{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261104-r03-absolute-convergence-of-series-0066",
  "pilot_number": "R0034-17916",
  "number": 17916,
  "family_id": "qf-mod35-absseries-absolute-partial",
  "title": {
    "zh-Hant": "第 17916 題：絕對收斂",
    "en": "Question 17916: A Finite Sum of Absolute Values Is Not an Infinite Certificate"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261104-r03-absolute-convergence-of-series-0066",
    "batch_id": "batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-absolute-convergence-of-series"
    ],
    "title_zh": "絕對收斂",
    "stem_zh": "partial sums與absolute-series exact convergence certificate。操作=absseries-absolute-partial；資料={\"terms\":[[1,17],[-1,18]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([1/17, -1/18], [1/17, 1/18], 35/306, finite_absolute_partial_sum_only)",
      "explanation_zh": "Wave5第四層核對=([1/17, -1/18], [1/17, 1/18], 35/306, finite_absolute_partial_sum_only)；依完整明示資料重算；absolute convergence核sum absolute values並推出原級數收斂；term趨零 bounded或alternation不足；conditional需雙證書。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "絕對收斂"
      ]
    },
    "variant": {
      "family_id": "qf-mod35-absseries-absolute-partial",
      "seed": 202611043066,
      "parameters": {
        "terms": [
          [
            1,
            17
          ],
          [
            -1,
            18
          ]
        ],
        "domain": "Exact_absolute_or_conditional_series",
        "operation": "absseries-absolute-partial",
        "semantic_template": "qf-mod35-absseries-absolute-partial/{\"terms\":[[1,17],[-1,18]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
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  "original_english_edition": {
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    "question_id": "q-20261014-advance-20261104-r03-absolute-convergence-of-series-0066",
    "batch_id": "batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
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      "family_id": "qf-mod35-absseries-absolute-partial",
      "seed": 202611043066,
      "semantic_template": "qf-mod35-absseries-absolute-partial/{\"terms\":[[1,17],[-1,18]]}"
    },
    "fields": {
      "title": "A Finite Sum of Absolute Values Is Not an Infinite Certificate",
      "stem": "Supplied ordered exact rational terms=[1/17, -1/18]. Give (terms, absolute_terms, absolute_partial_sum, boundary).",
      "answer": {
        "kind": "expression",
        "value": "([1/17, -1/18], [1/17, 1/18], 35/306, finite_absolute_partial_sum_only)",
        "explanation": "Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/17, -1/18], [1/17, 1/18], 35/306, finite_absolute_partial_sum_only)."
      },
      "skills": [
        "A Finite Sum of Absolute Values Is Not an Infinite Certificate"
      ],
      "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": "partial sums與absolute-series exact convergence certificate。操作=absseries-absolute-partial；資料={\"terms\":[[1,17],[-1,18]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Supplied ordered exact rational terms=[1/17, -1/18]. Give (terms, absolute_terms, absolute_partial_sum, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=absseries-absolute-partial;original question snapshot retained;source presentation interface resolved in this page",
      "after": "EXPLICIT ordered output fields=[terms, absolute_terms, absolute_partial_sum, boundary]",
      "relation": "implies",
      "rule_id": "absseries_explicit_source_interface_fields_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題快照保留歷史介面標記，這裡直接列完整欄位，不需瀏覽器JavaScript或外部查表。欄位名只是presentation metadata，不是數學運算；term-zero的第四個否是『條件是否充分』，不是誤標成另一個converges旗標。",
        "en": "Preserve the original snapshot's historical interface marker while listing the full ordered fields here,without browser JavaScript or an external lookup.Field labels are presentation metadata,not mathematical evaluation.The fourth term-zero No marks insufficiency,not a mislabeled second convergence flag."
      },
      "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": "original ordered signed term index=0;value=1/17",
      "after": "absolute term at SAME index=1/17",
      "relation": "implies",
      "rule_id": "absseries_each_term_absolute_before_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先逐原項取絕對值，維持順序；不能等正負抵消後只取整個和的絕對值。",
        "en": "Take the absolute value of each original term in place;do not cancel signs first and take only the absolute value of the total."
      },
      "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": "finite absolute term1 sign reversal := mul(-1, -1/18)",
      "after": "finite absolute term1 sign reversal = 1/18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite absolute term1 sign reversal，不以小數近似替代。",
          "en": "Recompute finite absolute term1 sign reversal with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-1/18"
        ],
        "output": "1/18",
        "label": "finite absolute term1 sign reversal"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "original ordered signed term index=1;value=-1/18",
      "after": "absolute term at SAME index=1/18",
      "relation": "implies",
      "rule_id": "absseries_each_term_absolute_before_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先逐原項取絕對值，維持順序；不能等正負抵消後只取整個和的絕對值。",
        "en": "Take the absolute value of each original term in place;do not cancel signs first and take only the absolute value of the total."
      },
      "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": "finite absolute partial sum := sum(1/17, 1/18)",
      "after": "finite absolute partial sum = 35/306",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite absolute partial sum，不以小數近似替代。",
          "en": "Recompute finite absolute partial sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/17",
          "1/18"
        ],
        "output": "35/306",
        "label": "finite absolute partial sum"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "finite original signed sum for distinction := sum(1/17, -1/18)",
      "after": "finite original signed sum for distinction = 1/306",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite original signed sum for distinction，不以小數近似替代。",
          "en": "Recompute finite original signed sum for distinction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/17",
          "-1/18"
        ],
        "output": "1/306",
        "label": "finite original signed sum for distinction"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "finite signed terms=[1/17, -1/18];same-order absolute terms=[1/17, 1/18]",
      "after": "sum absolute terms=35/306;absolute signed sum=1/306<=that;finite_absolute_partial_sum_only",
      "relation": "implies",
      "rule_id": "absseries_finite_absolute_sum_not_absolute_signed_sum_or_infinite_tail",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個量可不同；空表兩個空和皆0。這是指定有限資料，不給未知無限延續的絕對收斂證書。",
        "en": "These quantities may differ;both empty sums are0.This is the supplied finite data,not an absolute-convergence certificate for an unspecified infinite continuation."
      },
      "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": [
      [
        "1/17",
        "-1/18"
      ],
      [
        "1/17",
        "1/18"
      ],
      "35/306",
      "finite_absolute_partial_sum_only"
    ]
  },
  "conclusion": {
    "canonical_value": "([1/17, -1/18], [1/17, 1/18], 35/306, finite_absolute_partial_sum_only)",
    "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."
  },
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      "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."
    },
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}