{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261011-advance-20261015-r05-sequence-0027",
  "pilot_number": "R0013-07102",
  "number": 7102,
  "family_id": "qf-density15-sequence-affine-recurrence-window",
  "title": {
    "zh-Hant": "第 7102 題：明示遞迴的有限窗口",
    "en": "Question 7102: Ordered Recurrence Window and Adjacent Differences"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261011-advance-20261015-r05-sequence-0027",
    "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+(7)，窗口start=0、length=1。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。",
    "answer": {
      "kind": "expression",
      "value": "([0], [0], [])",
      "explanation_zh": "明示遞迴窗口與差列=([0], [0], [])。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": 202610155027,
      "parameters": {
        "s": 0,
        "d": 7,
        "start": 0,
        "length": 1,
        "domain": "finite_zero_based_sequence_protocol",
        "operation": "density15_sequence_affine_recurrence_window",
        "semantic_template": "qf-density15-sequence-affine-recurrence-window/[[0],[\"0\"]]"
      }
    },
    "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-0027",
    "batch_id": "batch-20261011-advance-20261015-r05-sequence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "119d588a660ee5a753b4fe18eda2466436a8e8b62640ed67060da33173c0c0ed",
      "family_id": "qf-density15-sequence-affine-recurrence-window",
      "seed": 202610155027,
      "semantic_template": "qf-density15-sequence-affine-recurrence-window/[[0],[\"0\"]]"
    },
    "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+(7), start=0, length=1. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.",
      "answer": {
        "kind": "expression",
        "value": "([0], [0], [])",
        "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 ([0], [0], [])."
      },
      "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+(7)，窗口start=0、length=1。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+(7), start=0, length=1. 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": "a(0)=0;a(n+1)=a(n)+(7)",
      "after": "a(n)=0+n·(7) by this supplied affine recurrence",
      "relation": "implies",
      "rule_id": "affine_recurrence_closed_form_derivation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "基礎n=0成立；若a(n)=s+nd，則再加d得s+(n+1)d。這是由明示遞推推出，不是觀察少數項猜無限規則。",
        "en": "The base n=0 holds;if a(n)=s+nd,adding d gives s+(n+1)d.This follows from the explicitly given recurrence,not guessing an infinite rule from a few observations."
      },
      "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": "window start increment := mul(0, 7)",
      "after": "window start increment = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用已推導閉式求到窗口起點所累積的步量。",
        "en": "Use the derived form for the accumulated increment at the window start."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 window start increment，不以小數近似替代。",
          "en": "Recompute window start increment with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "7"
        ],
        "output": "0",
        "label": "window start increment"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "window start value := sum(0, 0)",
      "after": "window start value = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "初值加步量得到第一個窗口項。",
        "en": "Add the increment to the initial value to obtain the first window term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 window start value，不以小數近似替代。",
          "en": "Recompute window start value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "window start value"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "window indices=[0];values=[0]",
      "after": "adjacent deltas=[]",
      "relation": "implies",
      "rule_id": "ordered_recurrence_window_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "索引與項值一一對齊，length=1沒有相鄰差；不對項值去重或要求它們為自然數。",
        "en": "Align each index with its term;length=1 has no adjacent differences.Do not deduplicate term values or require them to be natural numbers."
      },
      "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": [
      [
        0
      ],
      [
        "0"
      ],
      []
    ]
  },
  "conclusion": {
    "canonical_value": "([0], [0], [])",
    "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": 27,
      "original_line_sha256": "4793e6ad969b43bb80f43863fe8d50eb7b1de77b92e0267558729341949f84ba",
      "question_compact_sha256": "119d588a660ee5a753b4fe18eda2466436a8e8b62640ed67060da33173c0c0ed",
      "english_row_compact_sha256": "130f940937199f047ed23ab27bb0f49d04eebd13567c094c37e2c60bc3094cde",
      "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": "0aaa15f7b3c49522068467c7dd94cf4f90450fe4862d4e360f32f7bd6079eb35",
    "original_candidate_line_sha256": "0aaa15f7b3c49522068467c7dd94cf4f90450fe4862d4e360f32f7bd6079eb35",
    "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": 5,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "168ac02887e262f8f9099b9698cda605540ad30200b90c54b6455a5006564ad4",
      "arithmetic_steps": 2
    },
    "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."
  }
}