{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-real-exponential-function-01",
  "mko_id": "mko-real-exponential-function",
  "title": {
    "zh-Hant": "全級數的加法律與嚴格正性",
    "en": "Addition and strict positivity of the full exponential series"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-real-exponential-function-01",
    "source": {
      "mko_id": "mko-real-exponential-function",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "7b0736c512a2b1e27562551d8a9de463446e4b54c5998868cfe7573aee0b4f17",
      "english_sha256": "1fafde29fc55b9df97ad047683d8c6f01be2efa1ac103b831e7d23ec97eafb33"
    },
    "mode": "theorem_proof",
    "audience_prerequisites": [
      "algebra",
      "epsilon",
      "cauchy_product",
      "real_complete"
    ],
    "goal": {
      "zh-Hant": "從 E(x)=sum x^n/n! 證全實數絕對收斂、加法律、逆元與正性。",
      "en": "From E(x)=sum x^n/n!, prove absolute convergence for all real x, addition, inverses and positivity."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "E(x) 定義為 n≥0 的全級數 x^n/n!；x,y 為任意固定實數。",
          "en": "E(x) is the whole series x^n/n! over n≥0; x,y are arbitrary fixed real numbers."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "u_n=|x|^n/n!,\\quad x\\ne0",
        "after_tex": "u_{n+1}/u_n=|x|/(n+1)",
        "relation": "case_split",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先固定 x 計算完整相鄰絕對比；x=0 則首項為 1、其餘全零。",
          "en": "Fix x and compute the full absolute ratio; at x=0 only the initial term is one and all later terms vanish."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "不在 x=0 的零項上做除零比值。",
            "en": "No ratio division by a zero term is performed at x=0."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S02",
        "uses": [
          "S01"
        ],
        "before_tex": "N>2|x|,\\quad n\\ge N",
        "after_tex": "u_{n+1}\\le\\tfrac12u_n,\\quad u_{N+j}\\le2^{-j}u_N",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [
          "N\\in\\mathbb N,\\ N>2|x|"
        ],
        "explanation": {
          "zh-Hant": "選整數 N 後歸納幾何控制整個尾項。",
          "en": "After selecting integer N, induction geometrically controls the entire tail."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01"
          ],
          "explanation": {
            "zh-Hant": "N 可依固定 x，但 j 的界是全稱。",
            "en": "N may depend on the fixed x, but the bound covers every j."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S03",
        "uses": [
          "S02",
          "real_complete"
        ],
        "before_tex": "\\sum_{j=0}^{M}u_{N+j}\\le2u_N",
        "after_tex": "\\sum_{n=0}^{\\infty}|x|^n/n!<\\infty\\quad\\forall x\\in\\mathbb R",
        "relation": "implies",
        "rule": "real_complete",
        "conditions": [
          "\\text{bounded monotone real sequences converge}"
        ],
        "explanation": {
          "zh-Hant": "非負尾部分和單調有界（實數完備性），有限前綴不影響收斂，故全實數絕對收斂。",
          "en": "Nonnegative tail partial sums are monotone and bounded by 2u_N; real completeness and a finite prefix give absolute convergence for every real x."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "real_complete"
          ],
          "explanation": {
            "zh-Hant": "需實數單調有界收斂性作此步前提，而非有限 M 抽樣。",
            "en": "This step assumes real bounded-monotone convergence, not sampled finite M values."
          }
        },
        "leaf_basis": "real_complete"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "cauchy_product"
        ],
        "before_tex": "\\sum_{k=0}^{m}\\frac{x^ky^{m-k}}{k!(m-k)!}",
        "after_tex": "\\frac1{m!}\\sum_{k=0}^{m}\\binom mkx^ky^{m-k}=\\frac{(x+y)^m}{m!}",
        "relation": "equivalent",
        "rule": "cauchy_product",
        "conditions": [
          "m\\in\\mathbb N_0"
        ],
        "explanation": {
          "zh-Hant": "對任意 degree m 以二項式定理核全體係數恆等式，不只一個選定 m。",
          "en": "For arbitrary degree m, the binomial theorem proves the coefficient identity for every degree, not just one selected m."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1",
            "cauchy_product"
          ],
          "explanation": {
            "zh-Hant": "factorial 分母滿足 0≤k≤m，所以均合法。",
            "en": "The factorial denominators are valid because 0≤k≤m."
          }
        },
        "leaf_basis": "cauchy_product"
      },
      {
        "id": "S05",
        "uses": [
          "S03",
          "S04"
        ],
        "before_tex": "E(x)E(y)=\\sum_{m=0}^{\\infty}\\sum_{k=0}^{m}\\frac{x^ky^{m-k}}{k!(m-k)!}",
        "after_tex": "E(x)E(y)=E(x+y)",
        "relation": "theorem_application",
        "rule": "cauchy_product",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩級數絕對收斂先允許 Cauchy product 與按總次數重排，再用所有 degree 的恆等式。",
          "en": "Absolute convergence first licenses the Cauchy product and regrouping by total degree; then use the identity at every degree."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "係數恆等式與無窮重排是不同責任，都已列前提。",
            "en": "The coefficient identity and infinite regrouping are separate obligations, both supplied."
          }
        },
        "leaf_basis": "cauchy_product"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05"
        ],
        "before_tex": "E(0)=1,\\quad y=-x",
        "after_tex": "E(x)E(-x)=1",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "零值由唯一非零首項給出，再以加法律得乘法逆元。",
          "en": "The zero value comes from the sole nonzero initial term; addition then gives multiplicative inverses."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "有限部分和通常沒有此精確逆元性。",
            "en": "Finite partial sums generally do not have this exact inverse property."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S07",
        "uses": [
          "S06"
        ],
        "before_tex": "E(x/2)E(-x/2)=1",
        "after_tex": "E(x/2)\\ne0",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先以半點逆元性排除零，之後才能證嚴格正而不只是非負。",
          "en": "First exclude zero using the half-point inverse identity, enabling strict positivity rather than only nonnegativity."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S06"
          ],
          "explanation": {
            "zh-Hant": "不能從平方非負單獨推出嚴格正。",
            "en": "A nonnegative square alone does not establish strict positivity."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S08",
        "uses": [
          "S05",
          "S07"
        ],
        "before_tex": "E(x)=E(x/2+x/2)=E(x/2)^2",
        "after_tex": "E(x)>0\\quad\\forall x\\in\\mathbb R",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "實數非零平方嚴格正，完成全函數的結論。",
          "en": "A nonzero real square is strictly positive, completing the full-function conclusion."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S05",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "此證明沒有將非零有理點的有限部分和稱為 exact E(x)。",
            "en": "No finite partial sum at a nonzero rational point has been called exact E(x)."
          }
        },
        "leaf_basis": "algebra"
      }
    ],
    "result": {
      "zh-Hant": "E 全實數絕對收斂，滿足 E(x+y)=E(x)E(y)、E(0)=1、E(x)E(-x)=1 且 E(x)>0。",
      "en": "E converges absolutely on R and satisfies addition, E(0)=1, E(x)E(-x)=1 and E(x)>0."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "有限部分和等於一般 exact E(x)。",
          "en": "A finite partial sum is the general exact E(x)."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "仍缺非零無窮尾，除特值另證外不能如此等同。",
          "en": "It omits an infinite tail; equality requires a separately proved special value."
        }
      },
      {
        "claim": {
          "zh-Hant": "驗 m=3 係數足以證加法律。",
          "en": "Checking degree three alone proves addition."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "需每個 m 的二項式恆等式與絕對重排依據。",
          "en": "Every degree identity and absolute regrouping are required."
        }
      },
      {
        "claim": {
          "zh-Hant": "E(x)=E(x/2)^2 單獨就證 >0。",
          "en": "The square identity alone proves strict positivity."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "必先排除 E(x/2)=0。",
          "en": "E(x/2)=0 must first be excluded."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "完整級數在 x 處的項為 x^n/n!，而冪級數係數為 1/n!；係數相鄰比 1/(n+1) 趨 0，因此 radius 為 infinity 且對每個實 x 絕對收斂。Addition law 需對所有 degree 使用二項式係數合併 Cauchy product，且重排責任由絕對收斂支撐；單一 declared degree 只核一個係數。E(0)=1；E(x)E(-x)=1，且 E(x/2)E(-x/2)=1 先給 E(x/2) 非零，再由 E(x)=E(x/2)^2 推出嚴格正。有限 partial sum 不是 E(x)，也不由本節點自動推出導數、log、ODE 唯一性、injectivity 或非零有理點的 exact rational value。",
          "en": "Terms at x are x^n/n!, while the power-series coefficients 1/n! have ratio1/(n+1) tending to 0, giving infinite radius and absolute convergence for every fixed real x. The addition law needs the binomial coefficient identity at every degree and absolute convergence for Cauchy-product rearrangement; one declared degree checks only one coefficient. E(0)=1 and E(x)E(−x)=1. The half inverse identity makes E(x/2) nonzero before E(x)=E(x/2)^2 proves strict positivity. A finite partial-sum computation is not the whole exponential function, even where a special value coincides. Derivatives, logarithms, ODE uniqueness, injectivity and exact rational values at nonzero rational inputs are not automatic claims of this node."
        },
        "source": {
          "mko_id": "mko-real-exponential-function",
          "sha256": "7b0736c512a2b1e27562551d8a9de463446e4b54c5998868cfe7573aee0b4f17",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "algebra",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "實數域的等式運算；除法前分母必非零，不等式乘正數保持方向。",
          "en": "Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities."
        }
      },
      {
        "id": "epsilon",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "收斂表示每個正 epsilon 有一個整數 N，使全部 n≥N 的誤差嚴格小於 epsilon；使用實數 Archimedean 性。",
          "en": "Convergence requires for every positive epsilon an integer N controlling every n≥N by a strict error bound; use the Archimedean property of the reals."
        }
      },
      {
        "id": "cauchy_product",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "兩個絕對收斂級數的乘積等於其 Cauchy product，且可依總次數重排；二項式定理對每個非負整數 degree 成立。",
          "en": "The product of two absolutely convergent series equals their Cauchy product and may be grouped by total degree; the binomial theorem holds for every nonnegative integer degree."
        }
      },
      {
        "id": "real_complete",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "非空有上界的實集合有 supremum，故單調有界實數列收斂；這也適用非負級數有界部分和。",
          "en": "Every nonempty bounded-above real set has a supremum, so bounded monotone real sequences converge, including bounded partial sums of nonnegative series."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "degree-three",
        "lhs": "1/factorial(3)+1/factorial(2)+1/factorial(2)+1/factorial(3)",
        "rhs": "2**3/factorial(3)",
        "step": "S04"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "不引入導數、log、ODE、單射或超越性。",
      "en": "No derivative, logarithm, ODE, injectivity or transcendence claim."
    },
    "md_symbols": [
      [
        "E(x)",
        "完整冪級數函數",
        "whole power-series function"
      ],
      [
        "u_n",
        "絕對項",
        "absolute term"
      ],
      [
        "m",
        "任意總次數",
        "arbitrary total degree"
      ]
    ],
    "md_error_controls": [
      {
        "title": "全級數 / Whole series",
        "zh": "區分完整 E 與有限 S_N。",
        "en": "Separate the full E from finite S_N."
      },
      {
        "title": "先非零 / Nonzero first",
        "zh": "用逆元性再用平方推出嚴格正。",
        "en": "Use inverses to exclude zero before deriving positivity from a square."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "a1687161d95b6c66cce720bd37cdfa330f5a9bfd717247f0fa87c5f638f6ca35",
    "original_source_sha256": "ea4e23ef55650baa4c70e04bfc52f0293ed62cfd1b2f52bb15cc8695fe035c75",
    "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-S186",
      "BR142-P060"
    ],
    "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."
  }
}