{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261108-r06-unique-prime-factorization-0078",
  "pilot_number": "R0041-21378",
  "number": 21378,
  "family_id": "qf-mod39-upf-unit-sign-boundary",
  "title": {
    "zh-Hant": "第 21378 題：整數唯一質因數分解",
    "en": "Question 21378: Separate Zero, Sign Units and the Positive Prime Multiset"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261108-r06-unique-prime-factorization-0078",
    "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-unit-sign-boundary；資料={\"n\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(-1, 否, -1, [], units_have_empty_prime_multiset)",
      "explanation_zh": "Wave6第二層核對=(-1, 否, -1, [], units_have_empty_prime_multiset)；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": 1,
      "skills": [
        "整數唯一質因數分解"
      ]
    },
    "variant": {
      "family_id": "qf-mod39-upf-unit-sign-boundary",
      "seed": 202611086078,
      "parameters": {
        "n": -1,
        "domain": "Exact_integer_unique_prime_factorization",
        "operation": "upf-unit-sign-boundary",
        "semantic_template": "qf-mod39-upf-unit-sign-boundary/{\"n\":-1}"
      }
    },
    "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-0078",
    "batch_id": "batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "b0f1ded881d68138a07295cfbcb15d1f531d9bc70452316f00f69e6105b00662",
      "family_id": "qf-mod39-upf-unit-sign-boundary",
      "seed": 202611086078,
      "semantic_template": "qf-mod39-upf-unit-sign-boundary/{\"n\":-1}"
    },
    "fields": {
      "title": "Separate Zero, Sign Units and the Positive Prime Multiset",
      "stem": "Use original bounded integer n=-1. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(-1, No, -1, [], units_have_empty_prime_multiset)",
        "explanation": "Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-1, No, -1, [], units_have_empty_prime_multiset)."
      },
      "skills": [
        "Separate Zero, Sign Units and the Positive Prime Multiset"
      ],
      "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-unit-sign-boundary；資料={\"n\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use original bounded integer n=-1. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=upf-unit-sign-boundary;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary]",
      "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": "ORIGINAL signed integer=-1;actual sign=-1;positive prime multiset=[]",
      "after": "ordinary_positive_form_claim fixed No;zero has no finite prime product;units use empty product1;nonzero sign is separated from |n| prime factors",
      "relation": "implies",
      "rule_id": "upf_zero_units_sign_nonclaim_not_positive_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "零不是有限prime乘積；±1是units、prime多重集空；其餘先分離符號。固定No不測n是否正數，正合數亦可在此boundary保No，不否定其分解存在。",
        "en": "Zero is not a finite prime product;±1 are units with an empty prime multiset.Other nonzero integers separate sign from positive factors.The fixed No is not a positivity test or a denial of factorization for a positive integer."
      },
      "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,
      false,
      -1,
      [],
      "units_have_empty_prime_multiset"
    ]
  },
  "conclusion": {
    "canonical_value": "(-1, 否, -1, [], units_have_empty_prime_multiset)",
    "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": 78,
      "original_line_sha256": "0d5f03e40787328cae80100ae7853b16fb8581fb997b21707c1a4fb4b4359653",
      "question_compact_sha256": "b0f1ded881d68138a07295cfbcb15d1f531d9bc70452316f00f69e6105b00662",
      "english_row_compact_sha256": "0d854f80876c7216ac3cf860d8faa57d79ca5ce491c50b8fcda22561c6fdce35",
      "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": "32846a86025c9cfb499941de9ca80d454c3c6b83a0b650813946b067475a2e84",
    "original_candidate_line_sha256": "32846a86025c9cfb499941de9ca80d454c3c6b83a0b650813946b067475a2e84",
    "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": 2,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "46f7ad059b98a524c48ba691aeab3c790f5cb2689bff0595c433f3f7b8c68113",
      "arithmetic_steps": 0
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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."
  }
}