{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261012-r02-counting-principle-0060",
  "pilot_number": "R0009-05310",
  "number": 5310,
  "family_id": "qf-density12-counting-conditional-candidate-defect",
  "title": {
    "zh-Hant": "第 5310 題：條件分支的等寬資格與計數缺陷",
    "en": "Question 5310: Actual Branch Widths versus a Candidate Width"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261012-r02-counting-principle-0060",
    "batch_id": "batch-20261008-advance-20261012-r02-counting-principle-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-counting-principle"
    ],
    "title_zh": "條件分支的等寬資格與計數缺陷",
    "stem_zh": "有限選擇協議；第一階段各選擇的第二階段數依序counts=[0, 1]，n=2為分支數，待核共同寬度h=1。依序答 (實際總數, 假定n·h總數, 每個分支是否都等於h, n·h−實際總數)。",
    "answer": {
      "kind": "expression",
      "value": "(1, 2, 否, 1)",
      "explanation_zh": "條件分支資格與缺陷=(1, 2, 否, 1)。先加實際分支並逐一核對寬度；總數相同不保證等寬，空第一階段條件全稱為是，非概率獨立假設。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 2,
      "skills": [
        "條件分支的等寬資格與計數缺陷"
      ]
    },
    "variant": {
      "family_id": "qf-density12-counting-conditional-candidate-defect",
      "seed": 202610122060,
      "parameters": {
        "counts": [
          0,
          1
        ],
        "h": 1,
        "operation": "density12_counting_conditional_candidate_defect",
        "domain": "finite_tagged_counting_protocol",
        "semantic_template": "qf-density12-counting-conditional-candidate-defect/[2,1,1,false]"
      }
    },
    "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": "2026-10-08",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261008-advance-20261012-r02-counting-principle-0060",
    "batch_id": "batch-20261008-advance-20261012-r02-counting-principle-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "427dd8cfb809d915a892950982425e9d96883231d56c732f8f1bcd1f8089b444",
      "family_id": "qf-density12-counting-conditional-candidate-defect",
      "seed": 202610122060,
      "semantic_template": "qf-density12-counting-conditional-candidate-defect/[2,1,1,false]"
    },
    "fields": {
      "title": "Actual Branch Widths versus a Candidate Width",
      "stem": "Ordered second-stage branch counts=[0, 1], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).",
      "answer": {
        "kind": "expression",
        "value": "(1, 2, No, 1)",
        "explanation": "Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (1, 2, No, 1)."
      },
      "skills": [
        "Actual Branch Widths versus a Candidate Width"
      ],
      "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": "有限選擇協議；第一階段各選擇的第二階段數依序counts=[0, 1]，n=2為分支數，待核共同寬度h=1。依序答 (實際總數, 假定n·h總數, 每個分支是否都等於h, n·h−實際總數)。",
        "en": "Ordered second-stage branch counts=[0, 1], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "declared finite tagged choices or full-product protocol",
      "after": "combinatorial counts,not probabilities or independence",
      "relation": "implies",
      "rule_id": "finite_counting_protocol_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法需互斥標籤，乘法需題目明示允許完整跨階段組合；都不是概率獨立假設。",
        "en": "Addition uses disjoint tags and multiplication uses the declared full cross-stage protocol;neither is a probability-independence assumption."
      },
      "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": "actual branch total := sum(0, 1)",
      "after": "actual branch total = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "條件分支按各自實際寬度相加。",
        "en": "Add the actual widths of the conditional branches."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual branch total，不以小數近似替代。",
          "en": "Recompute actual branch total with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "actual branch total"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "assumed common-width total := mul(2, 1)",
      "after": "assumed common-width total = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "另算分支數n乘待核寬度h，尚未認定每支真的同寬。",
        "en": "Separately compute n times candidate width h without yet asserting equal branch widths."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 assumed common-width total，不以小數近似替代。",
          "en": "Recompute assumed common-width total with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "assumed common-width total"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "branch0 width=0;candidate h=1",
      "after": "matches=否",
      "relation": "implies",
      "rule_id": "each_branch_common_width_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分支查等寬；合計數相同仍可能由不同寬度抵消。",
        "en": "Audit every branch width;matching aggregate totals can still hide unequal widths."
      },
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "branch1 width=1;candidate h=1",
      "after": "matches=是",
      "relation": "implies",
      "rule_id": "each_branch_common_width_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分支查等寬；合計數相同仍可能由不同寬度抵消。",
        "en": "Audit every branch width;matching aggregate totals can still hide unequal widths."
      },
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "assumed minus actual total := sub(2, 1)",
      "after": "assumed minus actual total = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "缺陷方向固定為n·h減實際總數。",
        "en": "The signed defect is assumed n·h minus the actual total."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 assumed minus actual total，不以小數近似替代。",
          "en": "Recompute assumed minus actual total with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "1"
        ],
        "output": "1",
        "label": "assumed minus actual total"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "branch comparisons=[否, 是]",
      "after": "all equal h=否",
      "relation": "implies",
      "rule_id": "uniform_width_not_aggregate_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "空第一階段沒有分支，等寬全稱空真但實際結果數0；這與零階段空直積的一個空序列不同。",
        "en": "With no first-stage branches,the width universal is vacuously true but the actual count is0;this differs from a zero-stage full product's single empty sequence."
      },
      "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": [
      "1",
      "2",
      false,
      "1"
    ]
  },
  "conclusion": {
    "canonical_value": "(1, 2, 否, 1)",
    "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-20261008-advance-20261012-r02-counting-principle-foundations",
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      "batch_sha256": "0fe31383b4044b85bc1f62d7e619a7dc1b3a078203cd4e7413dfad935f5e17e9",
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      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261012-r02-counting-principle-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261012-r02-counting-principle-foundations.en.jsonl"
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    "mko_bindings": {
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        "original_path": "public/data/mko/counting-principle.json",
        "english_path": "public/data/i18n/editions/mko-counting-principle.en.json"
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    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "3132b1200be7a5de38191badc52fb1727d4dd9d4c2f26818de2e8a76426d2d3f",
    "original_candidate_sha256": "4e16ef21dba4060a7c7b6258408d00f80db9c1c4feffef24f9e36460f38501b4",
    "original_candidate_line_sha256": "4e16ef21dba4060a7c7b6258408d00f80db9c1c4feffef24f9e36460f38501b4"
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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-p10",
    "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": "b0834d0a02f4464b6e262e4135c492eee37abc78e08fd430a50600556c952a1f",
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    },
    "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."
  }
}