{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261028-r02-finite-two-sided-ideal-0022",
  "pilot_number": "R0026-13747",
  "number": 13747,
  "family_id": "qf-mod28-ideal-qualification",
  "title": {
    "zh-Hant": "第 13747 題：有限雙邊理想",
    "en": "Question 13747: Two-Sided Ideals Require Three Complete Qualifications"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261028-r02-finite-two-sided-ideal-0022",
    "batch_id": "batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-two-sided-ideal"
    ],
    "title_zh": "有限雙邊理想",
    "stem_zh": "子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"subset\":[0,7]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([0, 7], 0, 是, [], [], 是)",
      "explanation_zh": "Wave4第二層核對=([0, 7], 0, 是, [], [], 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；加法子群與左右吸收三責任分開；不混同單側理想子環或抽樣吸收。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有限雙邊理想"
      ]
    },
    "variant": {
      "family_id": "qf-mod28-ideal-qualification",
      "seed": 202610282022,
      "parameters": {
        "n": 8,
        "addition": [
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          [
            1,
            2,
            3,
            7,
            0,
            6,
            4,
            5
          ],
          [
            2,
            3,
            7,
            5,
            1,
            4,
            0,
            6
          ],
          [
            3,
            7,
            5,
            6,
            2,
            0,
            1,
            4
          ],
          [
            4,
            0,
            1,
            2,
            6,
            7,
            5,
            3
          ],
          [
            5,
            6,
            4,
            0,
            7,
            2,
            3,
            1
          ],
          [
            6,
            4,
            0,
            1,
            5,
            3,
            7,
            2
          ],
          [
            7,
            5,
            6,
            4,
            3,
            1,
            2,
            0
          ]
        ],
        "multiplication": [
          [
            0,
            0,
            0,
            0,
            0,
            0,
            0,
            0
          ],
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          [
            0,
            2,
            7,
            6,
            6,
            2,
            7,
            0
          ],
          [
            0,
            3,
            6,
            1,
            5,
            4,
            2,
            7
          ],
          [
            0,
            4,
            6,
            5,
            1,
            3,
            2,
            7
          ],
          [
            0,
            5,
            2,
            4,
            3,
            1,
            6,
            7
          ],
          [
            0,
            6,
            7,
            2,
            2,
            6,
            7,
            0
          ],
          [
            0,
            7,
            0,
            7,
            7,
            7,
            0,
            0
          ]
        ],
        "subset": [
          0,
          7
        ],
        "domain": "Complete_finite_two_sided_ideal",
        "operation": "ideal-qualification",
        "semantic_template": "qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"n\":8,\"subset\":[0,7]}"
      }
    },
    "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-r02-finite-two-sided-ideal-0022",
    "batch_id": "batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "5778ef6f38ecba87cbcb3f2291ce78b43d6d85cff9cc61853384d886e520be4d",
      "family_id": "qf-mod28-ideal-qualification",
      "seed": 202610282022,
      "semantic_template": "qf-mod28-ideal-qualification/{\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"n\":8,\"subset\":[0,7]}"
    },
    "fields": {
      "title": "Two-Sided Ideals Require Three Complete Qualifications",
      "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, 7, 0, 6, 4, 5], [2, 3, 7, 5, 1, 4, 0, 6], [3, 7, 5, 6, 2, 0, 1, 4], [4, 0, 1, 2, 6, 7, 5, 3], [5, 6, 4, 0, 7, 2, 3, 1], [6, 4, 0, 1, 5, 3, 7, 2], [7, 5, 6, 4, 3, 1, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 6, 6, 2, 7, 0], [0, 3, 6, 1, 5, 4, 2, 7], [0, 4, 6, 5, 1, 3, 2, 7], [0, 5, 2, 4, 3, 1, 6, 7], [0, 6, 7, 2, 2, 6, 7, 0], [0, 7, 0, 7, 7, 7, 0, 0]] 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. Normalize subset=[0, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.",
      "answer": {
        "kind": "expression",
        "value": "([0, 7], 0, Yes, [], [], Yes)",
        "explanation": "Check nonempty additive-subgroup closure, including zero and inverses, before conjoining both absorption audits over every parent element. A one-sided ideal or subring is not automatically two-sided. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([0, 7], 0, Yes, [], [], Yes)."
      },
      "skills": [
        "Two-Sided Ideals Require Three Complete Qualifications"
      ],
      "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": "子集綁定同一完整有限父環。操作=ideal-qualification；資料={\"n\":8,\"addition\":[[0,1,2,3,4,5,6,7],[1,2,3,7,0,6,4,5],[2,3,7,5,1,4,0,6],[3,7,5,6,2,0,1,4],[4,0,1,2,6,7,5,3],[5,6,4,0,7,2,3,1],[6,4,0,1,5,3,7,2],[7,5,6,4,3,1,2,0]],\"multiplication\":[[0,0,0,0,0,0,0,0],[0,1,2,3,4,5,6,7],[0,2,7,6,6,2,7,0],[0,3,6,1,5,4,2,7],[0,4,6,5,1,3,2,7],[0,5,2,4,3,1,6,7],[0,6,7,2,2,6,7,0],[0,7,0,7,7,7,0,0]],\"subset\":[0,7]}。請依 [subset,zero,additive_subgroup,missing_left,missing_right,two_sided_ideal] 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, 7, 0, 6, 4, 5], [2, 3, 7, 5, 1, 4, 0, 6], [3, 7, 5, 6, 2, 0, 1, 4], [4, 0, 1, 2, 6, 7, 5, 3], [5, 6, 4, 0, 7, 2, 3, 1], [6, 4, 0, 1, 5, 3, 7, 2], [7, 5, 6, 4, 3, 1, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 6, 6, 2, 7, 0], [0, 3, 6, 1, 5, 4, 2, 7], [0, 4, 6, 5, 1, 3, 2, 7], [0, 5, 2, 4, 3, 1, 6, 7], [0, 6, 7, 2, 2, 6, 7, 0], [0, 7, 0, 7, 7, 7, 0, 0]] 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. Normalize subset=[0, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "same parent carrier Fin(8);addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 0, 6, 4, 5], [2, 3, 7, 5, 1, 4, 0, 6], [3, 7, 5, 6, 2, 0, 1, 4], [4, 0, 1, 2, 6, 7, 5, 3], [5, 6, 4, 0, 7, 2, 3, 1], [6, 4, 0, 1, 5, 3, 7, 2], [7, 5, 6, 4, 3, 1, 2, 0]];multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 6, 6, 2, 7, 0], [0, 3, 6, 1, 5, 4, 2, 7], [0, 4, 6, 5, 1, 3, 2, 7], [0, 5, 2, 4, 3, 1, 6, 7], [0, 6, 7, 2, 2, 6, 7, 0], [0, 7, 0, 7, 7, 7, 0, 0]]",
      "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, 4, 6, 5, 1, 3, 2, 7]",
      "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": "raw subset=[0, 7];same qualified parent zero=0",
      "after": "sorted distinct parent labels=[0, 7]",
      "relation": "implies",
      "rule_id": "ideal_subset_same_parent_normalization",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此子集按集合去重排序，但所有運算與逆元都來自同一父環，不另換加乘或把標籤當整數。",
        "en": "Normalize the subset as a sorted set,but obtain all operations and inverses from the same parent ring.Do not substitute new operations or integer arithmetic on 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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "subset element=0;parent additive inverse=0",
      "after": "inverse belongs to subset=是",
      "relation": "implies",
      "rule_id": "ideal_each_parent_additive_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法子群需每個成員的父環加法逆元仍在子集中；不是乘法逆元條件。",
        "en": "An additive subgroup needs each member's parent-ring additive inverse to remain in the subset,not a multiplicative inverse."
      },
      "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": "subset ordered pair=[0, 0];parent sum=0",
      "after": "sum belongs to subset=是",
      "relation": "implies",
      "rule_id": "ideal_all_additive_closure_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一核所有有序加法對，沒有抽樣替代閉包。",
        "en": "Check every ordered addition pair;sampling does not replace closure."
      },
      "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": "subset ordered pair=[0, 7];parent sum=7",
      "after": "sum belongs to subset=是",
      "relation": "implies",
      "rule_id": "ideal_all_additive_closure_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一核所有有序加法對，沒有抽樣替代閉包。",
        "en": "Check every ordered addition pair;sampling does not replace closure."
      },
      "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": "subset element=7;parent additive inverse=7",
      "after": "inverse belongs to subset=是",
      "relation": "implies",
      "rule_id": "ideal_each_parent_additive_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法子群需每個成員的父環加法逆元仍在子集中；不是乘法逆元條件。",
        "en": "An additive subgroup needs each member's parent-ring additive inverse to remain in the subset,not a multiplicative inverse."
      },
      "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": "subset ordered pair=[7, 0];parent sum=7",
      "after": "sum belongs to subset=是",
      "relation": "implies",
      "rule_id": "ideal_all_additive_closure_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一核所有有序加法對，沒有抽樣替代閉包。",
        "en": "Check every ordered addition pair;sampling does not replace closure."
      },
      "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": "subset ordered pair=[7, 7];parent sum=0",
      "after": "sum belongs to subset=是",
      "relation": "implies",
      "rule_id": "ideal_all_additive_closure_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一核所有有序加法對，沒有抽樣替代閉包。",
        "en": "Check every ordered addition pair;sampling does not replace closure."
      },
      "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": "subset nonempty=是;actual zero belongs=是;missing sums=[];missing inverses=[]",
      "after": "additive subgroup=是",
      "relation": "implies",
      "rule_id": "ideal_additive_subgroup_nonempty_zero_and_closure",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非空、含實際零、加法閉包與逆元閉包同時成立才合格；空集合不能因閉包空泛成立而成加法子群。",
        "en": "Nonemptiness,the actual zero,addition closure and inverse closure must all hold.An empty subset does not become an additive subgroup through vacuous closure checks."
      },
      "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": "parent r=0;subset x=0;left product r*x=0;right product x*r=0",
      "after": "left absorbed=是;right absorbed=是",
      "relation": "implies",
      "rule_id": "ideal_both_absorptions_all_parent_elements",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個父環r與子集x都分左右測試；非交換父環可能只通過一側。失敗清單保留r外層、x內層的原協定次序。",
        "en": "Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists."
      },
      "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": "parent r=0;subset x=7;left product r*x=0;right product x*r=0",
      "after": "left absorbed=是;right absorbed=是",
      "relation": "implies",
      "rule_id": "ideal_both_absorptions_all_parent_elements",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個父環r與子集x都分左右測試；非交換父環可能只通過一側。失敗清單保留r外層、x內層的原協定次序。",
        "en": "Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists."
      },
      "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": "parent r=1;subset x=0;left product r*x=0;right product x*r=0",
      "after": "left absorbed=是;right absorbed=是",
      "relation": "implies",
      "rule_id": "ideal_both_absorptions_all_parent_elements",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個父環r與子集x都分左右測試；非交換父環可能只通過一側。失敗清單保留r外層、x內層的原協定次序。",
        "en": "Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists."
      },
      "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": "parent r=1;subset x=7;left product r*x=7;right product x*r=7",
      "after": "left absorbed=是;right absorbed=是",
      "relation": "implies",
      "rule_id": "ideal_both_absorptions_all_parent_elements",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個父環r與子集x都分左右測試；非交換父環可能只通過一側。失敗清單保留r外層、x內層的原協定次序。",
        "en": "Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists."
      },
      "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": "parent r=2;subset x=0;left product r*x=0;right product x*r=0",
      "after": "left absorbed=是;right absorbed=是",
      "relation": "implies",
      "rule_id": "ideal_both_absorptions_all_parent_elements",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個父環r與子集x都分左右測試；非交換父環可能只通過一側。失敗清單保留r外層、x內層的原協定次序。",
        "en": "Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists."
      },
      "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": "parent r=2;subset x=7;left product r*x=0;right product x*r=0",
      "after": "left absorbed=是;right absorbed=是",
      "relation": "implies",
      "rule_id": "ideal_both_absorptions_all_parent_elements",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個父環r與子集x都分左右測試；非交換父環可能只通過一側。失敗清單保留r外層、x內層的原協定次序。",
        "en": "Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists."
      },
      "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": "parent r=3;subset x=0;left product r*x=0;right product x*r=0",
      "after": "left absorbed=是;right absorbed=是",
      "relation": "implies",
      "rule_id": "ideal_both_absorptions_all_parent_elements",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個父環r與子集x都分左右測試；非交換父環可能只通過一側。失敗清單保留r外層、x內層的原協定次序。",
        "en": "Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists."
      },
      "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": "parent r=3;subset x=7;left product r*x=7;right product x*r=7",
      "after": "left absorbed=是;right absorbed=是",
      "relation": "implies",
      "rule_id": "ideal_both_absorptions_all_parent_elements",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個父環r與子集x都分左右測試；非交換父環可能只通過一側。失敗清單保留r外層、x內層的原協定次序。",
        "en": "Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists."
      },
      "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": "parent r=4;subset x=0;left product r*x=0;right product x*r=0",
      "after": "left absorbed=是;right absorbed=是",
      "relation": "implies",
      "rule_id": "ideal_both_absorptions_all_parent_elements",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個父環r與子集x都分左右測試；非交換父環可能只通過一側。失敗清單保留r外層、x內層的原協定次序。",
        "en": "Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists."
      },
      "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": "parent r=4;subset x=7;left product r*x=7;right product x*r=7",
      "after": "left absorbed=是;right absorbed=是",
      "relation": "implies",
      "rule_id": "ideal_both_absorptions_all_parent_elements",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個父環r與子集x都分左右測試；非交換父環可能只通過一側。失敗清單保留r外層、x內層的原協定次序。",
        "en": "Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists."
      },
      "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": "parent r=5;subset x=0;left product r*x=0;right product x*r=0",
      "after": "left absorbed=是;right absorbed=是",
      "relation": "implies",
      "rule_id": "ideal_both_absorptions_all_parent_elements",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個父環r與子集x都分左右測試；非交換父環可能只通過一側。失敗清單保留r外層、x內層的原協定次序。",
        "en": "Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists."
      },
      "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": "parent r=5;subset x=7;left product r*x=7;right product x*r=7",
      "after": "left absorbed=是;right absorbed=是",
      "relation": "implies",
      "rule_id": "ideal_both_absorptions_all_parent_elements",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個父環r與子集x都分左右測試；非交換父環可能只通過一側。失敗清單保留r外層、x內層的原協定次序。",
        "en": "Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists."
      },
      "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": "parent r=6;subset x=0;left product r*x=0;right product x*r=0",
      "after": "left absorbed=是;right absorbed=是",
      "relation": "implies",
      "rule_id": "ideal_both_absorptions_all_parent_elements",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個父環r與子集x都分左右測試；非交換父環可能只通過一側。失敗清單保留r外層、x內層的原協定次序。",
        "en": "Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists."
      },
      "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": "parent r=6;subset x=7;left product r*x=0;right product x*r=0",
      "after": "left absorbed=是;right absorbed=是",
      "relation": "implies",
      "rule_id": "ideal_both_absorptions_all_parent_elements",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個父環r與子集x都分左右測試；非交換父環可能只通過一側。失敗清單保留r外層、x內層的原協定次序。",
        "en": "Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists."
      },
      "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": "parent r=7;subset x=0;left product r*x=0;right product x*r=0",
      "after": "left absorbed=是;right absorbed=是",
      "relation": "implies",
      "rule_id": "ideal_both_absorptions_all_parent_elements",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個父環r與子集x都分左右測試；非交換父環可能只通過一側。失敗清單保留r外層、x內層的原協定次序。",
        "en": "Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists."
      },
      "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": "parent r=7;subset x=7;left product r*x=0;right product x*r=0",
      "after": "left absorbed=是;right absorbed=是",
      "relation": "implies",
      "rule_id": "ideal_both_absorptions_all_parent_elements",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個父環r與子集x都分左右測試；非交換父環可能只通過一側。失敗清單保留r外層、x內層的原協定次序。",
        "en": "Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists."
      },
      "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": "missing left [r,x,rx]=[];missing right [x,r,xr]=[]",
      "after": "left absorbing=是;right absorbing=是;empty subset absorption is vacuous",
      "relation": "implies",
      "rule_id": "ideal_absorption_not_additive_subgroup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "吸收性兩責任各自保留，空子集兩側可空泛通過，但沒有因此得到加法子群或理想。",
        "en": "Retain both absorption obligations separately.An empty subset can pass them vacuously without becoming an additive subgroup or ideal."
      },
      "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": "additive subgroup=是;left absorbing=是;right absorbing=是",
      "after": "two-sided ideal=是",
      "relation": "implies",
      "rule_id": "ideal_three_complete_obligations_conjunction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法子群、左吸收、右吸收三條合取，不把子環或單側理想當雙側理想。",
        "en": "Conjoin additive subgroup,left absorption and right absorption.A subring or one-sided ideal is not automatically a two-sided ideal."
      },
      "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,
        7
      ],
      0,
      true,
      [],
      [],
      true
    ]
  },
  "conclusion": {
    "canonical_value": "([0, 7], 0, 是, [], [], 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-r02-finite-two-sided-ideal-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations/questions.jsonl",
      "batch_sha256": "e7677ee0f65af1b284b8f6afac5ce39c9ad7c470bc389638306c26f5b640c9b5",
      "original_line": 22,
      "original_line_sha256": "dfcc32b4cfee4adbcf784e2a88876899ce8051ea56cee5345215b4bd813b35d8",
      "question_compact_sha256": "5778ef6f38ecba87cbcb3f2291ce78b43d6d85cff9cc61853384d886e520be4d",
      "english_row_compact_sha256": "c1815046819b97debd87aecfb19db48f6f68ce216270c4a694479fcda85a5e29",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-finite-two-sided-ideal": {
        "sha256": "56473b0d9ce61008676f55869388fbd0ce017b0555d5a29cc9a088626d1ef36c",
        "english_sha256": "0500394e305af84ba3defda4666f536eb3db5c4d769350cc73df22321811454e",
        "original_path": "public/data/mko/finite-two-sided-ideal.json",
        "english_path": "public/data/i18n/editions/mko-finite-two-sided-ideal.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "e9c0ae2888b9f0f177728712cf7eb267a13a2a12f233a4c64ea50cb104f58bb0",
    "original_candidate_sha256": "ee9326267428a4783ec30956a227b79dfd23afee37e44e7f69cdb379afcbdff3",
    "original_candidate_line_sha256": "ee9326267428a4783ec30956a227b79dfd23afee37e44e7f69cdb379afcbdff3"
  },
  "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": 30,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "0abb8b1ea208203710b8dd8786d3d9f62252522717be4055c0fdf7549d3805ad",
      "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."
  }
}