{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-chebyshev-inequality-01",
  "mko_id": "mko-chebyshev-inequality",
  "title": {
    "zh-Hant": "平方偏差的Chebyshev推導",
    "en": "Chebyshev from squared deviations"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-chebyshev-inequality-01",
    "source": {
      "mko_id": "mko-chebyshev-inequality",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "53f883dd0d24e24dd2c178a77c932cdf6caf9b65e2b443bee96a237be01b73e3",
      "english_sha256": "af0da60b30b2be710d729655f44307db4c48210cbe727eded9d899eefcd264a5"
    },
    "mode": "theorem_proof",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "由Markov證Chebyshev並算exact例。",
      "en": "Derive Chebyshev from Markov and compute an exact instance."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "X取-2、0、2，質量1/4、1/2、1/4；epsilon=1。",
          "en": "X takes -2,0,2 with masses 1/4,1/2,1/4; epsilon=1."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "1/4+1/2+1/4=1",
        "after_tex": "\\mu=E[X]=-2/4+2/4=0",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整歸一law給mean零。",
          "en": "The complete normalized law gives mean zero."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "負值本身不妨礙Chebyshev，後面平方才套Markov。",
            "en": "Signed X is allowed; Markov will apply to its square."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "Y=(X-\\mu)^2",
        "after_tex": "Y\\in\\{4,0,4\\},\\quad Y\\ge0",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先建立非負變數Y而非直接對X套Markov。",
          "en": "Construct nonnegative Y rather than applying Markov directly to X."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "Y的質量仍來自原X law。",
            "en": "Y retains the masses from the original X law."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "E[Y]=4/4+0/2+4/4",
        "after_tex": "\\operatorname{Var}(X)=E[Y]=2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "中心二次矩逐support計算。",
          "en": "Compute the centered second moment across the support."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "中心為μ=0，不是觀測值或其他law的mean。",
            "en": "Center at mu=0, not an observation or another law's mean."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "\\varepsilon>0",
        "after_tex": "|X-\\mu|\\ge\\varepsilon\\iff Y\\ge\\varepsilon^2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "非負絕對偏差可平方而保持閾值等價。",
          "en": "Squaring nonnegative absolute deviations preserves threshold equivalence."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "epsilon須正，否則除epsilon平方失去資格。",
            "en": "Positive epsilon is necessary before division by its square."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "P(Y\\ge\\varepsilon^2)\\le E[Y]/\\varepsilon^2",
        "after_tex": "P(|X-\\mu|\\ge\\varepsilon)\\le\\operatorname{Var}(X)/\\varepsilon^2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "對Y用Markov，再代入事件等價與variance定義。",
          "en": "Apply Markov to Y, then substitute the event equivalence and variance."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "Markov前提Y≥0與有限期望均已建立。",
            "en": "Y nonnegative and finite expectation have already been established."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "\\varepsilon=1",
        "after_tex": "\\operatorname{Var}(X)/\\varepsilon^2=2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "本例原始Chebyshev界為2，允許大於1但很鬆。",
          "en": "The raw bound is 2; a valid upper bound may exceed1 and be loose."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "若需要機率範圍可另取min(1,2)，不可宣稱tail=2。",
            "en": "One may clip to min(1,2), but cannot claim a tail of 2."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "\\{|X|\\ge1\\}=\\{-2,2\\}",
        "after_tex": "P(|X|\\ge1)=1/2\\le2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "exact tail由兩端質量得1/2。",
          "en": "The two endpoint masses give exact tail 1/2."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "中間0不在tail，端點事件採≥。",
            "en": "Zero is excluded; the event retains≥."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "\\varepsilon=2",
        "after_tex": "P(|X|\\ge2)=1/2=2/2^2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "另一閾值達等號，說明界可緊也可鬆。",
          "en": "At another threshold equality holds, showing the bound can be sharp or loose."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "這只是本law的等號例，不把等號推成一般事實。",
            "en": "This is equality for this law, not a universal equality claim."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "Chebyshev由平方偏差的Markov界得到；epsilon=1的exact tail為1/2。",
      "en": "Chebyshev follows by Markov on squared deviations; at epsilon=1 the exact tail is 1/2."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "Chebyshev需要多個變數互相獨立。",
          "en": "Chebyshev needs independent variables."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "此推導只用一個X的二次矩。",
          "en": "The proof uses only one X and its second moment."
        }
      },
      {
        "claim": {
          "zh-Hant": "本例tail=2。",
          "en": "The tail is 2 here."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "2是鬆上界，tail為1/2。",
          "en": "Two is a loose bound; the tail is 1/2."
        }
      },
      {
        "claim": {
          "zh-Hant": "epsilon=0可照用公式。",
          "en": "The formula applies at epsilon=0."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "分母epsilon平方不得零。",
          "en": "Its denominator epsilon squared cannot be zero."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "epsilon>0且mean、variance有限。事件|X-mu|>=epsilon等同(X-mu)^2>=epsilon^2，對非負squared deviation套Markov。上界不等於actual tail，不需要independence；zero mass atoms與zero variance按a.s.語義處理。",
          "en": "Require positive epsilon and finite mean/variance. The event |X−mean|≥epsilon equals (X−mean)^2≥epsilon^2, allowing Markov on the nonnegative squared deviation. The raw bound is not actual tail equality and needs no independence; zero-mass atoms and zero variance retain their almost-sure meaning."
        },
        "source": {
          "mko_id": "mko-chebyshev-inequality",
          "sha256": "53f883dd0d24e24dd2c178a77c932cdf6caf9b65e2b443bee96a237be01b73e3",
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        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "Markov為已知前提：非負Y且a>0時P(Y≥a)≤E[Y]/a；variance定義為中心二次矩。",
          "en": "Assumed prerequisite Markov: for nonnegative Y and a>0, P(Y≥a)≤E[Y]/a; variance is the centered second moment."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "4/4+4/4",
        "rhs": "2"
      },
      {
        "id": "N02",
        "lhs": "1/4+1/4",
        "rhs": "1/2"
      },
      {
        "id": "N03",
        "lhs": "2/(2**2)",
        "rhs": "1/2"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "有限完整law；不需independence。",
      "en": "A complete finite law; independence is not required."
    },
    "md_symbols": [
      [
        "μ",
        "此law的mean",
        "mean under this law"
      ],
      [
        "Y",
        "平方偏差",
        "squared deviation"
      ],
      [
        "ε",
        "正門檻",
        "positive threshold"
      ]
    ],
    "md_error_controls": [
      {
        "title": "平方事件 / Squared event",
        "zh": "先證|X-μ|≥ε與Y≥ε²等價再套Markov。",
        "en": "Establish the event equivalence before invoking Markov."
      },
      {
        "title": "同law中心 / Same-law center",
        "zh": "mean、Y與variance全部使用同一完整law。",
        "en": "Use the same complete law for the mean, Y and variance."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "collection": "coverage-20261021",
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    "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": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
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      "BR142-S131",
      "BR142-S132",
      "BR142-S130",
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    "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."
  }
}