{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-standard-normal-distribution-01",
  "mko_id": "mko-standard-normal-distribution",
  "title": {
    "zh-Hant": "標準常態的對稱推導與named facts",
    "en": "Standard-normal symmetry with explicit named facts"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-standard-normal-distribution-01",
    "source": {
      "mko_id": "mko-standard-normal-distribution",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "5dac4a8d8eb1b38c2dbe86853a836e63d55563a544cfe67d5e6b4095fdaa9ada",
      "english_sha256": "1aebcb3737fd2d8647b6558c8a8272337f768dad39c6fdb34b76210d9375fdb4"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "清楚分開Gaussian前提與由它們推得的CDF對稱性。",
      "en": "Separate Gaussian prerequisites from derived CDF symmetry."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "φ(x)=(2π)^(-1/2)exp(-x²/2)，Z具有此density，Φ(t)=P(Z≤t)。",
          "en": "phi(x)=(2pi)^(-1/2)exp(-x²/2), Z has this density, and Phi(t)=P(Z≤t)."
        }
      },
      {
        "id": "G2",
        "statement": {
          "zh-Hant": "明列named theorem facts：Gaussian積分為sqrt(2π)，因此density歸一；E[Z]=0，E[Z²]=1。這些是前提，不在本例證明。",
          "en": "Explicit named theorem facts: the Gaussian integral is sqrt(2pi), hence normalization; E[Z]=0 and E[Z²]=1. These are premises, not proved here."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "G2",
          "work"
        ],
        "before_tex": "\\varphi(x)=\\frac{e^{-x^2/2}}{\\sqrt{2\\pi}}",
        "after_tex": "\\varphi(x)>0\\quad(x\\in\\mathbb R)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "exp正且常數分母正，density非負資格成立。",
          "en": "The exponential and constant denominator are positive, qualifying nonnegativity."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "work"
          ],
          "explanation": {
            "zh-Hant": "非負性本身不證積分等於一。",
            "en": "Nonnegativity alone does not prove normalization."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "G2",
          "S01",
          "work"
        ],
        "before_tex": "\\int_{\\mathbb R}e^{-x^2/2}\\,dx=\\sqrt{2\\pi}\\quad\\text{(assumed)}",
        "after_tex": "\\int_{\\mathbb R}\\varphi(x)\\,dx=1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "只用已宣告Gaussian積分前提乘上常數。",
          "en": "Multiply the explicitly assumed Gaussian integral by the normalization constant."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "這不是從幾個數值積分點回推整個積分。",
            "en": "No finite quadrature points are being used to infer the integral."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "G2",
          "S02",
          "work"
        ],
        "before_tex": "(-x)^2=x^2",
        "after_tex": "\\varphi(-x)=\\varphi(x)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "平方的偶性逐點給density偶性。",
          "en": "Evenness of the square gives pointwise evenness of the density."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "對稱中心是0，沒有引入未知μ。",
            "en": "The center is 0; no unknown mu has been introduced."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "G2",
          "S03",
          "work"
        ],
        "before_tex": "\\varphi(0)=1/\\sqrt{2\\pi}",
        "after_tex": "\\varphi(x)/\\varphi(0)=e^{-x^2/2}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "除去共同正normalization常數得到精確比值。",
          "en": "Cancel the common positive normalization constant to obtain the exact ratio."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "這是density比值，不是CDF比值。",
            "en": "This is a density ratio, not a CDF ratio."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "G2",
          "S04",
          "work"
        ],
        "before_tex": "\\Phi(-t)=\\int_{-\\infty}^{-t}\\varphi(u)\\,du",
        "after_tex": "\\Phi(-t)=\\int_t^\\infty\\varphi(v)\\,dv=1-\\Phi(t)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "代u=-v並用偶性、歸一；density law無singleton mass。",
          "en": "Substitute u=-v, using evenness and normalization; a density law has no singleton mass."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "補事件的端點差因無原子為零。",
            "en": "The endpoint discrepancy in the complement has zero mass."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "G2",
          "S05",
          "work"
        ],
        "before_tex": "t=0,\\quad\\Phi(0)=1-\\Phi(0)",
        "after_tex": "\\Phi(0)=1/2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "對稱式在零點化為兩倍CDF等於一。",
          "en": "At zero the symmetry identity gives twice the CDF equal to one."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "此exact值來自對稱與歸一，非approximation。",
            "en": "This exact value follows from symmetry and normalization, not approximation."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "G2",
          "S06",
          "work"
        ],
        "before_tex": "E[Z]=0,\\quad E[Z^2]=1\\quad\\text{(assumed)}",
        "after_tex": "\\operatorname{Var}(Z)=E[Z^2]-E[Z]^2=1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "用已宣告moments前提代variance identity。",
          "en": "Substitute the explicitly assumed moments into the variance identity."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不把moment前提重新標成此例完成的積分證明。",
            "en": "Do not relabel the moment premises as an integral proof completed here."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "G2",
          "S07",
          "work"
        ],
        "before_tex": "t=1",
        "after_tex": "\\Phi(-1)=1-\\Phi(1),\\quad\\varphi(1)/\\varphi(0)=e^{-1/2}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "給一個非零點的精確符號關係，不捏造elementary CDF數值。",
          "en": "Give exact symbolic relations at a nonzero point without inventing an elementary CDF value."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "Φ(1)未被本推導求成有理數，有限integration也不證前提。",
            "en": "Phi(1) has not been evaluated rationally, and finite integration proves none of the premises."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "在明示Gaussian前提下，Φ(-t)=1-Φ(t)、Φ(0)=1/2、Var(Z)=1。",
      "en": "Under the explicit Gaussian prerequisites, Phi(-t)=1-Phi(t), Phi(0)=1/2 and Var(Z)=1."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "取幾個quadrature值就證歸一。",
          "en": "A few quadrature values prove normalization."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "有限數值不是整實軸積分證明。",
          "en": "Finite values do not prove an integral over the whole line."
        }
      },
      {
        "claim": {
          "zh-Hant": "Φ(1)=φ(1)。",
          "en": "Phi(1)=phi(1)."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "φ正故Φ(1)>Φ(0)=1/2；而φ(1)<φ(0)=1/sqrt(2π)<1/2（π>2），故不等。",
          "en": "Positive phi gives Phi(1)>Phi(0)=1/2, whereas phi(1)<phi(0)=1/sqrt(2pi)<1/2 using pi>2, so they differ."
        }
      },
      {
        "claim": {
          "zh-Hant": "任意連續CDF都是standard normal。",
          "en": "Every continuous CDF is standard normal."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "均勻分布的連續CDF即不同。",
          "en": "A uniform distribution has a different continuous CDF."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "本節點把 Gaussian integral normalization、E[Z]=0 與 Var(Z)=1 明列為 named theorem facts，不以有限 quadrature 偽造完整推導。題庫可精確處理 density symbolic form、evenness、phi(x)/phi(0)=exp(-x^2/2)，以及由 declared normalization 與 symmetry 推出 Phi(0)=1/2 與 Phi(-x)=1-Phi(x)。本節不計算一般非零 x 的 elementary exact CDF 數值；有限數值積分不證 normalization 或 moments，continuous CDF 也不自動有 density 或是 normal law。",
          "en": "Gaussian-integral normalization, E[Z]=0 and Var(Z)=1 are explicitly named theorem facts, not derivations manufactured from finite quadrature. Exact families handle symbolic density, evenness and phi(x)/phi(0)=exp(−x^2/2), plus Phi(0)=1/2 and Phi(−x)=1−Phi(x) from declared normalization and symmetry. No general nonzero elementary CDF value is computed. Finite numerical integration proves neither normalization nor moments; a continuous CDF need not have a density or be normal."
        },
        "source": {
          "mko_id": "mko-standard-normal-distribution",
          "sha256": "5dac4a8d8eb1b38c2dbe86853a836e63d55563a544cfe67d5e6b4095fdaa9ada",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "Gaussian積分與moments為明示named theorem前提；另假設積分線性、變數代換、density law無原子及variance二次矩identity。",
          "en": "The Gaussian integral and moments are explicit named theorem prerequisites; also assume integral linearity, substitution, atomlessness of density laws and the second-moment variance identity."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "1-0**2",
        "rhs": "1"
      },
      {
        "id": "N02",
        "lhs": "1/2+1/2",
        "rhs": "1"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "不以有限quadrature證Gaussian積分或moments。",
      "en": "No finite-quadrature proof of the Gaussian integral or moments."
    },
    "md_symbols": [
      [
        "φ",
        "標準常態density",
        "standard-normal density"
      ],
      [
        "Φ",
        "標準常態CDF",
        "standard-normal CDF"
      ],
      [
        "Z",
        "實值隨機變數",
        "real random variable"
      ]
    ],
    "md_error_controls": [
      {
        "title": "前提標記 / Premise labeling",
        "zh": "歸一與moments保持named facts，不宣稱本例已證。",
        "en": "Keep normalization and moments labeled as named facts, not newly proved results."
      },
      {
        "title": "事件與門檻 / Event and threshold",
        "zh": "Φ(t)=P({ω:Z(ω)≤t})，t是標量門檻。",
        "en": "Phi(t)=P({omega:Z(omega)≤t}); t is a scalar threshold."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "ce6e79a0422b1524888abb53a308a69a70552c942a0768637ab742837af70ae2",
    "original_source_sha256": "9be2b5a4eeeb9649020eca8625ccd6a76eef2d3e32963e884d8eadc5a4490a58",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [
      "mko_definition"
    ],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
      "BR142-P068",
      "BR142-S199"
    ],
    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}