{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-boundary-value-problem",
  "semantic_family_id": "sf-boundary-value-problem",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "邊界值問題"
  },
  "summary": {
    "zh-Hant": "邊界值問題把定義域內的微分方程責任與邊界上的值或導數資料同時指定；候選通過不等於適定性。"
  },
  "statement": {
    "zh-Hant": "本輪固定矩形域上的二次多項式候選u。內部Poisson責任以符號Laplacian恆等核對，Dirichlet資料則把u精確限制到四條邊並逐係數比較；兩類責任分開記錄後才可說候選滿足該宣告BVP。"
  },
  "formula": {
    "tex": "B(u)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>B</mi><mo>(</mo><mi>u</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "B",
        "glyph": "B"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "u",
          "glyph": "u"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "50f04c7423bc4516ac8178a8d706e92adcd887c8024f88648551d9d40ac38be6"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "矩形邊界按left/right/bottom/top完整列出，邊界資料為對應單變數二次多項式係數。"
    },
    {
      "id": "a2",
      "zh-Hant": "符號恆等只核指定候選與指定方程／邊界；不證一般解存在、唯一、穩定、正則或適定。"
    }
  ],
  "symbols": [
    {
      "token": "B",
      "role": "boundary_value_record",
      "role_zh": "邊界值問題核對紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "u",
      "role": "candidate_function",
      "role_zh": "候選函數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-partial-differential-equation",
      "title_zh": "偏微分方程",
      "reason_zh": "內部責任由PDE算子指定。"
    },
    {
      "id": "mko-domain-codomain-range",
      "title_zh": "定義域、陪域和值域",
      "reason_zh": "矩形域與邊界必須明示。"
    },
    {
      "id": "mko-equation",
      "title_zh": "方程式",
      "reason_zh": "內部方程與邊界等式分別核對。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "內部殘差與邊界資料是兩組不同義務。",
      "整條邊的多項式係數相等是全邊恆等，而不是有限點抽樣。",
      "候選滿足宣告BVP不提供解的存在唯一或對資料擾動的穩定性。"
    ],
    "common_misconception_zh": "只核內部PDE或幾個邊界點就宣稱整個BVP已解且適定。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只核矩形域二次多項式候選的符號義務；不宣稱適定性。",
    "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": "加入弱解、函數空間、存在唯一與穩定性證書。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-boundary-value-problem.md 整理；2026-10-13 脈衝 R01（使用者手動接續 2026-10-25 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"
  }
}