marginal path0 bucket update := sum(0, 1/4)",
      "after": "marginal path0 bucket update = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 marginal path0 bucket update，不以小數近似替代。",
          "en": "Recompute marginal path0 bucket update with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/4"
        ],
        "output": "1/4",
        "label": "marginal path0 bucket update"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "support index=0;time=0;state=0;original mass=1/4;previous bucket=0",
      "after": "updated state mass=1/4",
      "relation": "implies",
      "rule_id": "marginal_all_paths_including_zero_mass",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一時刻狀態相同才聚合，原路徑質量不換位；即使總質量0，已列狀態的桶仍保留。",
        "en": "Aggregate paths with the same state at this time without reassigning masses.Retain a listed state's bucket even when its total is0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "marginal path1 bucket update := sum(1/4, 1/4)",
      "after": "marginal path1 bucket update = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 marginal path1 bucket update，不以小數近似替代。",
          "en": "Recompute marginal path1 bucket update with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/4",
          "1/4"
        ],
        "output": "1/2",
        "label": "marginal path1 bucket update"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "support index=1;time=0;state=0;original mass=1/4;previous bucket=1/4",
      "after": "updated state mass=1/2",
      "relation": "implies",
      "rule_id": "marginal_all_paths_including_zero_mass",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一時刻狀態相同才聚合，原路徑質量不換位；即使總質量0，已列狀態的桶仍保留。",
        "en": "Aggregate paths with the same state at this time without reassigning masses.Retain a listed state's bucket even when its total is0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "marginal path2 bucket update := sum(0, 1/2)",
      "after": "marginal path2 bucket update = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 marginal path2 bucket update，不以小數近似替代。",
          "en": "Recompute marginal path2 bucket update with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/2"
        ],
        "output": "1/2",
        "label": "marginal path2 bucket update"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "support index=2;time=0;state=1;original mass=1/2;previous bucket=0",
      "after": "updated state mass=1/2",
      "relation": "implies",
      "rule_id": "marginal_all_paths_including_zero_mass",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一時刻狀態相同才聚合，原路徑質量不換位；即使總質量0，已列狀態的桶仍保留。",
        "en": "Aggregate paths with the same state at this time without reassigning masses.Retain a listed state's bucket even when its total is0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "requested time=0;all state buckets=[[0, 1/2], [1, 1/2]]",
      "after": "marginal only;no joint or Markov conclusion",
      "relation": "implies",
      "rule_id": "marginal_not_joint_or_Markov_certificate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "輸出按數值狀態排序；邊際表不決定兩時刻依賴、聯合律或Markov性。",
        "en": "Order marginal states numerically.A marginal table determines neither two-time dependence,the joint law nor Markov behavior."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      0,
      [
        [
          0,
          "1/2"
        ],
        [
          1,
          "1/2"
        ]
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "(0, [[0, 1/2], [1, 1/2]])",
    "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."
    }
  },
  "definition_note": {
    "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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    "original_candidate_sha256": "ab800aac1a66eb1e2530ca9a7bc4ee2434baef66c7b2d181d72ee7cd878143c1",
    "original_candidate_line_sha256": "ab800aac1a66eb1e2530ca9a7bc4ee2434baef66c7b2d181d72ee7cd878143c1"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "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."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-006-p21",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598f",
      "question_first_witness_sha256": "3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 12,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "41e6701868090ec828ef819bbf7bdea6e7e02b10cd2db4db98fe486a4b1ed250",
      "arithmetic_steps": 4
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}