{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-harmonic-oscillator-model",
  "semantic_family_id": "sf-harmonic-oscillator-model",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "線性簡諧振子模型"
  },
  "summary": {
    "zh-Hant": "無阻尼線性簡諧振子以m x''+k x=0及m,k>0定義；本輪不宣稱三角函數閉式解。"
  },
  "statement": {
    "zh-Hant": "線性簡諧振子模型由正質量m與正彈性係數k組成，位移x滿足m x''+k x=0。比值k/m是角頻率平方候選；只有另有三角函數及導數語義時才能建立一般閉式解。"
  },
  "formula": {
    "tex": "H(x)=0",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>H</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "H",
          "glyph": "H"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          }
        ]
      },
      "rhs": {
        "type": "number",
        "value": 0
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "228d01927ad4d2548847594d75984e85d05cb7a2946c10c5c1e7c4bfa21eedda"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "m與k為正有理數；題庫只核模型資格、k/m、多項式候選的指定點殘差與單一狀態能量式。"
    },
    {
      "id": "a2",
      "zh-Hant": "單點能量值不證沿軌道守恆；本輪不執行sin/cos閉式解或穩定性證明。"
    }
  ],
  "symbols": [
    {
      "token": "H",
      "role": "oscillator_residual",
      "role_zh": "振子方程殘差",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "displacement",
      "role_zh": "位移函數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-ordinary-differential-equation",
      "title_zh": "常微分方程",
      "reason_zh": "振子方程是二階常微分方程。"
    },
    {
      "id": "mko-derivative-rules",
      "title_zh": "多項式與仿射函數的導數規則",
      "reason_zh": "候選殘差需要二階導數。"
    },
    {
      "id": "mko-power-root",
      "title_zh": "冪與根式",
      "reason_zh": "角頻率平方與能量平方項使用冪。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "m與k的正性是本模型資格；k/m可精確計算，但本輪只稱角頻率平方。",
      "多項式候選可在指定點重算x與x''後得到殘差，單點為零不等於整個區間為解。",
      "狀態能量式E=mv²/2+kx²/2可作單點量值；守恆需另證方程沿完整軌道的導數為零。"
    ],
    "common_misconception_zh": "未有三角函數導數支援就直接宣稱一般sin/cos解，或以單點能量冒充守恆定理。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只核模型及精確有限切片；未宣稱閉式解、穩定性或能量守恆已證。",
    "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": "加入三角函數導數、一般閉式解、相位空間與能量不變量。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-harmonic-oscillator-model.md 整理；2026-10-13 脈衝 R01（使用者手動接續 2026-10-22 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"
  }
}