{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261030-r01-finite-field-0055",
  "pilot_number": "R0029-15105",
  "number": 15105,
  "family_id": "qf-mod30-field-division",
  "title": {
    "zh-Hant": "第 15105 題：有限域",
    "en": "Question 15105: Field Division Uses a Nonzero Denominator and the Same Table"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261030-r01-finite-field-0055",
    "batch_id": "batch-20261014-advance-20261030-r01-finite-field-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-field"
    ],
    "title_zh": "有限域",
    "stem_zh": "完整有限交換環雙表與全部非零逆元。操作=field-division；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],\"numerator\":0,\"denominator\":1}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(0, 1, 1, 0, 0, 是)",
      "explanation_zh": "Wave4第四層核對=(0, 1, 1, 0, 0, 是)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；永不除以零且有限整環到域保留有限性步驟；域階為質數冪不必是質數。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "有限域"
      ]
    },
    "variant": {
      "family_id": "qf-mod30-field-division",
      "seed": 202610301055,
      "parameters": {
        "n": 5,
        "addition": [
          [
            0,
            1,
            2,
            3,
            4
          ],
          [
            1,
            4,
            3,
            0,
            2
          ],
          [
            2,
            3,
            1,
            4,
            0
          ],
          [
            3,
            0,
            4,
            2,
            1
          ],
          [
            4,
            2,
            0,
            1,
            3
          ]
        ],
        "multiplication": [
          [
            0,
            0,
            0,
            0,
            0
          ],
          [
            0,
            1,
            2,
            3,
            4
          ],
          [
            0,
            2,
            3,
            4,
            1
          ],
          [
            0,
            3,
            4,
            1,
            2
          ],
          [
            0,
            4,
            1,
            2,
            3
          ]
        ],
        "numerator": 0,
        "denominator": 1,
        "domain": "Complete_finite_field_table",
        "operation": "field-division",
        "semantic_template": "qf-mod30-field-division/{\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"denominator\":1,\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],\"n\":5,\"numerator\":0}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_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、一般域／Hilbert／martingale／無限基數／一般收斂、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-20261030-r01-finite-field-0055",
    "batch_id": "batch-20261014-advance-20261030-r01-finite-field-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "3de371c01eba31515f28c7b2de743a2a2e9b78cdfe49ff0ce0ce4370b2dfc422",
      "family_id": "qf-mod30-field-division",
      "seed": 202610301055,
      "semantic_template": "qf-mod30-field-division/{\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"denominator\":1,\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],\"n\":5,\"numerator\":0}"
    },
    "fields": {
      "title": "Field Division Uses a Nonzero Denominator and the Same Table",
      "stem": "Use the same already qualified complete finite commutative unital-ring tables on Fin(5)={0,…,4}, addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3]] in carrier-label order. The actual zero and one are distinct and need not be labels 0 and 1. The inverse audit is additional to, not a replacement for, those ring axioms. Division by zero is never defined. A finite field has prime-power order, not necessarily prime order. Under field qualification, numerator=0, denominator=1 is nonzero. Give (numerator,denominator,inverse,quotient,check,pass?).",
      "answer": {
        "kind": "expression",
        "value": "(0, 1, 1, 0, 0, Yes)",
        "explanation": "Multiply the numerator by the denominator's inverse, then multiply the quotient by the denominator in the same table to recover the numerator. Ordinary arithmetic on the carrier labels is not this operation. The exact result, retaining the requested order, is (0, 1, 1, 0, 0, Yes)."
      },
      "skills": [
        "Field Division Uses a Nonzero Denominator and the Same Table"
      ],
      "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": "完整有限交換環雙表與全部非零逆元。操作=field-division；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,4,3,0,2],[2,3,1,4,0],[3,0,4,2,1],[4,2,0,1,3]],\"multiplication\":[[0,0,0,0,0],[0,1,2,3,4],[0,2,3,4,1],[0,3,4,1,2],[0,4,1,2,3]],\"numerator\":0,\"denominator\":1}。請依 [numerator,denominator,inverse,quotient,check,pass] 欄序回答。",
        "en": "Use the same already qualified complete finite commutative unital-ring tables on Fin(5)={0,…,4}, addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3]] in carrier-label order. The actual zero and one are distinct and need not be labels 0 and 1. The inverse audit is additional to, not a replacement for, those ring axioms. Division by zero is never defined. A finite field has prime-power order, not necessarily prime order. Under field qualification, numerator=0, denominator=1 is nonzero. Give (numerator,denominator,inverse,quotient,check,pass?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "same complete carrier Fin(5);addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]];multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3]]",
      "after": "actual zero=0;one=1;additive inverses=[0, 3, 4, 1, 2];additive commute/associate=[是, 是];multiplicative commute/associate=[是, 是];left/right distributive=[是, 是]",
      "relation": "implies",
      "rule_id": "field_stated_full_commutative_parent_not_raw_oracle_shortcut",
      "conditions": [],
      "explanation": {
        "zh-Hant": "即使原始inverse計算器guard較寬，也先獨立核來源英文所宣告的完整交換含一父環，包含所有群／分配公理與不同0、1；不靠名字或只找逆元就跳過。",
        "en": "Even where the raw inverse oracle has looser guards,independently qualify the complete commutative unital parent declared by the source English scope,including every additive/distributive axiom and distinct identities.Do not skip them by name or inverse search alone."
      },
      "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": "element label=0;actual zero=0;one=1;two-sided multiplicative inverse candidates=[]",
      "after": "selected inverse witness=null;nonzero inverse obligation=否",
      "relation": "implies",
      "rule_id": "field_all_nonzero_inverse_obligations_actual_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整載體找兩側皆為實際one的逆元，非零性按實際zero判，不把標籤0自動排除；zero不負擔逆元義務。",
        "en": "Search the whole carrier for both products equaling the actual one.Nonzero is judged against the actual zero,not label0;zero has no inverse 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": "element label=1;actual zero=0;one=1;two-sided multiplicative inverse candidates=[1]",
      "after": "selected inverse witness=1;nonzero inverse obligation=是",
      "relation": "implies",
      "rule_id": "field_all_nonzero_inverse_obligations_actual_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整載體找兩側皆為實際one的逆元，非零性按實際zero判，不把標籤0自動排除；zero不負擔逆元義務。",
        "en": "Search the whole carrier for both products equaling the actual one.Nonzero is judged against the actual zero,not label0;zero has no inverse 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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "element label=2;actual zero=0;one=1;two-sided multiplicative inverse candidates=[4]",
      "after": "selected inverse witness=4;nonzero inverse obligation=是",
      "relation": "implies",
      "rule_id": "field_all_nonzero_inverse_obligations_actual_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整載體找兩側皆為實際one的逆元，非零性按實際zero判，不把標籤0自動排除；zero不負擔逆元義務。",
        "en": "Search the whole carrier for both products equaling the actual one.Nonzero is judged against the actual zero,not label0;zero has no inverse 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": "element label=3;actual zero=0;one=1;two-sided multiplicative inverse candidates=[3]",
      "after": "selected inverse witness=3;nonzero inverse obligation=是",
      "relation": "implies",
      "rule_id": "field_all_nonzero_inverse_obligations_actual_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整載體找兩側皆為實際one的逆元，非零性按實際zero判，不把標籤0自動排除；zero不負擔逆元義務。",
        "en": "Search the whole carrier for both products equaling the actual one.Nonzero is judged against the actual zero,not label0;zero has no inverse 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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "element label=4;actual zero=0;one=1;two-sided multiplicative inverse candidates=[2]",
      "after": "selected inverse witness=2;nonzero inverse obligation=是",
      "relation": "implies",
      "rule_id": "field_all_nonzero_inverse_obligations_actual_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整載體找兩側皆為實際one的逆元，非零性按實際zero判，不把標籤0自動排除；zero不負擔逆元義務。",
        "en": "Search the whole carrier for both products equaling the actual one.Nonzero is judged against the actual zero,not label0;zero has no inverse 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": "qualified field;raw numerator label=0;nonzero denominator label=1;inverse=1;multiplication row=[0, 0, 0, 0, 0]",
      "after": "quotient label=0;quotient*denominator=0=numerator",
      "relation": "implies",
      "rule_id": "field_division_same_table_multiply_back",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一乘法表作num乘den逆元，再乘回非零den檢查；分子可為實際零，不用標籤整數除法，也不定義除以零。",
        "en": "Use the same multiplication table for numerator times denominator inverse,then multiply back by the nonzero denominator.The numerator may be actual zero.No integer division of labels or division by zero is defined."
      },
      "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": [
      0,
      1,
      1,
      0,
      0,
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(0, 1, 1, 0, 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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    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "54de60dca082018923bf089f2b1eb0a6666cc401916115af792056c8bae935db",
    "original_candidate_sha256": "d98ae495c492ec763be27c402813be298513f61d91904b63196b464b65a81235",
    "original_candidate_line_sha256": "d98ae495c492ec763be27c402813be298513f61d91904b63196b464b65a81235"
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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-20261026-007-p30",
    "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": {
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      "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."
  }
}