{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261011-advance-20261015-r05-sequence-0022",
  "pilot_number": "R0013-07197",
  "number": 7197,
  "family_id": "qf-density15-sequence-window-boundary-queries",
  "title": {
    "zh-Hant": "第 7197 題：有限數列的索引邊界",
    "en": "Question 7197: Finite Sequence Query Bounds before Evaluation"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261011-advance-20261015-r05-sequence-0022",
    "batch_id": "batch-20261011-advance-20261015-r05-sequence-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-sequence"
    ],
    "title_zh": "有限數列的索引邊界",
    "stem_zh": "N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=9+(-1)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。",
    "answer": {
      "kind": "expression",
      "value": "(4, [是, 是, 否], [9, 6, 不合索引])",
      "explanation_zh": "有限數列索引資格與值=(4, [是, 是, 否], [9, 6, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "有限數列的索引邊界"
      ]
    },
    "variant": {
      "family_id": "qf-density15-sequence-window-boundary-queries",
      "seed": 202610155022,
      "parameters": {
        "s": 9,
        "d": -1,
        "L": 4,
        "queries": [
          0,
          3,
          4
        ],
        "domain": "finite_zero_based_sequence_protocol",
        "operation": "density15_sequence_window_boundary_queries",
        "semantic_template": "qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"9\"],[\"typed\",\"6\"],[\"untyped\"]]]"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-11",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261011-advance-20261015-r05-sequence-0022",
    "batch_id": "batch-20261011-advance-20261015-r05-sequence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "96ff3d03bc94d31cc7105c10d0f87e8a7fe1ea91842d91fd07c185cc6aa88df1",
      "family_id": "qf-density15-sequence-window-boundary-queries",
      "seed": 202610155022,
      "semantic_template": "qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"9\"],[\"typed\",\"6\"],[\"untyped\"]]]"
    },
    "fields": {
      "title": "Finite Sequence Query Bounds before Evaluation",
      "stem": "N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=9+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).",
      "answer": {
        "kind": "expression",
        "value": "(4, [Yes, Yes, No], [9, 6, invalid-index])",
        "explanation": "Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [9, 6, invalid-index])."
      },
      "skills": [
        "Finite Sequence Query Bounds before Evaluation"
      ],
      "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": "N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=9+(-1)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。",
        "en": "N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=9+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "N starts at0;ordered finite indices and repeated values retained",
      "after": "given recurrences differ from guessing an infinite rule from observations",
      "relation": "implies",
      "rule_id": "finite_ordered_sequence_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "數列有序且可重複，有限前綴本身不能決定唯一無限規則；若題目另明示遞推，則只依該規則推導，不擅自補規則。",
        "en": "Sequences are ordered and may repeat.A finite observed prefix alone does not determine a unique infinite rule;when a recurrence is explicitly supplied,use that rule rather than inventing one."
      },
      "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": "query0 d times index := mul(-1, 0)",
      "after": "query0 d times index = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只有在有限允許索引內才代入項公式。",
        "en": "Substitute into the term formula only inside the permitted finite index domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query0 d times index，不以小數近似替代。",
          "en": "Recompute query0 d times index with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "0"
        ],
        "output": "0",
        "label": "query0 d times index"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "query0 legal sequence value := sum(9, 0)",
      "after": "query0 legal sequence value = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "初值加有號步量乘索引，負項仍合法。",
        "en": "Add the initial value to signed increment times index;negative term values are allowed."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query0 legal sequence value，不以小數近似替代。",
          "en": "Recompute query0 legal sequence value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "0"
        ],
        "output": "9",
        "label": "query0 legal sequence value"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "candidate index=0;0≤index<4=是",
      "after": "value/status=9",
      "relation": "implies",
      "rule_id": "finite_sequence_index_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "域外候選不讀成已給項值，即使可以把數字形式代入某個外推式；L=0時沒有合法索引。",
        "en": "An outside candidate is not an already supplied term,even if a number could be inserted into an extrapolated formula.L=0 has no legal indices."
      },
      "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": "query1 d times index := mul(-1, 3)",
      "after": "query1 d times index = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只有在有限允許索引內才代入項公式。",
        "en": "Substitute into the term formula only inside the permitted finite index domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query1 d times index，不以小數近似替代。",
          "en": "Recompute query1 d times index with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "3"
        ],
        "output": "-3",
        "label": "query1 d times index"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "query1 legal sequence value := sum(9, -3)",
      "after": "query1 legal sequence value = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "初值加有號步量乘索引，負項仍合法。",
        "en": "Add the initial value to signed increment times index;negative term values are allowed."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query1 legal sequence value，不以小數近似替代。",
          "en": "Recompute query1 legal sequence value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "-3"
        ],
        "output": "6",
        "label": "query1 legal sequence value"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "candidate index=3;0≤index<4=是",
      "after": "value/status=6",
      "relation": "implies",
      "rule_id": "finite_sequence_index_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "域外候選不讀成已給項值，即使可以把數字形式代入某個外推式；L=0時沒有合法索引。",
        "en": "An outside candidate is not an already supplied term,even if a number could be inserted into an extrapolated formula.L=0 has no legal indices."
      },
      "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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "candidate index=4;0≤index<4=否",
      "after": "value/status=不合索引",
      "relation": "implies",
      "rule_id": "finite_sequence_index_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "域外候選不讀成已給項值，即使可以把數字形式代入某個外推式；L=0時沒有合法索引。",
        "en": "An outside candidate is not an already supplied term,even if a number could be inserted into an extrapolated formula.L=0 has no legal indices."
      },
      "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,
      [
        true,
        true,
        false
      ],
      [
        "9",
        "6",
        "不合索引"
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "(4, [是, 是, 否], [9, 6, 不合索引])",
    "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-20261011-advance-20261015-r05-sequence-foundations",
      "batch_path": "public/data/questions/batches/batch-20261011-advance-20261015-r05-sequence-foundations/questions.jsonl",
      "batch_sha256": "e8a28d58ec052ac711318299c789c1b43033dbf1739c5b39fa88b9544e87f61b",
      "original_line": 22,
      "original_line_sha256": "8842a93d5e25435e0f94cf2d233471635c8e208267c496a9f1a3c01764d1c299",
      "question_compact_sha256": "96ff3d03bc94d31cc7105c10d0f87e8a7fe1ea91842d91fd07c185cc6aa88df1",
      "english_row_compact_sha256": "4482c2aa8e8ca4f28cd01b4f38409bb91053847a698a90327074291d1a22875a",
      "english_manifest_path": "public/data/i18n/editions/batch-20261011-advance-20261015-r05-sequence-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261011-advance-20261015-r05-sequence-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-sequence": {
        "sha256": "61e652778c86fb6151fba32a0e61c914e4f99a1c496dd1c3d933c1f9297782f7",
        "english_sha256": "24722e5fa3e44eb82dbc3fccbd220761d9b4aa84bdf79ad9d04bad0d83621dde",
        "original_path": "public/data/mko/sequence.json",
        "english_path": "public/data/i18n/editions/mko-sequence.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "944e5a6e7ddc77eb28f9280c1b90da6c7083069b4790cbeb7488d009328efce8",
    "original_candidate_sha256": "2eef02194bd800ccf193c4c4718b76470f13b3545409ea93b25e893f26b88cbc",
    "original_candidate_line_sha256": "2eef02194bd800ccf193c4c4718b76470f13b3545409ea93b25e893f26b88cbc",
    "mko_binding_revision": {
      "id": "edition-symbol-bindings-20261101",
      "path": "content/edition-symbol-bindings-20261101/revision.json",
      "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c",
      "previous_english_sha256": {
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      }
    }
  },
  "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-20261026-005-p14",
    "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": "e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59",
      "question_first_witness_sha256": "0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 8,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "cf56120f22b7772f8c3fd011f4dcd6ae922c7bc785f573cf12d69956fb802385",
      "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."
  }
}