{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261105-r05-homeomorphism-0073", "pilot_number": "R0037-19048", "number": 19048, "family_id": "qf-mod36-homeo-finite-discrete-bijection", "title": { "zh-Hant": "第 19048 題：同胚", "en": "Question 19048: A Finite Discrete Bijection Is a Homeomorphism within That Scope" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261105-r05-homeomorphism-0073", "batch_id": "batch-20261014-advance-20261105-r05-homeomorphism-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-homeomorphism" ], "title_zh": "同胚", "stem_zh": "完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-discrete-bijection；資料={\"n\":5,\"map\":[0,4,3,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(5, [0, 4, 3, 1, 2], 是, 是, 是, finite_discrete_scope_only)", "explanation_zh": "Wave5第五層核對=(5, [0, 4, 3, 1, 2], 是, 是, 是, finite_discrete_scope_only)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "同胚" ] }, "variant": { "family_id": "qf-mod36-homeo-finite-discrete-bijection", "seed": 202611055073, "parameters": { "n": 5, "map": [ 0, 4, 3, 1, 2 ], "domain": "Complete_finite_topological_spaces", "operation": "homeo-finite-discrete-bijection", "semantic_template": "qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,4,3,1,2],\"n\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0073", "batch_id": "batch-20261014-advance-20261105-r05-homeomorphism-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "e9ee0890be97db2c244e68aaa4d4a2c11cb37c75f05e6748a1e46a2c1acf0e1c", "family_id": "qf-mod36-homeo-finite-discrete-bijection", "seed": 202611055073, "semantic_template": "qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,4,3,1,2],\"n\":5}" }, "fields": { "title": "A Finite Discrete Bijection Is a Homeomorphism within That Scope", "stem": "Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 4, 3, 1, 2]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).", "answer": { "kind": "expression", "value": "(5, [0, 4, 3, 1, 2], Yes, Yes, Yes, finite_discrete_scope_only)", "explanation": "Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 4, 3, 1, 2], Yes, Yes, Yes, finite_discrete_scope_only)." }, "skills": [ "A Finite Discrete Bijection Is a Homeomorphism within That Scope" ], "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": "完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-discrete-bijection；資料={\"n\":5,\"map\":[0,4,3,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 4, 3, 1, 2]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=homeo-finite-discrete-bijection;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary]", "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；本頁直接列原始欄位順序，不需JavaScript或查外部表。欄位名字本身不自動是數學證據，No依各題的聲稱範圍解讀。", "en": "Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field names are not mathematical evidence,and No retains the operation's specific 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 carrier Fin(5);original TOTAL source-indexed candidate map=[0, 4, 3, 1, 2]", "after": "all target values distinct and complete;explicit inverse=[0, 3, 4, 2, 1];inverse(map(x))=x and map(inverse(y))=y for EVERY carrier point", "relation": "implies", "rule_id": "homeo_complete_bijection_not_name_assumption", "conditions": [], "explanation": { "zh-Hant": "完整source列表每個target出現恰一次才接受雙射；逐原索引填inverse，保兩方向真的互逆，不因題族叫bijection就跳過資格。", "en": "Accept a bijection only when every target appears exactly once in the complete source list.Build the inverse by original indices and retain both inverse identities rather than trusting a family name." }, "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 finite carriers Fin(5);complete bijection=[0, 4, 3, 1, 2] and inverse=[0, 3, 4, 2, 1];BOTH topologies declared discrete", "after": "for ANY subset on either target,its preimage is a subset and therefore open;both map and inverse continuous;homeomorphism in this finite discrete scope", "relation": "implies", "rule_id": "homeo_discrete_all_subsets_two_way_not_general_bijection", "conditions": [], "explanation": { "zh-Hant": "離散拓撲有所有子集開，故兩方向對任意子集逆像都開。此論證依明示兩邊discrete，不把任意拓撲雙射直接升格homeomorphism，也不推metric或可微結構。", "en": "Every subset is open in a discrete topology,so preimages of arbitrary subsets are open in both directions.This uses both declared discrete topologies,not an arbitrary-topology bijection,and supplies no metric or differentiable conclusion." }, "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": [ 5, [ 0, 4, 3, 1, 2 ], true, true, true, "finite_discrete_scope_only" ] }, "conclusion": { "canonical_value": "(5, [0, 4, 3, 1, 2], 是, 是, 是, finite_discrete_scope_only)", "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-20261105-r05-homeomorphism-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261105-r05-homeomorphism-foundations/questions.jsonl", "batch_sha256": "434f3d979cb84cd573c41747c6bc152ba1ef46aa635fca24d2fb9981f5bc293f", "original_line": 73, "original_line_sha256": "ead35c00cbab6f5618b7f829a0183be0668117728562d923c4f65a66b1e2b8b8", "question_compact_sha256": "e9ee0890be97db2c244e68aaa4d4a2c11cb37c75f05e6748a1e46a2c1acf0e1c", "english_row_compact_sha256": "daf9561a03b1392bceb86ecaeb804320e38656e92bd59ade4bb218e65129b04e", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261105-r05-homeomorphism-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261105-r05-homeomorphism-foundations.en.jsonl" }, "mko_bindings": { "mko-homeomorphism": { "sha256": "2869b47d1ae828f6ec6ed61a0f7aa9d72ef812e946cb7b04184b3592d2f384de", "english_sha256": "d5057b40272d1d558ba56008bae40cf826c2405ad5dd379b5d22dd4562bcbc89", "original_path": "public/data/mko/homeomorphism.json", "english_path": "public/data/i18n/editions/mko-homeomorphism.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "7be57d88eceb13736454f82dd18bb3c4ec9d30c0367f974dcefdf8990dc4e3cb", "original_candidate_sha256": "7b568292563d313cbdda8683746bbafa0d8d1658091c98ca1ba184af1c977440", "original_candidate_line_sha256": "7b568292563d313cbdda8683746bbafa0d8d1658091c98ca1ba184af1c977440" }, "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-p38", "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": "065507b602ceee60cf47a700d387a51940e0c6724c35a3336a7a7948a199ed80", "question_first_witness_sha256": "7203f33b030792559673b0fee520a8c0e70903c4dd970b35293aa40852f333d6" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 3, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "de9fbb0ec02933639d35664b64915ab8b66ed2a15b6e90387df0ae498b2d1a45", "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." } }