{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20260928-absolute-value-piecewise-branch-0014",
  "pilot_number": "R0001-01039",
  "number": 1039,
  "family_id": "qf-absolute-value-piecewise-branch",
  "title": {
    "zh-Hant": "第 1039 題：絕對值分段判定",
    "en": "Question 1039: Piecewise Branch of Absolute Value"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20260928-absolute-value-piecewise-branch-0014",
    "batch_id": "batch-20260928-absolute-value-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-absolute-value"
    ],
    "title_zh": "絕對值分段判定",
    "stem_zh": "對實數 x=-1，應使用 |x| 分段定義中的哪個分支？請回答「非負分支 x」或「負分支 -x」。",
    "answer": {
      "kind": "short_text",
      "value": "負分支 -x",
      "explanation_zh": "x=-1<0，所以使用 negative 分支，|x|=-x=1。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 3,
      "skills": [
        "零的邊界",
        "分段函數"
      ]
    },
    "variant": {
      "family_id": "qf-absolute-value-piecewise-branch",
      "seed": 20261042,
      "parameters": {
        "x": -1,
        "domain": "reals",
        "operation": "piecewise_branch",
        "semantic_template": "qf-absolute-value-piecewise-branch/{\"domain\":\"reals\",\"operation\":\"piecewise_branch\",\"x\":-1}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "json_schema",
        "independent_answer_recomputation"
      ],
      "warning_zh": "本題為 candidate；有限整數案例的精確重算不是全部實數性質的形式證明或人工審查。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "Codex GPT-5 (single-context degraded run)",
      "generated_at": "2026-09-28",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20260928-absolute-value-piecewise-branch-0014",
    "batch_id": "batch-20260928-absolute-value-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "88c46e689740ba253b0e341f1936416b51f5892946fbc46d85cd3658722117a8",
      "family_id": "qf-absolute-value-piecewise-branch",
      "seed": 20261042,
      "semantic_template": "qf-absolute-value-piecewise-branch/{\"domain\":\"reals\",\"operation\":\"piecewise_branch\",\"x\":-1}"
    },
    "fields": {
      "title": "Piecewise Branch of Absolute Value",
      "stem": "For the real number x=-1, which branch of the piecewise definition of |x| applies?",
      "answer": {
        "kind": "short_text",
        "value": "negative branch -x",
        "explanation": "x=-1<0, so use the negative branch -x and obtain |x|=1."
      },
      "skills": [
        "zero boundary",
        "piecewise functions"
      ],
      "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": "對實數 x=-1，應使用 |x| 分段定義中的哪個分支？請回答「非負分支 x」或「負分支 -x」。",
        "en": "For the real number x=-1, which branch of the piecewise definition of |x| applies?"
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "x∈R",
      "after": "|x|=x if x≥0; |x|=−x if x<0",
      "relation": "implies",
      "rule_id": "absolute_definition",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個符號分支覆蓋全部實數，且0歸非負分支。",
        "en": "The sign branches cover every real number, with0 in the nonnegative branch."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "確認使用≥0而不是>0，不能把0遺漏。",
          "en": "Use ≥0 rather than >0 so that0 is not omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "x=-1",
      "after": "x≥0: 否",
      "relation": "implies",
      "rule_id": "sign_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先依實際給定值判斷符號，不從圖形猜測。",
        "en": "Determine the sign of the given value rather than guessing from a picture."
      },
      "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": "x=-1",
      "after": "負分支 -x",
      "relation": "implies",
      "rule_id": "select_absolute_branch",
      "conditions": [],
      "explanation": {
        "zh-Hant": "零也符合非負條件；負數才使用−x。",
        "en": "Zero satisfies the nonnegative condition; only negative values use −x."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "absolute value := mul(-1, -1)",
      "after": "absolute value = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依選定分支乘正1或負1，求出非負值。",
        "en": "Multiply by1 or −1 according to the selected branch to obtain a nonnegative value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 absolute value，不以小數近似替代。",
          "en": "Recompute absolute value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "1",
        "label": "absolute value"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "|x|=1",
      "after": "result≥0: 是",
      "relation": "implies",
      "rule_id": "absolute_result_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "核對結果非負，避免把負分支誤作負的絕對值。",
        "en": "Verify nonnegativity to avoid a negative result from the negative branch."
      },
      "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": {
    "x": -1,
    "nonnegative": false,
    "absolute": "1",
    "branch": "負分支 -x"
  },
  "conclusion": {
    "canonical_value": "負分支 -x",
    "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-20260928-absolute-value-foundations",
      "batch_path": "public/data/questions/batches/batch-20260928-absolute-value-foundations/questions.jsonl",
      "batch_sha256": "49b8ee3d45e2794c0becbb27af857e4727b0e8bb396a02f7a2366076a4788da8",
      "original_line": 39,
      "original_line_sha256": "af103a3b0f3bd51fe43d395e83eb882e74ee2277c9796e74a76f94be61fad976",
      "question_compact_sha256": "88c46e689740ba253b0e341f1936416b51f5892946fbc46d85cd3658722117a8",
      "english_row_compact_sha256": "828c991ec28a5f31385c700b30d370fbf0b42240ac78bb94d9e4af0e6adcee3d",
      "english_manifest_path": "public/data/i18n/editions/batch-20260928-absolute-value-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20260928-absolute-value-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-absolute-value": {
        "sha256": "7df0aed9ca41a0e67b84a464f450422348753d1171802220b41a103488fb5f95",
        "english_sha256": "7f191b7dbbd24ac7ec2a252a08d3178f961ef31414f76098c51f5f01d63473f6",
        "original_path": "public/data/mko/absolute-value.json",
        "english_path": "public/data/i18n/editions/mko-absolute-value.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b9fe70a7a264c90c20349be0449b2bd1a1b4d0e0a97092bc9fbcb7ca36ce0694",
    "original_candidate_sha256": "e31e4b052680ba1ec70950dfcfb134fb0dfeee709abd3658cb6efbed33973e37",
    "original_candidate_line_sha256": "e31e4b052680ba1ec70950dfcfb134fb0dfeee709abd3658cb6efbed33973e37"
  },
  "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-20261026-004-p2",
    "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": "c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496cc",
      "question_first_witness_sha256": "3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7ad"
    },
    "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": "464e7a7e428dd52f52a3e8a89bda96a7ec19b45dc58a0a5ec42e5628f2825782",
      "arithmetic_steps": 1
    },
    "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."
  }
}