{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261011-advance-20261015-r01-fraction-0032",
  "pilot_number": "R0012-06907",
  "number": 6907,
  "family_id": "qf-density15-fraction-adjacent-difference-chain",
  "title": {
    "zh-Hant": "第 6907 題：相鄰差鏈與端點差",
    "en": "Question 6907: Ordered Fraction Differences and Endpoint Check"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261011-advance-20261015-r01-fraction-0032",
    "batch_id": "batch-20261011-advance-20261015-r01-fraction-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-fraction"
    ],
    "title_zh": "相鄰差鏈與端點差",
    "stem_zh": "合法分數表示，原分母非零。原序四值=[(-6)/(3), (0)/(1), (1)/(2), (7)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。",
    "answer": {
      "kind": "expression",
      "value": "([2/1, 1/2, 3/1], 11/2, 11/2, 是)",
      "explanation_zh": "相鄰分數差鏈與端點差=([2/1, 1/2, 3/1], 11/2, 11/2, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"
    },
    "difficulty": {
      "band": "primary_foundation",
      "level": 1,
      "skills": [
        "相鄰差鏈與端點差"
      ]
    },
    "variant": {
      "family_id": "qf-density15-fraction-adjacent-difference-chain",
      "seed": 202610151032,
      "parameters": {
        "values": [
          [
            -6,
            3
          ],
          [
            0,
            1
          ],
          [
            1,
            2
          ],
          [
            7,
            2
          ]
        ],
        "domain": "finite_fraction_representation_protocol",
        "operation": "density15_fraction_adjacent_difference_chain",
        "semantic_template": "qf-density15-fraction-adjacent-difference-chain/[\"2/1\",\"1/2\",\"3/1\"]"
      }
    },
    "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-r01-fraction-0032",
    "batch_id": "batch-20261011-advance-20261015-r01-fraction-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "4f0ffe713973ab4a84c3182763a173383eb61a57619ecb0be16051263490d9d7",
      "family_id": "qf-density15-fraction-adjacent-difference-chain",
      "seed": 202610151032,
      "semantic_template": "qf-density15-fraction-adjacent-difference-chain/[\"2/1\",\"1/2\",\"3/1\"]"
    },
    "fields": {
      "title": "Ordered Fraction Differences and Endpoint Check",
      "stem": "Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-6/3, 0/1, 1/2, 7/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.",
      "answer": {
        "kind": "expression",
        "value": "([2/1, 1/2, 3/1], 11/2, 11/2, Yes)",
        "explanation": "Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([2/1, 1/2, 3/1], 11/2, 11/2, Yes)."
      },
      "skills": [
        "Ordered Fraction Differences and Endpoint Check"
      ],
      "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": "合法分數表示，原分母非零。原序四值=[(-6)/(3), (0)/(1), (1)/(2), (7)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。",
        "en": "Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-6/3, 0/1, 1/2, 7/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "original integer numerator/denominator representations;original denominators nonzero",
      "after": "representation and rational value are distinct data",
      "relation": "implies",
      "rule_id": "fraction_representation_value_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "負分母表示合法，但約分後的值不取代原始表示；涉及原分子分母的候選算法要用原資料。此有限核對不是商集良定性的一般證明。",
        "en": "Negative denominators are valid,but reduced values do not replace original representations.Candidate algorithms about raw numerators/denominators must use those raw data.This finite audit is not a general proof of quotient well-definedness."
      },
      "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": "ordered adjacent difference0 := sub(0, -2)",
      "after": "ordered adjacent difference0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原序取後值減前值，不排序、不省略重複值。",
        "en": "Subtract the preceding value from the next in original order,without sorting or deleting repeats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ordered adjacent difference0，不以小數近似替代。",
          "en": "Recompute ordered adjacent difference0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "-2"
        ],
        "output": "2",
        "label": "ordered adjacent difference0"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "ordered adjacent difference1 := sub(1/2, 0)",
      "after": "ordered adjacent difference1 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原序取後值減前值，不排序、不省略重複值。",
        "en": "Subtract the preceding value from the next in original order,without sorting or deleting repeats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ordered adjacent difference1，不以小數近似替代。",
          "en": "Recompute ordered adjacent difference1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1/2",
          "0"
        ],
        "output": "1/2",
        "label": "ordered adjacent difference1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "ordered adjacent difference2 := sub(7/2, 1/2)",
      "after": "ordered adjacent difference2 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原序取後值減前值，不排序、不省略重複值。",
        "en": "Subtract the preceding value from the next in original order,without sorting or deleting repeats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ordered adjacent difference2，不以小數近似替代。",
          "en": "Recompute ordered adjacent difference2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "7/2",
          "1/2"
        ],
        "output": "3",
        "label": "ordered adjacent difference2"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "adjacent difference chain sum := sum(2, 1/2, 3)",
      "after": "adjacent difference chain sum = 11/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把三個有號差完整加總。",
        "en": "Add all three signed adjacent differences."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 adjacent difference chain sum，不以小數近似替代。",
          "en": "Recompute adjacent difference chain sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "1/2",
          "3"
        ],
        "output": "11/2",
        "label": "adjacent difference chain sum"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "direct final minus initial := sub(7/2, -2)",
      "after": "direct final minus initial = 11/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "另由末值減初值，獨立核對端點差。",
        "en": "Separately subtract initial from final value to audit the endpoint difference."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 direct final minus initial，不以小數近似替代。",
          "en": "Recompute direct final minus initial with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "7/2",
          "-2"
        ],
        "output": "11/2",
        "label": "direct final minus initial"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "ordered values=[-2/1, 0/1, 1/2, 7/2];differences=[2/1, 1/2, 3/1]",
      "after": "chain sum=11/2;direct endpoint difference=11/2",
      "relation": "implies",
      "rule_id": "ordered_fraction_telescope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "(v1−v0)+(v2−v1)+(v3−v2)的中間項相消，保留方向；這不是把值集合排序後再求差。",
        "en": "The intermediate terms cancel in(v1−v0)+(v2−v1)+(v3−v2),retaining direction.This is not a difference chain of a sorted value set."
      },
      "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": [
      [
        "2",
        "1/2",
        "3"
      ],
      "11/2",
      "11/2",
      true
    ]
  },
  "conclusion": {
    "canonical_value": "([2/1, 1/2, 3/1], 11/2, 11/2, 是)",
    "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-r01-fraction-foundations",
      "batch_path": "public/data/questions/batches/batch-20261011-advance-20261015-r01-fraction-foundations/questions.jsonl",
      "batch_sha256": "33621867facfc78fb3d0ff06bd432e33a58b1514c00596b9334e3a66c8e322fa",
      "original_line": 32,
      "original_line_sha256": "6a97dd616f9bdadcb39fa2521e9d36f8010d95984afa5690620ff2418e7db5f1",
      "question_compact_sha256": "4f0ffe713973ab4a84c3182763a173383eb61a57619ecb0be16051263490d9d7",
      "english_row_compact_sha256": "075162e9ca528e7f1c0b3c3bbbf8ff3e1e705b5566ac5a771fb0ba1e825c98e7",
      "english_manifest_path": "public/data/i18n/editions/batch-20261011-advance-20261015-r01-fraction-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261011-advance-20261015-r01-fraction-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-fraction": {
        "sha256": "aa89902ad405685be331872579647cec33e7ab6792273fe60ad8fefe6129f919",
        "english_sha256": "ea8b15d28832a77ca243cb2c04a33ac67244aca632c895666d9a33fa9b4982d4",
        "original_path": "public/data/mko/fraction.json",
        "english_path": "public/data/i18n/editions/mko-fraction.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "c5c612ca91797c328f4489939ec6f7d885385e63ea21b334c09c421f17933d99",
    "original_candidate_sha256": "c577b67d1113eedd45425d0d36a3b812f8f42032725d9176294687225dc29228",
    "original_candidate_line_sha256": "c577b67d1113eedd45425d0d36a3b812f8f42032725d9176294687225dc29228"
  },
  "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-p13",
    "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": "90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00",
      "question_first_witness_sha256": "b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 7,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "222ff54c6153c535216027710756d0e32dc79d804464adc9f126ad05477fb9ef",
      "arithmetic_steps": 5
    },
    "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."
  }
}