{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261108-r06-unique-prime-factorization-0065",
  "pilot_number": "R0041-21365",
  "number": 21365,
  "family_id": "qf-mod39-upf-uniqueness-audit",
  "title": {
    "zh-Hant": "第 21365 題：整數唯一質因數分解",
    "en": "Question 21365: Only Two Valid Prime Products Can Supply This Uniqueness Certificate"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261108-r06-unique-prime-factorization-0065",
    "batch_id": "batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-unique-prime-factorization"
    ],
    "title_zh": "整數唯一質因數分解",
    "stem_zh": "完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":282,\"factors_left\":[2,3,47],\"factors_right\":[47,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(是, 是, [2, 3, 47], [2, 3, 47], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [47, 是, 0, 47, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)",
      "explanation_zh": "Wave6第二層核對=(是, 是, [2, 3, 47], [2, 3, 47], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [47, 是, 0, 47, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "整數唯一質因數分解"
      ]
    },
    "variant": {
      "family_id": "qf-mod39-upf-uniqueness-audit",
      "seed": 202611086065,
      "parameters": {
        "n": 282,
        "factors_left": [
          2,
          3,
          47
        ],
        "factors_right": [
          47,
          2,
          3
        ],
        "domain": "Exact_integer_unique_prime_factorization",
        "operation": "upf-uniqueness-audit",
        "semantic_template": "qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,47],\"factors_right\":[2,3,47],\"n\":282}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261108-r06-unique-prime-factorization-0065",
    "batch_id": "batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "c2939a397714a361ac397d9317a7042fed3020cc37df6e5522d7632e68f7e555",
      "family_id": "qf-mod39-upf-uniqueness-audit",
      "seed": 202611086065,
      "semantic_template": "qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,47],\"factors_right\":[2,3,47],\"n\":282}"
    },
    "fields": {
      "title": "Only Two Valid Prime Products Can Supply This Uniqueness Certificate",
      "stem": "Use positive integer n=282>1, candidate factors_left=[2, 3, 47] and factors_right=[47, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(Yes, Yes, [2, 3, 47], [2, 3, 47], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [47, Yes, 0, 47, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)",
        "explanation": "Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 47], [2, 3, 47], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [47, Yes, 0, 47, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."
      },
      "skills": [
        "Only Two Valid Prime Products Can Supply This Uniqueness Certificate"
      ],
      "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": "完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":282,\"factors_left\":[2,3,47],\"factors_right\":[47,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use positive integer n=282>1, candidate factors_left=[2, 3, 47] and factors_right=[47, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=upf-uniqueness-audit;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題歷史介面標記保留；此處直接列有序輸出欄位，不依賴JavaScript。欄位或tag不是額外數學證據，固定No保留原聲稱範圍。",
        "en": "Retain the historical interface marker while listing ordered output fields without JavaScript.Fields and tags are not extra mathematical evidence;fixed No markers retain their claim scope."
      },
      "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": "upf LEFT factor index0:ORIGINAL candidate factor=2;ALL possible trial divisors2..floor(sqrt(v))=[]",
      "after": "ALL [divisor,remainder] rows=[];prime=是",
      "relation": "implies",
      "rule_id": "upf_every_trial_divisor_primality_not_product_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合數若大於1必有不超過平方根的因數；完整區間無整除才證prime，2的區間為空仍是prime。乘積相等不能替代此責任。",
        "en": "A composite integer greater than1 has a divisor at most its square root.Complete nondivisibility over that interval certifies primality,including the empty interval for2.Matching a product cannot replace this obligation."
      },
      "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": "upf LEFT complete product at original factor0 := mul(1, 2)",
      "after": "upf LEFT complete product at original factor0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf LEFT complete product at original factor0，不以小數近似替代。",
          "en": "Recompute upf LEFT complete product at original factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "upf LEFT complete product at original factor0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "upf LEFT factor index1:ORIGINAL candidate factor=3;ALL possible trial divisors2..floor(sqrt(v))=[]",
      "after": "ALL [divisor,remainder] rows=[];prime=是",
      "relation": "implies",
      "rule_id": "upf_every_trial_divisor_primality_not_product_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合數若大於1必有不超過平方根的因數；完整區間無整除才證prime，2的區間為空仍是prime。乘積相等不能替代此責任。",
        "en": "A composite integer greater than1 has a divisor at most its square root.Complete nondivisibility over that interval certifies primality,including the empty interval for2.Matching a product cannot replace this obligation."
      },
      "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": "upf LEFT complete product at original factor1 := mul(2, 3)",
      "after": "upf LEFT complete product at original factor1 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf LEFT complete product at original factor1，不以小數近似替代。",
          "en": "Recompute upf LEFT complete product at original factor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "3"
        ],
        "output": "6",
        "label": "upf LEFT complete product at original factor1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "upf LEFT factor index2:ORIGINAL candidate factor=47;ALL possible trial divisors2..floor(sqrt(v))=[2, 3, 4, 5, 6]",
      "after": "ALL [divisor,remainder] rows=[[2, 1], [3, 2], [4, 3], [5, 2], [6, 5]];prime=是",
      "relation": "implies",
      "rule_id": "upf_every_trial_divisor_primality_not_product_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合數若大於1必有不超過平方根的因數；完整區間無整除才證prime，2的區間為空仍是prime。乘積相等不能替代此責任。",
        "en": "A composite integer greater than1 has a divisor at most its square root.Complete nondivisibility over that interval certifies primality,including the empty interval for2.Matching a product cannot replace this obligation."
      },
      "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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "upf LEFT complete product at original factor2 := mul(6, 47)",
      "after": "upf LEFT complete product at original factor2 = 282",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf LEFT complete product at original factor2，不以小數近似替代。",
          "en": "Recompute upf LEFT complete product at original factor2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "47"
        ],
        "output": "282",
        "label": "upf LEFT complete product at original factor2"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "upf LEFT:ORIGINAL factors=[2, 3, 47];ALL primality flags=[是, 是, 是];original n=282",
      "after": "empty product convention1;actual complete product=282;all_prime=是;valid_prime_product=是",
      "relation": "implies",
      "rule_id": "upf_candidate_prime_and_complete_product_separate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原factor順序。空列表每項為prime是vacuous True但乘積1不等n>1；合數乘積即使吻合亦無prime factorization證書。SAFE是來源投影限制，不是定理真假判斷。",
        "en": "Keep original factor order.An empty list vacuously has all-prime entries,but its product1 differs from n>1.A matching composite product is not a prime factorization.The SAFE guard concerns source projection,not theorem truth."
      },
      "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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "upf RIGHT factor index0:ORIGINAL candidate factor=47;ALL possible trial divisors2..floor(sqrt(v))=[2, 3, 4, 5, 6]",
      "after": "ALL [divisor,remainder] rows=[[2, 1], [3, 2], [4, 3], [5, 2], [6, 5]];prime=是",
      "relation": "implies",
      "rule_id": "upf_every_trial_divisor_primality_not_product_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合數若大於1必有不超過平方根的因數；完整區間無整除才證prime，2的區間為空仍是prime。乘積相等不能替代此責任。",
        "en": "A composite integer greater than1 has a divisor at most its square root.Complete nondivisibility over that interval certifies primality,including the empty interval for2.Matching a product cannot replace this obligation."
      },
      "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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "upf RIGHT complete product at original factor0 := mul(1, 47)",
      "after": "upf RIGHT complete product at original factor0 = 47",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf RIGHT complete product at original factor0，不以小數近似替代。",
          "en": "Recompute upf RIGHT complete product at original factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "47"
        ],
        "output": "47",
        "label": "upf RIGHT complete product at original factor0"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "upf RIGHT factor index1:ORIGINAL candidate factor=2;ALL possible trial divisors2..floor(sqrt(v))=[]",
      "after": "ALL [divisor,remainder] rows=[];prime=是",
      "relation": "implies",
      "rule_id": "upf_every_trial_divisor_primality_not_product_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合數若大於1必有不超過平方根的因數；完整區間無整除才證prime，2的區間為空仍是prime。乘積相等不能替代此責任。",
        "en": "A composite integer greater than1 has a divisor at most its square root.Complete nondivisibility over that interval certifies primality,including the empty interval for2.Matching a product cannot replace this obligation."
      },
      "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": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "upf RIGHT complete product at original factor1 := mul(47, 2)",
      "after": "upf RIGHT complete product at original factor1 = 94",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf RIGHT complete product at original factor1，不以小數近似替代。",
          "en": "Recompute upf RIGHT complete product at original factor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "47",
          "2"
        ],
        "output": "94",
        "label": "upf RIGHT complete product at original factor1"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "upf RIGHT factor index2:ORIGINAL candidate factor=3;ALL possible trial divisors2..floor(sqrt(v))=[]",
      "after": "ALL [divisor,remainder] rows=[];prime=是",
      "relation": "implies",
      "rule_id": "upf_every_trial_divisor_primality_not_product_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合數若大於1必有不超過平方根的因數；完整區間無整除才證prime，2的區間為空仍是prime。乘積相等不能替代此責任。",
        "en": "A composite integer greater than1 has a divisor at most its square root.Complete nondivisibility over that interval certifies primality,including the empty interval for2.Matching a product cannot replace this obligation."
      },
      "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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "upf RIGHT complete product at original factor2 := mul(94, 3)",
      "after": "upf RIGHT complete product at original factor2 = 282",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf RIGHT complete product at original factor2，不以小數近似替代。",
          "en": "Recompute upf RIGHT complete product at original factor2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "94",
          "3"
        ],
        "output": "282",
        "label": "upf RIGHT complete product at original factor2"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "upf RIGHT:ORIGINAL factors=[47, 2, 3];ALL primality flags=[是, 是, 是];original n=282",
      "after": "empty product convention1;actual complete product=282;all_prime=是;valid_prime_product=是",
      "relation": "implies",
      "rule_id": "upf_candidate_prime_and_complete_product_separate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原factor順序。空列表每項為prime是vacuous True但乘積1不等n>1；合數乘積即使吻合亦無prime factorization證書。SAFE是來源投影限制，不是定理真假判斷。",
        "en": "Keep original factor order.An empty list vacuously has all-prime entries,but its product1 differs from n>1.A matching composite product is not a prime factorization.The SAFE guard concerns source projection,not theorem truth."
      },
      "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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "SORTED left occurrence index=0,factor=2;ORIGINAL right factors=[47, 2, 3];full right product=282",
      "after": "[p,p_divides_product,FIRST original right index,right factor,right factor equals p]=[2, 是, 1, 2, 是];same right index MAY be reused",
      "relation": "implies",
      "rule_id": "upf_local_first_right_witness_not_occurrence_matching",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列按原right輸入順序取首整除因數，不消耗該occurrence。若candidate含合數，這只是診斷列，不對合數p宣稱Euclid lemma；整體證書還需兩側有效與完整重數相等。",
        "en": "Choose the first divisible factor in original right order without consuming an occurrence.Invalid composite candidates give diagnostic rows,not Euclid-lemma claims for composite p.Joint validity still needs two valid prime products and all multiplicities equal."
      },
      "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": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "SORTED left occurrence index=1,factor=3;ORIGINAL right factors=[47, 2, 3];full right product=282",
      "after": "[p,p_divides_product,FIRST original right index,right factor,right factor equals p]=[3, 是, 2, 3, 是];same right index MAY be reused",
      "relation": "implies",
      "rule_id": "upf_local_first_right_witness_not_occurrence_matching",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列按原right輸入順序取首整除因數，不消耗該occurrence。若candidate含合數，這只是診斷列，不對合數p宣稱Euclid lemma；整體證書還需兩側有效與完整重數相等。",
        "en": "Choose the first divisible factor in original right order without consuming an occurrence.Invalid composite candidates give diagnostic rows,not Euclid-lemma claims for composite p.Joint validity still needs two valid prime products and all multiplicities equal."
      },
      "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": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "SORTED left occurrence index=2,factor=47;ORIGINAL right factors=[47, 2, 3];full right product=282",
      "after": "[p,p_divides_product,FIRST original right index,right factor,right factor equals p]=[47, 是, 0, 47, 是];same right index MAY be reused",
      "relation": "implies",
      "rule_id": "upf_local_first_right_witness_not_occurrence_matching",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列按原right輸入順序取首整除因數，不消耗該occurrence。若candidate含合數，這只是診斷列，不對合數p宣稱Euclid lemma；整體證書還需兩側有效與完整重數相等。",
        "en": "Choose the first divisible factor in original right order without consuming an occurrence.Invalid composite candidates give diagnostic rows,not Euclid-lemma claims for composite p.Joint validity still needs two valid prime products and all multiplicities equal."
      },
      "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": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "ORIGINAL n=282;left valid=是,right valid=是;sorted multisets=[[2, 3, 47], [2, 3, 47]];local witness rows=[[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [47, 是, 0, 47, 是]]",
      "after": "joint uniqueness certificate=是;general proof:prime p divides other product;Euclid lemma gives equal prime factor;match ONE occurrence,cancel nonzero p,then induct on remaining factor count",
      "relation": "implies",
      "rule_id": "upf_general_uniqueness_Euclid_nonzero_cancel_induction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "Euclid lemma可由gcd(p,a)=1時的Bezout等式乘b證：p|ab就得p|b；反覆用於有限乘積。匹配一個相同prime後以非零整數消去，歸納直到空積1；剩餘prime皆≥2不可乘成1。來源可重用的局部index本身不證全域重數匹配。",
        "en": "Euclid's lemma follows from multiplying a Bezout identity when gcd(p,a)=1:if p divides ab,it divides b.Iterate over the finite product,match one equal prime,cancel that nonzero factor,and induct to the empty product1;remaining primes≥2 cannot multiply to1.Reused local indices alone do not prove global multiplicity matching."
      },
      "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,
      true,
      [
        2,
        3,
        47
      ],
      [
        2,
        3,
        47
      ],
      [
        [
          2,
          true,
          1,
          2,
          true
        ],
        [
          3,
          true,
          2,
          3,
          true
        ],
        [
          47,
          true,
          0,
          47,
          true
        ]
      ],
      true,
      "Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor"
    ]
  },
  "conclusion": {
    "canonical_value": "(是, 是, [2, 3, 47], [2, 3, 47], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [47, 是, 0, 47, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)",
    "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-20261014-advance-20261108-r06-unique-prime-factorization-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations/questions.jsonl",
      "batch_sha256": "51fa926e07cc6c20069db4e8080bcf373df4bf3de100f279fae9cd21c30f207d",
      "original_line": 65,
      "original_line_sha256": "4c34b222349ce7dc1bedf0fede07cb7eee000c98eb687f33e25dfa7cb1dfde2e",
      "question_compact_sha256": "c2939a397714a361ac397d9317a7042fed3020cc37df6e5522d7632e68f7e555",
      "english_row_compact_sha256": "b0e5673750d406d92509249160cda8037fe2a3d8c0a0e5fa17dbe30b0153cad1",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-unique-prime-factorization": {
        "sha256": "0f01853edccbfb8cdf298587be277da27032f1f31ab87712ee5745644aa66cff",
        "english_sha256": "daa563be289db97eeabc811b523c607b1c88cb0a7c4b7b320a62dee46bcc27ac",
        "original_path": "public/data/mko/source13-20261018-unique-prime-factorization.json",
        "english_path": "public/data/i18n/editions/mko-unique-prime-factorization.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "2bda8f45e99ed8658b697be77ed3e5ec90ceff1b4f72fa7aa6a47d1a0c6254cf",
    "original_candidate_sha256": "e5582021093ae479dd9826bd508216d933dfeef5a1e809fbe45b26eac27fdfd2",
    "original_candidate_line_sha256": "e5582021093ae479dd9826bd508216d933dfeef5a1e809fbe45b26eac27fdfd2",
    "mko_binding_revision": {
      "id": "edition-symbol-bindings-20261101",
      "path": "content/edition-symbol-bindings-20261101/revision.json",
      "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c",
      "previous_english_sha256": {
        "mko-unique-prime-factorization": "3efccc30de9556546dd853570ad8776c24689cd9507cd911b9139b0b825e9b76"
      }
    }
  },
  "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-20261026-009-p42",
    "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": "8f134403487694af4c70d2e00ec84bab542043ef4b3c54517c04f3c409c0262a",
      "question_first_witness_sha256": "ad772845088ed8e99f87f90d4264b05ea97be35775305a02011af5ee35f0f4ce"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 19,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "e1dc18c399fe089a596514790618b9bee86ba8381237f7132e383a5aff139174",
      "arithmetic_steps": 6
    },
    "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."
  }
}