{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261105-r07-weak-law-large-numbers-0013",
  "pilot_number": "R0037-19288",
  "number": 19288,
  "family_id": "qf-mod36-wlln-variance-average",
  "title": {
    "zh-Hant": "第 19288 題：弱大數律",
    "en": "Question 19288: Independence Gives Variance of the Average, Not a Finite-Sample Limit"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261105-r07-weak-law-large-numbers-0013",
    "batch_id": "batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-weak-law-large-numbers"
    ],
    "title_zh": "弱大數律",
    "stem_zh": "IID通用證書、共同有限mean與variance及exact Chebyshev tail。操作=wlln-variance-average；資料={\"mean\":[12,13],\"variance\":[13,14],\"n\":17,\"iid_certificate\":\"symbolic_all_finite_subfamilies\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(12/13, 13/14, 17, 13/238, independence_gives_variance_sigma_squared_over_n)",
      "explanation_zh": "Wave5第五層核對=(12/13, 13/14, 17, 13/238, independence_gives_variance_sigma_squared_over_n)；依完整明示資料獨立重算；independence給Var(sample mean)=sigma squared/n並以forall n大於等於N證依機率收斂；有限樣本或模擬不足，且不宣稱almost sure、CLT或有限n已等於極限。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "弱大數律"
      ]
    },
    "variant": {
      "family_id": "qf-mod36-wlln-variance-average",
      "seed": 202611057013,
      "parameters": {
        "mean": [
          12,
          13
        ],
        "variance": [
          13,
          14
        ],
        "n": 17,
        "iid_certificate": "symbolic_all_finite_subfamilies",
        "domain": "Symbolic_iid_Chebyshev_WLLN",
        "operation": "wlln-variance-average",
        "semantic_template": "qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[12,13],\"n\":17,\"variance\":[13,14]}"
      }
    },
    "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；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261105-r07-weak-law-large-numbers-0013",
    "batch_id": "batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "e5170cdaf35e0c74be3133e3f0c1f8f4e81eb37919b0964baea1bc0054a6c17e",
      "family_id": "qf-mod36-wlln-variance-average",
      "seed": 202611057013,
      "semantic_template": "qf-mod36-wlln-variance-average/{\"iid_certificate\":\"symbolic_all_finite_subfamilies\",\"mean\":[12,13],\"n\":17,\"variance\":[13,14]}"
    },
    "fields": {
      "title": "Independence Gives Variance of the Average, Not a Finite-Sample Limit",
      "stem": "Use declared mean=12/13, positive finite variance=13/14, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=17. Give (mean, variance, n, average_variance, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(12/13, 13/14, 17, 13/238, independence_gives_variance_sigma_squared_over_n)",
        "explanation": "Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (12/13, 13/14, 17, 13/238, independence_gives_variance_sigma_squared_over_n)."
      },
      "skills": [
        "Independence Gives Variance of the Average, Not a Finite-Sample 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": "IID通用證書、共同有限mean與variance及exact Chebyshev tail。操作=wlln-variance-average；資料={\"mean\":[12,13],\"variance\":[13,14],\"n\":17,\"iid_certificate\":\"symbolic_all_finite_subfamilies\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use declared mean=12/13, positive finite variance=13/14, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=17. Give (mean, variance, n, average_variance, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=wlln-variance-average;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[mean, variance, n, average_variance, certificate]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題快照保留歷史介面標記；本頁直接列原始欄位順序，不需JavaScript或查外部表。欄位名字本身不自動是數學證據，No依各題的聲稱範圍解讀。",
        "en": "Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field names are not mathematical evidence,and No retains the operation's specific claim scope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "original common finite mean=12/13;POSITIVE finite variance=13/14;whole-sequence iid_certificate=symbolic_all_finite_subfamilies",
      "after": "ASSUME for EVERY finite subfamily the declared independence and identical law/common moments;marker declares a premise,NOT independent evidence that a real sequence satisfies it",
      "relation": "implies",
      "rule_id": "wlln_whole_IID_declared_premise_not_token_proof",
      "conditions": [],
      "explanation": {
        "zh-Hant": "標記是本題前提而非靠字串或有限資料證IID。以下是若全序列真的具此前提才成立的條件式推導；不新增Lean證明或實際資料認證。",
        "en": "The marker is a premise of this problem,not proof of IID by a string or finite data.What follows is conditional on a whole sequence satisfying that premise,not a Lean proof or certification of empirical data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "for ANY positive integer n:sample_mean_n=(X1+...+Xn)/n;every Xi has the SAME mean and variance;distinct Xi,Xj independent",
      "after": "linearity gives E(sample_mean_n)=mean;centered products have zero expectation for i!=j;Var(sample_mean_n)=(n*variance)/n^2=variance/n",
      "relation": "implies",
      "rule_id": "wlln_universal_variance_sum_independence_not_one_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限方差保乘積可積；獨立讓交叉中心化期望分解成0·0，只有n個對角variance項。再乘1/n平方得variance/n，這是任意n的代數證書，不從已觀察平均值等於mean推。",
        "en": "Finite variances ensure integrability of the products.Independence factors every distinct centered cross term into0*0,leaving n diagonal variance terms.Scaling by1/n squared gives variance/n for arbitrary n,not a deduction from one observed average matching the mean."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "WLLN selected sum of diagonal variances := mul(17, 13/14)",
      "after": "WLLN selected sum of diagonal variances = 221/14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留參數、樣本、目標與分母，不交換未授權的聲稱。",
        "en": "Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 WLLN selected sum of diagonal variances，不以小數近似替代。",
          "en": "Recompute WLLN selected sum of diagonal variances with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "17",
          "13/14"
        ],
        "output": "221/14",
        "label": "WLLN selected sum of diagonal variances"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "WLLN selected average denominator squared := mul(17, 17)",
      "after": "WLLN selected average denominator squared = 289",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留參數、樣本、目標與分母，不交換未授權的聲稱。",
        "en": "Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 WLLN selected average denominator squared，不以小數近似替代。",
          "en": "Recompute WLLN selected average denominator squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "17",
          "17"
        ],
        "output": "289",
        "label": "WLLN selected average denominator squared"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "WLLN selected average variance := div(221/14, 289)",
      "after": "WLLN selected average variance = 13/238",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留參數、樣本、目標與分母，不交換未授權的聲稱。",
        "en": "Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 WLLN selected average variance，不以小數近似替代。",
          "en": "Recompute WLLN selected average variance with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "221/14",
          "289"
        ],
        "output": "13/238",
        "label": "WLLN selected average variance"
      }
    }
  ],
  "math_model": {
    "result": [
      "12/13",
      "13/14",
      17,
      "13/238",
      "independence_gives_variance_sigma_squared_over_n"
    ]
  },
  "conclusion": {
    "canonical_value": "(12/13, 13/14, 17, 13/238, independence_gives_variance_sigma_squared_over_n)",
    "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-20261014-advance-20261105-r07-weak-law-large-numbers-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations/questions.jsonl",
      "batch_sha256": "cb37c1b02120639bde918f12fac038639408d2229daab60e5de8ed9750046d5b",
      "original_line": 13,
      "original_line_sha256": "74f8e8a1d2300e0432d0e33ee3cab4935b3e070df1426b733c515ccecc27d54c",
      "question_compact_sha256": "e5170cdaf35e0c74be3133e3f0c1f8f4e81eb37919b0964baea1bc0054a6c17e",
      "english_row_compact_sha256": "148f4415d4790f239c99d340332281c5a291aa660221485012f10572ade69f34",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-weak-law-large-numbers": {
        "sha256": "7eb5ea7326a95ec1926f661b129d1f6ed74f95c53574aea5f3878111f80bb55c",
        "english_sha256": "cbabd753c9362d2298518ae3dd4332e7dc4007f7b436746fdb53dd41565de8a3",
        "original_path": "public/data/mko/source13-20261018-weak-law-large-numbers.json",
        "english_path": "public/data/i18n/editions/mko-weak-law-large-numbers.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "7be57d88eceb13736454f82dd18bb3c4ec9d30c0367f974dcefdf8990dc4e3cb",
    "original_candidate_sha256": "eb87103130058ed98e4e93522e5391599b764a0882a8ac2694bd4ab411c0060e",
    "original_candidate_line_sha256": "eb87103130058ed98e4e93522e5391599b764a0882a8ac2694bd4ab411c0060e"
  },
  "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-20261026-008-p38",
    "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": "065507b602ceee60cf47a700d387a51940e0c6724c35a3336a7a7948a199ed80",
      "question_first_witness_sha256": "7203f33b030792559673b0fee520a8c0e70903c4dd970b35293aa40852f333d6"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "fd30cf15c2673c8d847e56fb6ba93ceef6815596e85e5a246ac717ec4cff147c",
      "arithmetic_steps": 3
    },
    "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."
  }
}