{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261025-r02-convex-function-0082",
  "pilot_number": "R0022-11857",
  "number": 11857,
  "family_id": "qf-mod25-convexfn-jensen-instance-audit",
  "title": {
    "zh-Hant": "第 11857 題：凸函數",
    "en": "Question 11857: One Exact Jensen Instance: a Counterexample or Only an Example"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261025-r02-convex-function-0082",
    "batch_id": "batch-20261013-advance-20261025-r02-convex-function-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-convex-function"
    ],
    "title_zh": "凸函數",
    "stem_zh": "函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[-1,-6,3],\"interval\":[-5,5],\"x\":[-1,1],\"y\":[3,1],\"lambda\":[7,9]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([-1, -6, 3], [-5, 5], -1, 3, 7/9, -1/9, 296/81, 8/9, -224/81, 否, nonconvexity_witness)",
      "explanation_zh": "收束波核對=([-1, -6, 3], [-5, 5], -1, 3, 7/9, -1/9, 296/81, 8/9, -224/81, 否, nonconvexity_witness)；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；有限Jensen通過只是一個instance；正證書必須覆蓋所有端點與所有lambda。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "凸函數"
      ]
    },
    "variant": {
      "family_id": "qf-mod25-convexfn-jensen-instance-audit",
      "seed": 202610252082,
      "parameters": {
        "coefficients": [
          -1,
          -6,
          3
        ],
        "interval": [
          -5,
          5
        ],
        "x": [
          -1,
          1
        ],
        "y": [
          3,
          1
        ],
        "lambda": [
          7,
          9
        ],
        "domain": "Exact_univariate_quadratic_convexity",
        "operation": "convexfn-jensen-instance-audit",
        "semantic_template": "qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[-1,-6,3],\"interval\":[-5,5],\"lambda\":[7,9],\"x\":[-1,1],\"y\":[3,1]}"
      }
    },
    "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"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261025-r02-convex-function-0082",
    "batch_id": "batch-20261013-advance-20261025-r02-convex-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "0668c3366e4497420ef6c9682921affbac7726ce82ee667ce5848d067dcc61be",
      "family_id": "qf-mod25-convexfn-jensen-instance-audit",
      "seed": 202610252082,
      "semantic_template": "qf-mod25-convexfn-jensen-instance-audit/{\"coefficients\":[-1,-6,3],\"interval\":[-5,5],\"lambda\":[7,9],\"x\":[-1,1],\"y\":[3,1]}"
    },
    "fields": {
      "title": "One Exact Jensen Instance: a Counterexample or Only an Example",
      "stem": "Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, -6, 3], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-1/1, y=3/1, lambda=7/9 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.",
      "answer": {
        "kind": "expression",
        "value": "([-1, -6, 3], [-5, 5], -1, 3, 7/9, -1/9, 296/81, 8/9, -224/81, No, nonconvexity_witness)",
        "explanation": "Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -6, 3], [-5, 5], -1, 3, 7/9, -1/9, 296/81, 8/9, -224/81, No, nonconvexity_witness)."
      },
      "skills": [
        "One Exact Jensen Instance: a Counterexample or Only an Example"
      ],
      "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": "函數必須配合明示凸定義域。操作=convexfn-jensen-instance-audit；資料={\"coefficients\":[-1,-6,3],\"interval\":[-5,5],\"x\":[-1,1],\"y\":[3,1],\"lambda\":[7,9]}。請依 [coefficients,interval,x,y,lambda,point,lhs,rhs,gap,passes,scope] 欄序回答。",
        "en": "Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, -6, 3], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-1/1, y=3/1, lambda=7/9 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "declared interval record=[-5, 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": "arbitrary x,y in [-5,5];0<=lambda<=1;z=lambda*x+(1-lambda)*y",
      "after": "z--5=lambda*(x--5)+(1-lambda)*(y--5)>=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": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "univariate quadratic coefficients=[-1, -6, 3];order=[a,b,c]",
      "after": "f(t)=(-1)*t^2+(-6)*t+(3)",
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "f(t)=(-1)*t^2+(-6)*t+(3);z=lambda*x+(1-lambda)*y",
      "after": "rhs-lhs=(-1)*(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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "lambda*x^2+(1-lambda)*y^2-(lambda*x+(1-lambda)*y)^2",
      "after": "lambda*(1-lambda)*(x-y)^2;full gap=(-1)*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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Jensen complementary weight := sub(1, 7/9)",
      "after": "Jensen complementary weight = 2/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen complementary weight，不以小數近似替代。",
          "en": "Recompute Jensen complementary weight with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "7/9"
        ],
        "output": "2/9",
        "label": "Jensen complementary weight"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Jensen weighted x := mul(7/9, -1)",
      "after": "Jensen weighted x = -7/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen weighted x，不以小數近似替代。",
          "en": "Recompute Jensen weighted x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7/9",
          "-1"
        ],
        "output": "-7/9",
        "label": "Jensen weighted x"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Jensen weighted y := mul(2/9, 3)",
      "after": "Jensen weighted y = 2/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen weighted y，不以小數近似替代。",
          "en": "Recompute Jensen weighted y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2/9",
          "3"
        ],
        "output": "2/3",
        "label": "Jensen weighted y"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Jensen weighted input := sum(-7/9, 2/3)",
      "after": "Jensen weighted input = -1/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen weighted input，不以小數近似替代。",
          "en": "Recompute Jensen weighted input with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-7/9",
          "2/3"
        ],
        "output": "-1/9",
        "label": "Jensen weighted input"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Jensen lhs point squared := mul(-1/9, -1/9)",
      "after": "Jensen lhs point squared = 1/81",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen lhs point squared，不以小數近似替代。",
          "en": "Recompute Jensen lhs point squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1/9",
          "-1/9"
        ],
        "output": "1/81",
        "label": "Jensen lhs point squared"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Jensen lhs quadratic term := mul(-1, 1/81)",
      "after": "Jensen lhs quadratic term = -1/81",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen lhs quadratic term，不以小數近似替代。",
          "en": "Recompute Jensen lhs quadratic term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1/81"
        ],
        "output": "-1/81",
        "label": "Jensen lhs quadratic term"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Jensen lhs linear term := mul(-6, -1/9)",
      "after": "Jensen lhs linear term = 2/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen lhs linear term，不以小數近似替代。",
          "en": "Recompute Jensen lhs linear term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-6",
          "-1/9"
        ],
        "output": "2/3",
        "label": "Jensen lhs linear term"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Jensen lhs polynomial value := sum(-1/81, 2/3, 3)",
      "after": "Jensen lhs polynomial value = 296/81",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen lhs polynomial value，不以小數近似替代。",
          "en": "Recompute Jensen lhs polynomial value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1/81",
          "2/3",
          "3"
        ],
        "output": "296/81",
        "label": "Jensen lhs polynomial value"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Jensen left endpoint point squared := mul(-1, -1)",
      "after": "Jensen left endpoint point squared = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen left endpoint point squared，不以小數近似替代。",
          "en": "Recompute Jensen left endpoint point squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "1",
        "label": "Jensen left endpoint point squared"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Jensen left endpoint quadratic term := mul(-1, 1)",
      "after": "Jensen left endpoint quadratic term = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen left endpoint quadratic term，不以小數近似替代。",
          "en": "Recompute Jensen left endpoint quadratic term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-1",
        "label": "Jensen left endpoint quadratic term"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Jensen left endpoint linear term := mul(-6, -1)",
      "after": "Jensen left endpoint linear term = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen left endpoint linear term，不以小數近似替代。",
          "en": "Recompute Jensen left endpoint linear term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-6",
          "-1"
        ],
        "output": "6",
        "label": "Jensen left endpoint linear term"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Jensen left endpoint polynomial value := sum(-1, 6, 3)",
      "after": "Jensen left endpoint polynomial value = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen left endpoint polynomial value，不以小數近似替代。",
          "en": "Recompute Jensen left endpoint polynomial value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1",
          "6",
          "3"
        ],
        "output": "8",
        "label": "Jensen left endpoint polynomial value"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Jensen right endpoint point squared := mul(3, 3)",
      "after": "Jensen right endpoint point squared = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen right endpoint point squared，不以小數近似替代。",
          "en": "Recompute Jensen right endpoint point squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "3"
        ],
        "output": "9",
        "label": "Jensen right endpoint point squared"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Jensen right endpoint quadratic term := mul(-1, 9)",
      "after": "Jensen right endpoint quadratic term = -9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen right endpoint quadratic term，不以小數近似替代。",
          "en": "Recompute Jensen right endpoint quadratic term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "9"
        ],
        "output": "-9",
        "label": "Jensen right endpoint quadratic term"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Jensen right endpoint linear term := mul(-6, 3)",
      "after": "Jensen right endpoint linear term = -18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen right endpoint linear term，不以小數近似替代。",
          "en": "Recompute Jensen right endpoint linear term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-6",
          "3"
        ],
        "output": "-18",
        "label": "Jensen right endpoint linear term"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Jensen right endpoint polynomial value := sum(-9, -18, 3)",
      "after": "Jensen right endpoint polynomial value = -24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen right endpoint polynomial value，不以小數近似替代。",
          "en": "Recompute Jensen right endpoint polynomial value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-9",
          "-18",
          "3"
        ],
        "output": "-24",
        "label": "Jensen right endpoint polynomial value"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Jensen weighted left value := mul(7/9, 8)",
      "after": "Jensen weighted left value = 56/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen weighted left value，不以小數近似替代。",
          "en": "Recompute Jensen weighted left value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7/9",
          "8"
        ],
        "output": "56/9",
        "label": "Jensen weighted left value"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Jensen weighted right value := mul(2/9, -24)",
      "after": "Jensen weighted right value = -16/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen weighted right value，不以小數近似替代。",
          "en": "Recompute Jensen weighted right value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2/9",
          "-24"
        ],
        "output": "-16/3",
        "label": "Jensen weighted right value"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Jensen rhs := sum(56/9, -16/3)",
      "after": "Jensen rhs = 8/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen rhs，不以小數近似替代。",
          "en": "Recompute Jensen rhs with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "56/9",
          "-16/3"
        ],
        "output": "8/9",
        "label": "Jensen rhs"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Jensen rhs minus lhs := sub(8/9, 296/81)",
      "after": "Jensen rhs minus lhs = -224/81",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。",
        "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Jensen rhs minus lhs，不以小數近似替代。",
          "en": "Recompute Jensen rhs minus lhs with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "8/9",
          "296/81"
        ],
        "output": "-224/81",
        "label": "Jensen rhs minus lhs"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "qualified x=-1;y=3;lambda=7/9;lhs=296/81;rhs=8/9;gap=-224/81",
      "after": "Jensen instance passes=否;scope=nonconvexity_witness",
      "relation": "implies",
      "rule_id": "one_Jensen_instance_not_universal_positive_proof",
      "conditions": [],
      "explanation": {
        "zh-Hant": "負差給一個合法端點與權重的反例即足以否證；非負只通過這一例，尤其lambda=0或x=y可使非凸二次式也通過。",
        "en": "A negative gap at qualified endpoints and weight is a counterexample.Nonnegative proves only this instance;lambda0 or x=y can pass even for a nonconvex quadratic."
      },
      "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,
        -6,
        3
      ],
      [
        -5,
        5
      ],
      "-1",
      "3",
      "7/9",
      "-1/9",
      "296/81",
      "8/9",
      "-224/81",
      false,
      "nonconvexity_witness"
    ]
  },
  "conclusion": {
    "canonical_value": "([-1, -6, 3], [-5, 5], -1, 3, 7/9, -1/9, 296/81, 8/9, -224/81, 否, nonconvexity_witness)",
    "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-20261025-r02-convex-function-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261025-r02-convex-function-foundations/questions.jsonl",
      "batch_sha256": "91fa2b23bb979ddb7e7ef2a72c68b25b9f5d01b1197b550d3401ef5b32867613",
      "original_line": 82,
      "original_line_sha256": "ddfc12d9950d6cd73b3b64f4b9f2a6f08bd7a20d9beb624ec55903cbe3135a83",
      "question_compact_sha256": "0668c3366e4497420ef6c9682921affbac7726ce82ee667ce5848d067dcc61be",
      "english_row_compact_sha256": "6254031d8a4298f57d90f0ac3660189821b24a70aaaa262313c8221def46ca25",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261025-r02-convex-function-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261025-r02-convex-function-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-convex-function": {
        "sha256": "d5af460bbe41c8e967949f9398bde368d76471cd2b4a24e4f49f14ad90a05fed",
        "english_sha256": "147a00f6af20512702925d6854df7d6be01eeefa9aca4e6b2e1b268412020da4",
        "original_path": "public/data/mko/convex-function.json",
        "english_path": "public/data/i18n/editions/mko-convex-function.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "844102be97423689e3a5e0c418b86b56773cf737eab263267081710eff80cbea",
    "original_candidate_sha256": "040ab4f8ef128ddcb4c6b2788c526ab30cf4e8249e9dcc8c55fbf8d85d5e196d",
    "original_candidate_line_sha256": "040ab4f8ef128ddcb4c6b2788c526ab30cf4e8249e9dcc8c55fbf8d85d5e196d"
  },
  "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-p23",
    "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": "94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8fac",
      "question_first_witness_sha256": "0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01b"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 26,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "142c4ac6837b5141b6417bcf2dea60da9b25775c8a7c8d9572a4752d36cf89ec",
      "arithmetic_steps": 20
    },
    "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."
  }
}