{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-pulse-20261119-r07-0036",
  "pilot_number": "P1000-0861",
  "number": 861,
  "family_id": "qf-mod50-optdeep-improvement-witness",
  "title": {
    "zh-Hant": "第 861 題：有限最佳化",
    "en": "Question 861: One Strictly Better Feasible Point Refutes Candidate Global Optimality"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-pulse-20261119-r07-0036",
    "batch_id": "batch-20261015-advance-20261119-r07-optimization-problem-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-optimization-problem"
    ],
    "title_zh": "有限最佳化",
    "stem_zh": "完整有限feasible domain、明示neighborhood與exact objective table。操作=optdeep-improvement-witness；明示資料={\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[33,11,44,55]}；請依 exact_ordered_tuple_declared_by_interface 回答。",
    "answer": {
      "kind": "expression",
      "value": "([[A, 33], [B, 11], [C, 44], [D, 55]], A, 33, B, 11, 是, 否, one_strictly_better_feasible_point_refutes_candidate_global_optimality)",
      "explanation_zh": "Wave6第十三層核對=([[A, 33], [B, 11], [C, 44], [D, 55]], A, 33, B, 11, 是, 否, one_strictly_better_feasible_point_refutes_candidate_global_optimality)；依完整明示資料精確重算；local只比較全部neighbors，global比較全部feasible points；lower bound須完整且attained；strict improvement只是否定證書；empty/incomplete domain與ties不得冒充存在、global或unique optimum。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "有限最佳化"
      ]
    },
    "variant": {
      "family_id": "qf-mod50-optdeep-improvement-witness",
      "seed": 202611197036,
      "parameters": {
        "points": [
          "A",
          "B",
          "C",
          "D"
        ],
        "values": [
          33,
          11,
          44,
          55
        ],
        "candidate": "A",
        "challenger": "B",
        "domain": "Exact_finite_optimization_density",
        "operation": "optdeep-improvement-witness",
        "semantic_template": "optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[33,1],[11,1],[44,1],[55,1]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_contract_recomputation",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "GPT-5.6 Terra",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-pulse-20261119-r07-0036",
    "batch_id": "batch-20261015-advance-20261119-r07-optimization-problem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "1451db270d336acac0c64d8aaf43ef4b3016a28e888e5b72f4981178db621d14",
      "family_id": "qf-mod50-optdeep-improvement-witness",
      "seed": 202611197036,
      "semantic_template": "optdeep-improvement-witness|{\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[33,1],[11,1],[44,1],[55,1]]}"
    },
    "fields": {
      "title": "One Strictly Better Feasible Point Refutes Candidate Global Optimality",
      "stem": "Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[33, 11, 44, 55], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope).",
      "answer": {
        "kind": "expression",
        "value": "([[A, 33], [B, 11], [C, 44], [D, 55]], A, 33, B, 11, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)",
        "explanation": "Verify the challenger is distinct and has a strictly lower exact objective value. This is a refutation of the candidate global claim, not a proof that the challenger is globally optimal or that the candidate fails every neighborhood-local condition. The exact result, retaining the requested order, is ([[A, 33], [B, 11], [C, 44], [D, 55]], A, 33, B, 11, Yes, No, one_strictly_better_feasible_point_refutes_candidate_global_optimality)."
      },
      "skills": [
        "One Strictly Better Feasible Point Refutes Candidate Global Optimality"
      ],
      "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": "完整有限feasible domain、明示neighborhood與exact objective table。操作=optdeep-improvement-witness；明示資料={\"candidate\":\"A\",\"challenger\":\"B\",\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[33,11,44,55]}；請依 exact_ordered_tuple_declared_by_interface 回答。",
        "en": "Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[33, 11, 44, 55], candidate=A. The declared feasible challenger is=B. Give (complete_objective_table, candidate, candidate_value, challenger, challenger_value, strict_feasible_improvement, candidate_global_optimality_claim, refutation_scope)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "complete feasible domain",
      "after": "[[A, 33], [B, 11], [C, 44], [D, 55]]",
      "relation": "implies",
      "rule_id": "complete_objective_data",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整表與最小化方向都先固定，不能只抽樣幾點當全域證書。",
        "en": "Fix the complete table and minimization direction; sampling a few points is not a global certificate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "完整域、鄰域、達界與唯一性是不同責任，不能從局部資料偷換全域結論。",
          "en": "Completeness, neighborhoods, attainment and uniqueness are separate obligations; local data do not license a global conclusion."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "candidate=A; challenger=B",
      "after": "both feasible; f(candidate)=33, f(challenger)=11",
      "relation": "implies",
      "rule_id": "feasible_challenger",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先確認挑戰點在feasible domain且不同於候選點。",
        "en": "First check that the challenger is a distinct feasible point."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "完整域、鄰域、達界與唯一性是不同責任，不能從局部資料偷換全域結論。",
          "en": "Completeness, neighborhoods, attainment and uniqueness are separate obligations; local data do not license a global conclusion."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "challenger minus candidate := sub(11, 33)",
      "after": "challenger minus candidate = -22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "精確相減，嚴格負值才是改善。",
        "en": "Subtract exactly; a strictly negative difference certifies improvement."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 challenger minus candidate，不以小數近似替代。",
          "en": "Recompute challenger minus candidate with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "33"
        ],
        "output": "-22",
        "label": "challenger minus candidate"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "difference=-22<0",
      "after": "candidate cannot minimize over all feasible points",
      "relation": "implies",
      "rule_id": "strict_improvement_obstruction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全域最小要求不大於每個feasible值，一個嚴格更小的反例就足夠否定。",
        "en": "Global minimality requires a value no greater than every feasible value; one strictly better point refutes it."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "完整域、鄰域、達界與唯一性是不同責任，不能從局部資料偷換全域結論。",
          "en": "Completeness, neighborhoods, attainment and uniqueness are separate obligations; local data do not license a global conclusion."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        [
          "A",
          "33"
        ],
        [
          "B",
          "11"
        ],
        [
          "C",
          "44"
        ],
        [
          "D",
          "55"
        ]
      ],
      "A",
      "33",
      "B",
      "11",
      true,
      false,
      "one_strictly_better_feasible_point_refutes_candidate_global_optimality"
    ]
  },
  "conclusion": {
    "canonical_value": "([[A, 33], [B, 11], [C, 44], [D, 55]], A, 33, B, 11, 是, 否, one_strictly_better_feasible_point_refutes_candidate_global_optimality)",
    "explanation": {
      "zh-Hant": "上述完整代值與條件核對得到原題答案；欄位次序及真假標記保持原始合約。",
      "en": "The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings."
    }
  },
  "definition_note": {
    "zh-Hant": "完整有限可行集合上的最小化；一個嚴格更小的可行點即否定候選點的全域最優。",
    "en": "Minimization over a complete finite feasible set; one strictly smaller feasible point refutes the candidate's global optimality."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261015-advance-20261119-r07-optimization-problem-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261119-r07-optimization-problem-foundations/questions.jsonl",
      "batch_sha256": "7ca0a9388bb0b3ed98a619610814bbde72bb9ea5459f6f9f7e636bc2261b7d2d",
      "original_line": 36,
      "original_line_sha256": "433ccb65083a2c0df38977184c1338831baee4774f2e2370ece4fa3ee2c78770",
      "question_compact_sha256": "1451db270d336acac0c64d8aaf43ef4b3016a28e888e5b72f4981178db621d14",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261119-r07-optimization-problem-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261119-r07-optimization-problem-foundations.en.jsonl",
      "english_row_compact_sha256": "15ec4650b301c6a77c38bc692e6a577631ca9b9bebb5e89395ed5d02af63e406"
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    "mko_bindings": {
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        "sha256": "55c8accf83a0b3a16cb86f438bdc6575f8dcec45e670f324e863e79a1a233514",
        "english_sha256": "b6379fa21189b83b931814f4a2acdc266e8f73e8fa2fad6e9d2190456191279e",
        "original_path": "public/data/mko/optimization-problem.json",
        "english_path": "public/data/i18n/editions/mko-optimization-problem.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "afa7c0e9f7084cc5929124dcffe9929d5db7def32e40047056d19ea25535873e",
    "original_candidate_sha256": "9782fa067e8b2adfd85e2f90a90325eff444382c2d53844fa07415972bbea863",
    "original_candidate_line_sha256": "9782fa067e8b2adfd85e2f90a90325eff444382c2d53844fa07415972bbea863"
  },
  "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": "來源綁定的教學推導候選；精確算術及有限列舉不是獨立AI審查或一般形式化證明。原題、答案、MKO、公式、AST及歷史保持不變。",
      "en": "Source-bound pedagogical derivation candidate: exact arithmetic and finite enumeration are not independent AI 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-20261026-003-03",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "Thirty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct sentence shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full answer tuples, model results 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": {
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      "question_first_witness_sha256": "77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b"
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    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 4,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "f20f9ccdc304d309b2fc29d304071f4e86a2dfab295baf9c607329e593c008bc",
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    "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."
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}