{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-pulse-20261119-r07-0001",
  "pilot_number": "P250-0226",
  "number": 226,
  "family_id": "qf-mod50-optdeep-local-global",
  "title": {
    "zh-Hant": "第 226 題：有限最佳化",
    "en": "Question 226: Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-pulse-20261119-r07-0001",
    "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-local-global；明示資料={\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[1,2,3,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。",
    "answer": {
      "kind": "expression",
      "value": "([[A, 1], [B, 2], [C, 3], [D, 4]], [[A, B], [A, C], [B, D], [C, D]], A, 1, [[B, 2], [C, 3]], 是, 是, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)",
      "explanation_zh": "Wave6第十三層核對=([[A, 1], [B, 2], [C, 3], [D, 4]], [[A, B], [A, C], [B, D], [C, D]], A, 1, [[B, 2], [C, 3]], 是, 是, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)；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-local-global",
      "seed": 202611197001,
      "parameters": {
        "points": [
          "A",
          "B",
          "C",
          "D"
        ],
        "values": [
          1,
          2,
          3,
          4
        ],
        "edges": [
          [
            "A",
            "B"
          ],
          [
            "A",
            "C"
          ],
          [
            "B",
            "D"
          ],
          [
            "C",
            "D"
          ]
        ],
        "candidate": "A",
        "domain": "Exact_finite_optimization_density",
        "operation": "optdeep-local-global",
        "semantic_template": "optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[1,1],[2,1],[3,1],[4,1]]}"
      }
    },
    "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-r07-0001",
    "batch_id": "batch-20261015-advance-20261119-r07-optimization-problem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "0599d5089c076eab426e7cf41651049c6e19a73c4d2ba59d227d17b42c219a19",
      "family_id": "qf-mod50-optdeep-local-global",
      "seed": 202611197001,
      "semantic_template": "optdeep-local-global|{\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[1,1],[2,1],[3,1],[4,1]]}"
    },
    "fields": {
      "title": "Declared-Neighborhood Locality and Whole-Domain Globality Are Separate",
      "stem": "Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[1, 2, 3, 4], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope).",
      "answer": {
        "kind": "expression",
        "value": "([[A, 1], [B, 2], [C, 3], [D, 4]], [[A, B], [A, C], [B, D], [C, D]], A, 1, [[B, 2], [C, 3]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)",
        "explanation": "Compare the candidate with every declared neighbor for graph-local minimality and with every feasible point for global minimality. Retain every global minimizer, including ties. Graph-locality is not a local optimum in an unspecified continuous topology, and a partial list is not a whole-domain certificate. The exact result, retaining the requested order, is ([[A, 1], [B, 2], [C, 3], [D, 4]], [[A, B], [A, C], [B, D], [C, D]], A, 1, [[B, 2], [C, 3]], Yes, Yes, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)."
      },
      "skills": [
        "Declared-Neighborhood Locality and Whole-Domain Globality Are Separate"
      ],
      "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-local-global；明示資料={\"candidate\":\"A\",\"edges\":[[\"A\",\"B\"],[\"A\",\"C\"],[\"B\",\"D\"],[\"C\",\"D\"]],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[1,2,3,4]}；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=[1, 2, 3, 4], candidate=A. Complete undirected neighborhood edges=[[A, B], [A, C], [B, D], [C, D]]. Give (complete_objective_table, undirected_neighborhood_edges, candidate, candidate_value, complete_neighbor_values, neighborhood_local_minimum, global_minimum, all_global_minimizers, comparison_scope)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "feasible points=[A, B, C, D]",
      "after": "[[A, 1], [B, 2], [C, 3], [D, 4]]",
      "relation": "implies",
      "rule_id": "complete_objective_table",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先固定完整 feasible domain 及每一點的目標值，本題採最小化。",
        "en": "Fix the complete feasible domain and each objective value; this task is minimization."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "有限表的完整性是本題前提，不外推未列出的點。",
          "en": "Completeness of the finite table is a premise here, not a statement about unlisted points."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "candidate=A; edges=[[A, B], [A, C], [B, D], [C, D]]",
      "after": "neighbors(A)=[B, C]; f(A)=1",
      "relation": "implies",
      "rule_id": "declared_neighborhood",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從已宣告的無向鄰接關係列出候選點的全部鄰居；這不是把每個 feasible point 都當鄰居。",
        "en": "List every neighbor of the candidate from the declared undirected relation; not every feasible point is automatically a neighbor."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "local 使用鄰域，global 使用完整 feasible domain，責任不同。",
          "en": "Local uses the neighborhood; global uses the entire feasible domain."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "f(B)−f(A) := sub(2, 1)",
      "after": "f(B)−f(A) = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐鄰居比較與候選點的目標差。",
        "en": "Compare the objective difference at every neighbor."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f(B)−f(A)，不以小數近似替代。",
          "en": "Recompute f(B)−f(A) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "1"
        ],
        "output": "1",
        "label": "f(B)−f(A)"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "f(C)−f(A) := sub(3, 1)",
      "after": "f(C)−f(A) = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐鄰居比較與候選點的目標差。",
        "en": "Compare the objective difference at every neighbor."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f(C)−f(A)，不以小數近似替代。",
          "en": "Recompute f(C)−f(A) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "1"
        ],
        "output": "2",
        "label": "f(C)−f(A)"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "[[B, 1, 是], [C, 2, 是]]",
      "after": "local minimum=是",
      "relation": "implies",
      "rule_id": "local_minimum_definition",
      "conditions": [],
      "explanation": {
        "zh-Hant": "所有鄰居的差皆非負時，沒有鄰居嚴格更好；相等值允許非嚴格局部最小。",
        "en": "If all neighbor differences are nonnegative, no neighbor is strictly better; equal values are allowed for a non-strict local minimum."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "這個測試本身沒有比較非鄰居。",
          "en": "This test alone has not compared nonneighbors."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "global f(A)−f(A) := sub(1, 1)",
      "after": "global f(A)−f(A) = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全域判斷逐一比較每個 feasible point，包括候選點自己。",
        "en": "For global optimality compare every feasible point, including the candidate itself."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 global f(A)−f(A)，不以小數近似替代。",
          "en": "Recompute global f(A)−f(A) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "1"
        ],
        "output": "0",
        "label": "global f(A)−f(A)"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "global f(B)−f(A) := sub(2, 1)",
      "after": "global f(B)−f(A) = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全域判斷逐一比較每個 feasible point，包括候選點自己。",
        "en": "For global optimality compare every feasible point, including the candidate itself."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 global f(B)−f(A)，不以小數近似替代。",
          "en": "Recompute global f(B)−f(A) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "1"
        ],
        "output": "1",
        "label": "global f(B)−f(A)"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "global f(C)−f(A) := sub(3, 1)",
      "after": "global f(C)−f(A) = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全域判斷逐一比較每個 feasible point，包括候選點自己。",
        "en": "For global optimality compare every feasible point, including the candidate itself."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 global f(C)−f(A)，不以小數近似替代。",
          "en": "Recompute global f(C)−f(A) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "1"
        ],
        "output": "2",
        "label": "global f(C)−f(A)"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "global f(D)−f(A) := sub(4, 1)",
      "after": "global f(D)−f(A) = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全域判斷逐一比較每個 feasible point，包括候選點自己。",
        "en": "For global optimality compare every feasible point, including the candidate itself."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 global f(D)−f(A)，不以小數近似替代。",
          "en": "Recompute global f(D)−f(A) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "4",
          "1"
        ],
        "output": "3",
        "label": "global f(D)−f(A)"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "[[A, 0, 是], [B, 1, 是], [C, 2, 是], [D, 3, 是]]",
      "after": "global minimum=是; minimum value=1; all minimizers=[A]",
      "relation": "implies",
      "rule_id": "global_minimum_definition",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐點差的符號給全域資格；再保留所有達到最小值的點，不能任意丟掉 ties。",
        "en": "The pointwise differences determine global optimality; retain every point attaining the minimum, including ties."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "一個更好的 feasible point 就可否定全域最優，但 local 通過不能單獨證明 global。",
          "en": "One strictly better feasible point refutes global optimality, while a local pass alone does not prove global optimality."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "table": [
      [
        "A",
        "1"
      ],
      [
        "B",
        "2"
      ],
      [
        "C",
        "3"
      ],
      [
        "D",
        "4"
      ]
    ],
    "neighbors": [
      "B",
      "C"
    ],
    "candidate": "A",
    "local_tests": [
      [
        "B",
        "1",
        true
      ],
      [
        "C",
        "2",
        true
      ]
    ],
    "global_tests": [
      [
        "A",
        "0",
        true
      ],
      [
        "B",
        "1",
        true
      ],
      [
        "C",
        "2",
        true
      ],
      [
        "D",
        "3",
        true
      ]
    ],
    "local": true,
    "global": true,
    "minimum": "1",
    "minimizers": [
      "A"
    ]
  },
  "conclusion": {
    "canonical_value": "([[A, 1], [B, 2], [C, 3], [D, 4]], [[A, B], [A, C], [B, D], [C, D]], A, 1, [[B, 2], [C, 3]], 是, 是, [A], local_uses_complete_declared_neighborhood_global_uses_complete_feasible_domain)",
    "explanation": {
      "zh-Hant": "候選點 A：local=是，global=是；全部最小化點=[A]。",
      "en": "Candidate A: local=Yes, global=Yes; all minimizers=[A]."
    }
  },
  "definition_note": {
    "zh-Hant": "完整有限可行集合及無向鄰域；局部比較所有鄰居，全域比較所有可行點。",
    "en": "Use the complete finite feasible carrier and declared undirected neighborhood. Local checks cover every neighbor; global checks cover every feasible point."
  },
  "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": 1,
      "original_line_sha256": "cc9837b3f36d8d059ee72f752b0ee361c14d3b511f372113fc0cc4965388f63d",
      "question_compact_sha256": "0599d5089c076eab426e7cf41651049c6e19a73c4d2ba59d227d17b42c219a19",
      "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": "8ae1b02177940f9fa5c51bc4fb42b6b0effcef4190b2101c3be3bf5189a87cf5"
    },
    "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": "fb325b42cfcaa25ddbae992459c2684afd767fe218028bc0b9c0823e67c83b7c",
    "original_candidate_sha256": "7802aeea0fbbf144564085c85ea7856fa02db1569931e1511c357c679385ae73",
    "original_candidate_line_sha256": "7802aeea0fbbf144564085c85ea7856fa02db1569931e1511c357c679385ae73"
  },
  "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": "本稿是來源綁定的教學推導候選；精確重算不等於獨立數學審查或形式化證明。原題、答案、公式、AST 與歷史未被修改。",
      "en": "This is a source-bound pedagogical derivation candidate. Exact recomputation is not independent mathematical review or formal proof. Original questions, answers, formulas, ASTs and history are unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261024-002",
    "review_date": "2026-10-24",
    "owner_authorized_reader_publication": true,
    "scope": "Family prototypes and mathematical/bilingual reasoning variants reviewed in this existing context; parameter substitutions, all original/source bindings, full answer tuples and arithmetic ASTs independently reconstructed per question. Numeric binomial text slots are validated against independently regenerated n,k,C(n,k),n-k values. 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": "4727d1bd957a9b7f6cadb7972ddb5ab80a31f5259dc5394c1dff2b71944ed36b",
      "question_first_witness_sha256": "51e18649133238ac992229f6e16542a5f3af53ca73b2262ab0524508ea937dee"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 10,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "ec488371a9fe51c0510c90b6f29ea37302b33d591633f8cc0172f8b5622ccfef",
      "arithmetic_steps": 6
    },
    "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."
  }
}