{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0038", "pilot_number": "R0041-21488", "number": 21488, "family_id": "qf-mod40-bw-periodic-subsequence", "title": { "zh-Hant": "第 21488 題：實數列Bolzano－Weierstrass定理", "en": "Question 21488: A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0038", "batch_id": "batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-bolzano-weierstrass-theorem-real" ], "title_zh": "實數列Bolzano－Weierstrass定理", "stem_zh": "whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[13,-14,[15,2]],\"residue\":0,\"j\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(0, 13, 39, 13, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)", "explanation_zh": "Wave6第三層核對=(0, 13, 39, 13, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "實數列Bolzano－Weierstrass定理" ] }, "variant": { "family_id": "qf-mod40-bw-periodic-subsequence", "seed": 202611092038, "parameters": { "period_values": [ 13, -14, [ 15, 2 ] ], "residue": 0, "j": 13, "domain": "Symbolic_bounded_real_sequence_subsequence", "operation": "bw-periodic-subsequence", "semantic_template": "qf-mod40-bw-periodic-subsequence/{\"j\":13,\"period_values\":[13,-14,[15,2]],\"residue\":0}" } }, "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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0038", "batch_id": "batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "1870168da1b1b4cbf5b139c2bd114e4917bfd3b7a9a8c7f503379716a4aef528", "family_id": "qf-mod40-bw-periodic-subsequence", "seed": 202611092038, "semantic_template": "qf-mod40-bw-periodic-subsequence/{\"j\":13,\"period_values\":[13,-14,[15,2]],\"residue\":0}" }, "fields": { "title": "A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence", "stem": "Use the whole infinite real periodic sequence with nonempty period_values=[13, -14, 15/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=0 and subsequence parameter j=13≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).", "answer": { "kind": "expression", "value": "(0, 13, 39, 13, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)", "explanation": "With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (0, 13, 39, 13, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)." }, "skills": [ "A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence" ], "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": "whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[13,-14,[15,2]],\"residue\":0,\"j\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the whole infinite real periodic sequence with nonempty period_values=[13, -14, 15/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=0 and subsequence parameter j=13≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=bw-periodic-subsequence;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[residue, j, k_j, value, constant_subsequence_certificate, 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 period index=0;value=13;absolute value=13", "after": "prefix absolute maximum after this original entry=13", "relation": "implies", "rule_id": "bw_every_period_entry_absolute_bound", "conditions": [], "explanation": { "zh-Hant": "逐完整週期取絕對值最大，不只最大原值；負極值可能決定界。週期內重複值仍保留原索引。", "en": "Scan absolute values over the full period rather than the maximum signed value.A negative entry may determine the bound;retain original positions and duplicates." }, "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 period index=1;value=-14;absolute value=14", "after": "prefix absolute maximum after this original entry=14", "relation": "implies", "rule_id": "bw_every_period_entry_absolute_bound", "conditions": [], "explanation": { "zh-Hant": "逐完整週期取絕對值最大，不只最大原值；負極值可能決定界。週期內重複值仍保留原索引。", "en": "Scan absolute values over the full period rather than the maximum signed value.A negative entry may determine the bound;retain original positions and duplicates." }, "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 period index=2;value=15/2;absolute value=15/2", "after": "prefix absolute maximum after this original entry=14", "relation": "implies", "rule_id": "bw_every_period_entry_absolute_bound", "conditions": [], "explanation": { "zh-Hant": "逐完整週期取絕對值最大，不只最大原值；負極值可能決定界。週期內重複值仍保留原索引。", "en": "Scan absolute values over the full period rather than the maximum signed value.A negative entry may determine the bound;retain original positions and duplicates." }, "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 NONEMPTY full period=[13, -14, 15/2];period length=3", "after": "for EVERY n>=0,unique n=3q+r with0<=r<3 gives a_n=period_values[r];therefore |a_n|<=14", "relation": "implies", "rule_id": "bw_periodic_law_for_all_n_not_finite_observation", "conditions": [], "explanation": { "zh-Hant": "全稱有界來自明示永遠重複的law與除法算法，不是由幾個觀察值推定未來。這證指定模型，沒有把所有有界實數列的Bolzano-Weierstrass定理化成有限取樣。", "en": "Whole-sequence boundedness follows from the declared infinite periodic law and division with remainder,not observed future behavior.This proves the stated model,not the general Bolzano-Weierstrass theorem from finite samples." }, "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": "bw period times ORIGINAL j := mul(3, 13)", "after": "bw period times ORIGINAL j = 39", "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 bw period times ORIGINAL j，不以小數近似替代。", "en": "Recompute bw period times ORIGINAL j with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "13" ], "output": "39", "label": "bw period times ORIGINAL j" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "bw ORIGINAL residue subsequence evaluated index := sum(0, 39)", "after": "bw ORIGINAL residue subsequence evaluated index = 39", "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 bw ORIGINAL residue subsequence evaluated index，不以小數近似替代。", "en": "Recompute bw ORIGINAL residue subsequence evaluated index with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "39" ], "output": "39", "label": "bw ORIGINAL residue subsequence evaluated index" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "ORIGINAL residue=0,period=3;k(j)=0+3j for EVERY j>=0;displayed j=13,k_j=39", "after": "for ALL j,k(j+1)-k(j)=3>0;remainder always0;a_k(j)=13;for EVERY epsilon>0 error0