{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0008", "pilot_number": "R0041-21458", "number": 21458, "family_id": "qf-mod40-bw-periodic-boundedness", "title": { "zh-Hant": "第 21458 題：實數列Bolzano－Weierstrass定理", "en": "Question 21458: A Whole Declared Periodic Sequence Has an Exact Absolute Bound" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0008", "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-boundedness；資料={\"period_values\":[8,-9,[10,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([8, -9, 5], 9, 是, whole_infinite_periodic_sequence_bounded)", "explanation_zh": "Wave6第三層核對=([8, -9, 5], 9, 是, whole_infinite_periodic_sequence_bounded)；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": 1, "skills": [ "實數列Bolzano－Weierstrass定理" ] }, "variant": { "family_id": "qf-mod40-bw-periodic-boundedness", "seed": 202611092008, "parameters": { "period_values": [ 8, -9, [ 10, 2 ] ], "domain": "Symbolic_bounded_real_sequence_subsequence", "operation": "bw-periodic-boundedness", "semantic_template": "qf-mod40-bw-periodic-boundedness/{\"period_values\":[8,-9,[10,2]]}" } }, "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-0008", "batch_id": "batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "b2123965c27060e93cc5f8e8b1210eff0341ecf21b6ba2095ff97a2f902fc206", "family_id": "qf-mod40-bw-periodic-boundedness", "seed": 202611092008, "semantic_template": "qf-mod40-bw-periodic-boundedness/{\"period_values\":[8,-9,[10,2]]}" }, "fields": { "title": "A Whole Declared Periodic Sequence Has an Exact Absolute Bound", "stem": "Use the whole infinite real periodic sequence with nonempty period_values=[8, -9, 10/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).", "answer": { "kind": "expression", "value": "([8, -9, 5], 9, Yes, whole_infinite_periodic_sequence_bounded)", "explanation": "Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([8, -9, 5], 9, Yes, whole_infinite_periodic_sequence_bounded)." }, "skills": [ "A Whole Declared Periodic Sequence Has an Exact Absolute Bound" ], "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-boundedness；資料={\"period_values\":[8,-9,[10,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the whole infinite real periodic sequence with nonempty period_values=[8, -9, 10/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=bw-periodic-boundedness;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[period_values, absolute_bound, whole_sequence_bounded, 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=8;absolute value=8", "after": "prefix absolute maximum after this original entry=8", "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=-9;absolute value=9", "after": "prefix absolute maximum after this original entry=9", "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=5;absolute value=5", "after": "prefix absolute maximum after this original entry=9", "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=[8, -9, 5];period length=3", "after": "for EVERY n>=0,unique n=3q+r with0<=r<3 gives a_n=period_values[r];therefore |a_n|<=9", "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 } ], "math_model": { "result": [ [ "8", "-9", "5" ], "9", true, "whole_infinite_periodic_sequence_bounded" ] }, "conclusion": { "canonical_value": "([8, -9, 5], 9, 是, whole_infinite_periodic_sequence_bounded)", "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-20261109-r02-bolzano-weierstrass-theorem-real-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations/questions.jsonl", "batch_sha256": "cd386460b627b8305543a8fb6674bbe6aba93e951a5d4ce260bdc15d107b10c1", "original_line": 8, "original_line_sha256": "4d9f019991cfd042b15ea363e797f22cf8f03ebf2279a9328131540175c38845", "question_compact_sha256": "b2123965c27060e93cc5f8e8b1210eff0341ecf21b6ba2095ff97a2f902fc206", "english_row_compact_sha256": "ff3919c30dfdf11a4f7561ebb4f18b14254f552cc3be0bd52171ee51a2d623f8", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations.en.jsonl" }, "mko_bindings": { "mko-bolzano-weierstrass-theorem-real": { "sha256": "68509f22469c3dc3548c8128faec469688cb7a72f7b9fca9c4539568db8e44ea", "english_sha256": "484aa7b44b137b24493d954852772e36bbac420461a485eeb02b763116637432", "original_path": "public/data/mko/bolzano-weierstrass-theorem-real.json", "english_path": "public/data/i18n/editions/mko-bolzano-weierstrass-theorem-real.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "2bda8f45e99ed8658b697be77ed3e5ec90ceff1b4f72fa7aa6a47d1a0c6254cf", "original_candidate_sha256": "32438939b9d4b14c106d6f9fb566b05701c90ddea1c1bd1b8ec5df2383679c23", "original_candidate_line_sha256": "32438939b9d4b14c106d6f9fb566b05701c90ddea1c1bd1b8ec5df2383679c23" }, "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-20261026-009-p42", "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": "8f134403487694af4c70d2e00ec84bab542043ef4b3c54517c04f3c409c0262a", "question_first_witness_sha256": "ad772845088ed8e99f87f90d4264b05ea97be35775305a02011af5ee35f0f4ce" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 5, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "0ba573b52e688fc8aef3bbb9dac638f41d06d9cf11cfbbc17e692d9c1fcb9e12", "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." } }