{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261028-r01-finite-commutative-ring-0077",
  "pilot_number": "R0026-13502",
  "number": 13502,
  "family_id": "qf-mod28-commring-binomial-square",
  "title": {
    "zh-Hant": "第 13502 題：有限交換環",
    "en": "Question 13502: Compute a Binomial Square Using Ring Operations, Not Label Arithmetic"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261028-r01-finite-commutative-ring-0077",
    "batch_id": "batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-commutative-ring"
    ],
    "title_zh": "有限交換環",
    "stem_zh": "同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"a\":2,\"b\":5}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(2, 5, 6, 1, 4, 2, 1, 1, 是)",
      "explanation_zh": "Wave4第二層核對=(2, 5, 6, 1, 4, 2, 1, 1, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；乘法交換律逐全部元素對核對；交換環不自動是整環或域且可有零因子。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "有限交換環"
      ]
    },
    "variant": {
      "family_id": "qf-mod28-commring-binomial-square",
      "seed": 202610281077,
      "parameters": {
        "n": 8,
        "addition": [
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          [
            1,
            2,
            3,
            4,
            5,
            7,
            0,
            6
          ],
          [
            2,
            3,
            4,
            5,
            7,
            6,
            1,
            0
          ],
          [
            3,
            4,
            5,
            7,
            6,
            0,
            2,
            1
          ],
          [
            4,
            5,
            7,
            6,
            0,
            1,
            3,
            2
          ],
          [
            5,
            7,
            6,
            0,
            1,
            2,
            4,
            3
          ],
          [
            6,
            0,
            1,
            2,
            3,
            4,
            7,
            5
          ],
          [
            7,
            6,
            0,
            1,
            2,
            3,
            5,
            4
          ]
        ],
        "multiplication": [
          [
            0,
            0,
            0,
            0,
            0,
            0,
            0,
            0
          ],
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          [
            0,
            2,
            4,
            7,
            0,
            2,
            7,
            4
          ],
          [
            0,
            3,
            7,
            1,
            4,
            6,
            5,
            2
          ],
          [
            0,
            4,
            0,
            4,
            0,
            4,
            4,
            0
          ],
          [
            0,
            5,
            2,
            6,
            4,
            1,
            3,
            7
          ],
          [
            0,
            6,
            7,
            5,
            4,
            3,
            1,
            2
          ],
          [
            0,
            7,
            4,
            2,
            0,
            7,
            2,
            4
          ]
        ],
        "a": 2,
        "b": 5,
        "domain": "Complete_finite_commutative_ring_tables",
        "operation": "commring-binomial-square",
        "semantic_template": "qf-mod28-commring-binomial-square/{\"a\":2,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"b\":5,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"n\":8}"
      }
    },
    "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、整環／域、Hilbert、無限維算子、mutual independence、有向連通、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-20261028-r01-finite-commutative-ring-0077",
    "batch_id": "batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2988d80dd1c8fe2044b0024edd06f86825705b2f11b2010cdc06142b2b0411f9",
      "family_id": "qf-mod28-commring-binomial-square",
      "seed": 202610281077,
      "semantic_template": "qf-mod28-commring-binomial-square/{\"a\":2,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"b\":5,\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"n\":8}"
    },
    "fields": {
      "title": "Compute a Binomial Square Using Ring Operations, Not Label Arithmetic",
      "stem": "Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 7, 0, 6], [2, 3, 4, 5, 7, 6, 1, 0], [3, 4, 5, 7, 6, 0, 2, 1], [4, 5, 7, 6, 0, 1, 3, 2], [5, 7, 6, 0, 1, 2, 4, 3], [6, 0, 1, 2, 3, 4, 7, 5], [7, 6, 0, 1, 2, 3, 5, 4]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 7, 0, 2, 7, 4], [0, 3, 7, 1, 4, 6, 5, 2], [0, 4, 0, 4, 0, 4, 4, 0], [0, 5, 2, 6, 4, 1, 3, 7], [0, 6, 7, 5, 4, 3, 1, 2], [0, 7, 4, 2, 0, 7, 2, 4]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Require commutativity. a=2, b=5. Give (a,b,sum=a+b,lhs=sum*sum,a2,ab,b2,rhs=a2+ab+ab+b2,equal?).",
      "answer": {
        "kind": "expression",
        "value": "(2, 5, 6, 1, 4, 2, 1, 1, Yes)",
        "explanation": "Use the addition and multiplication tables throughout. The two copies of ab are added in the ring; do not substitute a carrier label called 2 for repeated addition. Zero divisors do not invalidate this identity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, 5, 6, 1, 4, 2, 1, 1, Yes)."
      },
      "skills": [
        "Compute a Binomial Square Using Ring Operations, Not Label Arithmetic"
      ],
      "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": "同一完整有限含一環的加法表與乘法表。操作=commring-binomial-square；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,7,0,6],[2,3,4,5,7,6,1,0],[3,4,5,7,6,0,2,1],[4,5,7,6,0,1,3,2],[5,7,6,0,1,2,4,3],[6,0,1,2,3,4,7,5],[7,6,0,1,2,3,5,4]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,4,7,0,2,7,4],[0,3,7,1,4,6,5,2],[0,4,0,4,0,4,4,0],[0,5,2,6,4,1,3,7],[0,6,7,5,4,3,1,2],[0,7,4,2,0,7,2,4]],\"a\":2,\"b\":5}。請依 [a,b,sum,lhs,a2,ab,b2,rhs,equal] 欄序回答。",
        "en": "Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 7, 0, 6], [2, 3, 4, 5, 7, 6, 1, 0], [3, 4, 5, 7, 6, 0, 2, 1], [4, 5, 7, 6, 0, 1, 3, 2], [5, 7, 6, 0, 1, 2, 4, 3], [6, 0, 1, 2, 3, 4, 7, 5], [7, 6, 0, 1, 2, 3, 5, 4]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 7, 0, 2, 7, 4], [0, 3, 7, 1, 4, 6, 5, 2], [0, 4, 0, 4, 0, 4, 4, 0], [0, 5, 2, 6, 4, 1, 3, 7], [0, 6, 7, 5, 4, 3, 1, 2], [0, 7, 4, 2, 0, 7, 2, 4]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Require commutativity. a=2, b=5. Give (a,b,sum=a+b,lhs=sum*sum,a2,ab,b2,rhs=a2+ab+ab+b2,equal?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "same parent carrier Fin(8);addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 7, 0, 6], [2, 3, 4, 5, 7, 6, 1, 0], [3, 4, 5, 7, 6, 0, 2, 1], [4, 5, 7, 6, 0, 1, 3, 2], [5, 7, 6, 0, 1, 2, 4, 3], [6, 0, 1, 2, 3, 4, 7, 5], [7, 6, 0, 1, 2, 3, 5, 4]];multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 7, 0, 2, 7, 4], [0, 3, 7, 1, 4, 6, 5, 2], [0, 4, 0, 4, 0, 4, 4, 0], [0, 5, 2, 6, 4, 1, 3, 7], [0, 6, 7, 5, 4, 3, 1, 2], [0, 7, 4, 2, 0, 7, 2, 4]]",
      "after": "both complete operation tables typed and closed",
      "relation": "implies",
      "rule_id": "parent_ring_original_complete_tables",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以下理想或交換律綁定同一父環；此處保留完整原表，不把元素標籤當成整數值。",
        "en": "Bind subsequent ideals or commutativity to this same parent ring.Preserve the complete original tables;carrier labels are not integer arithmetic values."
      },
      "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": "every additive triple checked;first associativity failure=null;all additive pairs commute=是",
      "after": "additive zero candidates=[0];actual zero=0;two-sided additive inverses in label order=[0, 6, 7, 5, 4, 3, 1, 2]",
      "relation": "implies",
      "rule_id": "parent_additive_abelian_group_complete_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整載體核加法結合、交換、兩側零與每個逆元；父環是題目使用的已合格前提，核查完整有限條件，不靠幾個樣本。",
        "en": "Audit additive associativity,commutativity,the two-sided zero and each inverse over the complete carrier.This qualified parent is the problem's premise,checked by complete finite conditions rather than samples."
      },
      "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": "every multiplicative triple checked;first associativity failure=null",
      "after": "multiplicative identity candidates=[1];actual one=1",
      "relation": "implies",
      "rule_id": "parent_multiplicative_monoid_complete_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "乘法只先要求結合與兩側one，不在此假設交換或非零元素可逆。",
        "en": "Multiplication here requires associativity and a two-sided one,not commutativity or inverses of every nonzero element."
      },
      "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": "all 512 left and 512 right distributivity triples checked",
      "after": "first left failure=null;first right failure=null;zero=0;one=1;distinct=是",
      "relation": "implies",
      "rule_id": "parent_both_distributivities_and_distinct_identities",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左右分配責任分開完整核對，0與1必是不同實際元素；不因含一就宣稱交換環、整環或域。",
        "en": "Audit both distributive obligations completely and separately.Zero and one must be distinct actual elements.Unital alone implies neither a commutative ring,domain nor field."
      },
      "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": "unordered parent elements=[0, 1];a*b=0;b*a=0",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "unordered parent elements=[0, 2];a*b=0;b*a=0",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "unordered parent elements=[0, 3];a*b=0;b*a=0",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "unordered parent elements=[0, 4];a*b=0;b*a=0",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "unordered parent elements=[0, 5];a*b=0;b*a=0",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "unordered parent elements=[0, 6];a*b=0;b*a=0",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "unordered parent elements=[0, 7];a*b=0;b*a=0",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "unordered parent elements=[1, 2];a*b=2;b*a=2",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "unordered parent elements=[1, 3];a*b=3;b*a=3",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "unordered parent elements=[1, 4];a*b=4;b*a=4",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "unordered parent elements=[1, 5];a*b=5;b*a=5",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "unordered parent elements=[1, 6];a*b=6;b*a=6",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "unordered parent elements=[1, 7];a*b=7;b*a=7",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "unordered parent elements=[2, 3];a*b=7;b*a=7",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "unordered parent elements=[2, 4];a*b=0;b*a=0",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "unordered parent elements=[2, 5];a*b=2;b*a=2",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "unordered parent elements=[2, 6];a*b=7;b*a=7",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "unordered parent elements=[2, 7];a*b=4;b*a=4",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "unordered parent elements=[3, 4];a*b=4;b*a=4",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "unordered parent elements=[3, 5];a*b=6;b*a=6",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "unordered parent elements=[3, 6];a*b=5;b*a=5",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "unordered parent elements=[3, 7];a*b=2;b*a=2",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "unordered parent elements=[4, 5];a*b=4;b*a=4",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "unordered parent elements=[4, 6];a*b=4;b*a=4",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "unordered parent elements=[4, 7];a*b=0;b*a=0",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "unordered parent elements=[5, 6];a*b=3;b*a=3",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "unordered parent elements=[5, 7];a*b=7;b*a=7",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "unordered parent elements=[6, 7];a*b=2;b*a=2",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "whole_ring_multiplicative_commutativity_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出所有不同元素對的兩個乘積，對角自動相等；單一相等對不足以認證全環交換。",
        "en": "Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity."
      },
      "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": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "all multiplicative noncommuting records=[]",
      "after": "parent ring commutative=是;zero divisors not excluded",
      "relation": "implies",
      "rule_id": "commutative_ring_extra_obligation_not_field",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這一步只加乘法交換資格，不排除零因子，也不保證非零元素乘法可逆。",
        "en": "This adds multiplicative commutativity only.It excludes neither zero divisors nor lack of multiplicative inverses."
      },
      "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": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "a times b:operation=multiplication;left label=2;right label=5;table row=[0, 2, 4, 7, 0, 2, 7, 4]",
      "after": "result label=2",
      "relation": "implies",
      "rule_id": "parent_table_operation_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原操作表與指定次序查值，結果仍是載體標籤，不用標籤的整數加乘替代。",
        "en": "Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels."
      },
      "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": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "binomial sum:operation=addition;left label=2;right label=5;table row=[2, 3, 4, 5, 7, 6, 1, 0]",
      "after": "result label=6",
      "relation": "implies",
      "rule_id": "parent_table_operation_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原操作表與指定次序查值，結果仍是載體標籤，不用標籤的整數加乘替代。",
        "en": "Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels."
      },
      "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": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "square of sum:operation=multiplication;left label=6;right label=6;table row=[0, 6, 7, 5, 4, 3, 1, 2]",
      "after": "result label=1",
      "relation": "implies",
      "rule_id": "parent_table_operation_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原操作表與指定次序查值，結果仍是載體標籤，不用標籤的整數加乘替代。",
        "en": "Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels."
      },
      "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": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "a square:operation=multiplication;left label=2;right label=2;table row=[0, 2, 4, 7, 0, 2, 7, 4]",
      "after": "result label=4",
      "relation": "implies",
      "rule_id": "parent_table_operation_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原操作表與指定次序查值，結果仍是載體標籤，不用標籤的整數加乘替代。",
        "en": "Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels."
      },
      "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": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "b square:operation=multiplication;left label=5;right label=5;table row=[0, 5, 2, 6, 4, 1, 3, 7]",
      "after": "result label=1",
      "relation": "implies",
      "rule_id": "parent_table_operation_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原操作表與指定次序查值，結果仍是載體標籤，不用標籤的整數加乘替代。",
        "en": "Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels."
      },
      "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": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "a square plus ab:operation=addition;left label=4;right label=2;table row=[4, 5, 7, 6, 0, 1, 3, 2]",
      "after": "result label=7",
      "relation": "implies",
      "rule_id": "parent_table_operation_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原操作表與指定次序查值，結果仍是載體標籤，不用標籤的整數加乘替代。",
        "en": "Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels."
      },
      "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": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "add second ab:operation=addition;left label=7;right label=2;table row=[7, 6, 0, 1, 2, 3, 5, 4]",
      "after": "result label=0",
      "relation": "implies",
      "rule_id": "parent_table_operation_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原操作表與指定次序查值，結果仍是載體標籤，不用標籤的整數加乘替代。",
        "en": "Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels."
      },
      "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": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "add b square:operation=addition;left label=0;right label=1;table row=[0, 1, 2, 3, 4, 5, 6, 7]",
      "after": "result label=1",
      "relation": "implies",
      "rule_id": "parent_table_operation_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原操作表與指定次序查值，結果仍是載體標籤，不用標籤的整數加乘替代。",
        "en": "Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels."
      },
      "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": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "sum label=6;lhs=1;a2=4;ab=2;b2=1;rhs=1",
      "after": "(a+b)^2=a2+ab+ab+b2 using both distributive laws and ba=ab",
      "relation": "implies",
      "rule_id": "binomial_two_copies_not_integer_label2",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先用左右分配展開a²+ab+ba+b²，再用已合格交換律把ba改ab；兩個ab透過環加法相加，不能拿標籤2直接乘ab。零因子不妨礙此恆等式。",
        "en": "Expand by both distributive laws to a²+ab+ba+b²,then use qualified commutativity to replace ba by ab.Add the two ab terms with ring addition,not by multiplying with a label named2.Zero divisors do not invalidate this identity."
      },
      "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": [
      2,
      5,
      6,
      1,
      4,
      2,
      1,
      1,
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(2, 5, 6, 1, 4, 2, 1, 1, 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": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations/questions.jsonl",
      "batch_sha256": "4eaedd63a5863f88680f5be7af99ae3ffa6d1c576c9da48338052890f175ab64",
      "original_line": 77,
      "original_line_sha256": "3ee0e71b79f777d187023dadc16da6878ca272a4ea312caae898ca9655760d18",
      "question_compact_sha256": "2988d80dd1c8fe2044b0024edd06f86825705b2f11b2010cdc06142b2b0411f9",
      "english_row_compact_sha256": "3f10b460a7bb26656ab80ad85f1cbd91cce2ec1a386c13d6f9f0630259f18d1f",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-finite-commutative-ring": {
        "sha256": "5a488ffcb874034a7b2144c7585ccef8981c9d6bc0d2959107aa85650113234a",
        "english_sha256": "99c36775e32ec1ae2f5f4db3a9f3366e90e9bbb558ea3d9b1c720c3bd62b70fb",
        "original_path": "public/data/mko/finite-commutative-ring.json",
        "english_path": "public/data/i18n/editions/mko-finite-commutative-ring.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "e9c0ae2888b9f0f177728712cf7eb267a13a2a12f233a4c64ea50cb104f58bb0",
    "original_candidate_sha256": "d6a782c28dfaa3e050060781d418ce6d8bf0f43c98712231ef4e78d1f15dd79e",
    "original_candidate_line_sha256": "d6a782c28dfaa3e050060781d418ce6d8bf0f43c98712231ef4e78d1f15dd79e"
  },
  "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-p27",
    "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": "f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798",
      "question_first_witness_sha256": "cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 42,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "c39573ff78ffed8ea046f50ada657b876269b83fc78eb99931a989400da74864",
      "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."
  }
}