{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0034",
  "pilot_number": "R0053-27284",
  "number": 27284,
  "family_id": "qf-mod49-clt-standardized-sum-moments",
  "title": {
    "zh-Hant": "第 27284 題：有限變異數IID中央極限定理",
    "en": "Question 27284: Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0034",
    "batch_id": "batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-central-limit-theorem-finite-variance-iid"
    ],
    "title_zh": "有限變異數IID中央極限定理",
    "stem_zh": "明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[1,2],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(2, [[2, 1/4], [3, 1/2], [4, 1/4]], 1, 3, 1/2, 3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)",
      "explanation_zh": "Wave6第十二層核對=(2, [[2, 1/4], [3, 1/2], [4, 1/4]], 1, 3, 1/2, 3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "有限變異數IID中央極限定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod49-clt-standardized-sum-moments",
      "seed": 202611181034,
      "parameters": {
        "values": [
          1,
          2
        ],
        "probabilities": [
          [
            1,
            2
          ],
          [
            1,
            2
          ]
        ],
        "process_certificate": "infinite_iid",
        "n": 2,
        "domain": "Exact_finite_IID_limit_theorem_boundary",
        "operation": "clt-standardized-sum-moments",
        "semantic_template": "clt-standardized-sum-moments|{law:[[1,[1,2]],[2,[1,2]]],n:2,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-r01-central-limit-theorem-finite-variance-iid-0034",
    "batch_id": "batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "3e5d253825bcacfe06c230d4596d995c458b8754a189c3f330e8de767c415180",
      "family_id": "qf-mod49-clt-standardized-sum-moments",
      "seed": 202611181034,
      "semantic_template": "clt-standardized-sum-moments|{law:[[1,[1,2]],[2,[1,2]]],n:2,process_certificate:\"infinite_iid\"}"
    },
    "fields": {
      "title": "Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit",
      "stem": "Use the aligned finite law values=[1, 2], 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 sample size n=2 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).",
      "answer": {
        "kind": "expression",
        "value": "(2, [[2, 1/4], [3, 1/2], [4, 1/4]], 1, 3, 1/2, 3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)",
        "explanation": "Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (2, [[2, 1/4], [3, 1/2], [4, 1/4]], 1, 3, 1/2, 3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)."
      },
      "skills": [
        "Complete Fixed-n Sum Moments Do Not Prove a Distributional 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": "明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[1,2],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use the aligned finite law values=[1, 2], 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 sample size n=2 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [clt-standardized-sum-moments]",
      "after": "derived = [[n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此頁直接列完整來源欄序與mode特定角色，不靠JavaScript或由答案外形猜介面；No、undefined、infinity各保原意。",
        "en": "List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [infinite_iid]",
      "after": "derived = [ASSUME infinite sequence is IID with this common marginal]",
      "relation": "implies",
      "rule_id": "CLT_infinite_iid_is_EXACT_GIVEN_premise_not_constructed_process",
      "conditions": [],
      "explanation": {
        "zh-Hant": "infinite_iid是明示無限過程前提token，並非這個有限計算已構造或驗證無限independence。後續使用該前提作條件式數學推論。",
        "en": "infinite_iid is an explicit infinite-process premise token,not a construction or finite verification of infinite independence.Subsequent reasoning is conditional on that premise."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "CLT ORIGINAL common marginal ORIGINAL law mass sum0 := sum(0, 1/2)",
      "after": "CLT ORIGINAL common marginal ORIGINAL law mass sum0 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL law mass sum0，不以小數近似替代。",
          "en": "Recompute CLT ORIGINAL common marginal ORIGINAL law mass sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/2"
        ],
        "output": "1/2",
        "label": "CLT ORIGINAL common marginal ORIGINAL law mass sum0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "CLT ORIGINAL common marginal ORIGINAL law mass sum1 := sum(1/2, 1/2)",
      "after": "CLT ORIGINAL common marginal ORIGINAL law mass sum1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL law mass sum1，不以小數近似替代。",
          "en": "Recompute CLT ORIGINAL common marginal ORIGINAL law mass sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/2",
          "1/2"
        ],
        "output": "1",
        "label": "CLT ORIGINAL common marginal ORIGINAL law mass sum1"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [CLT ORIGINAL common marginal, [1, 2], [[1, 2], [1, 2]]]",
      "after": "derived = [[[1, 1/2], [2, 1/2]], 1]",
      "relation": "implies",
      "rule_id": "CLT_ORIGINAL_value_probability_pairs_NUMERIC_order_normalized",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原始values與probabilities先配對再numeric排序，保留零機率label並核全部law質量1。這是有限marginal qualification，不是由samples證無限IID。",
        "en": "Pair ORIGINAL values/probabilities before numeric sorting,retaining zero-mass labels and checking total mass1.This qualifies a finite marginal,not an infinite IID process inferred from samples."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "CLT ORIGINAL common marginal ORIGINAL value times probability0 := mul(1, 1/2)",
      "after": "CLT ORIGINAL common marginal ORIGINAL value times probability0 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL value times probability0，不以小數近似替代。",
          "en": "Recompute CLT ORIGINAL common marginal ORIGINAL value times probability0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1/2"
        ],
        "output": "1/2",
        "label": "CLT ORIGINAL common marginal ORIGINAL value times probability0"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "CLT ORIGINAL common marginal ORIGINAL ascending mean sum0 := sum(0, 1/2)",
      "after": "CLT ORIGINAL common marginal ORIGINAL ascending mean sum0 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL ascending mean sum0，不以小數近似替代。",
          "en": "Recompute CLT ORIGINAL common marginal ORIGINAL ascending mean sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/2"
        ],
        "output": "1/2",
        "label": "CLT ORIGINAL common marginal ORIGINAL ascending mean sum0"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "CLT ORIGINAL common marginal exact value square0 := mul(1, 1)",
      "after": "CLT ORIGINAL common marginal exact value square0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT ORIGINAL common marginal exact value square0，不以小數近似替代。",
          "en": "Recompute CLT ORIGINAL common marginal exact value square0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "CLT ORIGINAL common marginal exact value square0"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "CLT ORIGINAL common marginal ORIGINAL square times probability0 := mul(1, 1/2)",
      "after": "CLT ORIGINAL common marginal ORIGINAL square times probability0 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL square times probability0，不以小數近似替代。",
          "en": "Recompute CLT ORIGINAL common marginal ORIGINAL square times probability0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1/2"
        ],
        "output": "1/2",
        "label": "CLT ORIGINAL common marginal ORIGINAL square times probability0"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "CLT ORIGINAL common marginal ORIGINAL ascending second moment sum0 := sum(0, 1/2)",
      "after": "CLT ORIGINAL common marginal ORIGINAL ascending second moment sum0 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL ascending second moment sum0，不以小數近似替代。",
          "en": "Recompute CLT ORIGINAL common marginal ORIGINAL ascending second moment sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/2"
        ],
        "output": "1/2",
        "label": "CLT ORIGINAL common marginal ORIGINAL ascending second moment sum0"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "CLT ORIGINAL common marginal ORIGINAL value times probability1 := mul(2, 1/2)",
      "after": "CLT ORIGINAL common marginal ORIGINAL value times probability1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL value times probability1，不以小數近似替代。",
          "en": "Recompute CLT ORIGINAL common marginal ORIGINAL value times probability1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1/2"
        ],
        "output": "1",
        "label": "CLT ORIGINAL common marginal ORIGINAL value times probability1"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "CLT ORIGINAL common marginal ORIGINAL ascending mean sum1 := sum(1/2, 1)",
      "after": "CLT ORIGINAL common marginal ORIGINAL ascending mean sum1 = 3/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL ascending mean sum1，不以小數近似替代。",
          "en": "Recompute CLT ORIGINAL common marginal ORIGINAL ascending mean sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/2",
          "1"
        ],
        "output": "3/2",
        "label": "CLT ORIGINAL common marginal ORIGINAL ascending mean sum1"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "CLT ORIGINAL common marginal exact value square1 := mul(2, 2)",
      "after": "CLT ORIGINAL common marginal exact value square1 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT ORIGINAL common marginal exact value square1，不以小數近似替代。",
          "en": "Recompute CLT ORIGINAL common marginal exact value square1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "CLT ORIGINAL common marginal exact value square1"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "CLT ORIGINAL common marginal ORIGINAL square times probability1 := mul(4, 1/2)",
      "after": "CLT ORIGINAL common marginal ORIGINAL square times probability1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL square times probability1，不以小數近似替代。",
          "en": "Recompute CLT ORIGINAL common marginal ORIGINAL square times probability1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "1/2"
        ],
        "output": "2",
        "label": "CLT ORIGINAL common marginal ORIGINAL square times probability1"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "CLT ORIGINAL common marginal ORIGINAL ascending second moment sum1 := sum(1/2, 2)",
      "after": "CLT ORIGINAL common marginal ORIGINAL ascending second moment sum1 = 5/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL ascending second moment sum1，不以小數近似替代。",
          "en": "Recompute CLT ORIGINAL common marginal ORIGINAL ascending second moment sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/2",
          "2"
        ],
        "output": "5/2",
        "label": "CLT ORIGINAL common marginal ORIGINAL ascending second moment sum1"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "CLT ORIGINAL common marginal ORIGINAL mean square := mul(3/2, 3/2)",
      "after": "CLT ORIGINAL common marginal ORIGINAL mean square = 9/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL mean square，不以小數近似替代。",
          "en": "Recompute CLT ORIGINAL common marginal ORIGINAL mean square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3/2",
          "3/2"
        ],
        "output": "9/4",
        "label": "CLT ORIGINAL common marginal ORIGINAL mean square"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "CLT ORIGINAL common marginal ORIGINAL second minus mean square := sub(5/2, 9/4)",
      "after": "CLT ORIGINAL common marginal ORIGINAL second minus mean square = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL second minus mean square，不以小數近似替代。",
          "en": "Recompute CLT ORIGINAL common marginal ORIGINAL second minus mean square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5/2",
          "9/4"
        ],
        "output": "1/4",
        "label": "CLT ORIGINAL common marginal ORIGINAL second minus mean square"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "source roles = [CLT ORIGINAL common marginal, [[1, 1/2], [2, 1/2]]]",
      "after": "derived = [3/2, 5/2, 1/4]",
      "relation": "implies",
      "rule_id": "CLT_ALL_finite_marginal_moments_not_a_process_or_limit_proof",
      "conditions": [],
      "explanation": {
        "zh-Hant": "E[X]、E[X²]與Var(X)=E[X²]-E[X]²都用同一完整law。value²先保精確教學量，再乘prob才照來源檢profile，不把null outcome的大value²額外拒絕。有限矩不證independence、無限過程或分布極限。",
        "en": "Use the SAME complete law for E[X],E[X²] and Var(X)=E[X²]-E[X]².Keep value² as an exact teaching intermediate and apply the source profile to its product with probability,not an extra rejection for a large zero-mass value square.Finite moments prove neither independence,an infinite process nor a distributional limit."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "CLT joint ORIGINAL word0 sum := sum(1, 1)",
      "after": "CLT joint ORIGINAL word0 sum = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT joint ORIGINAL word0 sum，不以小數近似替代。",
          "en": "Recompute CLT joint ORIGINAL word0 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "CLT joint ORIGINAL word0 sum"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "CLT joint ORIGINAL word0 probability factor0 := mul(1, 1/2)",
      "after": "CLT joint ORIGINAL word0 probability factor0 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT joint ORIGINAL word0 probability factor0，不以小數近似替代。",
          "en": "Recompute CLT joint ORIGINAL word0 probability factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1/2"
        ],
        "output": "1/2",
        "label": "CLT joint ORIGINAL word0 probability factor0"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "CLT joint ORIGINAL word0 probability factor1 := mul(1/2, 1/2)",
      "after": "CLT joint ORIGINAL word0 probability factor1 = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT joint ORIGINAL word0 probability factor1，不以小數近似替代。",
          "en": "Recompute CLT joint ORIGINAL word0 probability factor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "1/2"
        ],
        "output": "1/4",
        "label": "CLT joint ORIGINAL word0 probability factor1"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "CLT joint ORIGINAL sum2 bucket word0 := sum(0, 1/4)",
      "after": "CLT joint ORIGINAL sum2 bucket word0 = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT joint ORIGINAL sum2 bucket word0，不以小數近似替代。",
          "en": "Recompute CLT joint ORIGINAL sum2 bucket word0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/4"
        ],
        "output": "1/4",
        "label": "CLT joint ORIGINAL sum2 bucket word0"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "source roles = [2, 0, [1, 1], [1/2, 1/2]]",
      "after": "derived = [2, 1/4, 0, 1/4]",
      "relation": "implies",
      "rule_id": "CLT_EVERY_joint_word_original_coordinate_products_AND_full_sum_fibre",
      "conditions": [],
      "explanation": {
        "zh-Hant": "宣告IID給此有限word機率為各coordinate機率乘積；逐全部word依原source次序加入其真正sum fibre，包含零質量word／bucket，不從samples或矩直接猜分布。",
        "en": "The declared IID premise gives each finite word the product of coordinate probabilities.Enumerate EVERY word in source order and add it to its actual sum fibre,retaining zero-mass words/buckets.Do not guess the distribution from samples or moments."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "CLT joint ORIGINAL word1 sum := sum(1, 2)",
      "after": "CLT joint ORIGINAL word1 sum = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT joint ORIGINAL word1 sum，不以小數近似替代。",
          "en": "Recompute CLT joint ORIGINAL word1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "2"
        ],
        "output": "3",
        "label": "CLT joint ORIGINAL word1 sum"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "CLT joint ORIGINAL word1 probability factor0 := mul(1, 1/2)",
      "after": "CLT joint ORIGINAL word1 probability factor0 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT joint ORIGINAL word1 probability factor0，不以小數近似替代。",
          "en": "Recompute CLT joint ORIGINAL word1 probability factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1/2"
        ],
        "output": "1/2",
        "label": "CLT joint ORIGINAL word1 probability factor0"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "CLT joint ORIGINAL word1 probability factor1 := mul(1/2, 1/2)",
      "after": "CLT joint ORIGINAL word1 probability factor1 = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT joint ORIGINAL word1 probability factor1，不以小數近似替代。",
          "en": "Recompute CLT joint ORIGINAL word1 probability factor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "1/2"
        ],
        "output": "1/4",
        "label": "CLT joint ORIGINAL word1 probability factor1"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "CLT joint ORIGINAL sum3 bucket word1 := sum(0, 1/4)",
      "after": "CLT joint ORIGINAL sum3 bucket word1 = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT joint ORIGINAL sum3 bucket word1，不以小數近似替代。",
          "en": "Recompute CLT joint ORIGINAL sum3 bucket word1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/4"
        ],
        "output": "1/4",
        "label": "CLT joint ORIGINAL sum3 bucket word1"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "source roles = [2, 1, [1, 2], [1/2, 1/2]]",
      "after": "derived = [3, 1/4, 0, 1/4]",
      "relation": "implies",
      "rule_id": "CLT_EVERY_joint_word_original_coordinate_products_AND_full_sum_fibre",
      "conditions": [],
      "explanation": {
        "zh-Hant": "宣告IID給此有限word機率為各coordinate機率乘積；逐全部word依原source次序加入其真正sum fibre，包含零質量word／bucket，不從samples或矩直接猜分布。",
        "en": "The declared IID premise gives each finite word the product of coordinate probabilities.Enumerate EVERY word in source order and add it to its actual sum fibre,retaining zero-mass words/buckets.Do not guess the distribution from samples or moments."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "CLT joint ORIGINAL word2 sum := sum(2, 1)",
      "after": "CLT joint ORIGINAL word2 sum = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT joint ORIGINAL word2 sum，不以小數近似替代。",
          "en": "Recompute CLT joint ORIGINAL word2 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "1"
        ],
        "output": "3",
        "label": "CLT joint ORIGINAL word2 sum"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "CLT joint ORIGINAL word2 probability factor0 := mul(1, 1/2)",
      "after": "CLT joint ORIGINAL word2 probability factor0 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT joint ORIGINAL word2 probability factor0，不以小數近似替代。",
          "en": "Recompute CLT joint ORIGINAL word2 probability factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1/2"
        ],
        "output": "1/2",
        "label": "CLT joint ORIGINAL word2 probability factor0"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "CLT joint ORIGINAL word2 probability factor1 := mul(1/2, 1/2)",
      "after": "CLT joint ORIGINAL word2 probability factor1 = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT joint ORIGINAL word2 probability factor1，不以小數近似替代。",
          "en": "Recompute CLT joint ORIGINAL word2 probability factor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "1/2"
        ],
        "output": "1/4",
        "label": "CLT joint ORIGINAL word2 probability factor1"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "CLT joint ORIGINAL sum3 bucket word2 := sum(1/4, 1/4)",
      "after": "CLT joint ORIGINAL sum3 bucket word2 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT joint ORIGINAL sum3 bucket word2，不以小數近似替代。",
          "en": "Recompute CLT joint ORIGINAL sum3 bucket word2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/4",
          "1/4"
        ],
        "output": "1/2",
        "label": "CLT joint ORIGINAL sum3 bucket word2"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "source roles = [2, 2, [2, 1], [1/2, 1/2]]",
      "after": "derived = [3, 1/4, 1/4, 1/2]",
      "relation": "implies",
      "rule_id": "CLT_EVERY_joint_word_original_coordinate_products_AND_full_sum_fibre",
      "conditions": [],
      "explanation": {
        "zh-Hant": "宣告IID給此有限word機率為各coordinate機率乘積；逐全部word依原source次序加入其真正sum fibre，包含零質量word／bucket，不從samples或矩直接猜分布。",
        "en": "The declared IID premise gives each finite word the product of coordinate probabilities.Enumerate EVERY word in source order and add it to its actual sum fibre,retaining zero-mass words/buckets.Do not guess the distribution from samples or moments."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "CLT joint ORIGINAL word3 sum := sum(2, 2)",
      "after": "CLT joint ORIGINAL word3 sum = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT joint ORIGINAL word3 sum，不以小數近似替代。",
          "en": "Recompute CLT joint ORIGINAL word3 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "CLT joint ORIGINAL word3 sum"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "CLT joint ORIGINAL word3 probability factor0 := mul(1, 1/2)",
      "after": "CLT joint ORIGINAL word3 probability factor0 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT joint ORIGINAL word3 probability factor0，不以小數近似替代。",
          "en": "Recompute CLT joint ORIGINAL word3 probability factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1/2"
        ],
        "output": "1/2",
        "label": "CLT joint ORIGINAL word3 probability factor0"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "CLT joint ORIGINAL word3 probability factor1 := mul(1/2, 1/2)",
      "after": "CLT joint ORIGINAL word3 probability factor1 = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT joint ORIGINAL word3 probability factor1，不以小數近似替代。",
          "en": "Recompute CLT joint ORIGINAL word3 probability factor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "1/2"
        ],
        "output": "1/4",
        "label": "CLT joint ORIGINAL word3 probability factor1"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "CLT joint ORIGINAL sum4 bucket word3 := sum(0, 1/4)",
      "after": "CLT joint ORIGINAL sum4 bucket word3 = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT joint ORIGINAL sum4 bucket word3，不以小數近似替代。",
          "en": "Recompute CLT joint ORIGINAL sum4 bucket word3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/4"
        ],
        "output": "1/4",
        "label": "CLT joint ORIGINAL sum4 bucket word3"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "source roles = [2, 3, [2, 2], [1/2, 1/2]]",
      "after": "derived = [4, 1/4, 0, 1/4]",
      "relation": "implies",
      "rule_id": "CLT_EVERY_joint_word_original_coordinate_products_AND_full_sum_fibre",
      "conditions": [],
      "explanation": {
        "zh-Hant": "宣告IID給此有限word機率為各coordinate機率乘積；逐全部word依原source次序加入其真正sum fibre，包含零質量word／bucket，不從samples或矩直接猜分布。",
        "en": "The declared IID premise gives each finite word the product of coordinate probabilities.Enumerate EVERY word in source order and add it to its actual sum fibre,retaining zero-mass words/buckets.Do not guess the distribution from samples or moments."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "CLT SUM ORIGINAL total distribution mass0 := sum(0, 1/4)",
      "after": "CLT SUM ORIGINAL total distribution mass0 = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM ORIGINAL total distribution mass0，不以小數近似替代。",
          "en": "Recompute CLT SUM ORIGINAL total distribution mass0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/4"
        ],
        "output": "1/4",
        "label": "CLT SUM ORIGINAL total distribution mass0"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "CLT SUM ORIGINAL total distribution mass1 := sum(1/4, 1/2)",
      "after": "CLT SUM ORIGINAL total distribution mass1 = 3/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM ORIGINAL total distribution mass1，不以小數近似替代。",
          "en": "Recompute CLT SUM ORIGINAL total distribution mass1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/4",
          "1/2"
        ],
        "output": "3/4",
        "label": "CLT SUM ORIGINAL total distribution mass1"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "CLT SUM ORIGINAL total distribution mass2 := sum(3/4, 1/4)",
      "after": "CLT SUM ORIGINAL total distribution mass2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM ORIGINAL total distribution mass2，不以小數近似替代。",
          "en": "Recompute CLT SUM ORIGINAL total distribution mass2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3/4",
          "1/4"
        ],
        "output": "1",
        "label": "CLT SUM ORIGINAL total distribution mass2"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "CLT SUM ORIGINAL value times probability0 := mul(2, 1/4)",
      "after": "CLT SUM ORIGINAL value times probability0 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM ORIGINAL value times probability0，不以小數近似替代。",
          "en": "Recompute CLT SUM ORIGINAL value times probability0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1/4"
        ],
        "output": "1/2",
        "label": "CLT SUM ORIGINAL value times probability0"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "CLT SUM ORIGINAL ascending mean sum0 := sum(0, 1/2)",
      "after": "CLT SUM ORIGINAL ascending mean sum0 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM ORIGINAL ascending mean sum0，不以小數近似替代。",
          "en": "Recompute CLT SUM ORIGINAL ascending mean sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/2"
        ],
        "output": "1/2",
        "label": "CLT SUM ORIGINAL ascending mean sum0"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "CLT SUM exact value square0 := mul(2, 2)",
      "after": "CLT SUM exact value square0 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM exact value square0，不以小數近似替代。",
          "en": "Recompute CLT SUM exact value square0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "CLT SUM exact value square0"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "CLT SUM ORIGINAL square times probability0 := mul(4, 1/4)",
      "after": "CLT SUM ORIGINAL square times probability0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM ORIGINAL square times probability0，不以小數近似替代。",
          "en": "Recompute CLT SUM ORIGINAL square times probability0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "1/4"
        ],
        "output": "1",
        "label": "CLT SUM ORIGINAL square times probability0"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "CLT SUM ORIGINAL ascending second moment sum0 := sum(0, 1)",
      "after": "CLT SUM ORIGINAL ascending second moment sum0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM ORIGINAL ascending second moment sum0，不以小數近似替代。",
          "en": "Recompute CLT SUM ORIGINAL ascending second moment sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "CLT SUM ORIGINAL ascending second moment sum0"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "CLT SUM ORIGINAL value times probability1 := mul(3, 1/2)",
      "after": "CLT SUM ORIGINAL value times probability1 = 3/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM ORIGINAL value times probability1，不以小數近似替代。",
          "en": "Recompute CLT SUM ORIGINAL value times probability1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "1/2"
        ],
        "output": "3/2",
        "label": "CLT SUM ORIGINAL value times probability1"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "CLT SUM ORIGINAL ascending mean sum1 := sum(1/2, 3/2)",
      "after": "CLT SUM ORIGINAL ascending mean sum1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM ORIGINAL ascending mean sum1，不以小數近似替代。",
          "en": "Recompute CLT SUM ORIGINAL ascending mean sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/2",
          "3/2"
        ],
        "output": "2",
        "label": "CLT SUM ORIGINAL ascending mean sum1"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "CLT SUM exact value square1 := mul(3, 3)",
      "after": "CLT SUM exact value square1 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM exact value square1，不以小數近似替代。",
          "en": "Recompute CLT SUM exact value square1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "3"
        ],
        "output": "9",
        "label": "CLT SUM exact value square1"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "CLT SUM ORIGINAL square times probability1 := mul(9, 1/2)",
      "after": "CLT SUM ORIGINAL square times probability1 = 9/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM ORIGINAL square times probability1，不以小數近似替代。",
          "en": "Recompute CLT SUM ORIGINAL square times probability1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "1/2"
        ],
        "output": "9/2",
        "label": "CLT SUM ORIGINAL square times probability1"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "CLT SUM ORIGINAL ascending second moment sum1 := sum(1, 9/2)",
      "after": "CLT SUM ORIGINAL ascending second moment sum1 = 11/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM ORIGINAL ascending second moment sum1，不以小數近似替代。",
          "en": "Recompute CLT SUM ORIGINAL ascending second moment sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "9/2"
        ],
        "output": "11/2",
        "label": "CLT SUM ORIGINAL ascending second moment sum1"
      }
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "CLT SUM ORIGINAL value times probability2 := mul(4, 1/4)",
      "after": "CLT SUM ORIGINAL value times probability2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM ORIGINAL value times probability2，不以小數近似替代。",
          "en": "Recompute CLT SUM ORIGINAL value times probability2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "1/4"
        ],
        "output": "1",
        "label": "CLT SUM ORIGINAL value times probability2"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "CLT SUM ORIGINAL ascending mean sum2 := sum(2, 1)",
      "after": "CLT SUM ORIGINAL ascending mean sum2 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM ORIGINAL ascending mean sum2，不以小數近似替代。",
          "en": "Recompute CLT SUM ORIGINAL ascending mean sum2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "1"
        ],
        "output": "3",
        "label": "CLT SUM ORIGINAL ascending mean sum2"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "CLT SUM exact value square2 := mul(4, 4)",
      "after": "CLT SUM exact value square2 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM exact value square2，不以小數近似替代。",
          "en": "Recompute CLT SUM exact value square2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "4"
        ],
        "output": "16",
        "label": "CLT SUM exact value square2"
      }
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "CLT SUM ORIGINAL square times probability2 := mul(16, 1/4)",
      "after": "CLT SUM ORIGINAL square times probability2 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM ORIGINAL square times probability2，不以小數近似替代。",
          "en": "Recompute CLT SUM ORIGINAL square times probability2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "16",
          "1/4"
        ],
        "output": "4",
        "label": "CLT SUM ORIGINAL square times probability2"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "CLT SUM ORIGINAL ascending second moment sum2 := sum(11/2, 4)",
      "after": "CLT SUM ORIGINAL ascending second moment sum2 = 19/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM ORIGINAL ascending second moment sum2，不以小數近似替代。",
          "en": "Recompute CLT SUM ORIGINAL ascending second moment sum2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "11/2",
          "4"
        ],
        "output": "19/2",
        "label": "CLT SUM ORIGINAL ascending second moment sum2"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "CLT SUM ORIGINAL mean square := mul(3, 3)",
      "after": "CLT SUM ORIGINAL mean square = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM ORIGINAL mean square，不以小數近似替代。",
          "en": "Recompute CLT SUM ORIGINAL mean square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "3"
        ],
        "output": "9",
        "label": "CLT SUM ORIGINAL mean square"
      }
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "CLT SUM ORIGINAL second minus mean square := sub(19/2, 9)",
      "after": "CLT SUM ORIGINAL second minus mean square = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT SUM ORIGINAL second minus mean square，不以小數近似替代。",
          "en": "Recompute CLT SUM ORIGINAL second minus mean square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "19/2",
          "9"
        ],
        "output": "1/2",
        "label": "CLT SUM ORIGINAL second minus mean square"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "CLT ORIGINAL n times common mean := mul(2, 3/2)",
      "after": "CLT ORIGINAL n times common mean = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT ORIGINAL n times common mean，不以小數近似替代。",
          "en": "Recompute CLT ORIGINAL n times common mean with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "3/2"
        ],
        "output": "3",
        "label": "CLT ORIGINAL n times common mean"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "CLT ORIGINAL n times common variance := mul(2, 1/4)",
      "after": "CLT ORIGINAL n times common variance = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 CLT ORIGINAL n times common variance，不以小數近似替代。",
          "en": "Recompute CLT ORIGINAL n times common variance with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1/4"
        ],
        "output": "1/2",
        "label": "CLT ORIGINAL n times common variance"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "source roles = [2, 3/2, 1/4, [(2, 1/4), (3, 1/2), (4, 1/4)], 3, 1/2]",
      "after": "derived = [1, 3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n))]",
      "relation": "implies",
      "rule_id": "CLT_standardization_EXACT_0_1_moments_NOT_normality_or_limit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "sigma²>0且n>0，故標準化分母非0；E[Z]=(nmu-nmu)/(sigma sqrt(n))=0，Var(Z)=n sigma²/(sigma² n)=1。矩0、1只屬這個完整fixed-n law，不代表它已normal，也不構成一般CLT證明。",
        "en": "sigma²>0 and n>0 make the denominator nonzero.E[Z]=(nmu-nmu)/(sigma sqrt(n))=0 and Var(Z)=n sigma²/(sigma² n)=1.These exact moments belong to this complete FIXED-n law;they do not make it normal or prove the general CLT."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      2,
      [
        [
          2,
          "1/4"
        ],
        [
          3,
          "1/2"
        ],
        [
          4,
          "1/4"
        ]
      ],
      "1",
      "3",
      "1/2",
      "3",
      "1/2",
      0,
      1,
      "Z_n=(S_n-n*mu)/(sigma*sqrt(n))",
      "complete_joint_law_moments_do_not_prove_distributional_convergence"
    ]
  },
  "conclusion": {
    "canonical_value": "(2, [[2, 1/4], [3, 1/2], [4, 1/4]], 1, 3, 1/2, 3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)",
    "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-r01-central-limit-theorem-finite-variance-iid-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations/questions.jsonl",
      "batch_sha256": "6eaac14f97913d20b3a00dc8becf4943e2b1791210c8c5bbf3a89ecced69c6d1",
      "original_line": 34,
      "original_line_sha256": "d20974d1cdcc86af42cb13a57f8b0efd096ec5be4ee83b7f7b596b2f15c0a903",
      "question_compact_sha256": "3e5d253825bcacfe06c230d4596d995c458b8754a189c3f330e8de767c415180",
      "english_row_compact_sha256": "46a0af57772ca68b0e123e0f34aac1fdfa8532f3860e17ce97a5d3b0bdf6543f",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-central-limit-theorem-finite-variance-iid": {
        "sha256": "c37700292553db11f9702a86ba165e9287d7830d81ee8351eff28de1c07ea9fe",
        "english_sha256": "951e71ab789a556b03eeb44c3472b5e89e3ac252d23992bc8abf538f39a494cb",
        "original_path": "public/data/mko/central-limit-theorem-finite-variance-iid.json",
        "english_path": "public/data/i18n/editions/mko-central-limit-theorem-finite-variance-iid.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "a54409c0bc3f68730e1072c6d3d772970462c1bc7ffc203b4030a6136a610283",
    "original_candidate_sha256": "a91c33f05d7cc12b1dd9e638ae4abdc79fd360458de6bbf2177523c4ba86e4ed",
    "original_candidate_line_sha256": "a91c33f05d7cc12b1dd9e638ae4abdc79fd360458de6bbf2177523c4ba86e4ed"
  },
  "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-20261026-010-p54",
    "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": "1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62",
      "question_first_witness_sha256": "79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 61,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "4322ad69c99df8b1e7b2b4353ab188148f59b7decdaf71320e7d93b61fad0d02",
      "arithmetic_steps": 52
    },
    "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."
  }
}