{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-initial-value-problem",
  "semantic_family_id": "sf-initial-value-problem",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "初值問題"
  },
  "summary": {
    "zh-Hant": "初值問題在微分方程外指定某點的函數值或導數值；候選核對與存在唯一性是不同責任。"
  },
  "statement": {
    "zh-Hant": "初值問題由常微分方程及同一未知函數在指定點的初始資料組成。一階問題常給 y(x0)=y0；二階問題通常需 y(x0)=y0 與 y'(x0)=v0。候選解必須同時滿足方程與初值。"
  },
  "formula": {
    "tex": "y(a)=b",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>y</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mo>=</mo><mi>b</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "y",
          "glyph": "y"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "a",
            "glyph": "a"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "b",
        "glyph": "b"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "0ccd55b3c7e5f852c6a08cae00acc5eb770aff57016a251e44f232a47ce7325c"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫限整係數多項式候選、指定有限殘差網格與明示初值。"
    },
    {
      "id": "a2",
      "zh-Hant": "有限網格與初值相符只核對候選；不宣稱解存在、唯一或已求得一般解。"
    }
  ],
  "symbols": [
    {
      "token": "y",
      "role": "unknown_function",
      "role_zh": "未知函數",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "initial_point",
      "role_zh": "初始點",
      "scope": "representative_formula"
    },
    {
      "token": "b",
      "role": "initial_value",
      "role_zh": "初始函數值",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-ordinary-differential-equation",
      "title_zh": "常微分方程",
      "reason_zh": "初值問題以既定ODE為方程部分。"
    },
    {
      "id": "mko-domain-codomain-range",
      "title_zh": "定義域、陪域與值域",
      "reason_zh": "初始點必須在候選解與方程的共同定義域。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "初值是額外限制，不會單靠書寫就保證有解或唯一解。",
      "候選多項式要分開核對ODE殘差與初值；任一責任失敗就不能成為該IVP候選解。",
      "只在有限點殘差為零仍不是整個區間上的函數恆等證明。"
    ],
    "common_misconception_zh": "只核初值不核方程，或把有限採樣殘差為零當成全區間解。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪只核對候選與有限網格；存在唯一性與一般求解均未證。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "建立存在唯一條件、最大解區間與標準一階求解法。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-initial-value-problem.md 整理；2026-10-13 脈衝 R04（使用者手動接續 2026-10-21 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}