{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-series-comparison-test",
  "semantic_family_id": "sf-series-comparison-test",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "級數比較判別法"
  },
  "summary": {
    "zh-Hant": "非負項級數可由逐項或最終逐項上界與下界傳遞收斂或發散。"
  },
  "statement": {
    "zh-Hant": "對足夠大n明示0≤a_n≤b_n。若sum b_n收斂，則sum a_n收斂；若sum a_n發散，則sum b_n發散。題族以exact geometric tails、有限前綴與正下界證書核對；有限多項不影響級數收斂性。負項、反向不等式、只有有限多個sample或拿divergent upper bound推被比較級數發散，都不滿足判別責任。"
  },
  "formula": {
    "tex": "C(a,b)=c",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>C</mi><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mo>=</mo><mi>c</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "C",
          "glyph": "C"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "a",
            "glyph": "a"
          },
          {
            "type": "symbol",
            "name": "b",
            "glyph": "b"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "c",
        "glyph": "c"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "8a8b27978e615d5e8e1fe2efa1da8396545789c381872621e0141c28e516c388"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "比較自某個明示N起對每個n成立，且被比較項非負。"
    },
    {
      "id": "a2",
      "zh-Hant": "基準級數的收斂或發散性有獨立證書，不由有限前綴猜測。"
    }
  ],
  "symbols": [
    {
      "token": "C",
      "role": "series_comparison_operator",
      "role_zh": "級數比較算子",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "compared_series_terms",
      "role_zh": "被比較級數項",
      "scope": "representative_formula"
    },
    {
      "token": "b",
      "role": "reference_series_terms",
      "role_zh": "基準級數項",
      "scope": "representative_formula"
    },
    {
      "token": "c",
      "role": "comparison_conclusion",
      "role_zh": "比較結論",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-infinite-series",
      "title_zh": "無窮級數",
      "reason_zh": "級數與部分和語義。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "偏序與全序",
      "reason_zh": "逐項不等式。"
    },
    {
      "id": "mko-sequence-limit",
      "title_zh": "數列極限",
      "reason_zh": "部分和收斂與發散背景。"
    },
    {
      "id": "mko-monotone-sequence-convergence-theorem",
      "title_zh": "單調有界數列收斂定理",
      "reason_zh": "非負級數的單調部分和與上界收斂責任。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "先分清convergence upper comparison與divergence lower comparison的方向。",
      "eventual comparison允許任意有限前綴，但尾端責任必須是全稱。",
      "幾何尾與正下界只提供可重算instance，不取代一般定理證明。"
    ],
    "common_misconception_zh": "把有限多項抽樣、反向不等式或一個發散上界當成比較判別證書。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "exact geometric instances只證指定序列；一般比較判別仍需完整全稱論證。",
    "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": "加入極限比較判別與積分判別。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-series-comparison-test.md 整理；2026-10-15 脈衝 R01（使用者手動接續 2026-11-17 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}