{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261025-r03-convex-optimization-problem-0003",
  "pilot_number": "R0022-11978",
  "number": 11978,
  "family_id": "qf-mod25-convexopt-problem-qualification",
  "title": {
    "zh-Hant": "第 11978 題：凸最佳化問題",
    "en": "Question 11978: Separate a Quadratic Objective Certificate from Feasible-Interval Qualification"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261025-r03-convex-optimization-problem-0003",
    "batch_id": "batch-20261013-advance-20261025-r03-convex-optimization-problem-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-convex-optimization-problem"
    ],
    "title_zh": "凸最佳化問題",
    "stem_zh": "凸目標資格與凸可行域資格分開。操作=convexopt-problem-qualification；資料={\"coefficients\":[2,-10,-1],\"interval\":[-4,5]}。請依 [coefficients,interval,objective_convex,feasible_set_convex,qualified,scope] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([2, -10, -1], [-4, 5], 是, 是, 是, qualification_only_no_solution_claim)",
      "explanation_zh": "收束波核對=([2, -10, -1], [-4, 5], 是, 是, 是, qualification_only_no_solution_claim)；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；只核可行性求值或兩候選比較；不宣稱通用求解器全域最優存在唯一KKT或收斂。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "凸最佳化問題"
      ]
    },
    "variant": {
      "family_id": "qf-mod25-convexopt-problem-qualification",
      "seed": 202610253003,
      "parameters": {
        "coefficients": [
          2,
          -10,
          -1
        ],
        "interval": [
          -4,
          5
        ],
        "domain": "Exact_interval_convex_optimization",
        "operation": "convexopt-problem-qualification",
        "semantic_template": "qf-mod25-convexopt-problem-qualification/{\"coefficients\":[2,-10,-1],\"interval\":[-4,5]}"
      }
    },
    "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、BVP適定性、全域最佳解、Markov長期性、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"
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  "original_english_edition": {
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    "question_id": "q-20261013-advance-20261025-r03-convex-optimization-problem-0003",
    "batch_id": "batch-20261013-advance-20261025-r03-convex-optimization-problem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "625ef78c3da23538df485165a1e9ff687b54bd289db8f38c266ace65178b08a1",
      "family_id": "qf-mod25-convexopt-problem-qualification",
      "seed": 202610253003,
      "semantic_template": "qf-mod25-convexopt-problem-qualification/{\"coefficients\":[2,-10,-1],\"interval\":[-4,5]}"
    },
    "fields": {
      "title": "Separate a Quadratic Objective Certificate from Feasible-Interval Qualification",
      "stem": "Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -10, -1], not ascending-power order. The declared interval [lower,upper]=[-4, 5] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.",
      "answer": {
        "kind": "expression",
        "value": "([2, -10, -1], [-4, 5], Yes, Yes, Yes, qualification_only_no_solution_claim)",
        "explanation": "Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, -10, -1], [-4, 5], Yes, Yes, Yes, qualification_only_no_solution_claim)."
      },
      "skills": [
        "Separate a Quadratic Objective Certificate from Feasible-Interval Qualification"
      ],
      "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": "凸目標資格與凸可行域資格分開。操作=convexopt-problem-qualification；資料={\"coefficients\":[2,-10,-1],\"interval\":[-4,5]}。請依 [coefficients,interval,objective_convex,feasible_set_convex,qualified,scope] 欄序回答。",
        "en": "Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -10, -1], not ascending-power order. The declared interval [lower,upper]=[-4, 5] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "declared interval record=[-4, 5]",
      "after": "ordered_nonempty=是;nondegenerate=是",
      "relation": "implies",
      "rule_id": "interval_certificate_qualification_not_empty_set_nonconvexity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此非空區間協定用lower≤upper，非退化證書另需lower<upper。反向紀錄只是資格不合，不是把抽象空集判成非凸；單點區間本身仍凸。",
        "en": "This nonempty-interval protocol uses lower≤upper;the nondegenerate certificate additionally needs lower<upper.A reversed record is unqualified,not a declaration that the abstract empty set is nonconvex.A singleton interval is 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
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "univariate quadratic coefficients=[2, -10, -1];order=[a,b,c]",
      "after": "f(t)=(2)*t^2+(-10)*t+(-1)",
      "relation": "implies",
      "rule_id": "quadratic_descending_coefficient_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這裡按二次、一次、常數降冪排列，不是邊界限制多項式的常數在前次序；零首項保留。",
        "en": "These are quadratic,linear,constant coefficients in descending order,not the constant-first boundary-polynomial order.Retain zero leading coefficients."
      },
      "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": "f(t)=(2)*t^2+(-10)*t+(-1);z=lambda*x+(1-lambda)*y",
      "after": "rhs-lhs=(2)*(lambda*x^2+(1-lambda)*y^2-(lambda*x+(1-lambda)*y)^2)",
      "relation": "implies",
      "rule_id": "jensen_linear_and_constant_terms_cancel",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先展開Jensen兩邊，因權重和1，一次與常數項完全消去，只剩二次部分。",
        "en": "Expand both Jensen sides.The unit weight sum cancels the linear and constant terms exactly,leaving only the quadratic part."
      },
      "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": "lambda*x^2+(1-lambda)*y^2-(lambda*x+(1-lambda)*y)^2",
      "after": "lambda*(1-lambda)*(x-y)^2;full gap=(2)*lambda*(1-lambda)*(x-y)^2",
      "relation": "implies",
      "rule_id": "jensen_gap_full_symbolic_factorization",
      "conditions": [],
      "explanation": {
        "zh-Hant": "展開平方再收集x²、xy、y²，分別得到lambda(1−lambda)、−2lambda(1−lambda)、lambda(1−lambda)，故是(x−y)²倍數，覆蓋所有實數端點。",
        "en": "Expand the square and collect x²,xy,y².The coefficients are lambda(1−lambda),−2lambda(1−lambda),lambda(1−lambda),yielding the(x−y)^2 factor for all real endpoints."
      },
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "arbitrary x,y in [-4,5];0<=lambda<=1;z=lambda*x+(1-lambda)*y",
      "after": "z--4=lambda*(x--4)+(1-lambda)*(y--4)>=0;5-z=lambda*(5-x)+(1-lambda)*(5-y)>=0",
      "relation": "implies",
      "rule_id": "interval_all_segments_slack_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個差額均是非負和，故lower≤z≤upper，對每對端點及每個lambda都成立，不用抽樣；端點和單點情況包含在內。",
        "en": "Both slacks are nonnegative sums,hence lower≤z≤upper for every endpoint pair and every permitted lambda.No sampling is used;endpoints and singleton cases are included."
      },
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "global quadratic leading coefficient=2;interval record=[-4, 5]",
      "after": "global objective certificate=是;feasible interval certificate=是;qualified=是",
      "relation": "implies",
      "rule_id": "optimization_two_qualifications_not_solver",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標旗標是全實數二次函數的a≥0準則：a<0取0、1與半權重便有負差。域旗標另核非空有序區間。兩者合格只說此凸優化問題資格，不證有解、唯一、KKT或求解器收斂；不合全域準則也不判所有限制單點問題。",
        "en": "The objective flag is the global quadratic a≥0 criterion:for a<0,endpoints0,1 and half weights give a negative gap.The domain flag separately qualifies a nonempty ordered interval.Joint qualification proves no solution,uniqueness,KKT or solver convergence;failure of the global criterion does not classify every restricted singleton problem."
      },
      "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": [
      [
        2,
        -10,
        -1
      ],
      [
        -4,
        5
      ],
      true,
      true,
      true,
      "qualification_only_no_solution_claim"
    ]
  },
  "conclusion": {
    "canonical_value": "([2, -10, -1], [-4, 5], 是, 是, 是, qualification_only_no_solution_claim)",
    "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."
  },
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      "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."
    },
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}