{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261117-r01-series-comparison-test-0001",
  "pilot_number": "R0052-26976",
  "number": 26976,
  "family_id": "qf-mod48-series-upper-geometric",
  "title": {
    "zh-Hant": "第 26976 題：級數比較判別法",
    "en": "Question 26976: A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261117-r01-series-comparison-test-0001",
    "batch_id": "batch-20261015-advance-20261117-r01-series-comparison-test-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-series-comparison-test"
    ],
    "title_zh": "級數比較判別法",
    "stem_zh": "非負項、eventual forall比較方向與reference series狀態責任。操作=series-upper-geometric；資料={\"a_scale\":1,\"b_scale\":2,\"ratio\":[1,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(1, 2, 1/7, 7/6, 7/3, 是, upper_comparison_proves_convergence)",
      "explanation_zh": "Wave6第十一層核對=(1, 2, 1/7, 7/6, 7/3, 是, upper_comparison_proves_convergence)；依完整明示資料獨立重算；有限前綴不改變收斂性，但有限samples不證明eventual comparison；上界收斂與下界發散方向不得反轉；exact families不取代一般定理。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "級數比較判別法"
      ]
    },
    "variant": {
      "family_id": "qf-mod48-series-upper-geometric",
      "seed": 202611171001,
      "parameters": {
        "a_scale": 1,
        "b_scale": 2,
        "ratio": [
          1,
          7
        ],
        "domain": "Exact_nonnegative_series_comparison",
        "operation": "series-upper-geometric",
        "semantic_template": "series-upper-geometric|{a_scale:[1,1],b_scale:[2,1],ratio:[1,7]}"
      }
    },
    "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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "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-advance-20261117-r01-series-comparison-test-0001",
    "batch_id": "batch-20261015-advance-20261117-r01-series-comparison-test-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "c8dcdf0c3ddabb6f880d1c1369df1da0977872f61e0c784c54370f49a9cddb42",
      "family_id": "qf-mod48-series-upper-geometric",
      "seed": 202611171001,
      "semantic_template": "series-upper-geometric|{a_scale:[1,1],b_scale:[2,1],ratio:[1,7]}"
    },
    "fields": {
      "title": "A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series",
      "stem": "Use exact scales A=1,B=2, rational ratio r=1/7. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(1, 2, 1/7, 7/6, 7/3, Yes, upper_comparison_proves_convergence)",
        "explanation": "Compute both exact sums and compare them. The first term has exponent0, including r=0. The full nonnegative comparison and convergent reference are premises; finite samples do not prove the general theorem. The exact result, retaining the requested order, is (1, 2, 1/7, 7/6, 7/3, Yes, upper_comparison_proves_convergence)."
      },
      "skills": [
        "A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series"
      ],
      "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": "非負項、eventual forall比較方向與reference series狀態責任。操作=series-upper-geometric；資料={\"a_scale\":1,\"b_scale\":2,\"ratio\":[1,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use exact scales A=1,B=2, rational ratio r=1/7. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [series-upper-geometric]",
      "after": "derived = [[a_scale, b_scale, ratio, sum_a, sum_b, sums_ordered, certificate]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "直接列出此操作及mode的完整來源欄序，不需JavaScript；布林、字串與null角色保持區別，不補欄或以答案形狀猜介面。",
        "en": "List this operation/mode exact source field order without JavaScript.Keep Boolean,string and null roles distinct;do not pad fields or infer the interface from answer shape."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。",
          "en": "Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [1, 2, 1/7]",
      "after": "derived = [EVERY tail term is A*r^k and B*r^k for k>=0, r^0=1 including r=0, 是]",
      "relation": "implies",
      "rule_id": "Series_WHOLE_geometric_tail_premises_not_finite_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整條tail律是給定前提，k由0開始；0<=A<=B與r>=0給逐項比較，0<=r<1才可用geometric和。r=0的首項仍有r^0=1。",
        "en": "The whole tail law is given,with k starting at0.Nonnegative A<=B and r give termwise comparison;0<=r<1 licenses the geometric sum.At r=0 the first term still uses r^0=1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。",
          "en": "Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Series ORIGINAL one minus ratio := sub(1, 1/7)",
      "after": "Series ORIGINAL one minus ratio = 6/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色精確代值並保留符號運算；不用浮點近似，也不因額外教學運算縮小原題域。",
        "en": "Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Series ORIGINAL one minus ratio，不以小數近似替代。",
          "en": "Recompute Series ORIGINAL one minus ratio with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "1/7"
        ],
        "output": "6/7",
        "label": "Series ORIGINAL one minus ratio"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Series ORIGINAL geometric tailA := div(1, 6/7)",
      "after": "Series ORIGINAL geometric tailA = 7/6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色精確代值並保留符號運算；不用浮點近似，也不因額外教學運算縮小原題域。",
        "en": "Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Series ORIGINAL geometric tailA，不以小數近似替代。",
          "en": "Recompute Series ORIGINAL geometric tailA with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "6/7"
        ],
        "output": "7/6",
        "label": "Series ORIGINAL geometric tailA"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Series ORIGINAL geometric tailB := div(2, 6/7)",
      "after": "Series ORIGINAL geometric tailB = 7/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色精確代值並保留符號運算；不用浮點近似，也不因額外教學運算縮小原題域。",
        "en": "Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Series ORIGINAL geometric tailB，不以小數近似替代。",
          "en": "Recompute Series ORIGINAL geometric tailB with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "2",
          "6/7"
        ],
        "output": "7/3",
        "label": "Series ORIGINAL geometric tailB"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [1, 2, 1/7, 6/7]",
      "after": "derived = [7/6, 7/3, 是]",
      "relation": "implies",
      "rule_id": "Series_GENERAL_geometric_partial_sum_limit_and_upper_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限和S_N=A(1-r^N)/(1-r)，乘(1-r)可逐項相消；r^N→0給tail和A/(1-r)，B同理。非負a的部分和遞增且由已收斂b的和控制，故收斂；不是以有限項近似代替∀n。",
        "en": "For the finite sum,multiplication by(1-r) telescopes to A(1-r^N);r^N tends to0,hence the tail sum A/(1-r),and similarly for B.The nonnegative a partial sums increase and are bounded by the convergent b sum.This is not a finite-sample replacement for all n."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。",
          "en": "Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "1",
      "2",
      "1/7",
      "7/6",
      "7/3",
      true,
      "upper_comparison_proves_convergence"
    ]
  },
  "conclusion": {
    "canonical_value": "(1, 2, 1/7, 7/6, 7/3, 是, upper_comparison_proves_convergence)",
    "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": {
      "batch_id": "batch-20261015-advance-20261117-r01-series-comparison-test-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261117-r01-series-comparison-test-foundations/questions.jsonl",
      "batch_sha256": "c62071690ae3f92dd63f88f17bae18ae63a3b9197857ad9735879ab85b1a2824",
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      "question_compact_sha256": "c8dcdf0c3ddabb6f880d1c1369df1da0977872f61e0c784c54370f49a9cddb42",
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      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261117-r01-series-comparison-test-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261117-r01-series-comparison-test-foundations.en.jsonl"
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    "mko_bindings": {
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        "original_path": "public/data/mko/series-comparison-test.json",
        "english_path": "public/data/i18n/editions/mko-series-comparison-test.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "a3def7368821a4d3dc7757fefd3c4a099713f8de8fb8e6f3434025e9d7b5a6d0",
    "original_candidate_sha256": "1c5537582dff0cd22885489774448f8c2a7df4cef0227ff09aefb26f21d43404",
    "original_candidate_line_sha256": "1c5537582dff0cd22885489774448f8c2a7df4cef0227ff09aefb26f21d43404"
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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-20261026-010-p53",
    "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": "729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252",
      "question_first_witness_sha256": "d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "30b7f67d10f2ea527e842f3c644df24e878ae239bd6611cdd4edca5f57efc31d",
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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."
  }
}