{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261010-advance-20261014-r06-absolute-value-0051",
  "pilot_number": "R0059-30226",
  "number": 30226,
  "family_id": "qf-density14-absolute-closed-radius-mask",
  "title": {
    "zh-Hant": "第 30226 題：閉半徑的四點資格",
    "en": "Question 30226: Inclusive Absolute-Value Threshold Mask"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261010-advance-20261014-r06-absolute-value-0051",
    "batch_id": "batch-20261010-advance-20261014-r06-absolute-value-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-absolute-value"
    ],
    "title_zh": "閉半徑的四點資格",
    "stem_zh": "有理嵌入的絕對值協議；r=0/1≥0，points=[0/1, 1/1, -1/1, 0/-2]。依序列 (r, 四個abs(point), 四個abs(point)≤r資格)，包含端點。",
    "answer": {
      "kind": "expression",
      "value": "(0/1, [0/1, 1/1, 1/1, 0/1], [是, 否, 否, 是])",
      "explanation_zh": "閉半徑與絕對值資格=(0/1, [0/1, 1/1, 1/1, 0/1], [是, 否, 否, 是])。先取各絕對值，再用精確閉門檻比較，r=0合法；有限門檻代入不是一般不等式求解。有限分段核對非全部實數性質證明。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 1,
      "skills": [
        "閉半徑的四點資格"
      ]
    },
    "variant": {
      "family_id": "qf-density14-absolute-closed-radius-mask",
      "seed": 202610146051,
      "parameters": {
        "r": [
          0,
          1
        ],
        "points": [
          [
            0,
            1
          ],
          [
            1,
            1
          ],
          [
            -1,
            1
          ],
          [
            0,
            -2
          ]
        ],
        "domain": "finite_rational_absolute_protocol",
        "operation": "density14_absolute_closed_radius_mask",
        "semantic_template": "qf-density14-absolute-closed-radius-mask/[\"0/1\",[\"0/1\",\"1/1\",\"1/1\",\"0/1\"]]"
      }
    },
    "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": "2026-10-10",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261010-advance-20261014-r06-absolute-value-0051",
    "batch_id": "batch-20261010-advance-20261014-r06-absolute-value-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "301eb0006f4c560d1d0a2a65c0239f2d32d3a6b48af1956a7ecf0c6bd2a2eba5",
      "family_id": "qf-density14-absolute-closed-radius-mask",
      "seed": 202610146051,
      "semantic_template": "qf-density14-absolute-closed-radius-mask/[\"0/1\",[\"0/1\",\"1/1\",\"1/1\",\"0/1\"]]"
    },
    "fields": {
      "title": "Inclusive Absolute-Value Threshold Mask",
      "stem": "Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=0/1≥0, ordered points=[0/1, 1/1, -1/1, 0/-2]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.",
      "answer": {
        "kind": "expression",
        "value": "(0/1, [0/1, 1/1, 1/1, 0/1], [Yes, No, No, Yes])",
        "explanation": "Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (0/1, [0/1, 1/1, 1/1, 0/1], [Yes, No, No, Yes])."
      },
      "skills": [
        "Inclusive Absolute-Value Threshold Mask"
      ],
      "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": "有理嵌入的絕對值協議；r=0/1≥0，points=[0/1, 1/1, -1/1, 0/-2]。依序列 (r, 四個abs(point), 四個abs(point)≤r資格)，包含端點。",
        "en": "Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=0/1≥0, ordered points=[0/1, 1/1, -1/1, 0/-2]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-density14-absolute-closed-radius-mask, density14_absolute_closed_radius_mask, finite_rational_absolute_protocol]",
      "after": "derived = [[radius, ALL_absolute_values, ALL_inclusive_threshold_flags]]",
      "relation": "implies",
      "rule_id": "R59_EXACT_family_domain_FULLprefix_operation_and_answer_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原bank的完整operation前綴、domain及輸出欄序；frame只用orthogonal_equal_unit，角度unit只用degrees。分數整數保/1；不增不存在的參數，不改原中英來源。",
        "en": "Retain the ORIGINAL bank full operation prefix,domain and ordered answer roles.Use orthogonal_equal_unit only where a frame is declared,and degrees where an angle unit is declared.Integer fractions retain/1.Do not invent parameters or alter original bilingual sources."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。",
          "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [Absolute target r, [0, 1]]",
      "after": "derived = [0/1, 0, 1]",
      "relation": "implies",
      "rule_id": "Absolute_RAW_SAFE_pair_NONZERO_den_normalize_SIGN_and_gcd",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原pair兩數SAFE且den非0；這個bank允許負den，先正分母約分才按值選sign。不套primitive正den或STR22 component cap。",
        "en": "Require a RAW SAFE pair with nonzero denominator.This bank ALLOWS negative denominators;normalize sign/gcd BEFORE deciding the value sign.Do not import a primitive positive-denominator rule or a STR22 component cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。",
          "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source roles = [Absolute ORIGINAL points slot0, [0, 1]]",
      "after": "derived = [0/1, 0, 1]",
      "relation": "implies",
      "rule_id": "Absolute_RAW_SAFE_pair_NONZERO_den_normalize_SIGN_and_gcd",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原pair兩數SAFE且den非0；這個bank允許負den，先正分母約分才按值選sign。不套primitive正den或STR22 component cap。",
        "en": "Require a RAW SAFE pair with nonzero denominator.This bank ALLOWS negative denominators;normalize sign/gcd BEFORE deciding the value sign.Do not import a primitive positive-denominator rule or a STR22 component cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。",
          "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [Absolute ORIGINAL points slot1, [1, 1]]",
      "after": "derived = [1/1, 1, 1]",
      "relation": "implies",
      "rule_id": "Absolute_RAW_SAFE_pair_NONZERO_den_normalize_SIGN_and_gcd",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原pair兩數SAFE且den非0；這個bank允許負den，先正分母約分才按值選sign。不套primitive正den或STR22 component cap。",
        "en": "Require a RAW SAFE pair with nonzero denominator.This bank ALLOWS negative denominators;normalize sign/gcd BEFORE deciding the value sign.Do not import a primitive positive-denominator rule or a STR22 component cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。",
          "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [Absolute ORIGINAL points slot2, [-1, 1]]",
      "after": "derived = [-1/1, -1, 1]",
      "relation": "implies",
      "rule_id": "Absolute_RAW_SAFE_pair_NONZERO_den_normalize_SIGN_and_gcd",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原pair兩數SAFE且den非0；這個bank允許負den，先正分母約分才按值選sign。不套primitive正den或STR22 component cap。",
        "en": "Require a RAW SAFE pair with nonzero denominator.This bank ALLOWS negative denominators;normalize sign/gcd BEFORE deciding the value sign.Do not import a primitive positive-denominator rule or a STR22 component cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。",
          "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [Absolute ORIGINAL points slot3, [0, -2]]",
      "after": "derived = [0/1, 0, 1]",
      "relation": "implies",
      "rule_id": "Absolute_RAW_SAFE_pair_NONZERO_den_normalize_SIGN_and_gcd",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原pair兩數SAFE且den非0；這個bank允許負den，先正分母約分才按值選sign。不套primitive正den或STR22 component cap。",
        "en": "Require a RAW SAFE pair with nonzero denominator.This bank ALLOWS negative denominators;normalize sign/gcd BEFORE deciding the value sign.Do not import a primitive positive-denominator rule or a STR22 component cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。",
          "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "source roles = [Absolute ORIGINAL points slot0, 0]",
      "after": "derived = [是, 0]",
      "relation": "implies",
      "rule_id": "Absolute_value_normalized_sign_branch_ZERO_is_nonnegative",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以已正規化數值的sign選分段：x>=0保x，x<0取0-x；zero在非負分支，不因raw負分母而誤判。",
        "en": "Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。",
          "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "source roles = [Absolute ORIGINAL points slot1, 1]",
      "after": "derived = [是, 1]",
      "relation": "implies",
      "rule_id": "Absolute_value_normalized_sign_branch_ZERO_is_nonnegative",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以已正規化數值的sign選分段：x>=0保x，x<0取0-x；zero在非負分支，不因raw負分母而誤判。",
        "en": "Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。",
          "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Absolute ORIGINAL points slot2 ORIGINAL negative branch negation := sub(0, -1)",
      "after": "Absolute ORIGINAL points slot2 ORIGINAL negative branch negation = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。",
        "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Absolute ORIGINAL points slot2 ORIGINAL negative branch negation，不以小數近似替代。",
          "en": "Recompute Absolute ORIGINAL points slot2 ORIGINAL negative branch negation with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "-1"
        ],
        "output": "1",
        "label": "Absolute ORIGINAL points slot2 ORIGINAL negative branch negation"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "source roles = [Absolute ORIGINAL points slot2, -1]",
      "after": "derived = [否, 1]",
      "relation": "implies",
      "rule_id": "Absolute_value_normalized_sign_branch_ZERO_is_nonnegative",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以已正規化數值的sign選分段：x>=0保x，x<0取0-x；zero在非負分支，不因raw負分母而誤判。",
        "en": "Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。",
          "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "source roles = [Absolute ORIGINAL points slot3, 0]",
      "after": "derived = [是, 0]",
      "relation": "implies",
      "rule_id": "Absolute_value_normalized_sign_branch_ZERO_is_nonnegative",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以已正規化數值的sign選分段：x>=0保x，x<0取0-x；zero在非負分支，不因raw負分母而誤判。",
        "en": "Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。",
          "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Absolute ORIGINAL magnitude minus target slot0 := sub(0, 0)",
      "after": "Absolute ORIGINAL magnitude minus target slot0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。",
        "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Absolute ORIGINAL magnitude minus target slot0，不以小數近似替代。",
          "en": "Recompute Absolute ORIGINAL magnitude minus target slot0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "Absolute ORIGINAL magnitude minus target slot0"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Absolute ORIGINAL magnitude minus target slot1 := sub(1, 0)",
      "after": "Absolute ORIGINAL magnitude minus target slot1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。",
        "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Absolute ORIGINAL magnitude minus target slot1，不以小數近似替代。",
          "en": "Recompute Absolute ORIGINAL magnitude minus target slot1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "Absolute ORIGINAL magnitude minus target slot1"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Absolute ORIGINAL magnitude minus target slot2 := sub(1, 0)",
      "after": "Absolute ORIGINAL magnitude minus target slot2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。",
        "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Absolute ORIGINAL magnitude minus target slot2，不以小數近似替代。",
          "en": "Recompute Absolute ORIGINAL magnitude minus target slot2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "Absolute ORIGINAL magnitude minus target slot2"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Absolute ORIGINAL magnitude minus target slot3 := sub(0, 0)",
      "after": "Absolute ORIGINAL magnitude minus target slot3 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。",
        "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Absolute ORIGINAL magnitude minus target slot3，不以小數近似替代。",
          "en": "Recompute Absolute ORIGINAL magnitude minus target slot3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "Absolute ORIGINAL magnitude minus target slot3"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "source roles = [0/1, [0/1, 1/1, -1/1, 0/1]]",
      "after": "derived = [[0/1, 1/1, 1/1, 0/1], [0/1, 1/1, 1/1, 0/1], [是, 否, 否, 是]]",
      "relation": "implies",
      "rule_id": "Absolute_CLOSED_radius_FOUR_original_positions_INCLUSIVE",
      "conditions": [],
      "explanation": {
        "zh-Hant": "radius須非負，abs(point)<=r含等號；r0亦合法，只有零值過關。保全部四個原位置，不把有限mask說成一般不等式全解。",
        "en": "The radius must be nonnegative,and abs(point)<=r INCLUDES equality.At r=0 only a zero value qualifies.Retain ALL four original positions;this finite mask is not a general inequality solution."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。",
          "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "0/1",
      [
        "0/1",
        "1/1",
        "1/1",
        "0/1"
      ],
      [
        true,
        false,
        false,
        true
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "(0/1, [0/1, 1/1, 1/1, 0/1], [是, 否, 否, 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-20261010-advance-20261014-r06-absolute-value-foundations",
      "batch_path": "public/data/questions/batches/batch-20261010-advance-20261014-r06-absolute-value-foundations/questions.jsonl",
      "batch_sha256": "6f64d92445cd0622999a4aea7165b9a1bc243bbea1361e5440354586968dac04",
      "original_line": 51,
      "original_line_sha256": "c62a7f27ea5c3b1aebfa6b24e7ac65f4f2c7f9b11cf5d854d4458405fc938bb4",
      "question_compact_sha256": "301eb0006f4c560d1d0a2a65c0239f2d32d3a6b48af1956a7ecf0c6bd2a2eba5",
      "english_row_compact_sha256": "1c6a49118400f820ba67dc5273df6e6bcb6e3f7c0e2a6b567e1be8e8204e65e6",
      "english_manifest_path": "public/data/i18n/editions/batch-20261010-advance-20261014-r06-absolute-value-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261010-advance-20261014-r06-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": "17d593a551933064f195bd9277d5085fe2b6c98e11cbda0c8aa27d11b7b07949",
    "original_candidate_sha256": "53c34fe33d0f7d8a7e0e6e08699a4eea78f960eeb4638618b29c10e47d10cbe4",
    "original_candidate_line_sha256": "53c34fe33d0f7d8a7e0e6e08699a4eea78f960eeb4638618b29c10e47d10cbe4"
  },
  "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-20261026-011-p60",
    "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": "4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5",
      "question_first_witness_sha256": "a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 16,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "49f292eadff208cf781176323f48e1afa5414d4f1df5548dbe9312e8f3b61bd7",
      "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."
  }
}