{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261009-advance-20261013-r01-pythagorean-theorem-0077",
  "pilot_number": "R0011-06302",
  "number": 6302,
  "family_id": "qf-density13-pythagorean-candidate-residual",
  "title": {
    "zh-Hant": "第 6302 題：候選斜邊的正性與平方殘差",
    "en": "Question 6302: Hypotenuse Sign and Square Residual"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261009-advance-20261013-r01-pythagorean-theorem-0077",
    "batch_id": "batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-euclid-pythagorean-theorem"
    ],
    "title_zh": "候選斜邊的正性與平方殘差",
    "stem_zh": "歐氏直角邊長模型；正股長，a=3、b=4，待核候選c=5。依序答 (c是否正, c²−a²−b², 正且殘差零是否)，c只作候選。",
    "answer": {
      "kind": "expression",
      "value": "(是, 0, 是)",
      "explanation_zh": "斜邊候選資格與殘差=(是, 0, 是)。正性與平方殘差分開核對，候選非正斜邊不合資格；有限邊長檢查非新普遍證明，未完成任意仿射橋接。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 1,
      "skills": [
        "候選斜邊的正性與平方殘差"
      ]
    },
    "variant": {
      "family_id": "qf-density13-pythagorean-candidate-residual",
      "seed": 202610131077,
      "parameters": {
        "a": 3,
        "b": 4,
        "c": 5,
        "geometry": "euclidean_right_side_model",
        "operation": "density13_pythagorean_candidate_residual",
        "domain": "positive_integer_right_side_protocol",
        "semantic_template": "qf-density13-pythagorean-candidate-residual/[\"25\",5]"
      }
    },
    "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": "2026-10-09",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261009-advance-20261013-r01-pythagorean-theorem-0077",
    "batch_id": "batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "415a2f500b3ca50b56b4392b88dd7378c254a38059c2e0017e7bd4684066c8d3",
      "family_id": "qf-density13-pythagorean-candidate-residual",
      "seed": 202610131077,
      "semantic_template": "qf-density13-pythagorean-candidate-residual/[\"25\",5]"
    },
    "fields": {
      "title": "Hypotenuse Sign and Square Residual",
      "stem": "Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=3, b=4, candidate c=5. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.",
      "answer": {
        "kind": "expression",
        "value": "(Yes, 0, Yes)",
        "explanation": "Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, 0, Yes)."
      },
      "skills": [
        "Hypotenuse Sign and Square Residual"
      ],
      "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": "歐氏直角邊長模型；正股長，a=3、b=4，待核候選c=5。依序答 (c是否正, c²−a²−b², 正且殘差零是否)，c只作候選。",
        "en": "Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=3, b=4, candidate c=5. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "Euclidean right-side model;positive leg a=3",
      "after": "use the supplied right-angle premise:a²+b²=c²",
      "relation": "implies",
      "rule_id": "pythagorean_right_side_premise",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用已宣告的歐氏直角邊長模型及正股長前提；此有限數值核對不是畢氏定理的新普遍證明，也不建立任意仿射幾何橋接。",
        "en": "Use the declared Euclidean right-side model with positive legs.This finite numerical check is not a new universal proof of the Pythagorean theorem or an arbitrary affine-geometric bridge."
      },
      "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": "known leg a squared := square(3)",
      "after": "known leg a squared = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先完整平方已知正股長a。",
        "en": "Square the complete known positive leg a."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 known leg a squared，不以小數近似替代。",
          "en": "Recompute known leg a squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "3"
        ],
        "output": "9",
        "label": "known leg a squared"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "leg b squared := square(4)",
      "after": "leg b squared = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "另一正股長b獨立平方。",
        "en": "Square the other positive leg b separately."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 leg b squared，不以小數近似替代。",
          "en": "Recompute leg b squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "4"
        ],
        "output": "16",
        "label": "leg b squared"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "sum of leg squares := sum(9, 16)",
      "after": "sum of leg squares = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "股平方相加是模型中的斜邊平方，不是斜邊本身。",
        "en": "The sum of leg squares is the model's squared hypotenuse,not the hypotenuse itself."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum of leg squares，不以小數近似替代。",
          "en": "Recompute sum of leg squares with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "16"
        ],
        "output": "25",
        "label": "sum of leg squares"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "candidate c squared := square(5)",
      "after": "candidate c squared = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "候選可非正，平方檢查與正性資格分開。",
        "en": "A submitted candidate may be nonpositive;check its square separately from positive-length eligibility."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate c squared，不以小數近似替代。",
          "en": "Recompute candidate c squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "5"
        ],
        "output": "25",
        "label": "candidate c squared"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "candidate signed square residual := sub(25, 25)",
      "after": "candidate signed square residual = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "方向為c²−a²−b²，保留正負號。",
        "en": "The signed residual is c²−a²−b²;retain its sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate signed square residual，不以小數近似替代。",
          "en": "Recompute candidate signed square residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "25",
          "25"
        ],
        "output": "0",
        "label": "candidate signed square residual"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "candidate c=5;positive=是;residual=0",
      "after": "positive correct hypotenuse=是",
      "relation": "implies",
      "rule_id": "positive_hypotenuse_candidate_conjunction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正性與殘差0要同時成立；c=−5即使對3、4股給殘差0，也不是正斜邊。",
        "en": "Require both positivity and residual0;for legs3,4,the candidate−5 has residual0 but is not a positive hypotenuse."
      },
      "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": [
      true,
      "0",
      true
    ],
    "leg_square_sum": "25"
  },
  "conclusion": {
    "canonical_value": "(是, 0, 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": {
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    "question_binding": {
      "batch_id": "batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations",
      "batch_path": "public/data/questions/batches/batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations/questions.jsonl",
      "batch_sha256": "5f116273675c3b113e25fd685584bb2e75a21cdb3412eedd9a4ce581b6ff5ace",
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      "original_line_sha256": "ecd8ad7d0216547ba57b271652a4848d9510c027f9d0f49a08f4f96cd5f5b98e",
      "question_compact_sha256": "415a2f500b3ca50b56b4392b88dd7378c254a38059c2e0017e7bd4684066c8d3",
      "english_row_compact_sha256": "4bac4a9a13f73560a885aa41c4e5b4301a6ce773b65663d240b2e197828666eb",
      "english_manifest_path": "public/data/i18n/editions/batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations.en.jsonl"
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    "mko_bindings": {
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        "original_path": "public/data/mko/pythagorean-theorem.json",
        "english_path": "public/data/i18n/editions/mko-euclid-pythagorean-theorem.en.json"
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    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "5b2ce631281eb24b279908f47c6de66fa60d326520852879d324c4f3ac653970",
    "original_candidate_sha256": "6e526794c1bb104f98fddfdef6186458034e18c1d7e78c24cfa3d9094fd41c49",
    "original_candidate_line_sha256": "6e526794c1bb104f98fddfdef6186458034e18c1d7e78c24cfa3d9094fd41c49"
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  "historical_source_state": {
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      "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-p12",
    "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": {
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    },
    "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": "6b65a98b917c509018ad3a666d7369f9d6a363e701c4ff09a53d5b864b229a10",
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    "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."
  }
}