{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261023-r04-metric-convergence-0077",
  "pilot_number": "R0020-10552",
  "number": 10552,
  "family_id": "qf-mod23-metricconv-limit-uniqueness-audit",
  "title": {
    "zh-Hant": "第 10552 題：度量收斂證書",
    "en": "Question 10552: A Conditional Uniqueness Check Is Not Two Convergence Claims"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261023-r04-metric-convergence-0077",
    "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-limit-uniqueness-audit；資料={\"n\":4,\"distances\":[[0,2,5,9],[2,0,3,7],[5,3,0,4],[9,7,4,0]],\"prefix\":[1,2],\"tail_value\":3,\"candidate_a\":3,\"candidate_b\":0}。請依 [prefix,tail,candidate_a,candidate_b,distance,a_limit,b_limit,uniqueness_consistent] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([1, 2], 3, 3, 0, 9, 是, 否, 是)",
      "explanation_zh": "現代橋梁核對=([1, 2], 3, 3, 0, 9, 是, 否, 是)；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-limit-uniqueness-audit",
      "seed": 202610234077,
      "parameters": {
        "n": 4,
        "distances": [
          [
            0,
            2,
            5,
            9
          ],
          [
            2,
            0,
            3,
            7
          ],
          [
            5,
            3,
            0,
            4
          ],
          [
            9,
            7,
            4,
            0
          ]
        ],
        "prefix": [
          1,
          2
        ],
        "tail_value": 3,
        "candidate_a": 3,
        "candidate_b": 0,
        "domain": "Finite_metric_convergence_certificates",
        "operation": "metricconv-limit-uniqueness-audit",
        "semantic_template": "qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":3,\"candidate_b\":0,\"distances\":[[0,2,5,9],[2,0,3,7],[5,3,0,4],[9,7,4,0]],\"n\":4,\"prefix\":[1,2],\"tail_value\":3}"
      }
    },
    "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-0077",
    "batch_id": "batch-20261013-advance-20261023-r04-metric-convergence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "4fa3b7bba0d1d283f20bb2f4b5de703da108de256da633b1c6f66a5651f357fe",
      "family_id": "qf-mod23-metricconv-limit-uniqueness-audit",
      "seed": 202610234077,
      "semantic_template": "qf-mod23-metricconv-limit-uniqueness-audit/{\"candidate_a\":3,\"candidate_b\":0,\"distances\":[[0,2,5,9],[2,0,3,7],[5,3,0,4],[9,7,4,0]],\"n\":4,\"prefix\":[1,2],\"tail_value\":3}"
    },
    "fields": {
      "title": "A Conditional Uniqueness Check Is Not Two Convergence Claims",
      "stem": "Use the complete qualified finite metric table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 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. infinite sequence has prefix=[1, 2] and constant tail=3; candidates a=3, b=0. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?).",
      "answer": {
        "kind": "expression",
        "value": "([1, 2], 3, 3, 0, 9, Yes, No, Yes)",
        "explanation": "Check each candidate against the declared constant tail. The final implication may be true when its antecedent is false; it does not assert both candidates converge. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], 3, 3, 0, 9, Yes, No, Yes)."
      },
      "skills": [
        "A Conditional Uniqueness Check Is Not Two Convergence 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": "完整有限度量與明示序列模型。操作=metricconv-limit-uniqueness-audit；資料={\"n\":4,\"distances\":[[0,2,5,9],[2,0,3,7],[5,3,0,4],[9,7,4,0]],\"prefix\":[1,2],\"tail_value\":3,\"candidate_a\":3,\"candidate_b\":0}。請依 [prefix,tail,candidate_a,candidate_b,distance,a_limit,b_limit,uniqueness_consistent] 欄序回答。",
        "en": "Use the complete qualified finite metric table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 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. infinite sequence has prefix=[1, 2] and constant tail=3; candidates a=3, b=0. Give (prefix,tail,a,b,d(a,b),a is a limit?,b is a limit?,if both are limits then a=b?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "carrier Fin(4);complete distance table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 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": "phase=candidate a;prefix=[1, 2];all later terms explicitly equal=3",
      "after": "candidate=3;infinite constant tail begins at=2",
      "relation": "implies",
      "rule_id": "declared_infinite_tail_not_pattern_guess",
      "conditions": [],
      "explanation": {
        "zh-Hant": "題目明示所有後項等於tail，不是從有限prefix猜無限規律；前綴本身不改尾段極限。",
        "en": "The problem explicitly specifies every later term as tail;this is not guessing an infinite rule from a finite prefix.The prefix does not change the tail limit."
      },
      "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": "phase=candidate a;candidate=tail=3;distance=0",
      "after": "for every epsilon>0 choose N=2;every n>=N has distance0<epsilon",
      "relation": "implies",
      "rule_id": "constant_tail_full_quantified_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留∀epsilon>0存在N且所有n≥N的順序：固定N為prefix長度便對每個正epsilon成立。",
        "en": "Retain the order for every epsilon>0 there exists N such that every n≥N works.N equal to prefix length works for every positive epsilon."
      },
      "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": "phase=candidate b;prefix=[1, 2];all later terms explicitly equal=3",
      "after": "candidate=0;infinite constant tail begins at=2",
      "relation": "implies",
      "rule_id": "declared_infinite_tail_not_pattern_guess",
      "conditions": [],
      "explanation": {
        "zh-Hant": "題目明示所有後項等於tail，不是從有限prefix猜無限規律；前綴本身不改尾段極限。",
        "en": "The problem explicitly specifies every later term as tail;this is not guessing an infinite rule from a finite prefix.The prefix does not change the tail limit."
      },
      "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": "candidate b separating epsilon := div(9, 2)",
      "after": "candidate b separating epsilon = 9/2",
      "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 candidate b separating epsilon，不以小數近似替代。",
          "en": "Recompute candidate b separating epsilon with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "9",
          "2"
        ],
        "output": "9/2",
        "label": "candidate b separating epsilon"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "phase=candidate b;tail distance=9;epsilon=9/2",
      "after": "for any proposed N choose n=max(N,2);distance=9>=epsilon",
      "relation": "implies",
      "rule_id": "constant_tail_nonlimit_counterwitness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "度量分離給正距離；取其半為epsilon，任意N後仍有尾項距離原值，違反收斂的全尾段條件。",
        "en": "Metric separation gives positive distance.Choose half of it as epsilon;after every proposed N a tail term retains that distance and violates the full-tail convergence condition."
      },
      "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": "a limit=是;b limit=否;distance(a,b)=9",
      "after": "if both are limits then a=b=是",
      "relation": "implies",
      "rule_id": "limit_uniqueness_conditional_not_two_claims",
      "conditions": [],
      "explanation": {
        "zh-Hant": "分開核兩個極限資格，最後只是條件蘊涵。前提為假可使此欄真，不能說兩候選都收斂。",
        "en": "Audit both limit qualifications separately.The last field is conditional implication;it can be true with a false antecedent and does not assert both candidates converge."
      },
      "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": [
      [
        1,
        2
      ],
      3,
      3,
      0,
      9,
      true,
      false,
      true
    ]
  },
  "conclusion": {
    "canonical_value": "([1, 2], 3, 3, 0, 9, 是, 否, 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": {
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    "question_binding": {
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      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261023-r04-metric-convergence-foundations/questions.jsonl",
      "batch_sha256": "5c57223072961823be7eab0acfdf51b00c146ece44957874eb970794b6cdb5db",
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      "original_line_sha256": "c9a3f28f24387e5bdc77d99917040734862162ecdb0a9f75491a84b5e3a0bf3d",
      "question_compact_sha256": "4fa3b7bba0d1d283f20bb2f4b5de703da108de256da633b1c6f66a5651f357fe",
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      "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"
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    "mko_bindings": {
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        "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": "f94adfb1678e0c9bed141181f1068acbc4da3f17d472ce29fc376312cd7c3668",
    "original_candidate_line_sha256": "f94adfb1678e0c9bed141181f1068acbc4da3f17d472ce29fc376312cd7c3668"
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  "historical_source_state": {
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      "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": 7,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "16493c5bfc8fd60f8b09f2953eb90b56221c17af41eb54b8bcd287a7739386e8",
      "arithmetic_steps": 1
    },
    "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."
  }
}