{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261029-r01-finite-integral-domain-0100",
  "pilot_number": "R0028-14650",
  "number": 14650,
  "family_id": "qf-mod29-integraldomain-finite-inverse",
  "title": {
    "zh-Hant": "第 14650 題：有限整環",
    "en": "Question 14650: Keep the Finite Step from Integral Domain to Invertibility"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261029-r01-finite-integral-domain-0100",
    "batch_id": "batch-20261014-advance-20261029-r01-finite-integral-domain-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-integral-domain"
    ],
    "title_zh": "有限整環",
    "stem_zh": "同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],\"multiplication\":[[0,1,2,3,4],[1,1,1,1,1],[2,1,4,0,3],[3,1,0,4,2],[4,1,3,2,0]],\"element\":0}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(0, 0, 0, 是, finite_injective_multiplication_implies_surjective)",
      "explanation_zh": "Wave4第三層核對=(0, 0, 0, 是, finite_injective_multiplication_implies_surjective)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；非零零因子逐全部元素對核對；有限整環到域須保留有限性定理步驟。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有限整環"
      ]
    },
    "variant": {
      "family_id": "qf-mod29-integraldomain-finite-inverse",
      "seed": 202610291100,
      "parameters": {
        "n": 5,
        "addition": [
          [
            2,
            0,
            3,
            4,
            1
          ],
          [
            0,
            1,
            2,
            3,
            4
          ],
          [
            3,
            2,
            4,
            1,
            0
          ],
          [
            4,
            3,
            1,
            0,
            2
          ],
          [
            1,
            4,
            0,
            2,
            3
          ]
        ],
        "multiplication": [
          [
            0,
            1,
            2,
            3,
            4
          ],
          [
            1,
            1,
            1,
            1,
            1
          ],
          [
            2,
            1,
            4,
            0,
            3
          ],
          [
            3,
            1,
            0,
            4,
            2
          ],
          [
            4,
            1,
            3,
            2,
            0
          ]
        ],
        "element": 0,
        "domain": "Complete_finite_integral_domain",
        "operation": "integraldomain-finite-inverse",
        "semantic_template": "qf-mod29-integraldomain-finite-inverse/{\"addition\":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],\"element\":0,\"multiplication\":[[0,1,2,3,4],[1,1,1,1,1],[2,1,4,0,3],[3,1,0,4,2],[4,1,3,2,0]],\"n\":5}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式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-20261029-r01-finite-integral-domain-0100",
    "batch_id": "batch-20261014-advance-20261029-r01-finite-integral-domain-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "91fa07581557986256a756618b32fe88084b3eda9658d030c147f89a70e528eb",
      "family_id": "qf-mod29-integraldomain-finite-inverse",
      "seed": 202610291100,
      "semantic_template": "qf-mod29-integraldomain-finite-inverse/{\"addition\":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],\"element\":0,\"multiplication\":[[0,1,2,3,4],[1,1,1,1,1],[2,1,4,0,3],[3,1,0,4,2],[4,1,3,2,0]],\"n\":5}"
    },
    "fields": {
      "title": "Keep the Finite Step from Integral Domain to Invertibility",
      "stem": "Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]] and multiplication=[[0, 1, 2, 3, 4], [1, 1, 1, 1, 1], [2, 1, 4, 0, 3], [3, 1, 0, 4, 2], [4, 1, 3, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=0. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).",
      "answer": {
        "kind": "expression",
        "value": "(0, 0, 0, Yes, finite_injective_multiplication_implies_surjective)",
        "explanation": "Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (0, 0, 0, Yes, finite_injective_multiplication_implies_surjective)."
      },
      "skills": [
        "Keep the Finite Step from Integral Domain to Invertibility"
      ],
      "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": "同一完整有限交換含一環表。操作=integraldomain-finite-inverse；資料={\"n\":5,\"addition\":[[2,0,3,4,1],[0,1,2,3,4],[3,2,4,1,0],[4,3,1,0,2],[1,4,0,2,3]],\"multiplication\":[[0,1,2,3,4],[1,1,1,1,1],[2,1,4,0,3],[3,1,0,4,2],[4,1,3,2,0]],\"element\":0}。請依 [element,one,inverse,found,finite_theorem_step] 欄序回答。",
        "en": "Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]] and multiplication=[[0, 1, 2, 3, 4], [1, 1, 1, 1, 1], [2, 1, 4, 0, 3], [3, 1, 0, 4, 2], [4, 1, 3, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=0. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "same carrier Fin(5);addition=[[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]];multiplication=[[0, 1, 2, 3, 4], [1, 1, 1, 1, 1], [2, 1, 4, 0, 3], [3, 1, 0, 4, 2], [4, 1, 3, 2, 0]]",
      "after": "actual additive zero=1;actual multiplicative one=0;additive inverses=[4, 1, 3, 2, 0]",
      "relation": "implies",
      "rule_id": "commutative_parent_actual_tables_and_identities",
      "conditions": [],
      "explanation": {
        "zh-Hant": "所有係數與整環論證綁同一完整父表；單位元是實際元素，不預設標籤0或1。",
        "en": "Bind coefficients and domain reasoning to the same complete parent tables.Identities are actual elements,not presumed labels0 or1."
      },
      "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": "all additive/multiplicative pairs and triples checked;first failures=[null, null, null, null]",
      "after": "additive commute=是;multiplicative commute=是;zero differs from one=是",
      "relation": "implies",
      "rule_id": "complete_commutative_unital_parent_qualification",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核加法Abelian群、乘法結合與交換、左右分配及0≠1；整環與多項式責任在此合格前提之後，不用有限個正樣本冒充全表資格。",
        "en": "Qualify the whole additive abelian group,multiplicative associativity/commutativity,both distributive laws and distinct identities.Domain and polynomial obligations follow this premise,not sampled successes."
      },
      "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": "ordered nonzero labels=[0, 0];actual zero=1;product label=0",
      "after": "nonzero zero-divisor witness=否",
      "relation": "implies",
      "rule_id": "domain_all_ordered_nonzero_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整檢查所有有序非零元素對，包括交換後的另一個有序對；一個非零乘積為零的見證即否證整環。",
        "en": "Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain 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": "ordered nonzero labels=[0, 2];actual zero=1;product label=2",
      "after": "nonzero zero-divisor witness=否",
      "relation": "implies",
      "rule_id": "domain_all_ordered_nonzero_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整檢查所有有序非零元素對，包括交換後的另一個有序對；一個非零乘積為零的見證即否證整環。",
        "en": "Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain 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": "ordered nonzero labels=[0, 3];actual zero=1;product label=3",
      "after": "nonzero zero-divisor witness=否",
      "relation": "implies",
      "rule_id": "domain_all_ordered_nonzero_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整檢查所有有序非零元素對，包括交換後的另一個有序對；一個非零乘積為零的見證即否證整環。",
        "en": "Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain 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": "ordered nonzero labels=[0, 4];actual zero=1;product label=4",
      "after": "nonzero zero-divisor witness=否",
      "relation": "implies",
      "rule_id": "domain_all_ordered_nonzero_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整檢查所有有序非零元素對，包括交換後的另一個有序對；一個非零乘積為零的見證即否證整環。",
        "en": "Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain 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": "ordered nonzero labels=[2, 0];actual zero=1;product label=2",
      "after": "nonzero zero-divisor witness=否",
      "relation": "implies",
      "rule_id": "domain_all_ordered_nonzero_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整檢查所有有序非零元素對，包括交換後的另一個有序對；一個非零乘積為零的見證即否證整環。",
        "en": "Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain 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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "ordered nonzero labels=[2, 2];actual zero=1;product label=4",
      "after": "nonzero zero-divisor witness=否",
      "relation": "implies",
      "rule_id": "domain_all_ordered_nonzero_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整檢查所有有序非零元素對，包括交換後的另一個有序對；一個非零乘積為零的見證即否證整環。",
        "en": "Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain 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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "ordered nonzero labels=[2, 3];actual zero=1;product label=0",
      "after": "nonzero zero-divisor witness=否",
      "relation": "implies",
      "rule_id": "domain_all_ordered_nonzero_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整檢查所有有序非零元素對，包括交換後的另一個有序對；一個非零乘積為零的見證即否證整環。",
        "en": "Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain 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": "ordered nonzero labels=[2, 4];actual zero=1;product label=3",
      "after": "nonzero zero-divisor witness=否",
      "relation": "implies",
      "rule_id": "domain_all_ordered_nonzero_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整檢查所有有序非零元素對，包括交換後的另一個有序對；一個非零乘積為零的見證即否證整環。",
        "en": "Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain 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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "ordered nonzero labels=[3, 0];actual zero=1;product label=3",
      "after": "nonzero zero-divisor witness=否",
      "relation": "implies",
      "rule_id": "domain_all_ordered_nonzero_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整檢查所有有序非零元素對，包括交換後的另一個有序對；一個非零乘積為零的見證即否證整環。",
        "en": "Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain 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": "ordered nonzero labels=[3, 2];actual zero=1;product label=0",
      "after": "nonzero zero-divisor witness=否",
      "relation": "implies",
      "rule_id": "domain_all_ordered_nonzero_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整檢查所有有序非零元素對，包括交換後的另一個有序對；一個非零乘積為零的見證即否證整環。",
        "en": "Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain 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": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "ordered nonzero labels=[3, 3];actual zero=1;product label=4",
      "after": "nonzero zero-divisor witness=否",
      "relation": "implies",
      "rule_id": "domain_all_ordered_nonzero_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整檢查所有有序非零元素對，包括交換後的另一個有序對；一個非零乘積為零的見證即否證整環。",
        "en": "Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain 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": "ordered nonzero labels=[3, 4];actual zero=1;product label=2",
      "after": "nonzero zero-divisor witness=否",
      "relation": "implies",
      "rule_id": "domain_all_ordered_nonzero_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整檢查所有有序非零元素對，包括交換後的另一個有序對；一個非零乘積為零的見證即否證整環。",
        "en": "Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain 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": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "ordered nonzero labels=[4, 0];actual zero=1;product label=4",
      "after": "nonzero zero-divisor witness=否",
      "relation": "implies",
      "rule_id": "domain_all_ordered_nonzero_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整檢查所有有序非零元素對，包括交換後的另一個有序對；一個非零乘積為零的見證即否證整環。",
        "en": "Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain 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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "ordered nonzero labels=[4, 2];actual zero=1;product label=3",
      "after": "nonzero zero-divisor witness=否",
      "relation": "implies",
      "rule_id": "domain_all_ordered_nonzero_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整檢查所有有序非零元素對，包括交換後的另一個有序對；一個非零乘積為零的見證即否證整環。",
        "en": "Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain 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": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "ordered nonzero labels=[4, 3];actual zero=1;product label=2",
      "after": "nonzero zero-divisor witness=否",
      "relation": "implies",
      "rule_id": "domain_all_ordered_nonzero_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整檢查所有有序非零元素對，包括交換後的另一個有序對；一個非零乘積為零的見證即否證整環。",
        "en": "Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain 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": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "ordered nonzero labels=[4, 4];actual zero=1;product label=0",
      "after": "nonzero zero-divisor witness=否",
      "relation": "implies",
      "rule_id": "domain_all_ordered_nonzero_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整檢查所有有序非零元素對，包括交換後的另一個有序對；一個非零乘積為零的見證即否證整環。",
        "en": "Check every ordered pair of nonzero elements,including the reversed ordered pair.One nonzero pair with zero product refutes the domain 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": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "all ordered nonzero zero-divisor pairs=[]",
      "after": "integral domain=是;invertibility is a later finite theorem step",
      "relation": "implies",
      "rule_id": "domain_definition_not_inverse_definition",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整環此處是交換含一父環加無非零零因子；非零可逆不是額外定義，而是稍後需保留有限性步驟的結論。",
        "en": "Here a domain is a commutative unital parent with no nonzero zero divisors.Nonzero invertibility is not another defining clause;it follows later with a finite-carrier step."
      },
      "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": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "nonzero element=0;multiplication map on full carrier=[0, 1, 2, 3, 4];n=5",
      "after": "cancellation makes map injective;finite equal-size carrier makes it surjective",
      "relation": "implies",
      "rule_id": "finite_injective_to_surjective_indispensable_step",
      "conditions": [],
      "explanation": {
        "zh-Hant": "消去律先給單射；再用有限且來源目標是同一載體才得滿射。這個有限步不能推到任意無限整環，例如整數。",
        "en": "Cancellation first gives injectivity.Only then does finiteness of the same source/target carrier give surjectivity.This finite step does not extend to arbitrary infinite domains,such as the integers."
      },
      "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": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "actual one=0;preimage candidates in increasing label order=[0]",
      "after": "selected inverse=0;left product=0;right product=0",
      "relation": "implies",
      "rule_id": "finite_domain_identity_preimage_is_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "滿射保證one有原像，取最小標籤的見證；父環交換律使另一側也為one，不拿零當分母。",
        "en": "Surjectivity supplies a preimage of one;choose the least label witness.Commutativity makes the other product one too.Zero is never inverted."
      },
      "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,
      0,
      0,
      true,
      "finite_injective_multiplication_implies_surjective"
    ]
  },
  "conclusion": {
    "canonical_value": "(0, 0, 0, 是, finite_injective_multiplication_implies_surjective)",
    "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-20261029-r01-finite-integral-domain-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261029-r01-finite-integral-domain-foundations/questions.jsonl",
      "batch_sha256": "74eb9c04e4dbba6e62362ef674d4ac30479671c5931224a6b9a92cd89bf8557f",
      "original_line": 100,
      "original_line_sha256": "b08c725fe18c0ef97ac23f213f1d71688bfb60247f0f40ff990ade515a8fb604",
      "question_compact_sha256": "91fa07581557986256a756618b32fe88084b3eda9658d030c147f89a70e528eb",
      "english_row_compact_sha256": "5751d128db939938890eabca0bb5be9868db52a88856bcfb1a1a71cd712300e6",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261029-r01-finite-integral-domain-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261029-r01-finite-integral-domain-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-finite-integral-domain": {
        "sha256": "5aefd54faece27112b7f0cc473cb7465bf2c77336f1d8ebfa180261027ba56fc",
        "english_sha256": "c032ef1004cdaaf34161bef5e966394605373aef8940ce31a696453c59a97a93",
        "original_path": "public/data/mko/finite-integral-domain.json",
        "english_path": "public/data/i18n/editions/mko-finite-integral-domain.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "466f72389eb454e24322e59df2871ec206c351545a4a7b072d39fb5f6afc2697",
    "original_candidate_sha256": "4797940f3aba2915c8e9c6a0be267006164a437ddf6f4c9cda37020bc74f243e",
    "original_candidate_line_sha256": "4797940f3aba2915c8e9c6a0be267006164a437ddf6f4c9cda37020bc74f243e"
  },
  "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-20261026-007-p29",
    "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": "bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209ae",
      "question_first_witness_sha256": "61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 21,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "be3eea04ae3b6cc23578584de82be6e18dba67fa8b2e636997a56e204942b5b8",
      "arithmetic_steps": 0
    },
    "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."
  }
}