{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261024-r07-convex-set-0071",
  "pilot_number": "R0021-11171",
  "number": 11171,
  "family_id": "qf-mod24-convexset-halfspace-combination",
  "title": {
    "zh-Hant": "第 11171 題：凸集合證書與反例",
    "en": "Question 11171: Linear Halfspace Closure at Qualified Endpoints"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261024-r07-convex-set-0071",
    "batch_id": "batch-20261013-advance-20261024-r07-convex-set-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-convex-set"
    ],
    "title_zh": "凸集合證書與反例",
    "stem_zh": "量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":22,\"left\":[0,0],\"right\":[21,0],\"lambda\":[21,23]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([1, 1], 22, [0, 0], [21, 0], 0, 21, 21/23, [42/23, 0], 42/23, 是)",
      "explanation_zh": "依賴結構核對=([1, 1], 22, [0, 0], [21, 0], 0, 21, 21/23, [42/23, 0], 42/23, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "凸集合證書與反例"
      ]
    },
    "variant": {
      "family_id": "qf-mod24-convexset-halfspace-combination",
      "seed": 202610247071,
      "parameters": {
        "normal": [
          1,
          1
        ],
        "bound": 22,
        "left": [
          0,
          0
        ],
        "right": [
          21,
          0
        ],
        "lambda": [
          21,
          23
        ],
        "domain": "Exact_algebraic_convex_set_certificates",
        "operation": "convexset-halfspace-combination",
        "semantic_template": "qf-mod24-convexset-halfspace-combination/{\"bound\":22,\"lambda\":[21,23],\"left\":[0,0],\"normal\":[1,1],\"right\":[21,0]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261024-r07-convex-set-0071",
    "batch_id": "batch-20261013-advance-20261024-r07-convex-set-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "b8d159b34514cffe7f209406bcd23c8dbb8183386a10d5afcc61a2bddb819675",
      "family_id": "qf-mod24-convexset-halfspace-combination",
      "seed": 202610247071,
      "semantic_template": "qf-mod24-convexset-halfspace-combination/{\"bound\":22,\"lambda\":[21,23],\"left\":[0,0],\"normal\":[1,1],\"right\":[21,0]}"
    },
    "fields": {
      "title": "Linear Halfspace Closure at Qualified Endpoints",
      "stem": "Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=22, left=[0, 0], right=[21, 0], lambda=21/23∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.",
      "answer": {
        "kind": "expression",
        "value": "([1, 1], 22, [0, 0], [21, 0], 0, 21, 21/23, [42/23, 0], 42/23, Yes)",
        "explanation": "Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 22, [0, 0], [21, 0], 0, 21, 21/23, [42/23, 0], 42/23, Yes)."
      },
      "skills": [
        "Linear Halfspace Closure at Qualified Endpoints"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-halfspace-combination；資料={\"normal\":[1,1],\"bound\":22,\"left\":[0,0],\"right\":[21,0],\"lambda\":[21,23]}。請依 [normal,bound,left,right,left_dot,right_dot,lambda,point,point_dot,inside] 欄序回答。",
        "en": "Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=22, left=[0, 0], right=[21, 0], lambda=21/23∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "left endpoint dot term0 := mul(1, 0)",
      "after": "left endpoint dot term0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left endpoint dot term0，不以小數近似替代。",
          "en": "Recompute left endpoint dot term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "left endpoint dot term0"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "left endpoint dot term1 := mul(1, 0)",
      "after": "left endpoint dot term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left endpoint dot term1，不以小數近似替代。",
          "en": "Recompute left endpoint dot term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "left endpoint dot term1"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "left endpoint dot sum := sum(0, 0)",
      "after": "left endpoint dot sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left endpoint dot sum，不以小數近似替代。",
          "en": "Recompute left endpoint dot sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "left endpoint dot sum"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "right endpoint dot term0 := mul(1, 21)",
      "after": "right endpoint dot term0 = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 right endpoint dot term0，不以小數近似替代。",
          "en": "Recompute right endpoint dot term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "21"
        ],
        "output": "21",
        "label": "right endpoint dot term0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "right endpoint dot term1 := mul(1, 0)",
      "after": "right endpoint dot term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 right endpoint dot term1，不以小數近似替代。",
          "en": "Recompute right endpoint dot term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "right endpoint dot term1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "right endpoint dot sum := sum(21, 0)",
      "after": "right endpoint dot sum = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 right endpoint dot sum，不以小數近似替代。",
          "en": "Recompute right endpoint dot sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "0"
        ],
        "output": "21",
        "label": "right endpoint dot sum"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "normal=[1, 1];bound=22;left dot=0;right dot=21",
      "after": "both endpoints satisfy the closed halfspace inequality",
      "relation": "implies",
      "rule_id": "halfspace_endpoint_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核兩端點真的在半空間；不由要證的凸性倒推端點合格，法向量為零仍按同一不等式。",
        "en": "First qualify both endpoint memberships.Do not infer them from the desired convexity;the same inequality also handles a zero normal."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "halfspace point complementary weight := sub(1, 21/23)",
      "after": "halfspace point complementary weight = 2/23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 halfspace point complementary weight，不以小數近似替代。",
          "en": "Recompute halfspace point complementary weight with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "21/23"
        ],
        "output": "2/23",
        "label": "halfspace point complementary weight"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "halfspace point:raw lambda=[21, 23];lambda=21/23;one-minus-lambda=2/23",
      "after": "in_closed_unit_interval=是;lambda weights left",
      "relation": "implies",
      "rule_id": "two_point_direction_and_qualification",
      "conditions": [],
      "explanation": {
        "zh-Hant": "lambda乘左端，1−lambda乘右端；lambda=0選右，1選左。區間外仍可算仿射式，但不是凸組合，不截斷。",
        "en": "Lambda weights the left endpoint and1−lambda the right.Lambda0 selects right,1 selects left.Outside values still define the affine expression,but not a convex combination;do not clip."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "halfspace point:ordered vectors=[[0, 0], [21, 0]];aligned weights=[21/23, 2/23]",
      "after": "dimension=2;preserve duplicates and order",
      "relation": "implies",
      "rule_id": "ordered_vector_weight_pairing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留向量順序、重複項與係數配對；同一座標加完所有向量的貢獻。",
        "en": "Preserve vector order,duplicates and coefficient pairing.Sum every vector's contribution within the same coordinate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "halfspace point coordinate0 term0 := mul(21/23, 0)",
      "after": "halfspace point coordinate0 term0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 halfspace point coordinate0 term0，不以小數近似替代。",
          "en": "Recompute halfspace point coordinate0 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "21/23",
          "0"
        ],
        "output": "0",
        "label": "halfspace point coordinate0 term0"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "halfspace point coordinate0 term1 := mul(2/23, 21)",
      "after": "halfspace point coordinate0 term1 = 42/23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 halfspace point coordinate0 term1，不以小數近似替代。",
          "en": "Recompute halfspace point coordinate0 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2/23",
          "21"
        ],
        "output": "42/23",
        "label": "halfspace point coordinate0 term1"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "halfspace point coordinate0 sum := sum(0, 42/23)",
      "after": "halfspace point coordinate0 sum = 42/23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 halfspace point coordinate0 sum，不以小數近似替代。",
          "en": "Recompute halfspace point coordinate0 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "42/23"
        ],
        "output": "42/23",
        "label": "halfspace point coordinate0 sum"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "halfspace point coordinate1 term0 := mul(21/23, 0)",
      "after": "halfspace point coordinate1 term0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 halfspace point coordinate1 term0，不以小數近似替代。",
          "en": "Recompute halfspace point coordinate1 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "21/23",
          "0"
        ],
        "output": "0",
        "label": "halfspace point coordinate1 term0"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "halfspace point coordinate1 term1 := mul(2/23, 0)",
      "after": "halfspace point coordinate1 term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 halfspace point coordinate1 term1，不以小數近似替代。",
          "en": "Recompute halfspace point coordinate1 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2/23",
          "0"
        ],
        "output": "0",
        "label": "halfspace point coordinate1 term1"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "halfspace point coordinate1 sum := sum(0, 0)",
      "after": "halfspace point coordinate1 sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 halfspace point coordinate1 sum，不以小數近似替代。",
          "en": "Recompute halfspace point coordinate1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "halfspace point coordinate1 sum"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "combined point dot term0 := mul(1, 42/23)",
      "after": "combined point dot term0 = 42/23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 combined point dot term0，不以小數近似替代。",
          "en": "Recompute combined point dot term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "42/23"
        ],
        "output": "42/23",
        "label": "combined point dot term0"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "combined point dot term1 := mul(1, 0)",
      "after": "combined point dot term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 combined point dot term1，不以小數近似替代。",
          "en": "Recompute combined point dot term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "combined point dot term1"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "combined point dot sum := sum(42/23, 0)",
      "after": "combined point dot sum = 42/23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 combined point dot sum，不以小數近似替代。",
          "en": "Recompute combined point dot sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "42/23",
          "0"
        ],
        "output": "42/23",
        "label": "combined point dot sum"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "halfspace bound slack := sub(22, 42/23)",
      "after": "halfspace bound slack = 464/23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 halfspace bound slack，不以小數近似替代。",
          "en": "Recompute halfspace bound slack with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "22",
          "42/23"
        ],
        "output": "464/23",
        "label": "halfspace bound slack"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "normal dot z=42/23;endpoint dots=(0,21);lambda=21/23;bound=22",
      "after": "normal dot z=lambda*0+(1-lambda)*21<=22;inside=是",
      "relation": "implies",
      "rule_id": "halfspace_linear_universal_reason",
      "conditions": [],
      "explanation": {
        "zh-Hant": "內積線性使新點的值為兩端值的非負加權和，各≤bound且權重和1，故仍≤bound；這是完整代數理由，數值只是本題實例。",
        "en": "Linearity makes the new dot product a nonnegative unit-weighted sum of endpoint values.Each is≤bound,so the result is≤bound.This algebraic reason is universal;the numbers instantiate it."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        1,
        1
      ],
      22,
      [
        0,
        0
      ],
      [
        21,
        0
      ],
      0,
      21,
      "21/23",
      [
        "42/23",
        "0"
      ],
      "42/23",
      true
    ]
  },
  "conclusion": {
    "canonical_value": "([1, 1], 22, [0, 0], [21, 0], 0, 21, 21/23, [42/23, 0], 42/23, 是)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261013-advance-20261024-r07-convex-set-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261024-r07-convex-set-foundations/questions.jsonl",
      "batch_sha256": "51606fc9d1c6f335d18e50f6488d4d2b08c7343abef8be6c7d1fa99ccc3023b0",
      "original_line": 71,
      "original_line_sha256": "5fb9f6cd3d8728efff3f0d3670bc296fc1d5d34005a0591e27e04feb48edb907",
      "question_compact_sha256": "b8d159b34514cffe7f209406bcd23c8dbb8183386a10d5afcc61a2bddb819675",
      "english_row_compact_sha256": "57dff282a0fd739f788d3b0b7ba5ae60f14eb7d790f551e2f30107091a64e916",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261024-r07-convex-set-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261024-r07-convex-set-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-convex-set": {
        "sha256": "0d52be71dca290f856018d176bbb26dcd877c887031f690bd1d656db017ce02d",
        "english_sha256": "f81d0d2b42596c9a4b95349d75fc98970570af520e6edb9c1af988358ff84f0c",
        "original_path": "public/data/mko/convex-set.json",
        "english_path": "public/data/i18n/editions/mko-convex-set.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "18bbb89127e14ce3466c1a8730f1c3c6e22a8f59a9148433b2abc7f6ea274ee5",
    "original_candidate_sha256": "833454a26ac8da5fa0871f7ec72f095fc3a7428add8e1dabd4e3b3ddf3a35b17",
    "original_candidate_line_sha256": "833454a26ac8da5fa0871f7ec72f095fc3a7428add8e1dabd4e3b3ddf3a35b17"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-006-p22",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258",
      "question_first_witness_sha256": "a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 21,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "9b0cbea740f6b449777a8e241f1cff6e1b3c911ebdc596e27278e4cdd92cda1a",
      "arithmetic_steps": 17
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}