{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261024-r07-convex-set-0083",
  "pilot_number": "R0021-11133",
  "number": 11133,
  "family_id": "qf-mod24-convexset-finite-midpoint-counterexample",
  "title": {
    "zh-Hant": "第 11133 題：凸集合證書與反例",
    "en": "Question 11133: A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261024-r07-convex-set-0083",
    "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-finite-midpoint-counterexample；資料={\"points\":[[0,0],[16,0],[0,16]],\"left\":[16,0],\"right\":[0,16]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[0, 0], [16, 0], [0, 16]], [16, 0], [0, 16], [8, 8], 否, 是)",
      "explanation_zh": "依賴結構核對=([[0, 0], [16, 0], [0, 16]], [16, 0], [0, 16], [8, 8], 否, 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": 1,
      "skills": [
        "凸集合證書與反例"
      ]
    },
    "variant": {
      "family_id": "qf-mod24-convexset-finite-midpoint-counterexample",
      "seed": 202610247083,
      "parameters": {
        "points": [
          [
            0,
            0
          ],
          [
            16,
            0
          ],
          [
            0,
            16
          ]
        ],
        "left": [
          16,
          0
        ],
        "right": [
          0,
          16
        ],
        "domain": "Exact_algebraic_convex_set_certificates",
        "operation": "convexset-finite-midpoint-counterexample",
        "semantic_template": "qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[16,0],\"points\":[[0,0],[0,16],[16,0]],\"right\":[0,16]}"
      }
    },
    "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-0083",
    "batch_id": "batch-20261013-advance-20261024-r07-convex-set-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "8a596766bc4875b20ff6807294e6d00442c4cfe67ab95820832015794f9a4d7e",
      "family_id": "qf-mod24-convexset-finite-midpoint-counterexample",
      "seed": 202610247083,
      "semantic_template": "qf-mod24-convexset-finite-midpoint-counterexample/{\"left\":[16,0],\"points\":[[0,0],[0,16],[16,0]],\"right\":[0,16]}"
    },
    "fields": {
      "title": "A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It",
      "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. Distinct finite points=[[0, 0], [16, 0], [0, 16]] retain input order; endpoints left=[16, 0], right=[0, 16] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.",
      "answer": {
        "kind": "expression",
        "value": "([[0, 0], [16, 0], [0, 16]], [16, 0], [0, 16], [8, 8], No, Yes)",
        "explanation": "An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [16, 0], [0, 16]], [16, 0], [0, 16], [8, 8], No, Yes)."
      },
      "skills": [
        "A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It"
      ],
      "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-finite-midpoint-counterexample；資料={\"points\":[[0,0],[16,0],[0,16]],\"left\":[16,0],\"right\":[0,16]}。請依 [points,left,right,midpoint,midpoint_present,nonconvex_witness] 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. Distinct finite points=[[0, 0], [16, 0], [0, 16]] retain input order; endpoints left=[16, 0], right=[0, 16] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "ordered finite point set=[[0, 0], [16, 0], [0, 16]];left=[16, 0];right=[0, 16]",
      "after": "both endpoints belong;choose lambda=1/2",
      "relation": "implies",
      "rule_id": "midpoint_member_premises",
      "conditions": [],
      "explanation": {
        "zh-Hant": "反例需兩端點本來都屬集合；固定合法lambda=1/2算精確中點，不靠圖形目測。",
        "en": "A counterexample requires both endpoints to be members.Choose the permitted lambda1/2 and compute the exact midpoint,not a visual estimate."
      },
      "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": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "midpoint coordinate0 numerator := sum(16, 0)",
      "after": "midpoint coordinate0 numerator = 16",
      "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 midpoint coordinate0 numerator，不以小數近似替代。",
          "en": "Recompute midpoint coordinate0 numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "16",
          "0"
        ],
        "output": "16",
        "label": "midpoint coordinate0 numerator"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "midpoint coordinate0 := div(16, 2)",
      "after": "midpoint coordinate0 = 8",
      "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 midpoint coordinate0，不以小數近似替代。",
          "en": "Recompute midpoint coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "16",
          "2"
        ],
        "output": "8",
        "label": "midpoint coordinate0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "midpoint coordinate1 numerator := sum(0, 16)",
      "after": "midpoint coordinate1 numerator = 16",
      "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 midpoint coordinate1 numerator，不以小數近似替代。",
          "en": "Recompute midpoint coordinate1 numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "16"
        ],
        "output": "16",
        "label": "midpoint coordinate1 numerator"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "midpoint coordinate1 := div(16, 2)",
      "after": "midpoint coordinate1 = 8",
      "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 midpoint coordinate1，不以小數近似替代。",
          "en": "Recompute midpoint coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "16",
          "2"
        ],
        "output": "8",
        "label": "midpoint coordinate1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "candidate point index=0;point=[0, 0];midpoint=[8, 8]",
      "after": "all coordinates equal=否",
      "relation": "implies",
      "rule_id": "midpoint_complete_set_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一比集合所有點，全部座標同值才算同一點，不能只比一個座標。",
        "en": "Compare every set member;all coordinates must match,not merely one."
      },
      "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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "candidate point index=1;point=[16, 0];midpoint=[8, 8]",
      "after": "all coordinates equal=否",
      "relation": "implies",
      "rule_id": "midpoint_complete_set_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一比集合所有點，全部座標同值才算同一點，不能只比一個座標。",
        "en": "Compare every set member;all coordinates must match,not merely one."
      },
      "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": "candidate point index=2;point=[0, 16];midpoint=[8, 8]",
      "after": "all coordinates equal=否",
      "relation": "implies",
      "rule_id": "midpoint_complete_set_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一比集合所有點，全部座標同值才算同一點，不能只比一個座標。",
        "en": "Compare every set member;all coordinates must match,not merely one."
      },
      "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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "all point comparisons=[否, 否, 否]",
      "after": "midpoint present=否;this nonconvex witness=是",
      "relation": "implies",
      "rule_id": "midpoint_failed_counterexample_not_convexity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "中點缺席給一個精確反例即可否定凸性；中點存在只表示這次反例失敗，不證集合凸。",
        "en": "An absent midpoint is one exact counterexample refuting convexity.A present midpoint only defeats this counterexample;it does not prove the set convex."
      },
      "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": [
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          0,
          0
        ],
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          0
        ],
        [
          0,
          16
        ]
      ],
      [
        16,
        0
      ],
      [
        0,
        16
      ],
      [
        "8",
        "8"
      ],
      false,
      true
    ]
  },
  "conclusion": {
    "canonical_value": "([[0, 0], [16, 0], [0, 16]], [16, 0], [0, 16], [8, 8], 否, 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": {
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      "question_compact_sha256": "8a596766bc4875b20ff6807294e6d00442c4cfe67ab95820832015794f9a4d7e",
      "english_row_compact_sha256": "df59a8f86b858749082989a1ff3ec7d4417ec2244e5bf0aa1b0504ee6b8c747d",
      "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"
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    "mko_bindings": {
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        "original_path": "public/data/mko/convex-set.json",
        "english_path": "public/data/i18n/editions/mko-convex-set.en.json"
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    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "18bbb89127e14ce3466c1a8730f1c3c6e22a8f59a9148433b2abc7f6ea274ee5",
    "original_candidate_sha256": "24e80de6ad30f51ecdf605513b849f617e5026987147e458f085885091c340b7",
    "original_candidate_line_sha256": "24e80de6ad30f51ecdf605513b849f617e5026987147e458f085885091c340b7"
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  "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": 9,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "57cf5777ad2806a34958d9b77984ce3fd8086a7931f9d20be62c1637ffdcf388",
      "arithmetic_steps": 4
    },
    "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."
  }
}