{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-pulse-20261119-r07-0076",
  "pilot_number": "P1000-0901",
  "number": 901,
  "family_id": "qf-mod50-optdeep-boundary",
  "title": {
    "zh-Hant": "第 901 題：有限最佳化",
    "en": "Question 901: Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-pulse-20261119-r07-0076",
    "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-boundary；明示資料={\"mode\":\"empty_feasible\",\"model\":{}}；請依 exact_ordered_tuple_declared_by_interface 回答。",
    "answer": {
      "kind": "expression",
      "value": "(empty_feasible, [], 否, empty_feasible_set_has_no_optimizer)",
      "explanation_zh": "Wave6第十三層核對=(empty_feasible, [], 否, empty_feasible_set_has_no_optimizer)；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": 1,
      "skills": [
        "有限最佳化"
      ]
    },
    "variant": {
      "family_id": "qf-mod50-optdeep-boundary",
      "seed": 202611197076,
      "parameters": {
        "mode": "empty_feasible",
        "model": {},
        "domain": "Exact_finite_optimization_density",
        "operation": "optdeep-boundary",
        "semantic_template": "optdeep-boundary|{\"mode\":\"empty_feasible\",\"model\":{}}"
      }
    },
    "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-r07-0076",
    "batch_id": "batch-20261015-advance-20261119-r07-optimization-problem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "580f8292e474bc568998de90d0b491b3369208ce5f120bcf39145bae72196c78",
      "family_id": "qf-mod50-optdeep-boundary",
      "seed": 202611197076,
      "semantic_template": "optdeep-boundary|{\"mode\":\"empty_feasible\",\"model\":{}}"
    },
    "fields": {
      "title": "Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims",
      "stem": "Use mode=empty_feasible, complete model={}. Give (mode, feasible_set, optimizer_exists_claim, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(empty_feasible, [], No, empty_feasible_set_has_no_optimizer)",
        "explanation": "The empty feasible set has no optimizer. A list not established as complete cannot certify globality. Tied minima establish existence but not uniqueness. A graph-local minimum can fail globally; use the actual complete neighborhood and feasible objective table rather than conflating the two comparison scopes. The exact result, retaining the requested order, is (empty_feasible, [], No, empty_feasible_set_has_no_optimizer)."
      },
      "skills": [
        "Empty Domains, Partial Lists, Ties and Local-Only Witnesses Block Different Claims"
      ],
      "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-boundary；明示資料={\"mode\":\"empty_feasible\",\"model\":{}}；請依 exact_ordered_tuple_declared_by_interface 回答。",
        "en": "Use mode=empty_feasible, complete model={}. Give (mode, feasible_set, optimizer_exists_claim, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "mode=empty_feasible",
      "after": "identify the exact missing optimization obligation",
      "relation": "implies",
      "rule_id": "optimization_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先區分不存在、未證完整、不唯一與只達局部最小等不同情況。",
        "en": "Distinguish nonexistence, unverified completeness, nonuniqueness and merely local optimality."
      },
      "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": "feasible set=∅",
      "after": "no x belongs to the feasible set",
      "relation": "implies",
      "rule_id": "empty_feasible_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "最佳化解首先必須是feasible point；空域沒有可選元素。",
        "en": "An optimizer must first be feasible; the empty domain has no selectable element."
      },
      "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": "optimizer implies an existing feasible point",
      "after": "there is no optimizer",
      "relation": "implies",
      "rule_id": "empty_existence_obstruction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是否定存在性，不是把最佳值捏造成0或某個虛構點。",
        "en": "This denies existence, not assigns an invented value0 or a fictitious 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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "existence requirement fails",
      "after": "optimizer exists=false",
      "relation": "implies",
      "rule_id": "empty_boundary_result",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本題不另外指定空集合上的擴充實數infimum慣例。",
        "en": "This task does not introduce an extended-real infimum convention for the empty set."
      },
      "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": [
      "empty_feasible",
      [],
      false,
      "empty_feasible_set_has_no_optimizer"
    ]
  },
  "conclusion": {
    "canonical_value": "(empty_feasible, [], 否, empty_feasible_set_has_no_optimizer)",
    "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": "空集合無最優點；未證完整的清單不能支持全域結論；並列不唯一；局部最小不等於全域最小。",
    "en": "An empty set has no optimizer; an unverified partial list supports no global claim; ties are not unique; a local minimum need not be global."
  },
  "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": 76,
      "original_line_sha256": "00fa2d9a22d3f6ad4c0ed59d0bf1f9f341a07a2f36e34f57e92a20394cd0940e",
      "question_compact_sha256": "580f8292e474bc568998de90d0b491b3369208ce5f120bcf39145bae72196c78",
      "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": "21af19f2d4d805915fa5c0ac41df80350099ad9dc3c5110ada1b4d57320d29e3"
    },
    "mko_bindings": {
      "mko-optimization-problem": {
        "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": "04ec5f2dd29133e50240d9ce8795feef54c29ae75f54543474a974d281fd854b",
    "original_candidate_line_sha256": "04ec5f2dd29133e50240d9ce8795feef54c29ae75f54543474a974d281fd854b"
  },
  "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-20261026-003-04",
    "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": {
      "semantic_review_sha256": "6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772",
      "question_first_witness_sha256": "77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b"
    },
    "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": "1c7b4b7c2a42d1000e69acb23798b774bb1f8d74886df1af9f8396832720496a",
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    "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."
  }
}