{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261025-r02-convex-function-0073",
  "pilot_number": "R0022-11898",
  "number": 11898,
  "family_id": "qf-mod25-convexfn-quadratic-universal-certificate",
  "title": {
    "zh-Hant": "第 11898 題：凸函數",
    "en": "Question 11898: A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261025-r02-convex-function-0073",
    "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-quadratic-universal-certificate；資料={\"coefficients\":[2,10,-2],\"interval\":[-24,25]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([2, 10, -2], [-24, 25], 是, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 是)",
      "explanation_zh": "收束波核對=([2, 10, -2], [-24, 25], 是, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 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；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": 3,
      "skills": [
        "凸函數"
      ]
    },
    "variant": {
      "family_id": "qf-mod25-convexfn-quadratic-universal-certificate",
      "seed": 202610252073,
      "parameters": {
        "coefficients": [
          2,
          10,
          -2
        ],
        "interval": [
          -24,
          25
        ],
        "domain": "Exact_univariate_quadratic_convexity",
        "operation": "convexfn-quadratic-universal-certificate",
        "semantic_template": "qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[2,10,-2],\"interval\":[-24,25]}"
      }
    },
    "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-0073",
    "batch_id": "batch-20261013-advance-20261025-r02-convex-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "3a94518de73a150a8027a24938ee2a0514e6a521f887ce49e5a312232a28f632",
      "family_id": "qf-mod25-convexfn-quadratic-universal-certificate",
      "seed": 202610252073,
      "semantic_template": "qf-mod25-convexfn-quadratic-universal-certificate/{\"coefficients\":[2,10,-2],\"interval\":[-24,25]}"
    },
    "fields": {
      "title": "A Nondegenerate-Interval Quadratic Jensen-Gap Certificate",
      "stem": "Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 10, -2], not ascending-power order. The declared interval [lower,upper]=[-24, 25] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. 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": "([2, 10, -2], [-24, 25], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)",
        "explanation": "Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 10, -2], [-24, 25], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)."
      },
      "skills": [
        "A Nondegenerate-Interval Quadratic Jensen-Gap Certificate"
      ],
      "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-quadratic-universal-certificate；資料={\"coefficients\":[2,10,-2],\"interval\":[-24,25]}。請依 [coefficients,interval,nondegenerate,leading_coefficient,jensen_gap_formula,convex] 欄序回答。",
        "en": "Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 10, -2], not ascending-power order. The declared interval [lower,upper]=[-24, 25] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. 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=[-24, 25]",
      "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 [-24,25];0<=lambda<=1;z=lambda*x+(1-lambda)*y",
      "after": "z--24=lambda*(x--24)+(1-lambda)*(y--24)>=0;25-z=lambda*(25-x)+(1-lambda)*(25-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=[2, 10, -2];order=[a,b,c]",
      "after": "f(t)=(2)*t^2+(10)*t+(-2)",
      "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)=(2)*t^2+(10)*t+(-2);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": "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=(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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "interval=[-24, 25];nondegenerate=是;leading=2",
      "after": "qualified universal quadratic certificate=是",
      "relation": "implies",
      "rule_id": "quadratic_global_sign_and_nondegenerate_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若a≥0，lambda(1−lambda)與平方皆非負，故所有Jensen差非負。此題證書另要求非退化；單點或反向紀錄的No不是函數在單點上非凸。",
        "en": "If a≥0,the nonnegative weight product and square make every Jensen gap nonnegative.This certificate separately requires a nondegenerate interval.No for a singleton or reversed record does not classify a singleton function as nonconvex."
      },
      "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,
        -2
      ],
      [
        -24,
        25
      ],
      true,
      2,
      "rhs-lhs=a*lambda*(1-lambda)*(x-y)^2",
      true
    ]
  },
  "conclusion": {
    "canonical_value": "([2, 10, -2], [-24, 25], 是, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 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-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": 73,
      "original_line_sha256": "3d5eb74f759ad0decc154e4458906350a1a449c9531c72876fc28b9b6a36a5c3",
      "question_compact_sha256": "3a94518de73a150a8027a24938ee2a0514e6a521f887ce49e5a312232a28f632",
      "english_row_compact_sha256": "0cd1f9da638c65cfff587dd21d9ac84e5e847886cbbaeca053a3f607b4fc9fa7",
      "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": "5ff0d8f5ef00061428442f757fba5516aa9dcb9d74f1d429a5f5af16c164cf97",
    "original_candidate_line_sha256": "5ff0d8f5ef00061428442f757fba5516aa9dcb9d74f1d429a5f5af16c164cf97"
  },
  "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": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "58b534645ac8be5746cfb59309b3d6062c494b456e8a9139e9b06a181076e0ae",
      "arithmetic_steps": 0
    },
    "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."
  }
}