{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261114-r04-standard-normal-distribution-0030",
  "pilot_number": "R0049-25230",
  "number": 25230,
  "family_id": "qf-mod45-normaldist-evenness-ratio",
  "title": {
    "zh-Hant": "第 25230 題：標準常態分布",
    "en": "Question 25230: Evenness and the Density Ratio Follow from the Squared Argument"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261114-r04-standard-normal-distribution-0030",
    "batch_id": "batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-standard-normal-distribution"
    ],
    "title_zh": "標準常態分布",
    "stem_zh": "standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[5,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(5/6, -5/6, (2*pi)^(-1/2)*exp(-25/72), (2*pi)^(-1/2)*exp(-25/72), 是, phi(x)/phi(0)=exp(-25/72), evenness_follows_from_x_squared)",
      "explanation_zh": "Wave6第八層核對=(5/6, -5/6, (2*pi)^(-1/2)*exp(-25/72), (2*pi)^(-1/2)*exp(-25/72), 是, phi(x)/phi(0)=exp(-25/72), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "標準常態分布"
      ]
    },
    "variant": {
      "family_id": "qf-mod45-normaldist-evenness-ratio",
      "seed": 202611144030,
      "parameters": {
        "x": [
          5,
          6
        ],
        "domain": "Symbolic_standard_normal_distribution",
        "operation": "normaldist-evenness-ratio",
        "semantic_template": "qf-mod45-normaldist-evenness-ratio/{\"x\":[5,6]}"
      }
    },
    "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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、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-20261114-r04-standard-normal-distribution-0030",
    "batch_id": "batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "51ef6459a7af9358d4c48b94fe53a02252a3e1278c86ed6211c23aeed9eb0b15",
      "family_id": "qf-mod45-normaldist-evenness-ratio",
      "seed": 202611144030,
      "semantic_template": "qf-mod45-normaldist-evenness-ratio/{\"x\":[5,6]}"
    },
    "fields": {
      "title": "Evenness and the Density Ratio Follow from the Squared Argument",
      "stem": "Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=5/6 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(5/6, -5/6, (2*pi)^(-1/2)*exp(-25/72), (2*pi)^(-1/2)*exp(-25/72), Yes, phi(x)/phi(0)=exp(-25/72), evenness_follows_from_x_squared)",
        "explanation": "The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (5/6, -5/6, (2*pi)^(-1/2)*exp(-25/72), (2*pi)^(-1/2)*exp(-25/72), Yes, phi(x)/phi(0)=exp(-25/72), evenness_follows_from_x_squared)."
      },
      "skills": [
        "Evenness and the Density Ratio Follow from the Squared Argument"
      ],
      "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": "standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[5,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=5/6 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [normaldist-evenness-ratio]",
      "after": "derived = [[x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本頁列完整來源欄序，不需JavaScript；LP三種mode雖同為6欄，角色仍由原mode選，不由答案形狀猜。",
        "en": "List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape."
      },
      "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": "Gaussian ORIGINAL x squared := mul(5/6, 5/6)",
      "after": "Gaussian ORIGINAL x squared = 25/36",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。",
        "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gaussian ORIGINAL x squared，不以小數近似替代。",
          "en": "Recompute Gaussian ORIGINAL x squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5/6",
          "5/6"
        ],
        "output": "25/36",
        "label": "Gaussian ORIGINAL x squared"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Gaussian square times exact half := mul(25/36, 1/2)",
      "after": "Gaussian square times exact half = 25/72",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。",
        "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gaussian square times exact half，不以小數近似替代。",
          "en": "Recompute Gaussian square times exact half with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25/36",
          "1/2"
        ],
        "output": "25/72",
        "label": "Gaussian square times exact half"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [5/6, -25/72]",
      "after": "derived = [(2*pi)^(-1/2)*exp(-25/72), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization is a NAMED Gaussian integral premise]",
      "relation": "implies",
      "rule_id": "Gaussian_WHOLE_declared_density_symbolic_value_named_normalization",
      "conditions": [],
      "explanation": {
        "zh-Hant": "精確有理數只算指數−x²/2，exp與常數因子保符號。全Gaussian law與normalization是明示前提，不由一個density值證成。",
        "en": "Exact rational arithmetic computes only the exponent−x^2/2;retain exp and its normalization factor symbolically.The whole Gaussian law and normalization are declared premises,not proved by one density value."
      },
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Gaussian negative x squared := mul(-5/6, -5/6)",
      "after": "Gaussian negative x squared = 25/36",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。",
        "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gaussian negative x squared，不以小數近似替代。",
          "en": "Recompute Gaussian negative x squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-5/6",
          "-5/6"
        ],
        "output": "25/36",
        "label": "Gaussian negative x squared"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [5/6, -5/6, 25/36, 25/36]",
      "after": "derived = [(2*pi)^(-1/2)*exp(-25/72), (2*pi)^(-1/2)*exp(-25/72), 是, phi(x)/phi(0)=exp(-25/72)]",
      "relation": "implies",
      "rule_id": "Gaussian_evenness_and_ratio_cancel_positive_phi0",
      "conditions": [],
      "explanation": {
        "zh-Hant": "(−x)²=x²給evenness；phi(0)=(2pi)^−1/2>0，才能除並取消常數，留下exp(−x²/2)。這不是CDF或normalization求值。",
        "en": "Equality(−x)^2=x^2 gives evenness.Phi-density phi(0)=(2pi)^−1/2 is positive,so division cancels the factor and leaves exp(−x^2/2),not a CDF or normalization evaluation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "5/6",
      "-5/6",
      "(2*pi)^(-1/2)*exp(-25/72)",
      "(2*pi)^(-1/2)*exp(-25/72)",
      true,
      "phi(x)/phi(0)=exp(-25/72)",
      "evenness_follows_from_x_squared"
    ]
  },
  "conclusion": {
    "canonical_value": "(5/6, -5/6, (2*pi)^(-1/2)*exp(-25/72), (2*pi)^(-1/2)*exp(-25/72), 是, phi(x)/phi(0)=exp(-25/72), evenness_follows_from_x_squared)",
    "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-20261114-r04-standard-normal-distribution-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations/questions.jsonl",
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      "question_compact_sha256": "51ef6459a7af9358d4c48b94fe53a02252a3e1278c86ed6211c23aeed9eb0b15",
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      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations.en.jsonl"
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    "mko_bindings": {
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        "original_path": "public/data/mko/source13-20261018-standard-normal-distribution.json",
        "english_path": "public/data/i18n/editions/mko-standard-normal-distribution.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "991c73df066352bfcfb1142a2492e104cd1d20dafe7e9ecea8cff26a4f9d8e50",
    "original_candidate_sha256": "acf32f0276d0de6865eb746f5e359e29690483cd1d0211bccd76d3d23c211ece",
    "original_candidate_line_sha256": "acf32f0276d0de6865eb746f5e359e29690483cd1d0211bccd76d3d23c211ece"
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  "historical_source_state": {
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      "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-p50",
    "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": {
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    },
    "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": "e4bd7c6b3b5e3cdb138563f04e1ce20d49eb4a72c138a90c11beb27a81e8d713",
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    },
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    "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."
  }
}