{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261029-r01-finite-integral-domain-0058",
  "pilot_number": "R0028-14608",
  "number": 14608,
  "family_id": "qf-mod29-integraldomain-cancellation",
  "title": {
    "zh-Hant": "第 14608 題：有限整環",
    "en": "Question 14608: Nonzero Cancellation Has an Implication, Not an Equality Premise Flag"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261029-r01-finite-integral-domain-0058",
    "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-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],\"a\":3,\"b\":4,\"c\":4}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(3, 4, 4, 1, 1, 是, 是, nonzero_cancellation_in_domain)",
      "explanation_zh": "Wave4第三層核對=(3, 4, 4, 1, 1, 是, 是, nonzero_cancellation_in_domain)；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": 1,
      "skills": [
        "有限整環"
      ]
    },
    "variant": {
      "family_id": "qf-mod29-integraldomain-cancellation",
      "seed": 202610291058,
      "parameters": {
        "n": 5,
        "addition": [
          [
            0,
            1,
            2,
            3,
            4
          ],
          [
            1,
            3,
            4,
            2,
            0
          ],
          [
            2,
            4,
            1,
            0,
            3
          ],
          [
            3,
            2,
            0,
            4,
            1
          ],
          [
            4,
            0,
            3,
            1,
            2
          ]
        ],
        "multiplication": [
          [
            0,
            0,
            0,
            0,
            0
          ],
          [
            0,
            3,
            1,
            4,
            2
          ],
          [
            0,
            1,
            2,
            3,
            4
          ],
          [
            0,
            4,
            3,
            2,
            1
          ],
          [
            0,
            2,
            4,
            1,
            3
          ]
        ],
        "a": 3,
        "b": 4,
        "c": 4,
        "domain": "Complete_finite_integral_domain",
        "operation": "integraldomain-cancellation",
        "semantic_template": "qf-mod29-integraldomain-cancellation/{\"a\":3,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"b\":4,\"c\":4,\"multiplication\":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],\"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-0058",
    "batch_id": "batch-20261014-advance-20261029-r01-finite-integral-domain-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "156dea458277231d4ffc53be6e387cec65d865b21bc5738ae2bf8e64babcbe55",
      "family_id": "qf-mod29-integraldomain-cancellation",
      "seed": 202610291058,
      "semantic_template": "qf-mod29-integraldomain-cancellation/{\"a\":3,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"b\":4,\"c\":4,\"multiplication\":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],\"n\":5}"
    },
    "fields": {
      "title": "Nonzero Cancellation Has an Implication, Not an Equality Premise Flag",
      "stem": "Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 1, 4, 2], [0, 1, 2, 3, 4], [0, 4, 3, 2, 1], [0, 2, 4, 1, 3]] 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 a qualified integral domain and nonzero multiplier a=3, with b=4, c=4. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).",
      "answer": {
        "kind": "expression",
        "value": "(3, 4, 4, 1, 1, Yes, Yes, nonzero_cancellation_in_domain)",
        "explanation": "Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (3, 4, 4, 1, 1, Yes, Yes, nonzero_cancellation_in_domain)."
      },
      "skills": [
        "Nonzero Cancellation Has an Implication, Not an Equality Premise Flag"
      ],
      "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-cancellation；資料={\"n\":5,\"addition\":[[0,1,2,3,4],[1,3,4,2,0],[2,4,1,0,3],[3,2,0,4,1],[4,0,3,1,2]],\"multiplication\":[[0,0,0,0,0],[0,3,1,4,2],[0,1,2,3,4],[0,4,3,2,1],[0,2,4,1,3]],\"a\":3,\"b\":4,\"c\":4}。請依 [a,b,c,ab,ac,premise_equal,cancellation_valid,certificate] 欄序回答。",
        "en": "Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 1, 4, 2], [0, 1, 2, 3, 4], [0, 4, 3, 2, 1], [0, 2, 4, 1, 3]] 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 a qualified integral domain and nonzero multiplier a=3, with b=4, c=4. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "same carrier Fin(5);addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]];multiplication=[[0, 0, 0, 0, 0], [0, 3, 1, 4, 2], [0, 1, 2, 3, 4], [0, 4, 3, 2, 1], [0, 2, 4, 1, 3]]",
      "after": "actual additive zero=0;actual multiplicative one=2;additive inverses=[0, 4, 3, 2, 1]",
      "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=[1, 1];actual zero=0;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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "ordered nonzero labels=[1, 2];actual zero=0;product label=1",
      "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=[1, 3];actual zero=0;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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "ordered nonzero labels=[1, 4];actual zero=0;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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "ordered nonzero labels=[2, 1];actual zero=0;product label=1",
      "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=0;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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "ordered nonzero labels=[2, 3];actual zero=0;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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "ordered nonzero labels=[2, 4];actual zero=0;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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "ordered nonzero labels=[3, 1];actual zero=0;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": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "ordered nonzero labels=[3, 2];actual zero=0;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": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "ordered nonzero labels=[3, 3];actual zero=0;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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "ordered nonzero labels=[3, 4];actual zero=0;product label=1",
      "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, 1];actual zero=0;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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "ordered nonzero labels=[4, 2];actual zero=0;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": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "ordered nonzero labels=[4, 3];actual zero=0;product label=1",
      "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=0;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": "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": "a times b:operation=multiplication;left label=3;right label=4;original row=[0, 4, 3, 2, 1]",
      "after": "result label=1",
      "relation": "implies",
      "rule_id": "same_parent_operation_not_integer_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原操作與原次序查表；元素標籤不是整數加乘的值，不能用模數猜父環結構。",
        "en": "Use the original operation/order lookup.Labels are not integer sums/products,and modular guesses do not define the parent ring."
      },
      "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": "a times c:operation=multiplication;left label=3;right label=4;original row=[0, 4, 3, 2, 1]",
      "after": "result label=1",
      "relation": "implies",
      "rule_id": "same_parent_operation_not_integer_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原操作與原次序查表；元素標籤不是整數加乘的值，不能用模數猜父環結構。",
        "en": "Use the original operation/order lookup.Labels are not integer sums/products,and modular guesses do not define the parent ring."
      },
      "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": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "domain qualified;nonzero multiplier=3;actual products ab=1;ac=1",
      "after": "premise ab=ac=是;if equal then b=c=是",
      "relation": "implies",
      "rule_id": "nonzero_domain_cancellation_conditional",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若ab=ac，以分配律得a(b−c)=0；a非零且無非零零因子迫使b−c=0，所以b=c。若本題兩乘積不等，蘊涵空泛真，不宣稱此時b=c；零乘數則直接不合協定。",
        "en": "If ab=ac,distributivity gives a(b−c)=0.Nonzero a and absence of nonzero zero divisors force b−c=0,so b=c.If the reported products differ,the implication is vacuously true and does not assert b=c.A zero multiplier is outside this protocol."
      },
      "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": [
      3,
      4,
      4,
      1,
      1,
      true,
      true,
      "nonzero_cancellation_in_domain"
    ]
  },
  "conclusion": {
    "canonical_value": "(3, 4, 4, 1, 1, 是, 是, nonzero_cancellation_in_domain)",
    "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": 58,
      "original_line_sha256": "9360b6c37fc56a9b3062275161d5462ceeac04f7b7492da23280d065fdbd6e70",
      "question_compact_sha256": "156dea458277231d4ffc53be6e387cec65d865b21bc5738ae2bf8e64babcbe55",
      "english_row_compact_sha256": "ed7ddeb8aa2e370e576dc55c3de110a3518527a0f8e24269a0c76a164890c1c4",
      "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": "28083df76c98544a7e0f558422b34c2a1cc64a5e460c351b85287a300e91800e",
    "original_candidate_line_sha256": "28083df76c98544a7e0f558422b34c2a1cc64a5e460c351b85287a300e91800e"
  },
  "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": 22,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "f41eb42bfb427468b9847510ed9d8e6ea39c54a56b04da0a71dac8adafc10b7f",
      "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."
  }
}