{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-flow-network",
  "semantic_family_id": "sf-finite-flow-network",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限流網路"
  },
  "summary": {
    "zh-Hant": "有限有向網路中的非負流須逐邊不超過capacity，並在source與sink以外每個頂點滿足流量守恆。"
  },
  "statement": {
    "zh-Hant": "network使用finite loop-free simple directed graph，source s與sink t不同；antiparallel edges可同時存在但每個ordered pair最多一條。capacity與flow為exact nonnegative rationals。feasible flow逐邊滿足0≤f(e)≤c(e)，且每個internal vertex的完整inflow等於outflow。flow value同時由source net outflow與sink net inflow計算並須相等。允許zero capacity與disconnected vertices。feasible不等於maximum，不得提前宣稱min-cut equality或augmenting-path optimality。"
  },
  "formula": {
    "tex": "V(f,s,t)=z",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>V</mi><mo>(</mo><mi>f</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>z</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "V",
          "glyph": "V"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "f",
            "glyph": "f"
          },
          {
            "type": "symbol",
            "name": "s",
            "glyph": "s"
          },
          {
            "type": "symbol",
            "name": "t",
            "glyph": "t"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "z",
        "glyph": "z"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "e739d090a5693a93f8d21aa5bebef8dbed0a4046ea25fa76db85838930c3a5bf"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "vertices、edges、capacity與flow keys完整一一對應；不允許self-loop或duplicate directed edge。"
    },
    {
      "id": "a2",
      "zh-Hant": "所有加總以exact rational arithmetic進行。"
    }
  ],
  "symbols": [
    {
      "token": "V",
      "role": "flow_value_operator",
      "role_zh": "流值算子",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "finite_edge_flow",
      "role_zh": "有限邊流",
      "scope": "representative_formula"
    },
    {
      "token": "s",
      "role": "source_vertex",
      "role_zh": "源點",
      "scope": "representative_formula"
    },
    {
      "token": "t",
      "role": "sink_vertex",
      "role_zh": "匯點",
      "scope": "representative_formula"
    },
    {
      "token": "z",
      "role": "exact_flow_value",
      "role_zh": "精確流值",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-directed-graph",
      "title_zh": "有限有向圖",
      "reason_zh": "network carrier。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "capacity與flow的edge mapping。"
    },
    {
      "id": "mko-linear-programming",
      "title_zh": "線性規劃",
      "reason_zh": "capacity與conservation constraints。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "network資格先固定directed-edge convention。",
      "capacity與conservation為不同feasibility責任。",
      "source與sink兩種flow-value計算必須交叉核對。"
    ],
    "common_misconception_zh": "只看總流值不檢每條edge／internal vertex，或把任何feasible flow稱maximum。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪不建立max-flow min-cut theorem、residual network或optimality。",
    "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": "加入最大流最小割定理。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-flow-network.md 整理；2026-10-15 脈衝 R07（使用者手動接續 2026-11-13 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"
  }
}