{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0055",
  "pilot_number": "R0054-27605",
  "number": 27605,
  "family_id": "qf-mod49-slln-fixed-n-deviation",
  "title": {
    "zh-Hant": "第 27605 題：有限值IID強大數律",
    "en": "Question 27605: The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0055",
    "batch_id": "batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-strong-law-large-numbers-finite-valued-iid"
    ],
    "title_zh": "有限值IID強大數律",
    "stem_zh": "無限IID process certificate、finite marginal law與almost-sure limit責任。操作=slln-fixed-n-deviation；資料={\"values\":[2,3],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":1,\"epsilon\":[2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(1, 5/2, 1/2, 2, 1, fixed_n_deviation_event, 否, not_an_almost_sure_limit_certificate)",
      "explanation_zh": "Wave6第十二層核對=(1, 5/2, 1/2, 2, 1, fixed_n_deviation_event, 否, not_an_almost_sure_limit_certificate)；依完整明示資料獨立重算；固定prefix mean與deviation event只屬fixed-n audit；non-IID/nonfinite-support是scoped theorem不適用，finite prefix不證a.s. convergence。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "有限值IID強大數律"
      ]
    },
    "variant": {
      "family_id": "qf-mod49-slln-fixed-n-deviation",
      "seed": 202611182055,
      "parameters": {
        "values": [
          2,
          3
        ],
        "probabilities": [
          [
            1,
            2
          ],
          [
            1,
            2
          ]
        ],
        "process_certificate": "infinite_iid",
        "n": 1,
        "epsilon": [
          2,
          4
        ],
        "domain": "Exact_finite_valued_SLLN_boundary",
        "operation": "slln-fixed-n-deviation",
        "semantic_template": "slln-fixed-n-deviation|{epsilon:[1,2],law:[[2,[1,2]],[3,[1,2]]],n:1,process_certificate:\"infinite_iid\"}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-0055",
    "batch_id": "batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "e9ffd3f641a871054af546317a47394bb7d6b4e6e58ef9969224c2e7c232e8a6",
      "family_id": "qf-mod49-slln-fixed-n-deviation",
      "seed": 202611182055,
      "semantic_template": "slln-fixed-n-deviation|{epsilon:[1,2],law:[[2,[1,2]],[3,[1,2]]],n:1,process_certificate:\"infinite_iid\"}"
    },
    "fields": {
      "title": "The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit",
      "stem": "Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=1 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(1, 5/2, 1/2, 2, 1, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)",
        "explanation": "Count every qualifying joint word and separately sum its exact product probability. Zero-mass words still enter the combinatorial count. A fixed-n deviation event, even when fully enumerated, is not an almost-sure convergence certificate. The exact result, retaining the requested order, is (1, 5/2, 1/2, 2, 1, fixed_n_deviation_event, No, not_an_almost_sure_limit_certificate)."
      },
      "skills": [
        "The Complete Fixed-n Deviation Event Is Separate from an Almost-Sure Limit"
      ],
      "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": "無限IID process certificate、finite marginal law與almost-sure limit責任。操作=slln-fixed-n-deviation；資料={\"values\":[2,3],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":1,\"epsilon\":[2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use n=1 and epsilon=2/4>0; the event is |sample_mean−mu|≥epsilon. Give (n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [slln-fixed-n-deviation]",
      "after": "derived = [[n, common_mean, epsilon, deviating_joint_word_count, deviation_probability, event_scope, almost_sure_limit_proof_claim, boundary]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "直接明列此操作與mode完整欄序，不依JavaScript、不由答案形狀猜角色；布林、字串、null與未作聲稱保持不同。",
        "en": "List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [infinite_iid]",
      "after": "derived = [ASSUME whole infinite sequence is IID with this common marginal]",
      "relation": "implies",
      "rule_id": "SLLN_infinite_IID_EXACT_GIVEN_premise_not_finite_observation_inference",
      "conditions": [],
      "explanation": {
        "zh-Hant": "infinite_iid是題設無限過程前提，非有限資料或本程式已構造／證明的independence。",
        "en": "infinite_iid is a GIVEN whole-process premise,not independence constructed or proved by finite data/this program."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "SLLN ORIGINAL common marginal ORIGINAL mass sum0 := sum(0, 1/2)",
      "after": "SLLN ORIGINAL common marginal ORIGINAL mass sum0 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mass sum0，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL mass sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/2"
        ],
        "output": "1/2",
        "label": "SLLN ORIGINAL common marginal ORIGINAL mass sum0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "SLLN ORIGINAL common marginal ORIGINAL mass sum1 := sum(1/2, 1/2)",
      "after": "SLLN ORIGINAL common marginal ORIGINAL mass sum1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mass sum1，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL mass sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/2",
          "1/2"
        ],
        "output": "1",
        "label": "SLLN ORIGINAL common marginal ORIGINAL mass sum1"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "SLLN ORIGINAL common marginal ORIGINAL weighted value0 := mul(2, 1/2)",
      "after": "SLLN ORIGINAL common marginal ORIGINAL weighted value0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL weighted value0，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL weighted value0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1/2"
        ],
        "output": "1",
        "label": "SLLN ORIGINAL common marginal ORIGINAL weighted value0"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "SLLN ORIGINAL common marginal ORIGINAL mean sum0 := sum(0, 1)",
      "after": "SLLN ORIGINAL common marginal ORIGINAL mean sum0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mean sum0，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL mean sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "SLLN ORIGINAL common marginal ORIGINAL mean sum0"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "SLLN ORIGINAL common marginal exact unbounded value square0 := mul(2, 2)",
      "after": "SLLN ORIGINAL common marginal exact unbounded value square0 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal exact unbounded value square0，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL common marginal exact unbounded value square0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "SLLN ORIGINAL common marginal exact unbounded value square0"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "SLLN ORIGINAL common marginal ORIGINAL weighted square0 := mul(4, 1/2)",
      "after": "SLLN ORIGINAL common marginal ORIGINAL weighted square0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL weighted square0，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL weighted square0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "1/2"
        ],
        "output": "2",
        "label": "SLLN ORIGINAL common marginal ORIGINAL weighted square0"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "SLLN ORIGINAL common marginal ORIGINAL second moment sum0 := sum(0, 2)",
      "after": "SLLN ORIGINAL common marginal ORIGINAL second moment sum0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL second moment sum0，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL second moment sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "2"
        ],
        "output": "2",
        "label": "SLLN ORIGINAL common marginal ORIGINAL second moment sum0"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "SLLN ORIGINAL common marginal ORIGINAL weighted value1 := mul(3, 1/2)",
      "after": "SLLN ORIGINAL common marginal ORIGINAL weighted value1 = 3/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL weighted value1，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL weighted value1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "1/2"
        ],
        "output": "3/2",
        "label": "SLLN ORIGINAL common marginal ORIGINAL weighted value1"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "SLLN ORIGINAL common marginal ORIGINAL mean sum1 := sum(1, 3/2)",
      "after": "SLLN ORIGINAL common marginal ORIGINAL mean sum1 = 5/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mean sum1，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL mean sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "3/2"
        ],
        "output": "5/2",
        "label": "SLLN ORIGINAL common marginal ORIGINAL mean sum1"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "SLLN ORIGINAL common marginal exact unbounded value square1 := mul(3, 3)",
      "after": "SLLN ORIGINAL common marginal exact unbounded value square1 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal exact unbounded value square1，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL common marginal exact unbounded value square1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "3"
        ],
        "output": "9",
        "label": "SLLN ORIGINAL common marginal exact unbounded value square1"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "SLLN ORIGINAL common marginal ORIGINAL weighted square1 := mul(9, 1/2)",
      "after": "SLLN ORIGINAL common marginal ORIGINAL weighted square1 = 9/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL weighted square1，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL weighted square1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "1/2"
        ],
        "output": "9/2",
        "label": "SLLN ORIGINAL common marginal ORIGINAL weighted square1"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "SLLN ORIGINAL common marginal ORIGINAL second moment sum1 := sum(2, 9/2)",
      "after": "SLLN ORIGINAL common marginal ORIGINAL second moment sum1 = 13/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL second moment sum1，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL second moment sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "9/2"
        ],
        "output": "13/2",
        "label": "SLLN ORIGINAL common marginal ORIGINAL second moment sum1"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "SLLN ORIGINAL common marginal ORIGINAL mean squared := mul(5/2, 5/2)",
      "after": "SLLN ORIGINAL common marginal ORIGINAL mean squared = 25/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mean squared，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL mean squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5/2",
          "5/2"
        ],
        "output": "25/4",
        "label": "SLLN ORIGINAL common marginal ORIGINAL mean squared"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "SLLN ORIGINAL common marginal ORIGINAL second minus mean squared := sub(13/2, 25/4)",
      "after": "SLLN ORIGINAL common marginal ORIGINAL second minus mean squared = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL second minus mean squared，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL common marginal ORIGINAL second minus mean squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "13/2",
          "25/4"
        ],
        "output": "1/4",
        "label": "SLLN ORIGINAL common marginal ORIGINAL second minus mean squared"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "source roles = [SLLN ORIGINAL common marginal, [2, 3], [[1, 2], [1, 2]]]",
      "after": "derived = [[[2, 1/2], [3, 1/2]], 1, 5/2, 13/2, 1/4]",
      "relation": "implies",
      "rule_id": "SLLN_complete_ORIGINAL_numeric_value_probability_alignment_ALL_moments",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先綁原始value與probability再numeric排序，保零質量labels。完整law核非負與mass1，且來源仍計E[X²]-E[X]²；SLLN允許variance0，不照搬CLT的strict-positive條件。",
        "en": "Pair ORIGINAL values/probabilities before numeric sorting and retain zero-mass labels.Qualify the complete nonnegative mass1 law and its source second-minus-mean-square variance.SLLN allows variance0;do not import the CLT positive-variance condition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "SLLN ORIGINAL joint word0 integer sum := sum(2)",
      "after": "SLLN ORIGINAL joint word0 integer sum = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL joint word0 integer sum，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL joint word0 integer sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2"
        ],
        "output": "2",
        "label": "SLLN ORIGINAL joint word0 integer sum"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "SLLN ORIGINAL joint word0 mean := div(2, 1)",
      "after": "SLLN ORIGINAL joint word0 mean = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL joint word0 mean，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL joint word0 mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "SLLN ORIGINAL joint word0 mean"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "SLLN ORIGINAL joint word0 mean minus mu := sub(2, 5/2)",
      "after": "SLLN ORIGINAL joint word0 mean minus mu = -1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL joint word0 mean minus mu，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL joint word0 mean minus mu with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "5/2"
        ],
        "output": "-1/2",
        "label": "SLLN ORIGINAL joint word0 mean minus mu"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "SLLN ORIGINAL deviating word0 probability factor0 := mul(1, 1/2)",
      "after": "SLLN ORIGINAL deviating word0 probability factor0 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL deviating word0 probability factor0，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL deviating word0 probability factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1/2"
        ],
        "output": "1/2",
        "label": "SLLN ORIGINAL deviating word0 probability factor0"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "SLLN ORIGINAL deviation probability word0 := sum(0, 1/2)",
      "after": "SLLN ORIGINAL deviation probability word0 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL deviation probability word0，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL deviation probability word0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/2"
        ],
        "output": "1/2",
        "label": "SLLN ORIGINAL deviation probability word0"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "source roles = [1, 0, [2], 5/2, 1/2]",
      "after": "derived = [2, 1/2, 是, 1/2, 1, 1/2]",
      "relation": "implies",
      "rule_id": "SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐所有word驗|mean-mu|>=epsilon含等號；即使機率0，若deviate仍計入combinatorial count，weighted probability另加。未deviate依來源不計不必要probability product；有限event不作a.s. limit證書。",
        "en": "For EVERY word test |mean-mu|>=epsilon,equality included.A zero-mass deviating word still contributes to the combinatorial count,while probability is weighted separately.Non-deviating words need no unused probability product under the source.This finite event is not an almost-sure limit certificate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "SLLN ORIGINAL joint word1 integer sum := sum(3)",
      "after": "SLLN ORIGINAL joint word1 integer sum = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL joint word1 integer sum，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL joint word1 integer sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3"
        ],
        "output": "3",
        "label": "SLLN ORIGINAL joint word1 integer sum"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "SLLN ORIGINAL joint word1 mean := div(3, 1)",
      "after": "SLLN ORIGINAL joint word1 mean = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL joint word1 mean，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL joint word1 mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "3",
          "1"
        ],
        "output": "3",
        "label": "SLLN ORIGINAL joint word1 mean"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "SLLN ORIGINAL joint word1 mean minus mu := sub(3, 5/2)",
      "after": "SLLN ORIGINAL joint word1 mean minus mu = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL joint word1 mean minus mu，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL joint word1 mean minus mu with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "5/2"
        ],
        "output": "1/2",
        "label": "SLLN ORIGINAL joint word1 mean minus mu"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "SLLN ORIGINAL deviating word1 probability factor0 := mul(1, 1/2)",
      "after": "SLLN ORIGINAL deviating word1 probability factor0 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL deviating word1 probability factor0，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL deviating word1 probability factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1/2"
        ],
        "output": "1/2",
        "label": "SLLN ORIGINAL deviating word1 probability factor0"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "SLLN ORIGINAL deviation probability word1 := sum(1/2, 1/2)",
      "after": "SLLN ORIGINAL deviation probability word1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 SLLN ORIGINAL deviation probability word1，不以小數近似替代。",
          "en": "Recompute SLLN ORIGINAL deviation probability word1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/2",
          "1/2"
        ],
        "output": "1",
        "label": "SLLN ORIGINAL deviation probability word1"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "source roles = [1, 1, [3], 5/2, 1/2]",
      "after": "derived = [3, 1/2, 是, 1/2, 2, 1]",
      "relation": "implies",
      "rule_id": "SLLN_EVERY_joint_word_inclusive_event_count_SEPARATE_weighted_probability",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐所有word驗|mean-mu|>=epsilon含等號；即使機率0，若deviate仍計入combinatorial count，weighted probability另加。未deviate依來源不計不必要probability product；有限event不作a.s. limit證書。",
        "en": "For EVERY word test |mean-mu|>=epsilon,equality included.A zero-mass deviating word still contributes to the combinatorial count,while probability is weighted separately.Non-deviating words need no unused probability product under the source.This finite event is not an almost-sure limit certificate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      1,
      "5/2",
      "1/2",
      2,
      "1",
      "fixed_n_deviation_event",
      false,
      "not_an_almost_sure_limit_certificate"
    ]
  },
  "conclusion": {
    "canonical_value": "(1, 5/2, 1/2, 2, 1, fixed_n_deviation_event, 否, not_an_almost_sure_limit_certificate)",
    "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."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations/questions.jsonl",
      "batch_sha256": "ddb49de712a058f17353bffb58a7d023ec4a62dc04e72fd2aa4d83570a11cec2",
      "original_line": 55,
      "original_line_sha256": "e531b0497da557c933ba09b6b70d4779779dd441e8fe4a98d70d275d34b354ff",
      "question_compact_sha256": "e9ffd3f641a871054af546317a47394bb7d6b4e6e58ef9969224c2e7c232e8a6",
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      "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。",
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    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
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}