{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261022-r05-finite-group-0060", "pilot_number": "R0057-29260", "number": 29260, "family_id": "qf-str22-group-identity-inverse-inventory", "title": { "zh-Hant": "第 29260 題：完整有限群表", "en": "Question 29260: Two-Sided Identity and Two-Sided Inverse Inventories" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261022-r05-finite-group-0060", "batch_id": "batch-20261013-advance-20261022-r05-finite-group-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-finite-group" ], "title_zh": "完整有限群表", "stem_zh": "完整有限Cayley表。操作=group-identity-inverse-inventory；資料={\"n\":5,\"table\":[[3,2,4,1,0],[2,0,3,4,1],[4,3,1,0,2],[1,4,0,2,3],[0,1,2,3,4]]}。請依 [n,table,identities,inverse_rows,all_have_inverse] 欄序回答。", "answer": { "kind": "expression", "value": "(5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], [4], [[0, [2]], [1, [3]], [2, [0]], [3, [1]], [4, [4]]], 是)", "explanation_zh": "結構基礎核對=(5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], [4], [[0, [2]], [1, [3]], [2, [0]], [3, [1]], [4, [4]]], 是)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；結合律逐三元組核對；抽樣不證群公理。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "完整有限群表" ] }, "variant": { "family_id": "qf-str22-group-identity-inverse-inventory", "seed": 202610225060, "parameters": { "n": 5, "table": [ [ 3, 2, 4, 1, 0 ], [ 2, 0, 3, 4, 1 ], [ 4, 3, 1, 0, 2 ], [ 1, 4, 0, 2, 3 ], [ 0, 1, 2, 3, 4 ] ], "domain": "Complete_finite_operation_table", "operation": "group-identity-inverse-inventory", "semantic_template": "qf-str22-group-identity-inverse-inventory/{\"n\":5,\"table\":[[3,2,4,1,0],[2,0,3,4,1],[4,3,1,0,2],[1,4,0,2,3],[0,1,2,3,4]]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-13", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261013-advance-20261022-r05-finite-group-0060", "batch_id": "batch-20261013-advance-20261022-r05-finite-group-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "b2ecb79278b8709fbf1815bebc080fcb0b2e302f82312e43da61297aeb0732bd", "family_id": "qf-str22-group-identity-inverse-inventory", "seed": 202610225060, "semantic_template": "qf-str22-group-identity-inverse-inventory/{\"n\":5,\"table\":[[3,2,4,1,0],[2,0,3,4,1],[4,3,1,0,2],[1,4,0,2,3],[0,1,2,3,4]]}" }, "fields": { "title": "Two-Sided Identity and Two-Sided Inverse Inventories", "stem": "The complete finite Cayley table=[[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.", "answer": { "kind": "expression", "value": "(5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], [4], [[0, [2]], [1, [3]], [2, [0]], [3, [1]], [4, [4]]], Yes)", "explanation": "Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], [4], [[0, [2]], [1, [3]], [2, [0]], [3, [1]], [4, [4]]], Yes)." }, "skills": [ "Two-Sided Identity and Two-Sided Inverse Inventories" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "完整有限Cayley表。操作=group-identity-inverse-inventory；資料={\"n\":5,\"table\":[[3,2,4,1,0],[2,0,3,4,1],[4,3,1,0,2],[1,4,0,2,3],[0,1,2,3,4]]}。請依 [n,table,identities,inverse_rows,all_have_inverse] 欄序回答。", "en": "The complete finite Cayley table=[[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-str22-group-identity-inverse-inventory, group-identity-inverse-inventory]", "after": "derived = [[n, table, identities, inverse_rows, all_have_inverse]]", "relation": "implies", "rule_id": "STR22_exact_interface_and_source_ORDERED_tuple_roles", "conditions": [], "explanation": { "zh-Hant": "此銀行依原tuple欄序與hyphen operation。fraction約分、正分母且整數省略/1，與前輪舊契約不同；不改原中英快照或題意。", "en": "Use this bank exact ordered tuple and hyphenated operation.Rational outputs reduce with positive denominators and OMIT/1 for integers,unlike the preceding legacy bank.Do not change original bilingual snapshots or meaning." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [5, 0, [3, 2, 4, 1, 0]]", "after": "derived = [[是, 是, 是, 是, 是]]", "relation": "implies", "rule_id": "Group_EVERY_original_row_closure", "conditions": [], "explanation": { "zh-Hant": "對這個完整Cayley row每一格檢查結果仍在Fin(n)；保row/column標籤。closure只是可作有限binary operation，未證其他群公理。", "en": "Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [5, 1, [2, 0, 3, 4, 1]]", "after": "derived = [[是, 是, 是, 是, 是]]", "relation": "implies", "rule_id": "Group_EVERY_original_row_closure", "conditions": [], "explanation": { "zh-Hant": "對這個完整Cayley row每一格檢查結果仍在Fin(n)；保row/column標籤。closure只是可作有限binary operation，未證其他群公理。", "en": "Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [5, 2, [4, 3, 1, 0, 2]]", "after": "derived = [[是, 是, 是, 是, 是]]", "relation": "implies", "rule_id": "Group_EVERY_original_row_closure", "conditions": [], "explanation": { "zh-Hant": "對這個完整Cayley row每一格檢查結果仍在Fin(n)；保row/column標籤。closure只是可作有限binary operation，未證其他群公理。", "en": "Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [5, 3, [1, 4, 0, 2, 3]]", "after": "derived = [[是, 是, 是, 是, 是]]", "relation": "implies", "rule_id": "Group_EVERY_original_row_closure", "conditions": [], "explanation": { "zh-Hant": "對這個完整Cayley row每一格檢查結果仍在Fin(n)；保row/column標籤。closure只是可作有限binary operation，未證其他群公理。", "en": "Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [5, 4, [0, 1, 2, 3, 4]]", "after": "derived = [[是, 是, 是, 是, 是]]", "relation": "implies", "rule_id": "Group_EVERY_original_row_closure", "conditions": [], "explanation": { "zh-Hant": "對這個完整Cayley row每一格檢查結果仍在Fin(n)；保row/column標籤。closure只是可作有限binary operation，未證其他群公理。", "en": "Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], 0]", "after": "derived = [[3, 2, 4, 1, 0], [3, 2, 4, 1, 0], 否]", "relation": "implies", "rule_id": "Group_EACH_candidate_identity_BOTH_sides_ALL_elements", "conditions": [], "explanation": { "zh-Hant": "同一e對全部x同時滿足e*x=x與x*e=x才列identity，不從單邊或一列推兩邊。", "en": "The SAME e must satisfy e*x=x AND x*e=x for EVERY x.A one-sided condition or one row alone is insufficient." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], 1]", "after": "derived = [[2, 0, 3, 4, 1], [2, 0, 3, 4, 1], 否]", "relation": "implies", "rule_id": "Group_EACH_candidate_identity_BOTH_sides_ALL_elements", "conditions": [], "explanation": { "zh-Hant": "同一e對全部x同時滿足e*x=x與x*e=x才列identity，不從單邊或一列推兩邊。", "en": "The SAME e must satisfy e*x=x AND x*e=x for EVERY x.A one-sided condition or one row alone is insufficient." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], 2]", "after": "derived = [[4, 3, 1, 0, 2], [4, 3, 1, 0, 2], 否]", "relation": "implies", "rule_id": "Group_EACH_candidate_identity_BOTH_sides_ALL_elements", "conditions": [], "explanation": { "zh-Hant": "同一e對全部x同時滿足e*x=x與x*e=x才列identity，不從單邊或一列推兩邊。", "en": "The SAME e must satisfy e*x=x AND x*e=x for EVERY x.A one-sided condition or one row alone is insufficient." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "source roles = [5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], 3]", "after": "derived = [[1, 4, 0, 2, 3], [1, 4, 0, 2, 3], 否]", "relation": "implies", "rule_id": "Group_EACH_candidate_identity_BOTH_sides_ALL_elements", "conditions": [], "explanation": { "zh-Hant": "同一e對全部x同時滿足e*x=x與x*e=x才列identity，不從單邊或一列推兩邊。", "en": "The SAME e must satisfy e*x=x AND x*e=x for EVERY x.A one-sided condition or one row alone is insufficient." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "source roles = [5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], 4]", "after": "derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], 是]", "relation": "implies", "rule_id": "Group_EACH_candidate_identity_BOTH_sides_ALL_elements", "conditions": [], "explanation": { "zh-Hant": "同一e對全部x同時滿足e*x=x與x*e=x才列identity，不從單邊或一列推兩邊。", "en": "The SAME e must satisfy e*x=x AND x*e=x for EVERY x.A one-sided condition or one row alone is insufficient." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "source roles = [5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], 4, 0]", "after": "derived = [[[0, 3, 3], [1, 2, 2], [2, 4, 4], [3, 1, 1], [4, 0, 0]], [2]]", "relation": "implies", "rule_id": "Group_EVERY_element_ALL_two_sided_inverse_candidates", "conditions": [], "explanation": { "zh-Hant": "對這個x枚舉所有y，兩個乘積都須等同一e；保全部候選，尚未associative時不能先假設inverse唯一。", "en": "Enumerate EVERY y for this x,and require BOTH products equal the SAME e.Retain all candidates;without established associativity,do not assume inverse uniqueness." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "source roles = [5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], 4, 1]", "after": "derived = [[[0, 2, 2], [1, 0, 0], [2, 3, 3], [3, 4, 4], [4, 1, 1]], [3]]", "relation": "implies", "rule_id": "Group_EVERY_element_ALL_two_sided_inverse_candidates", "conditions": [], "explanation": { "zh-Hant": "對這個x枚舉所有y，兩個乘積都須等同一e；保全部候選，尚未associative時不能先假設inverse唯一。", "en": "Enumerate EVERY y for this x,and require BOTH products equal the SAME e.Retain all candidates;without established associativity,do not assume inverse uniqueness." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "source roles = [5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], 4, 2]", "after": "derived = [[[0, 4, 4], [1, 3, 3], [2, 1, 1], [3, 0, 0], [4, 2, 2]], [0]]", "relation": "implies", "rule_id": "Group_EVERY_element_ALL_two_sided_inverse_candidates", "conditions": [], "explanation": { "zh-Hant": "對這個x枚舉所有y，兩個乘積都須等同一e；保全部候選，尚未associative時不能先假設inverse唯一。", "en": "Enumerate EVERY y for this x,and require BOTH products equal the SAME e.Retain all candidates;without established associativity,do not assume inverse uniqueness." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "source roles = [5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], 4, 3]", "after": "derived = [[[0, 1, 1], [1, 4, 4], [2, 0, 0], [3, 2, 2], [4, 3, 3]], [1]]", "relation": "implies", "rule_id": "Group_EVERY_element_ALL_two_sided_inverse_candidates", "conditions": [], "explanation": { "zh-Hant": "對這個x枚舉所有y，兩個乘積都須等同一e；保全部候選，尚未associative時不能先假設inverse唯一。", "en": "Enumerate EVERY y for this x,and require BOTH products equal the SAME e.Retain all candidates;without established associativity,do not assume inverse uniqueness." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "source roles = [5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], 4, 4]", "after": "derived = [[[0, 0, 0], [1, 1, 1], [2, 2, 2], [3, 3, 3], [4, 4, 4]], [4]]", "relation": "implies", "rule_id": "Group_EVERY_element_ALL_two_sided_inverse_candidates", "conditions": [], "explanation": { "zh-Hant": "對這個x枚舉所有y，兩個乘積都須等同一e；保全部候選，尚未associative時不能先假設inverse唯一。", "en": "Enumerate EVERY y for this x,and require BOTH products equal the SAME e.Retain all candidates;without established associativity,do not assume inverse uniqueness." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "source roles = [5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]]]", "after": "derived = [[4], [[0, [2]], [1, [3]], [2, [0]], [3, [1]], [4, [4]]], 是]", "relation": "implies", "rule_id": "Group_unique_identity_required_NO_vacuous_inverse_claim", "conditions": [], "explanation": { "zh-Hant": "沒有unique two-sided e就沒有本題inverse rows，all-have-inverse回No，不對空rows作vacuous True；inventory仍不包含associativity。", "en": "Without a unique two-sided e,the inverse-row list is empty and all-have-inverse is No,not vacuous True.An identity/inverse inventory still does not include associativity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null } ], "math_model": { "result": [ 5, [ [ 3, 2, 4, 1, 0 ], [ 2, 0, 3, 4, 1 ], [ 4, 3, 1, 0, 2 ], [ 1, 4, 0, 2, 3 ], [ 0, 1, 2, 3, 4 ] ], [ 4 ], [ [ 0, [ 2 ] ], [ 1, [ 3 ] ], [ 2, [ 0 ] ], [ 3, [ 1 ] ], [ 4, [ 4 ] ] ], true ] }, "conclusion": { "canonical_value": "(5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], [4], [[0, [2]], [1, [3]], [2, [0]], [3, [1]], [4, [4]]], 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-20261013-advance-20261022-r05-finite-group-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261022-r05-finite-group-foundations/questions.jsonl", "batch_sha256": "77b65df8e9a0d3e23c7ae294869da93a08c0d74f9ac38b442deaa7b2a3673f58", "original_line": 60, "original_line_sha256": "434b1c4f28fef55427137e7de0a3701acb5907a79f35d28d202f5d2fd5723f0c", "question_compact_sha256": "b2ecb79278b8709fbf1815bebc080fcb0b2e302f82312e43da61297aeb0732bd", "english_row_compact_sha256": "78712fbac9d4d3885f1c3aa31ce9598eb422185d2a138749fc4faca663e326ed", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261022-r05-finite-group-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261022-r05-finite-group-foundations.en.jsonl" }, "mko_bindings": { "mko-finite-group": { "sha256": "491d28b1773ac105187d1f17664e8e55b1ab4831613c354693901e40fae75f01", "english_sha256": "dc483579d967de45b346797aa7d8b882baaf395c1c61b932fcf9072babf13bc2", "original_path": "public/data/mko/finite-group.json", "english_path": "public/data/i18n/editions/mko-finite-group.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "cd714b09e116fbc0088007895c4d956bc0f4fad5a61507e4a16f09649d05a3b0", "original_candidate_sha256": "73ad55af7efe4ba4cf37b3e37258a4a985bbc76be1086e53bc4a92eaa316ad63", "original_candidate_line_sha256": "73ad55af7efe4ba4cf37b3e37258a4a985bbc76be1086e53bc4a92eaa316ad63" }, "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-20261026-011-p58", "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": "db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6", "question_first_witness_sha256": "58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 17, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "49701fb15173f94125823968421078dd5e3f9e18bce4d76aac1f0ea2dceebb3a", "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." } }