{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261027-r03-complete-metric-space-0099", "pilot_number": "R0024-12849", "number": 12849, "family_id": "qf-mod27-complete-evidence-mode-audit", "title": { "zh-Hant": "第 12849 題：完備度量空間", "en": "Question 12849: A Space-Wide Certificate Is Not a List of Successful Sequences" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261027-r03-complete-metric-space-0099", "batch_id": "batch-20261014-advance-20261027-r03-complete-metric-space-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-complete-metric-space" ], "title_zh": "完備度量空間", "stem_zh": "完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。", "answer": { "kind": "expression", "value": "(4, 10, finite_minimum_separation, 是, universal_finite_certificate)", "explanation_zh": "Wave4根節點核對=(4, 10, finite_minimum_separation, 是, universal_finite_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "完備度量空間" ] }, "variant": { "family_id": "qf-mod27-complete-evidence-mode-audit", "seed": 202610273099, "parameters": { "n": 4, "metric": [ [ 0, 10, 11, 11 ], [ 10, 0, 13, 13 ], [ 11, 13, 0, 14 ], [ 11, 13, 14, 0 ] ], "evidence_mode": "finite_minimum_separation", "domain": "Certified_complete_finite_metric_space", "operation": "complete-evidence-mode-audit", "semantic_template": "qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],\"n\":4}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式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-20261027-r03-complete-metric-space-0099", "batch_id": "batch-20261014-advance-20261027-r03-complete-metric-space-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "2d352776eeffdaa495940a5421db61203ad7cdb98c9f3e238bb91b6c3ef3a6c0", "family_id": "qf-mod27-complete-evidence-mode-audit", "seed": 202610273099, "semantic_template": "qf-mod27-complete-evidence-mode-audit/{\"evidence_mode\":\"finite_minimum_separation\",\"metric\":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],\"n\":4}" }, "fields": { "title": "A Space-Wide Certificate Is Not a List of Successful Sequences", "stem": "Use the complete qualified finite metric table=[[0, 10, 11, 11], [10, 0, 13, 13], [11, 13, 0, 14], [11, 13, 14, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.", "answer": { "kind": "expression", "value": "(4, 10, finite_minimum_separation, Yes, universal_finite_certificate)", "explanation": "Compute the same exact separation in both modes, but distinguish the universal argument from a proposed sample-only justification. Insufficient evidence is not a claim that the qualified finite space is incomplete. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 10, finite_minimum_separation, Yes, universal_finite_certificate)." }, "skills": [ "A Space-Wide Certificate Is Not a List of Successful Sequences" ], "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": "完備性量化空間內每個Cauchy序列。操作=complete-evidence-mode-audit；資料={\"n\":4,\"metric\":[[0,10,11,11],[10,0,13,13],[11,13,0,14],[11,13,14,0]],\"evidence_mode\":\"finite_minimum_separation\"}。請依 [n,min_positive_distance,evidence_mode,sufficient,boundary] 欄序回答。", "en": "Use the complete qualified finite metric table=[[0, 10, 11, 11], [10, 0, 13, 13], [11, 13, 0, 14], [11, 13, 14, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. evidence_mode=finite_minimum_separation. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "carrier Fin(4);complete nonnegative integer metric candidate=[[0, 10, 11, 11], [10, 0, 13, 13], [11, 13, 0, 14], [11, 13, 14, 0]]", "after": "zero iff equal=是;symmetric=是", "relation": "implies", "rule_id": "complete_metric_separation_and_symmetry", "conditions": [], "explanation": { "zh-Hant": "先核整表的非負、對角與非對角分離、對稱，不只核本題用到的兩格；至少兩點是本最小正距離協定前提，不是單點空間不完備。", "en": "First qualify nonnegativity,diagonal/off-diagonal separation and symmetry on the entire table,not only queried cells.At least two points is a premise of this minimum-positive-distance protocol,not a claim that a singleton is incomplete." }, "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": "endpoints=[0, 0];direct distance=0;all intermediate labels=[0, 1, 2, 3]", "after": "all two-leg distances=[0, 20, 22, 22];direct<=every two-leg=是", "relation": "implies", "rule_id": "complete_metric_all_triangle_intermediates", "conditions": [], "explanation": { "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。", "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles." }, "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": "endpoints=[0, 1];direct distance=10;all intermediate labels=[0, 1, 2, 3]", "after": "all two-leg distances=[10, 10, 24, 24];direct<=every two-leg=是", "relation": "implies", "rule_id": "complete_metric_all_triangle_intermediates", "conditions": [], "explanation": { "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。", "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles." }, "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": "endpoints=[0, 2];direct distance=11;all intermediate labels=[0, 1, 2, 3]", "after": "all two-leg distances=[11, 23, 11, 25];direct<=every two-leg=是", "relation": "implies", "rule_id": "complete_metric_all_triangle_intermediates", "conditions": [], "explanation": { "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。", "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles." }, "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": "endpoints=[0, 3];direct distance=11;all intermediate labels=[0, 1, 2, 3]", "after": "all two-leg distances=[11, 23, 25, 11];direct<=every two-leg=是", "relation": "implies", "rule_id": "complete_metric_all_triangle_intermediates", "conditions": [], "explanation": { "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。", "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles." }, "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": "endpoints=[1, 0];direct distance=10;all intermediate labels=[0, 1, 2, 3]", "after": "all two-leg distances=[10, 10, 24, 24];direct<=every two-leg=是", "relation": "implies", "rule_id": "complete_metric_all_triangle_intermediates", "conditions": [], "explanation": { "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。", "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles." }, "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": "endpoints=[1, 1];direct distance=0;all intermediate labels=[0, 1, 2, 3]", "after": "all two-leg distances=[20, 0, 26, 26];direct<=every two-leg=是", "relation": "implies", "rule_id": "complete_metric_all_triangle_intermediates", "conditions": [], "explanation": { "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。", "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles." }, "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": "endpoints=[1, 2];direct distance=13;all intermediate labels=[0, 1, 2, 3]", "after": "all two-leg distances=[21, 13, 13, 27];direct<=every two-leg=是", "relation": "implies", "rule_id": "complete_metric_all_triangle_intermediates", "conditions": [], "explanation": { "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。", "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles." }, "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": "endpoints=[1, 3];direct distance=13;all intermediate labels=[0, 1, 2, 3]", "after": "all two-leg distances=[21, 13, 27, 13];direct<=every two-leg=是", "relation": "implies", "rule_id": "complete_metric_all_triangle_intermediates", "conditions": [], "explanation": { "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。", "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles." }, "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": "endpoints=[2, 0];direct distance=11;all intermediate labels=[0, 1, 2, 3]", "after": "all two-leg distances=[11, 23, 11, 25];direct<=every two-leg=是", "relation": "implies", "rule_id": "complete_metric_all_triangle_intermediates", "conditions": [], "explanation": { "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。", "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles." }, "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": "endpoints=[2, 1];direct distance=13;all intermediate labels=[0, 1, 2, 3]", "after": "all two-leg distances=[21, 13, 13, 27];direct<=every two-leg=是", "relation": "implies", "rule_id": "complete_metric_all_triangle_intermediates", "conditions": [], "explanation": { "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。", "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles." }, "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": "endpoints=[2, 2];direct distance=0;all intermediate labels=[0, 1, 2, 3]", "after": "all two-leg distances=[22, 26, 0, 28];direct<=every two-leg=是", "relation": "implies", "rule_id": "complete_metric_all_triangle_intermediates", "conditions": [], "explanation": { "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。", "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles." }, "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": "endpoints=[2, 3];direct distance=14;all intermediate labels=[0, 1, 2, 3]", "after": "all two-leg distances=[22, 26, 14, 14];direct<=every two-leg=是", "relation": "implies", "rule_id": "complete_metric_all_triangle_intermediates", "conditions": [], "explanation": { "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。", "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles." }, "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": "endpoints=[3, 0];direct distance=11;all intermediate labels=[0, 1, 2, 3]", "after": "all two-leg distances=[11, 23, 25, 11];direct<=every two-leg=是", "relation": "implies", "rule_id": "complete_metric_all_triangle_intermediates", "conditions": [], "explanation": { "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。", "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles." }, "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": "endpoints=[3, 1];direct distance=13;all intermediate labels=[0, 1, 2, 3]", "after": "all two-leg distances=[21, 13, 27, 13];direct<=every two-leg=是", "relation": "implies", "rule_id": "complete_metric_all_triangle_intermediates", "conditions": [], "explanation": { "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。", "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles." }, "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": "endpoints=[3, 2];direct distance=14;all intermediate labels=[0, 1, 2, 3]", "after": "all two-leg distances=[22, 26, 14, 14];direct<=every two-leg=是", "relation": "implies", "rule_id": "complete_metric_all_triangle_intermediates", "conditions": [], "explanation": { "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。", "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles." }, "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": "endpoints=[3, 3];direct distance=0;all intermediate labels=[0, 1, 2, 3]", "after": "all two-leg distances=[22, 26, 28, 0];direct<=every two-leg=是", "relation": "implies", "rule_id": "complete_metric_all_triangle_intermediates", "conditions": [], "explanation": { "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。", "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles." }, "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": "S018", "uses": [ "G001", "S017" ], "before": "all off-diagonal positive distances=[10, 11, 11, 10, 13, 13, 11, 13, 14, 11, 13, 14]", "after": "minimum positive distance delta=10>0", "relation": "implies", "rule_id": "finite_metric_positive_minimum_separation", "conditions": [], "explanation": { "zh-Hant": "有限非空正距離清單有正最小值，這是後續全稱Cauchy證書的空間尺度，不是從某條序列估出的門檻。", "en": "A finite nonempty list of positive distances has a positive minimum.This is the space-wide scale for the universal Cauchy certificate,not a threshold guessed from a sequence." }, "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": "arbitrary Cauchy sequence in Fin(4);choose epsilon=delta=10>0", "after": "some N has every m,k>=N with d(x_m,x_k)<10;minimum separation forces x_m=x_k", "relation": "implies", "rule_id": "every_Cauchy_sequence_eventually_constant", "conditions": [], "explanation": { "zh-Hant": "由Cauchy定義對epsilon=delta取得一個N。不同點的距離至少delta，但尾對嚴格小於delta，故任意尾對都相同，特別等於x_N。", "en": "Apply the Cauchy definition at epsilon=delta to obtain N.Distinct points have distance at least delta,whereas every tail pair has distance strictly below delta.Thus all tail terms equal x_N." }, "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": "all tail terms equal x_N;typed sequence guarantees x_N in Fin(4)", "after": "for every positive eta,all k>=N have distance(x_k,x_N)=0