{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261023-r04-metric-convergence-0003",
  "pilot_number": "R0020-10503",
  "number": 10503,
  "family_id": "qf-mod23-metricconv-ball-inventory",
  "title": {
    "zh-Hant": "第 10503 題：度量收斂證書",
    "en": "Question 10503: A Strict Open Ball in a Complete Finite Metric"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261023-r04-metric-convergence-0003",
    "batch_id": "batch-20261013-advance-20261023-r04-metric-convergence-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-metric-convergence"
    ],
    "title_zh": "度量收斂證書",
    "stem_zh": "完整有限度量與明示序列模型。操作=metricconv-ball-inventory；資料={\"n\":4,\"distances\":[[0,3,7,12],[3,0,4,9],[7,4,0,5],[12,9,5,0]],\"center\":2,\"epsilon\":[3,1]}。請依 [n,center,epsilon,ball] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(4, 2, 3, [2])",
      "explanation_zh": "現代橋梁核對=(4, 2, 3, [2])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；保留forall epsilon exists N forall n>=N；有限尾段本身不證一般收斂。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "度量收斂證書"
      ]
    },
    "variant": {
      "family_id": "qf-mod23-metricconv-ball-inventory",
      "seed": 202610234003,
      "parameters": {
        "n": 4,
        "distances": [
          [
            0,
            3,
            7,
            12
          ],
          [
            3,
            0,
            4,
            9
          ],
          [
            7,
            4,
            0,
            5
          ],
          [
            12,
            9,
            5,
            0
          ]
        ],
        "center": 2,
        "epsilon": [
          3,
          1
        ],
        "domain": "Finite_metric_convergence_certificates",
        "operation": "metricconv-ball-inventory",
        "semantic_template": "qf-mod23-metricconv-ball-inventory/{\"center\":2,\"distances\":[[0,3,7,12],[3,0,4,9],[7,4,0,5],[12,9,5,0]],\"epsilon\":[3,1],\"n\":4}"
      }
    },
    "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、Markov／可微聲明、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-20261023-r04-metric-convergence-0003",
    "batch_id": "batch-20261013-advance-20261023-r04-metric-convergence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "63575b5d2b5606758ed7db8a2a25db402cbf67ceaddcde14f8769c5a4bf51fd5",
      "family_id": "qf-mod23-metricconv-ball-inventory",
      "seed": 202610234003,
      "semantic_template": "qf-mod23-metricconv-ball-inventory/{\"center\":2,\"distances\":[[0,3,7,12],[3,0,4,9],[7,4,0,5],[12,9,5,0]],\"epsilon\":[3,1],\"n\":4}"
    },
    "fields": {
      "title": "A Strict Open Ball in a Complete Finite Metric",
      "stem": "Use the complete qualified finite metric table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=2, positive epsilon=3/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "(4, 2, 3, [2])",
        "explanation": "Compare exact distances strictly; a point at distance equal to epsilon is excluded. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (4, 2, 3, [2])."
      },
      "skills": [
        "A Strict Open Ball in a Complete Finite Metric"
      ],
      "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": "完整有限度量與明示序列模型。操作=metricconv-ball-inventory；資料={\"n\":4,\"distances\":[[0,3,7,12],[3,0,4,9],[7,4,0,5],[12,9,5,0]],\"center\":2,\"epsilon\":[3,1]}。請依 [n,center,epsilon,ball] 欄序回答。",
        "en": "Use the complete qualified finite metric table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]] on Fin(4)={0,…,3}. Preserve the convergence quantifiers: for every positive epsilon there exists N such that every n≥N satisfies d(x_n,candidate)<epsilon. A finite recorded tail alone does not certify this infinite statement. center=2, positive epsilon=3/1. Give (carrier size,center,normalized epsilon,every label with distance from center strictly less than epsilon in increasing order). Use exact reduced rational outputs with positive denominators, omitting /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "carrier Fin(4);complete distance table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]]",
      "after": "nonnegative typed entries=是;zero iff equal=是;symmetric=是;all triangle tests=是",
      "relation": "implies",
      "rule_id": "whole_qualified_finite_metric",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整表先核度量前提，不只驗本題查到的格；標籤不是距離值，對角0、非對角正且所有三角條件成立才可用後續論證。",
        "en": "Qualify the whole metric table,not only queried cells.Labels are not distances:diagonal zero,positive separation,symmetry and every triangle condition are premises for later reasoning."
      },
      "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": "raw epsilon=[3, 1]",
      "after": "normalized positive epsilon=3",
      "relation": "implies",
      "rule_id": "metric_positive_epsilon_original_denominator",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原分母非零，正規化後epsilon須正；負原分母可正規化，不以機器總化除法略過條件。",
        "en": "The original denominator is nonzero and normalized epsilon must be positive.A negative original denominator may normalize legitimately;do not bypass the guard by totalized machine division."
      },
      "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": "ball state0 distance times epsilon denominator := mul(7, 1)",
      "after": "ball state0 distance times epsilon denominator = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ball state0 distance times epsilon denominator，不以小數近似替代。",
          "en": "Recompute ball state0 distance times epsilon denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7",
          "1"
        ],
        "output": "7",
        "label": "ball state0 distance times epsilon denominator"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "ball state0:distance=7;epsilon=3;positive denominator=1",
      "after": "cross product=7;strictly below numerator=否",
      "relation": "implies",
      "rule_id": "strict_metric_distance_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正分母交乘後仍用嚴格小於；等於epsilon時不在開球內。",
        "en": "Cross-multiply by the positive denominator and retain strict inequality.Equality with epsilon is outside the open ball."
      },
      "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": "ball state1 distance times epsilon denominator := mul(4, 1)",
      "after": "ball state1 distance times epsilon denominator = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ball state1 distance times epsilon denominator，不以小數近似替代。",
          "en": "Recompute ball state1 distance times epsilon denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "1"
        ],
        "output": "4",
        "label": "ball state1 distance times epsilon denominator"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "ball state1:distance=4;epsilon=3;positive denominator=1",
      "after": "cross product=4;strictly below numerator=否",
      "relation": "implies",
      "rule_id": "strict_metric_distance_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正分母交乘後仍用嚴格小於；等於epsilon時不在開球內。",
        "en": "Cross-multiply by the positive denominator and retain strict inequality.Equality with epsilon is outside the open ball."
      },
      "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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "ball state2 distance times epsilon denominator := mul(0, 1)",
      "after": "ball state2 distance times epsilon denominator = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ball state2 distance times epsilon denominator，不以小數近似替代。",
          "en": "Recompute ball state2 distance times epsilon denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "ball state2 distance times epsilon denominator"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "ball state2:distance=0;epsilon=3;positive denominator=1",
      "after": "cross product=0;strictly below numerator=是",
      "relation": "implies",
      "rule_id": "strict_metric_distance_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正分母交乘後仍用嚴格小於；等於epsilon時不在開球內。",
        "en": "Cross-multiply by the positive denominator and retain strict inequality.Equality with epsilon is outside the open ball."
      },
      "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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "ball state3 distance times epsilon denominator := mul(5, 1)",
      "after": "ball state3 distance times epsilon denominator = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ball state3 distance times epsilon denominator，不以小數近似替代。",
          "en": "Recompute ball state3 distance times epsilon denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "1"
        ],
        "output": "5",
        "label": "ball state3 distance times epsilon denominator"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "ball state3:distance=5;epsilon=3;positive denominator=1",
      "after": "cross product=5;strictly below numerator=否",
      "relation": "implies",
      "rule_id": "strict_metric_distance_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正分母交乘後仍用嚴格小於；等於epsilon時不在開球內。",
        "en": "Cross-multiply by the positive denominator and retain strict inequality.Equality with epsilon is outside the open ball."
      },
      "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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "center=2;strict radius=3;all carrier labels=[0, 1, 2, 3]",
      "after": "open ball=[2]",
      "relation": "implies",
      "rule_id": "complete_strict_ball_inventory",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一核完整載體，保留中心並排除邊界，不能依標籤大小或圖形外觀猜球。",
        "en": "Inspect the complete carrier,including the center and excluding the boundary.Do not infer the ball from label sizes or a drawing."
      },
      "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": [
      4,
      2,
      "3",
      [
        2
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "(4, 2, 3, [2])",
    "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-20261023-r04-metric-convergence-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261023-r04-metric-convergence-foundations/questions.jsonl",
      "batch_sha256": "5c57223072961823be7eab0acfdf51b00c146ece44957874eb970794b6cdb5db",
      "original_line": 3,
      "original_line_sha256": "d176b278eac317d98b44267f1cfd86eecc52e87fcd995a91267418c669d1e781",
      "question_compact_sha256": "63575b5d2b5606758ed7db8a2a25db402cbf67ceaddcde14f8769c5a4bf51fd5",
      "english_row_compact_sha256": "a178909fe0ac96ddf0c8539112661cd51738ce95da52a94de87554f073aa4d76",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261023-r04-metric-convergence-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261023-r04-metric-convergence-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-metric-convergence": {
        "sha256": "185db7b6b678014d6791d0448e2eccc1efc643b288d9b95c2b7b91f3ea710969",
        "english_sha256": "8970fc590fd89152780c85bc3235042b210a5536388a57ef8b5ea934bc5eb1d5",
        "original_path": "public/data/mko/metric-convergence.json",
        "english_path": "public/data/i18n/editions/mko-metric-convergence.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "bd67681fffd9e29972c322d497cd945a668a794dd25e0c8cd26f1a6ce22e4dfb",
    "original_candidate_sha256": "e8f0cbb0f50d2e66e71577bad32064874bf74f8dcb633feb66730c3d4e552d03",
    "original_candidate_line_sha256": "e8f0cbb0f50d2e66e71577bad32064874bf74f8dcb633feb66730c3d4e552d03"
  },
  "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-p21",
    "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": "b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598f",
      "question_first_witness_sha256": "3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 11,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "edd108e5b0010fec89a6aff727484ca9311677c8375f0cae05f283bf46266464",
      "arithmetic_steps": 4
    },
    "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."
  }
}