{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261105-r05-homeomorphism-0079", "pilot_number": "R0037-19079", "number": 19079, "family_id": "qf-mod36-homeo-metric-nonimplication", "title": { "zh-Hant": "第 19079 題：同胚", "en": "Question 19079: Equal Scale Parameters Are Not an Unqualified Isometry Test" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261105-r05-homeomorphism-0079", "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-metric-nonimplication；資料={\"n\":5,\"map\":[0,1,3,4,2],\"source_scale\":4,\"target_scale\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(5, 是, 否, 4, 5, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)", "explanation_zh": "Wave5第五層核對=(5, 是, 否, 4, 5, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；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": 1, "skills": [ "同胚" ] }, "variant": { "family_id": "qf-mod36-homeo-metric-nonimplication", "seed": 202611055079, "parameters": { "n": 5, "map": [ 0, 1, 3, 4, 2 ], "source_scale": 4, "target_scale": 5, "domain": "Complete_finite_topological_spaces", "operation": "homeo-metric-nonimplication", "semantic_template": "qf-mod36-homeo-metric-nonimplication/{\"map\":[0,1,3,4,2],\"n\":5,\"source_scale\":4,\"target_scale\":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-0079", "batch_id": "batch-20261014-advance-20261105-r05-homeomorphism-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "60c6c39fabc8f0a032ebc664453297bbe14651c10a8d60bdad3400dd887a6a70", "family_id": "qf-mod36-homeo-metric-nonimplication", "seed": 202611055079, "semantic_template": "qf-mod36-homeo-metric-nonimplication/{\"map\":[0,1,3,4,2],\"n\":5,\"source_scale\":4,\"target_scale\":5}" }, "fields": { "title": "Equal Scale Parameters Are Not an Unqualified Isometry Test", "stem": "Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 1, 3, 4, 2], positive source_scale=4 and target_scale=5. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).", "answer": { "kind": "expression", "value": "(5, Yes, No, 4, 5, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)", "explanation": "The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 4, 5, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)." }, "skills": [ "Equal Scale Parameters Are Not an Unqualified Isometry Test" ], "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-metric-nonimplication；資料={\"n\":5,\"map\":[0,1,3,4,2],\"source_scale\":4,\"target_scale\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 1, 3, 4, 2], positive source_scale=4 and target_scale=5. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=homeo-metric-nonimplication;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, 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, 1, 3, 4, 2]", "after": "all target values distinct and complete;explicit inverse=[0, 1, 4, 2, 3];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": "homeo source singleton-ball radius := div(4, 2)", "after": "homeo source singleton-ball radius = 2", "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，保留參數、樣本、目標與分母，不交換未授權的聲稱。", "en": "Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 homeo source singleton-ball radius，不以小數近似替代。", "en": "Recompute homeo source singleton-ball radius with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "4", "2" ], "output": "2", "label": "homeo source singleton-ball radius" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "homeo target singleton-ball radius := div(5, 2)", "after": "homeo target singleton-ball radius = 5/2", "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，保留參數、樣本、目標與分母，不交換未授權的聲稱。", "en": "Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 homeo target singleton-ball radius，不以小數近似替代。", "en": "Recompute homeo target singleton-ball radius with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "5", "2" ], "output": "5/2", "label": "homeo target singleton-ball radius" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "actual n=5;positive source off-diagonal distance=4;target off-diagonal distance=5;complete bijection=[0, 1, 3, 4, 2]", "after": "positive radii=[2, 5/2] give singleton balls,so BOTH metric topologies discrete and map homeomorphic;equal SCALE parameters=否", "relation": "implies", "rule_id": "homeo_scaled_discrete_topology_not_unqualified_isometry_flag", "conditions": [], "explanation": { "zh-Hant": "各尺度的一半為正，ball只含中心，單點開而所有子集開，所以兩邊都離散且任意此雙射是homeomorphism。第三欄只逐字比較尺度，不把No當未附條件的等距否證。", "en": "Half each positive scale gives singleton balls,hence every subset is open and both metric topologies are discrete.This supplied bijection is a homeomorphism.The third field compares scale parameters only;No is not an unqualified disproof of isometry." }, "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": "actual distinct source pair[0,1];distinct images=[0, 1];source distance=4;target distance=5", "after": "in THIS at-least-two-point scaled-discrete model,isometry iff scales equal=否;no differentiability or congruence structure supplied", "relation": "implies", "rule_id": "homeo_actual_nonzero_distance_witness_not_extra_geometry", "conditions": [], "explanation": { "zh-Hant": "n至少2時這對點給非零距離見證，尺度不同就不是此model的等距；其他可微或幾何全等問題沒有資料，不自行作判決。", "en": "With at least two points this pair witnesses a nonzero distance.Unequal scales refute isometry in this model;no differentiability or geometric-congruence structure has been supplied." }, "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, true, false, "4", "5", "homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence" ] }, "conclusion": { "canonical_value": "(5, 是, 否, 4, 5, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)", "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": 79, "original_line_sha256": "3cdd2757155a864709c723cbfb74654ef417b61e71c01d1566f532a9d56100a3", "question_compact_sha256": "60c6c39fabc8f0a032ebc664453297bbe14651c10a8d60bdad3400dd887a6a70", "english_row_compact_sha256": "58b265d2e8b1fc6dcd90c8a6bda8d6e4943214cb6999798332372ce654020163", "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": "b558e748945cd1b23a876a5d95f1b7cb03d92d72087b450d05fce0bc76e213f3", "original_candidate_line_sha256": "b558e748945cd1b23a876a5d95f1b7cb03d92d72087b450d05fce0bc76e213f3" }, "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": 6, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "891e624f3269d94fc8a3ab2d5a61170b9795b8c0e2373abfffb3656df313cfaa", "arithmetic_steps": 2 }, "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." } }