{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261024-r05-finite-convex-combination-0061",
  "pilot_number": "R0021-11036",
  "number": 11036,
  "family_id": "qf-mod24-convcomb-two-point",
  "title": {
    "zh-Hant": "第 11036 題：有限凸組合",
    "en": "Question 11036: Audit lambda without Clipping an Affine Extension"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261024-r05-finite-convex-combination-0061",
    "batch_id": "batch-20261013-advance-20261024-r05-finite-convex-combination-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-convex-combination"
    ],
    "title_zh": "有限凸組合",
    "stem_zh": "有限非空向量列與精確有理權重。操作=convcomb-two-point；資料={\"left\":[-2,0],\"right\":[1,3],\"lambda\":[11,13]}。請依 [left,right,lambda,in_unit_interval,point,convex_combination] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([-2, 0], [1, 3], 11/13, 是, [-20/13, 6/13], 是)",
      "explanation_zh": "依賴結構核對=([-2, 0], [1, 3], 11/13, 是, [-20/13, 6/13], 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；凸權重允許零但不得為負且總和必須精確為1；不由凸組合主張座標唯一性。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "有限凸組合"
      ]
    },
    "variant": {
      "family_id": "qf-mod24-convcomb-two-point",
      "seed": 202610245061,
      "parameters": {
        "left": [
          -2,
          0
        ],
        "right": [
          1,
          3
        ],
        "lambda": [
          11,
          13
        ],
        "domain": "Exact_finite_rational_convex_combinations",
        "operation": "convcomb-two-point",
        "semantic_template": "qf-mod24-convcomb-two-point/{\"lambda\":[11,13],\"left\":[-2,0],\"right\":[1,3]}"
      }
    },
    "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-r05-finite-convex-combination-0061",
    "batch_id": "batch-20261013-advance-20261024-r05-finite-convex-combination-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "81f5f00f2b3c8642e139cdf002ee98c9312bf24ae2db6db1ef13e29f2f31cced",
      "family_id": "qf-mod24-convcomb-two-point",
      "seed": 202610245061,
      "semantic_template": "qf-mod24-convcomb-two-point/{\"lambda\":[11,13],\"left\":[-2,0],\"right\":[1,3]}"
    },
    "fields": {
      "title": "Audit lambda without Clipping an Affine Extension",
      "stem": "Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[-2, 0], right=[1, 3], lambda=11/13. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.",
      "answer": {
        "kind": "expression",
        "value": "([-2, 0], [1, 3], 11/13, Yes, [-20/13, 6/13], Yes)",
        "explanation": "Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 0], [1, 3], 11/13, Yes, [-20/13, 6/13], Yes)."
      },
      "skills": [
        "Audit lambda without Clipping an Affine Extension"
      ],
      "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": "有限非空向量列與精確有理權重。操作=convcomb-two-point；資料={\"left\":[-2,0],\"right\":[1,3],\"lambda\":[11,13]}。請依 [left,right,lambda,in_unit_interval,point,convex_combination] 欄序回答。",
        "en": "Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[-2, 0], right=[1, 3], lambda=11/13. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "two-point complementary weight := sub(1, 11/13)",
      "after": "two-point complementary weight = 2/13",
      "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 two-point complementary weight，不以小數近似替代。",
          "en": "Recompute two-point complementary weight with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "11/13"
        ],
        "output": "2/13",
        "label": "two-point complementary weight"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "two-point:raw lambda=[11, 13];lambda=11/13;one-minus-lambda=2/13",
      "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": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "two-point:ordered vectors=[[-2, 0], [1, 3]];aligned weights=[11/13, 2/13]",
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "two-point coordinate0 term0 := mul(11/13, -2)",
      "after": "two-point coordinate0 term0 = -22/13",
      "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 two-point coordinate0 term0，不以小數近似替代。",
          "en": "Recompute two-point coordinate0 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11/13",
          "-2"
        ],
        "output": "-22/13",
        "label": "two-point coordinate0 term0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "two-point coordinate0 term1 := mul(2/13, 1)",
      "after": "two-point coordinate0 term1 = 2/13",
      "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 two-point coordinate0 term1，不以小數近似替代。",
          "en": "Recompute two-point coordinate0 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2/13",
          "1"
        ],
        "output": "2/13",
        "label": "two-point coordinate0 term1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "two-point coordinate0 sum := sum(-22/13, 2/13)",
      "after": "two-point coordinate0 sum = -20/13",
      "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 two-point coordinate0 sum，不以小數近似替代。",
          "en": "Recompute two-point coordinate0 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-22/13",
          "2/13"
        ],
        "output": "-20/13",
        "label": "two-point coordinate0 sum"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "two-point coordinate1 term0 := mul(11/13, 0)",
      "after": "two-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 two-point coordinate1 term0，不以小數近似替代。",
          "en": "Recompute two-point coordinate1 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11/13",
          "0"
        ],
        "output": "0",
        "label": "two-point coordinate1 term0"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "two-point coordinate1 term1 := mul(2/13, 3)",
      "after": "two-point coordinate1 term1 = 6/13",
      "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 two-point coordinate1 term1，不以小數近似替代。",
          "en": "Recompute two-point coordinate1 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2/13",
          "3"
        ],
        "output": "6/13",
        "label": "two-point coordinate1 term1"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "two-point coordinate1 sum := sum(0, 6/13)",
      "after": "two-point coordinate1 sum = 6/13",
      "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 two-point coordinate1 sum，不以小數近似替代。",
          "en": "Recompute two-point coordinate1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "6/13"
        ],
        "output": "6/13",
        "label": "two-point coordinate1 sum"
      }
    }
  ],
  "math_model": {
    "result": [
      [
        -2,
        0
      ],
      [
        1,
        3
      ],
      "11/13",
      true,
      [
        "-20/13",
        "6/13"
      ],
      true
    ]
  },
  "conclusion": {
    "canonical_value": "([-2, 0], [1, 3], 11/13, 是, [-20/13, 6/13], 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-r05-finite-convex-combination-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261024-r05-finite-convex-combination-foundations/questions.jsonl",
      "batch_sha256": "34dbfab01e21f4256c93e3fddb4b9afa9eeb164d0ca88091cb39e9fb74a5a292",
      "original_line": 61,
      "original_line_sha256": "b22c274ee4690a19b3b8f452120a53732a664e92692fe79679323040f0286001",
      "question_compact_sha256": "81f5f00f2b3c8642e139cdf002ee98c9312bf24ae2db6db1ef13e29f2f31cced",
      "english_row_compact_sha256": "17aa6e1b0710b5b0af8485f64d091ebc7dc11cf253410f075983becc0ed125eb",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261024-r05-finite-convex-combination-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261024-r05-finite-convex-combination-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-finite-convex-combination": {
        "sha256": "84206c94ef6c80e08ea032783fce4ef3d1e4b15b55088556463d360b07c78d80",
        "english_sha256": "cea2a4c5c0d854cbc55dac8f42fc0521d0afbfa6d70790f79fe1f6b1326a7214",
        "original_path": "public/data/mko/finite-convex-combination.json",
        "english_path": "public/data/i18n/editions/mko-finite-convex-combination.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "18bbb89127e14ce3466c1a8730f1c3c6e22a8f59a9148433b2abc7f6ea274ee5",
    "original_candidate_sha256": "ee2a3c7c60858d1c52c2ab267cef14b6bc1c02b33b36d6d6a8903fa5d600dbe4",
    "original_candidate_line_sha256": "ee2a3c7c60858d1c52c2ab267cef14b6bc1c02b33b36d6d6a8903fa5d600dbe4"
  },
  "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": "96153cd853d62ab747dfeb55cc357e1771c9c8fd1b7bd85293b639ead37c62b8",
      "arithmetic_steps": 7
    },
    "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."
  }
}