{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261012-advance-20261016-r06-injective-0084", "pilot_number": "R0013-07484", "number": 7484, "family_id": "qf-density16-inj-restriction-subset-witness", "title": { "zh-Hant": "第 7484 題：子域限制的全部碰撞", "en": "Question 7484: Typed Source Restriction and Complete Collision Witnesses" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261012-advance-20261016-r06-injective-0084", "batch_id": "batch-20261012-advance-20261016-r06-injective-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-injective" ], "title_zh": "子域限制的全部碰撞", "stem_zh": "Fin型別從0起，完整總表；Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]，images=[2, 1, 0, 0, 1]。原序subset=[4, 2, 0, 1, 3]，已宣告為無重複合法來源標籤。求 (原images, 排序subset, 子域內全部碰撞, 限制單射資格)。", "answer": { "kind": "expression", "value": "([2, 1, 0, 0, 1], [0, 1, 2, 3, 4], [[1, 4], [2, 3]], 否)", "explanation_zh": "有限結構核對=([2, 1, 0, 0, 1], [0, 1, 2, 3, 4], [[1, 4], [2, 3]], 否)。先驗完整原表，再只比較子域標籤；排序不把不同輸入合併。有限精確核對不是一般形式證明。先資格後求值，空函數條件依宣告陪域。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "子域限制的全部碰撞" ] }, "variant": { "family_id": "qf-density16-inj-restriction-subset-witness", "seed": 202610166084, "parameters": { "n": 5, "m": 3, "images": [ 2, 1, 0, 0, 1 ], "subset": [ 4, 2, 0, 1, 3 ], "domain": "Fin_total_table", "operation": "inj-restriction-subset-witness", "semantic_template": "qf-density16-inj-restriction-subset-witness/{\"images\":[2,1,0,0,1],\"m\":3,\"n\":5,\"subset\":[0,1,2,3,4]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-12", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261012-advance-20261016-r06-injective-0084", "batch_id": "batch-20261012-advance-20261016-r06-injective-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "5c0033908bf563b5a1e3ebd8e15200d14fb4b169875cd33fe17688bd1453c492", "family_id": "qf-density16-inj-restriction-subset-witness", "seed": 202610166084, "semantic_template": "qf-density16-inj-restriction-subset-witness/{\"images\":[2,1,0,0,1],\"m\":3,\"n\":5,\"subset\":[0,1,2,3,4]}" }, "fields": { "title": "Typed Source Restriction and Complete Collision Witnesses", "stem": "Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).", "answer": { "kind": "expression", "value": "([2, 1, 0, 0, 1], [0, 1, 2, 3, 4], [[1, 4], [2, 3]], No)", "explanation": "Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 0, 1], [0, 1, 2, 3, 4], [[1, 4], [2, 3]], No)." }, "skills": [ "Typed Source Restriction and Complete Collision Witnesses" ], "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": "Fin型別從0起，完整總表；Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]，images=[2, 1, 0, 0, 1]。原序subset=[4, 2, 0, 1, 3]，已宣告為無重複合法來源標籤。求 (原images, 排序subset, 子域內全部碰撞, 限制單射資格)。", "en": "Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source index=0∈Fin(5)", "after": "image[0]=2∈Fin(3)", "relation": "implies", "rule_id": "injective_complete_original_table_row", "conditions": [], "explanation": { "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。", "en": "First type-check every cell of the complete original table,not only cells queried later." }, "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": "source index=1∈Fin(5)", "after": "image[1]=1∈Fin(3)", "relation": "implies", "rule_id": "injective_complete_original_table_row", "conditions": [], "explanation": { "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。", "en": "First type-check every cell of the complete original table,not only cells queried later." }, "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": "source index=2∈Fin(5)", "after": "image[2]=0∈Fin(3)", "relation": "implies", "rule_id": "injective_complete_original_table_row", "conditions": [], "explanation": { "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。", "en": "First type-check every cell of the complete original table,not only cells queried later." }, "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": "source index=3∈Fin(5)", "after": "image[3]=0∈Fin(3)", "relation": "implies", "rule_id": "injective_complete_original_table_row", "conditions": [], "explanation": { "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。", "en": "First type-check every cell of the complete original table,not only cells queried later." }, "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": "source index=4∈Fin(5)", "after": "image[4]=1∈Fin(3)", "relation": "implies", "rule_id": "injective_complete_original_table_row", "conditions": [], "explanation": { "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。", "en": "First type-check every cell of the complete original table,not only cells queried later." }, "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": "Fin(5)→Fin(3);original images=[2, 1, 0, 0, 1]", "after": "injectivity forbids equal outputs from distinct original input labels", "relation": "implies", "rule_id": "finite_injectivity_label_scope", "conditions": [], "explanation": { "zh-Hant": "單射只禁止不同輸入同輸出；空函數可空真，不能混同滿射。所有原表格先合法，後續限制或刪除保留原始標籤。", "en": "Injectivity forbids equal outputs for distinct inputs;the empty function is vacuously injective,which is not surjectivity.Type-check the whole original table and retain original labels under restriction or deletion." }, "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": "phase=restriction;inputs=(0,1);outputs=(2,1)", "after": "collision=否", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "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": "phase=restriction;inputs=(0,2);outputs=(2,0)", "after": "collision=否", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "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": "phase=restriction;inputs=(0,3);outputs=(2,0)", "after": "collision=否", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "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": "phase=restriction;inputs=(0,4);outputs=(2,1)", "after": "collision=否", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "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": "phase=restriction;inputs=(1,2);outputs=(1,0)", "after": "collision=否", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "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": "phase=restriction;inputs=(1,3);outputs=(1,0)", "after": "collision=否", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "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": "phase=restriction;inputs=(1,4);outputs=(1,1)", "after": "collision=是", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "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": "phase=restriction;inputs=(2,3);outputs=(0,0)", "after": "collision=是", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "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": "phase=restriction;inputs=(2,4);outputs=(0,1)", "after": "collision=否", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "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": "phase=restriction;inputs=(3,4);outputs=(0,1)", "after": "collision=否", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "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": "phase=restriction;all input labels=[0, 1, 2, 3, 4]", "after": "collisions=[[1, 4], [2, 3]];injective=否", "relation": "implies", "rule_id": "complete_pairwise_collision_inventory", "conditions": [], "explanation": { "zh-Hant": "完整比較全部x