{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261107-r01-finite-group-action-0035", "pilot_number": "R0039-20135", "number": 20135, "family_id": "qf-mod38-groupaction-identity-audit", "title": { "zh-Hant": "第 20135 題：有限群作用", "en": "Question 20135: Every Point Must Be Fixed by the Declared Identity Row" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261107-r01-finite-group-action-0035", "batch_id": "batch-20261014-advance-20261107-r01-finite-group-action-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-finite-group-action" ], "title_zh": "有限群作用", "stem_zh": "完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是]], 是)", "explanation_zh": "Wave6根層核對=(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是]], 是)；依完整明示資料獨立重算；全稱核e作用與(gh)作用相容性；有限抽樣pairs不足；每個群元素誘導排列，但有效作用不自動宣稱faithful、free或transitive。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "有限群作用" ] }, "variant": { "family_id": "qf-mod38-groupaction-identity-audit", "seed": 202611071035, "parameters": { "group_table": [ [ 0, 1, 2, 3, 4, 5 ], [ 1, 2, 3, 4, 5, 0 ], [ 2, 3, 4, 5, 0, 1 ], [ 3, 4, 5, 0, 1, 2 ], [ 4, 5, 0, 1, 2, 3 ], [ 5, 0, 1, 2, 3, 4 ] ], "identity": 0, "point_count": 7, "action_table": [ [ 0, 1, 2, 3, 4, 5, 6 ], [ 1, 2, 3, 4, 5, 0, 6 ], [ 2, 3, 4, 5, 0, 1, 6 ], [ 3, 4, 5, 0, 1, 2, 6 ], [ 4, 5, 0, 1, 2, 3, 6 ], [ 5, 0, 1, 2, 3, 4, 6 ] ], "domain": "Complete_finite_group_action_tables", "operation": "groupaction-identity-audit", "semantic_template": "qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":7}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或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-20261107-r01-finite-group-action-0035", "batch_id": "batch-20261014-advance-20261107-r01-finite-group-action-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "a345be8f868a20dc81bc433fab10437b40bfe0fdafaa737c03d63176101c0296", "family_id": "qf-mod38-groupaction-identity-audit", "seed": 202611071035, "semantic_template": "qf-mod38-groupaction-identity-audit/{\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]],\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":7}" }, "fields": { "title": "Every Point Must Be Fixed by the Declared Identity Row", "stem": "Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).", "answer": { "kind": "expression", "value": "(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], Yes)", "explanation": "List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], Yes)." }, "skills": [ "Every Point Must Be Fixed by the Declared Identity Row" ], "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": "完整有限群Cayley table、完整有限集合與total action table。操作=groupaction-identity-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"identity\":0,\"point_count\":7,\"action_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,0,6],[2,3,4,5,0,1,6],[3,4,5,0,1,2,6],[4,5,0,1,2,3,6],[5,0,1,2,3,4,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=groupaction-identity-audit;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[declared_identity, point_checks, identity_action]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。欄位標籤不是額外數學證據，No保持原題的聲稱範圍。", "en": "Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the source claim scope." }, "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": "ORIGINAL complete group-label carrier Fin(6);declared identity label=0;Cayley table row g,column h gives g*h=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]]", "after": "ORIGINAL complete point carrier Fin(7);TOTAL action table row g,column x gives g*x=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]];all entries in range,so structural closure/totality checked;GROUP AXIOMS not assumed from shape", "relation": "implies", "rule_id": "groupaction_two_table_roles_complete_candidate_not_group_assumption", "conditions": [], "explanation": { "zh-Hant": "兩張表的column角色不同，一張h是群標籤，一張x是作用點。in-range完整表只保closed／total，群候選可不滿足公理，不能因叫group_table就先填True。", "en": "The two tables have different column roles:group label h versus action point x.Complete in-range tables establish closure/totality only.A candidate group may fail its axioms;its table name does not certify them." }, "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": "ACTUAL declared identity label=0;ORIGINAL identity action row=[0, 1, 2, 3, 4, 5, 6];complete point carrier Fin(7)", "after": "ALL ordered [point,identity_image,equality] rows=[[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 是]];identity-action axiom=是", "relation": "implies", "rule_id": "groupaction_every_point_identity_action_not_group_identity", "conditions": [], "explanation": { "zh-Hant": "逐作用點驗e*x=x，e是來源宣告的真標籤，不預設0；此axiom與群乘法表的雙邊identity責任分開。", "en": "Check e*x=x at every action point,using the actual declared identity label rather than assuming0.This action axiom is distinct from the two-sided identity law in the group table." }, "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": "this component audited ONLY the complete identity action row", "after": "identity-action flag alone does NOT establish group qualification or action compatibility", "relation": "implies", "rule_id": "groupaction_identity_component_not_full_action_claim", "conditions": [], "explanation": { "zh-Hant": "本題輸出只回答identity-row責任；其他公理沒有被此旗標自動取代，不把一個component通過升格valid_action。", "en": "This output answers the identity-row obligation only;the flag does not replace the remaining axioms or promote one component to valid_action." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ 0, [ [ 0, 0, true ], [ 1, 1, true ], [ 2, 2, true ], [ 3, 3, true ], [ 4, 4, true ], [ 5, 5, true ], [ 6, 6, true ] ], true ] }, "conclusion": { "canonical_value": "(0, [[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 是], [5, 5, 是], [6, 6, 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-20261107-r01-finite-group-action-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261107-r01-finite-group-action-foundations/questions.jsonl", "batch_sha256": "fa5acca0cbac7a9c6e87334ae4261fa3e0ac4c1cb2ba0e64150ffc4f96129b7a", "original_line": 35, "original_line_sha256": "fb90e1837d67c3038e2c6b18b65a4ec32ca4178b9751b265046d1a0a609e32ce", "question_compact_sha256": "a345be8f868a20dc81bc433fab10437b40bfe0fdafaa737c03d63176101c0296", "english_row_compact_sha256": "d6863c117348f749108c9c03fcac901b6f0bd849115bcc68af44b4c03c0ee06e", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261107-r01-finite-group-action-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261107-r01-finite-group-action-foundations.en.jsonl" }, "mko_bindings": { "mko-finite-group-action": { "sha256": "2e31b8a871f6fd32317885aa68f63e2380a6c3a7d97c031b152b653c6a879de0", "english_sha256": "d2fb382c41e0c0388963c9f896b8d1cc26ad6fce451783c0262bb3ae5312cc51", "original_path": "public/data/mko/finite-group-action.json", "english_path": "public/data/i18n/editions/mko-finite-group-action.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "521929dfe9b0919224c38312ad78e5fa56457fc886faaa12b137edd83962251e", "original_candidate_sha256": "78725ff481c36b82e86007ce0a4a3a58fbf90b1b08487cf91de3ff1bf21e9516", "original_candidate_line_sha256": "78725ff481c36b82e86007ce0a4a3a58fbf90b1b08487cf91de3ff1bf21e9516" }, "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-20261026-008-p40", "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": "6e11e35fafe32f16900ce9b522d01acb7b2ee6b463ab179392076cfbe1af8034", "question_first_witness_sha256": "cc0fa361fc25151729e70264340d2eae94f179f6f2a2edfe80ba5704fbbd0682" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 4, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "24a4424ab6abe9644384c6abccaf9ec30129d538edb9eb27555ddd73a0373948", "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." } }