{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261108-r01-finite-group-orbit-0016", "pilot_number": "R0041-21141", "number": 21141, "family_id": "qf-mod39-orbit-orbit-of-point", "title": { "zh-Hant": "第 21141 題：有限群作用的軌道", "en": "Question 21141: An Orbit Uses All Group Elements, Not One Action Image" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261108-r01-finite-group-orbit-0016", "batch_id": "batch-20261014-advance-20261108-r01-finite-group-orbit-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-finite-group-orbit" ], "title_zh": "有限群作用的軌道", "stem_zh": "完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(0, [0, 1], 2, all_group_elements_enumerated)", "explanation_zh": "Wave6第二層核對=(0, [0, 1], 2, all_group_elements_enumerated)；依完整明示資料獨立重算；orbit為全部g作用像且x約y iff存在g送x到y；單一步不是orbit；orbits相等或不交並分割carrier；不混同stabilizer、時間軌道或單generator reachability。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "有限群作用的軌道" ] }, "variant": { "family_id": "qf-mod39-orbit-orbit-of-point", "seed": 202611081016, "parameters": { "group_table": [ [ 0, 1 ], [ 1, 0 ] ], "identity": 0, "point_count": 5, "action_table": [ [ 0, 1, 2, 3, 4 ], [ 1, 0, 2, 3, 4 ] ], "point": 0, "domain": "Complete_finite_group_action_orbits", "operation": "orbit-orbit-of-point", "semantic_template": "qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":0,\"point_count\":5}" } }, "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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、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-20261108-r01-finite-group-orbit-0016", "batch_id": "batch-20261014-advance-20261108-r01-finite-group-orbit-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "8ca7f15d711a77395c7d6e3d13118489464768fe7680c7b0a84ba0f9fd609685", "family_id": "qf-mod39-orbit-orbit-of-point", "seed": 202611081016, "semantic_template": "qf-mod39-orbit-orbit-of-point/{\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point\":0,\"point_count\":5}" }, "fields": { "title": "An Orbit Uses All Group Elements, Not One Action Image", "stem": "Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=0. Give (point, orbit, group_order, certificate).", "answer": { "kind": "expression", "value": "(0, [0, 1], 2, all_group_elements_enumerated)", "explanation": "Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (0, [0, 1], 2, all_group_elements_enumerated)." }, "skills": [ "An Orbit Uses All Group Elements, Not One Action Image" ], "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": "完整已認證有限群作用與全部群元素。操作=orbit-orbit-of-point；資料={\"group_table\":[[0,1],[1,0]],\"identity\":0,\"point_count\":5,\"action_table\":[[0,1,2,3,4],[1,0,2,3,4]],\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=0. Give (point, orbit, group_order, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=orbit-orbit-of-point;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[point, orbit, group_order, certificate]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題歷史介面標記保留；此處直接列有序輸出欄位，不依賴JavaScript。欄位或tag不是額外數學證據，固定No保留原聲稱範圍。", "en": "Retain the historical interface marker while listing ordered output fields without JavaScript.Fields and tags are not extra mathematical evidence;fixed No markers retain their 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(2);declared identity label=0;Cayley table row g,column h gives g*h=[[0, 1], [1, 0]]", "after": "ORIGINAL complete point carrier Fin(5);TOTAL action table row g,column x gives g*x=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]];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": "ORIGINAL closed Cayley table=[[0, 1], [1, 0]];ACTUAL declared group identity=0", "after": "ALL [g,e*g,g*e,both_equal_g] rows=[[0, 0, 0, 是], [1, 1, 1, 是]];two-sided group identity=是", "relation": "implies", "rule_id": "groupaction_group_identity_both_sides_not_action_identity", "conditions": [], "explanation": { "zh-Hant": "群表的e*g與g*e兩側都須等g；這裡不把作用點的e*x=x當群identity證書。", "en": "Both e*g and g*e in the group table must equal g.This is not certified by an identity-action row on points." }, "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": "ORIGINAL group pair(g,h)=[0, 0];check ALL k in Fin(2)", "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是]]", "relation": "implies", "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption", "conditions": [], "explanation": { "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。", "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition." }, "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": "ORIGINAL group pair(g,h)=[0, 1];check ALL k in Fin(2)", "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 0, 0, 是]]", "relation": "implies", "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption", "conditions": [], "explanation": { "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。", "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition." }, "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": "ORIGINAL group pair(g,h)=[1, 0];check ALL k in Fin(2)", "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, 是], [1, 0, 0, 是]]", "relation": "implies", "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption", "conditions": [], "explanation": { "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。", "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition." }, "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": "ORIGINAL group pair(g,h)=[1, 1];check ALL k in Fin(2)", "after": "ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是]]", "relation": "implies", "rule_id": "groupaction_complete_group_associativity_not_cyclic_assumption", "conditions": [], "explanation": { "zh-Hant": "逐完整原表算兩種括號，不假設群循環、交換或可以用整數加法代替乘法。", "en": "Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition." }, "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": "ORIGINAL group element g=0;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 0, 0, 是], [1, 1, 1, 否]]", "after": "first actual two-sided inverse witness=0;exists=是", "relation": "implies", "rule_id": "groupaction_each_element_actual_two_sided_inverse", "conditions": [], "explanation": { "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。", "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from 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": "S009", "uses": [ "G001", "S008" ], "before": "ORIGINAL group element g=1;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 1, 1, 否], [1, 0, 0, 是]]", "after": "first actual two-sided inverse witness=1;exists=是", "relation": "implies", "rule_id": "groupaction_each_element_actual_two_sided_inverse", "conditions": [], "explanation": { "zh-Hant": "每個g要存在同一個h同時滿足左右inverse；候選表不合格時可找不到，照實保None而非憑標籤公式猜inverse。", "en": "For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from 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": "S010", "uses": [ "G001", "S009" ], "before": "structurally closed COMPLETE group candidate;two-sided identity=是;ALL-triple associativity=是;ALL-element inverses=是", "after": "group_valid=是 is the CONJUNCTION;False remains a reported candidate result,not a silent repair", "relation": "implies", "rule_id": "groupaction_group_axiom_conjunction_candidate_may_fail", "conditions": [], "explanation": { "zh-Hant": "完整三項責任合取才是群資格。即使作用表看似排列，壞group table也不會因此變群；不自行換identity或修表。", "en": "Group qualification is the conjunction of these complete obligations.A permutation-looking action table cannot repair a bad group table;do not replace the identity or modify the 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": "S011", "uses": [ "G001", "S010" ], "before": "ACTUAL declared identity label=0;ORIGINAL identity action row=[0, 1, 2, 3, 4];complete point carrier Fin(5)", "after": "ALL ordered [point,identity_image,equality] rows=[[0, 0, 是], [1, 1, 是], [2, 2, 是], [3, 3, 是], [4, 4, 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": "S012", "uses": [ "G001", "S011" ], "before": "ORIGINAL ordered group pair(g,h)=[0, 0];actual group product=0;action table roles unchanged", "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是]];earliest failure retained,not overwritten", "relation": "implies", "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples", "conditions": [], "explanation": { "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。", "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure." }, "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": "ORIGINAL ordered group pair(g,h)=[0, 1];actual group product=1;action table roles unchanged", "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, 是], [1, 0, 0, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是]];earliest failure retained,not overwritten", "relation": "implies", "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples", "conditions": [], "explanation": { "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。", "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure." }, "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": "ORIGINAL ordered group pair(g,h)=[1, 0];actual group product=1;action table roles unchanged", "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, 是], [1, 0, 0, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是]];earliest failure retained,not overwritten", "relation": "implies", "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples", "conditions": [], "explanation": { "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。", "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure." }, "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": "ORIGINAL ordered group pair(g,h)=[1, 1];actual group product=0;action table roles unchanged", "after": "ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, 是], [1, 1, 1, 是], [2, 2, 2, 是], [3, 3, 3, 是], [4, 4, 4, 是]];earliest failure retained,not overwritten", "relation": "implies", "rule_id": "groupaction_all_ordered_compatibility_triples_not_samples", "conditions": [], "explanation": { "zh-Hant": "左邊先查group product再作用，右邊先h作用再g作用，不交換g,h。逐全部點及全部有序group pair核，失敗witness保g,h,x的首LEX順序。", "en": "For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure." }, "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": "group action ordered group-pair count := mul(2, 2)", "after": "group action ordered group-pair count = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 group action ordered group-pair count，不以小數近似替代。", "en": "Recompute group action ordered group-pair count with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "2" ], "output": "4", "label": "group action ordered group-pair count" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "group action complete triple-domain count := mul(4, 5)", "after": "group action complete triple-domain count = 20", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 group action complete triple-domain count，不以小數近似替代。", "en": "Recompute group action complete triple-domain count with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "5" ], "output": "20", "label": "group action complete triple-domain count" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "COMPLETE triple domain order^2*point_count=20;this derivation ACTUALLY checked all20 triples", "after": "compatibility=是;FIRST source-order failure[g,h,x,left,right]=null;null means NO failure anywhere", "relation": "implies", "rule_id": "groupaction_full_domain_count_first_failure_not_early_stop_measurement", "conditions": [], "explanation": { "zh-Hant": "triple_count是完整量詞域大小，不是把原型在第一錯誤處停止的迭代數冒充測量。本推導確實列了每對g,h的全部點，所以null給全域通過，首失敗則保確切資料與順序。", "en": "The triple_count is the complete quantified domain size,not a measurement of an early-stopping prototype.This derivation actually checks every point for every g,h pair,so null certifies full compatibility and a first failure retains exact source data/order." }, "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": "ORIGINAL group-row label=0;ORIGINAL point-indexed action row=[0, 1, 2, 3, 4]", "after": "sorted image MULTISET=[0, 1, 2, 3, 4];equals full point carrier=是;original row NOT reordered", "relation": "implies", "rule_id": "groupaction_every_row_permutation_actual_images_not_validity_converse", "conditions": [], "explanation": { "zh-Hant": "排序只核image多重集是否完整無重複，原row按點索引的資料不改。單獨每列是permutation仍不足完整action，須原group及兩個action axiom。", "en": "Sort only to test whether the image multiset covers the full carrier exactly once;do not reorder the original point-indexed row.Even all-permutation rows alone do not establish an action without the group and both action axioms." }, "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": "ORIGINAL group-row label=1;ORIGINAL point-indexed action row=[1, 0, 2, 3, 4]", "after": "sorted image MULTISET=[0, 1, 2, 3, 4];equals full point carrier=是;original row NOT reordered", "relation": "implies", "rule_id": "groupaction_every_row_permutation_actual_images_not_validity_converse", "conditions": [], "explanation": { "zh-Hant": "排序只核image多重集是否完整無重複，原row按點索引的資料不改。單獨每列是permutation仍不足完整action，須原group及兩個action axiom。", "en": "Sort only to test whether the image multiset covers the full carrier exactly once;do not reorder the original point-indexed row.Even all-permutation rows alone do not establish an action without the group and both action axioms." }, "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": "CURRENT original group/action qualified:group=是,identity_action=是,compatibility=是,permutations=是;actual inverses=[0, 1]", "after": "only NOW use the action axioms;row permutations ALONE are insufficient;invalid group/action data are rejected,not reported as an orbit", "relation": "implies", "rule_id": "orbit_reject_unqualified_tables_before_orbit_claims", "conditions": [], "explanation": { "zh-Hant": "這批先驗真群與兩個作用公理才作軌道論證，不能用排列列取代資格。重用舊版純數學函式時代入的仍是本題完整表與本題trace，不連結舊題當推導。", "en": "Qualify the actual group and both action axioms before using orbits.Permutation rows alone are insufficient.Reused frozen pure helpers receive this question's full tables and trace,not a link to an old derivation." }, "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": "CURRENT identity=0;full inverses=[0, 1];action table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]", "after": "reflexive:e*x=x;symmetric:y=g*x gives x=g_inverse*y;transitive:y=g*x,z=h*y gives z=(h*g)*x;therefore related iff the two full orbits are equal,and distinct orbits are disjoint", "relation": "implies", "rule_id": "orbit_equivalence_reflexive_symmetric_transitive_hg_order", "conditions": [], "explanation": { "zh-Hant": "對稱性使用g的雙邊逆元與作用相容；傳遞性先g再h，群乘積是h*g而非任意交換。若軌道共點，經對稱及傳遞互相包含，因此相等；否則不交。", "en": "Symmetry uses the actual inverse and action compatibility.Transitivity acts by g and then h,so the product is h*g,without assuming commutativity.If two orbits share a point,symmetry and transitivity give both inclusions;otherwise they are disjoint." }, "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": "orbit point0 distinct image count := sum(1, 1)", "after": "orbit point0 distinct image count = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。", "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 orbit point0 distinct image count，不以小數近似替代。", "en": "Recompute orbit point0 distinct image count with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1" ], "output": "2", "label": "orbit point0 distinct image count" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "ORIGINAL point=0;images in ALL group-index order=[0, 1];group order=2", "after": "full orbit sorted and deduplicated=[0, 1];orbit cardinality=2;group enumeration count=2;these counts need NOT agree", "relation": "implies", "rule_id": "orbit_all_original_group_rows_not_single_generator", "conditions": [], "explanation": { "zh-Hant": "逐全部群元素作用像取集合，不只一個g、不沿一個generator猜可達集，也不是stabilizer。多個g可送到同一點，群階與軌道大小分別保留。", "en": "Take images under every group element,not a single g,generator reachability or a stabilizer.Multiple elements may have the same image;keep group order distinct from orbit size." }, "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, 1 ], 2, "all_group_elements_enumerated" ] }, "conclusion": { "canonical_value": "(0, [0, 1], 2, all_group_elements_enumerated)", "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-20261108-r01-finite-group-orbit-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261108-r01-finite-group-orbit-foundations/questions.jsonl", "batch_sha256": "dcd2e6b603238a0bf2a153a0dbff2cf2d408944223663e5c56fdf0e9cd39ad05", "original_line": 16, "original_line_sha256": "1cd58e4a2956bfbf07a8a9d5d99896cf75136438d4e8e428c5474ec900456cce", "question_compact_sha256": "8ca7f15d711a77395c7d6e3d13118489464768fe7680c7b0a84ba0f9fd609685", "english_row_compact_sha256": "6775b48f63bb9c46de81a6ea056e1d025201d7c2792c4d98aa3bcee17f9aaeb2", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261108-r01-finite-group-orbit-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261108-r01-finite-group-orbit-foundations.en.jsonl" }, "mko_bindings": { "mko-finite-group-orbit": { "sha256": "090d257b39b4d7071dedd320490e40cf67e01ea92431b0b6dc042f6127849022", "english_sha256": "6f0d82e7b440502269691a0b3252ad89327583a2459e8dd927850ba70396e4d2", "original_path": "public/data/mko/finite-group-orbit.json", "english_path": "public/data/i18n/editions/mko-finite-group-orbit.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "2bda8f45e99ed8658b697be77ed3e5ec90ceff1b4f72fa7aa6a47d1a0c6254cf", "original_candidate_sha256": "ba248d1cae3181801654f8405332b9d857ef10937c3180ffa230c07a93d7ab8e", "original_candidate_line_sha256": "ba248d1cae3181801654f8405332b9d857ef10937c3180ffa230c07a93d7ab8e" }, "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-20261026-009-p42", "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": "8f134403487694af4c70d2e00ec84bab542043ef4b3c54517c04f3c409c0262a", "question_first_witness_sha256": "ad772845088ed8e99f87f90d4264b05ea97be35775305a02011af5ee35f0f4ce" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 24, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "fa1204538dbe6fe5ac41ed8cbbfa2f2e01c92fb09c34583b1e2a4868882afe7c", "arithmetic_steps": 3 }, "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." } }