{
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  "question_id": "q-20261013-advance-20261020-r05-dimensional-analysis-0086",
  "pilot_number": "R0001-01486",
  "number": 1486,
  "family_id": "qf-app20-dimension-equation-consistency",
  "title": {
    "zh-Hant": "第 1486 題：量綱指數向量",
    "en": "Question 1486: Equal Dimension Vectors Are Not a Physical Proof"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261020-r05-dimensional-analysis-0086",
    "batch_id": "batch-20261013-advance-20261020-r05-dimensional-analysis-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-dimensional-analysis"
    ],
    "title_zh": "量綱指數向量",
    "stem_zh": "基本量綱指數向量[L,M,T]。操作=dimension-equation-consistency；資料={\"left\":[0,0,3],\"right\":[0,0,3]}。請依 [left,right,consistent] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([0, 0, 3], [0, 0, 3], 是)",
      "explanation_zh": "應用基礎核對=([0, 0, 3], [0, 0, 3], 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；量綱一致是必要而非充分條件；不證物理模型正確。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "量綱指數向量"
      ]
    },
    "variant": {
      "family_id": "qf-app20-dimension-equation-consistency",
      "seed": 202610205086,
      "parameters": {
        "left": [
          0,
          0,
          3
        ],
        "right": [
          0,
          0,
          3
        ],
        "domain": "Dimension_exponent_vectors",
        "operation": "dimension-equation-consistency",
        "semantic_template": "qf-app20-dimension-equation-consistency/{\"left\":[0,0,3],\"right\":[0,0,3]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261020-r05-dimensional-analysis-0086",
    "batch_id": "batch-20261013-advance-20261020-r05-dimensional-analysis-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "6a9e8293f107a4f171af07b244faa23c36f9ae792982d9dddca58f20d75d8953",
      "family_id": "qf-app20-dimension-equation-consistency",
      "seed": 202610205086,
      "semantic_template": "qf-app20-dimension-equation-consistency/{\"left\":[0,0,3],\"right\":[0,0,3]}"
    },
    "fields": {
      "title": "Equal Dimension Vectors Are Not a Physical Proof",
      "stem": "Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, 0, 3], right=[0, 0, 3]. Give (left vector, right vector, identical?).",
      "answer": {
        "kind": "expression",
        "value": "([0, 0, 3], [0, 0, 3], Yes)",
        "explanation": "Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 0, 3], [0, 0, 3], Yes)."
      },
      "skills": [
        "Equal Dimension Vectors Are Not a Physical Proof"
      ],
      "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": "基本量綱指數向量[L,M,T]。操作=dimension-equation-consistency；資料={\"left\":[0,0,3],\"right\":[0,0,3]}。請依 [left,right,consistent] 欄序回答。",
        "en": "Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, 0, 3], right=[0, 0, 3]. Give (left vector, right vector, identical?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "dimension coordinate order=[L,M,T]",
      "after": "left/base=[0, 0, 3]; right=[0, 0, 3]",
      "relation": "implies",
      "rule_id": "dimension_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這些是量綱的指數，不是三個量值；順序固定為長度、質量、時間。",
        "en": "These are dimension exponents,not measured magnitudes;the order is length,mass,time."
      },
      "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": "L exponent difference := sub(0, 0)",
      "after": "L exponent difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "等式兩邊每一基本量綱的指數都必須相同。",
        "en": "The exponent of every base dimension must agree across an equation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 L exponent difference，不以小數近似替代。",
          "en": "Recompute L exponent difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "L exponent difference"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "M exponent difference := sub(0, 0)",
      "after": "M exponent difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "等式兩邊每一基本量綱的指數都必須相同。",
        "en": "The exponent of every base dimension must agree across an equation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M exponent difference，不以小數近似替代。",
          "en": "Recompute M exponent difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "M exponent difference"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "T exponent difference := sub(3, 3)",
      "after": "T exponent difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "等式兩邊每一基本量綱的指數都必須相同。",
        "en": "The exponent of every base dimension must agree across an equation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 T exponent difference，不以小數近似替代。",
          "en": "Recompute T exponent difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "3"
        ],
        "output": "0",
        "label": "T exponent difference"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "[是, 是, 是]",
      "after": "dimensionally consistent: 是",
      "relation": "implies",
      "rule_id": "all_dimension_equalities",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只核對量綱相容的必要條件；相容不代表數值方程真的成立。",
        "en": "This checks a necessary dimensional condition;consistency does not prove numerical equality."
      },
      "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": {
    "left": [
      0,
      0,
      3
    ],
    "right": [
      0,
      0,
      3
    ],
    "consistent": true
  },
  "conclusion": {
    "canonical_value": "([0, 0, 3], [0, 0, 3], 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."
  },
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      "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."
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