{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261011-advance-20261015-r05-sequence-0026",
  "pilot_number": "R0013-07101",
  "number": 7101,
  "family_id": "qf-density15-sequence-affine-recurrence-window",
  "title": {
    "zh-Hant": "第 7101 題：明示遞迴的有限窗口",
    "en": "Question 7101: Ordered Recurrence Window and Adjacent Differences"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261011-advance-20261015-r05-sequence-0026",
    "batch_id": "batch-20261011-advance-20261015-r05-sequence-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-sequence"
    ],
    "title_zh": "明示遞迴的有限窗口",
    "stem_zh": "N從0起，明示有限索引。a_0=0，a_(n+1)=a_n+(1)，窗口start=0、length=0。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。",
    "answer": {
      "kind": "expression",
      "value": "([], [], [])",
      "explanation_zh": "明示遞迴窗口與差列=([], [], [])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。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": 1,
      "skills": [
        "明示遞迴的有限窗口"
      ]
    },
    "variant": {
      "family_id": "qf-density15-sequence-affine-recurrence-window",
      "seed": 202610155026,
      "parameters": {
        "s": 0,
        "d": 1,
        "start": 0,
        "length": 0,
        "domain": "finite_zero_based_sequence_protocol",
        "operation": "density15_sequence_affine_recurrence_window",
        "semantic_template": "qf-density15-sequence-affine-recurrence-window/[\"empty_window\"]"
      }
    },
    "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-0026",
    "batch_id": "batch-20261011-advance-20261015-r05-sequence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "233e62b8efdda49aa136bc22533885ddb7aee173bb8b558b90f1ed9587fea77d",
      "family_id": "qf-density15-sequence-affine-recurrence-window",
      "seed": 202610155026,
      "semantic_template": "qf-density15-sequence-affine-recurrence-window/[\"empty_window\"]"
    },
    "fields": {
      "title": "Ordered Recurrence Window and Adjacent Differences",
      "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. a_0=0, a_(n+1)=a_n+(1), start=0, length=0. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.",
      "answer": {
        "kind": "expression",
        "value": "([], [], [])",
        "explanation": "Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([], [], [])."
      },
      "skills": [
        "Ordered Recurrence Window and Adjacent Differences"
      ],
      "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。a_0=0，a_(n+1)=a_n+(1)，窗口start=0、length=0。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。",
        "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. a_0=0, a_(n+1)=a_n+(1), start=0, length=0. Give (window indices, term values, adjacent differences); length=0 gives three empty lists."
      }
    }
  ],
  "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": "start=0;length=0",
      "after": "([], [], [])",
      "relation": "implies",
      "rule_id": "empty_recurrence_window",
      "conditions": [],
      "explanation": {
        "zh-Hant": "空窗口沒有要記錄的項，也沒有相鄰差；不為空輸出補造窗口項。",
        "en": "An empty window has no recorded terms or adjacent differences;do not invent entries for the empty output."
      },
      "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": [
      [],
      [],
      []
    ]
  },
  "conclusion": {
    "canonical_value": "([], [], [])",
    "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": 26,
      "original_line_sha256": "1243a21e51a03cf806225089ed8be6b4ec16b868cf9ec1d011d0298e3fe6bc01",
      "question_compact_sha256": "233e62b8efdda49aa136bc22533885ddb7aee173bb8b558b90f1ed9587fea77d",
      "english_row_compact_sha256": "091487370adf3c6e69b43c493b05d0c0deb59264bfa396d926bfad7be853e961",
      "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": "95ddbbd74099eee9806cb77e415f1161cc7f1aeec1e6ce46bb3a6289daddc250",
    "original_candidate_line_sha256": "95ddbbd74099eee9806cb77e415f1161cc7f1aeec1e6ce46bb3a6289daddc250",
    "mko_binding_revision": {
      "id": "edition-symbol-bindings-20261101",
      "path": "content/edition-symbol-bindings-20261101/revision.json",
      "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c",
      "previous_english_sha256": {
        "mko-sequence": "73ebcb30e2878e2fcb357eb7ed8f9eb08672f5b078965a636df48b982cf49fa3"
      }
    }
  },
  "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": 2,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "2a9e26f4b8c94e0162dd0dffcfa39a5cfa2345c2859cdb3be8b5c8f46694baf3",
      "arithmetic_steps": 0
    },
    "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."
  }
}