{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0076", "pilot_number": "R0041-21051", "number": 21051, "family_id": "qf-mod39-monoseq-hypothesis-boundary", "title": { "zh-Hant": "第 21051 題：單調有界數列收斂定理", "en": "Question 21051: Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0076", "batch_id": "batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-monotone-sequence-convergence-theorem" ], "title_zh": "單調有界數列收斂定理", "stem_zh": "實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"monotone_unbounded\",\"prefix_length\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(monotone_unbounded, [0, 1, 2], 是, 否, 否, monotone_without_bound_not_covered)", "explanation_zh": "Wave6第二層核對=(monotone_unbounded, [0, 1, 2], 是, 否, 否, monotone_without_bound_not_covered)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "單調有界數列收斂定理" ] }, "variant": { "family_id": "qf-mod39-monoseq-hypothesis-boundary", "seed": 202611082076, "parameters": { "mode": "monotone_unbounded", "prefix_length": 3, "domain": "Symbolic_monotone_bounded_real_sequence", "operation": "monoseq-hypothesis-boundary", "semantic_template": "qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":3}" } }, "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-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、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-20261108-r02-monotone-sequence-convergence-theorem-0076", "batch_id": "batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "7c756ff17beb78c3aed7d75a4dd74ee1e762b91b585c4dc19dc57e1d65eca3ef", "family_id": "qf-mod39-monoseq-hypothesis-boundary", "seed": 202611082076, "semantic_template": "qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":3}" }, "fields": { "title": "Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation", "stem": "Boundary mode=monotone_unbounded, displayed prefix_length=3. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).", "answer": { "kind": "expression", "value": "(monotone_unbounded, [0, 1, 2], Yes, No, No, monotone_without_bound_not_covered)", "explanation": "Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (monotone_unbounded, [0, 1, 2], Yes, No, No, monotone_without_bound_not_covered)." }, "skills": [ "Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation" ], "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": "實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"monotone_unbounded\",\"prefix_length\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Boundary mode=monotone_unbounded, displayed prefix_length=3. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=monoseq-hypothesis-boundary;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary]", "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 mode=monotone_unbounded;displayed prefix length=3;prefix_or_length=[0, 1, 2]", "after": "declared WHOLE a_n=n:adjacent difference1>0;for ANY real bound B choose natural n>B,so not bounded above and not a finite-limit sequence", "relation": "implies", "rule_id": "monoseq_boundary_whole_law_not_observed_prefix", "conditions": [], "explanation": { "zh-Hant": "先分清給的是全數列公式還是只給有限長度。前兩種的單調／有界／不收斂理由是全稱公式與明示反例，不由顯示的頭幾項推斷。", "en": "Distinguish a whole-sequence formula from a finite length.In the first two cases,monotonicity,bounds and divergence follow from the declared infinite law and explicit witnesses,not the displayed prefix." }, "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 boundary mode=monotone_unbounded;actual certificate flags=(是, 否, 否)", "after": "monotone convergence theorem not certified here;convergence in GENERAL does NOT imply monotonicity;unknown continuation not declared divergent", "relation": "implies", "rule_id": "monoseq_hypotheses_sufficient_not_convergence_necessity", "conditions": [], "explanation": { "zh-Hant": "單調有界是此定理的充分條件，不是所有收斂數列的必要條件。有限前綴的No不宣稱任何延伸都發散；bounded_nonmonotone這個指定交錯例的發散有自己的奇偶子序列證據。", "en": "Monotonicity and the matching bound are sufficient for this theorem,not necessary for every convergent sequence.A finite-prefix No does not make all extensions divergent;the specified alternating example has its own even/odd divergence witness." }, "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": [ "monotone_unbounded", [ 0, 1, 2 ], true, false, false, "monotone_without_bound_not_covered" ] }, "conclusion": { "canonical_value": "(monotone_unbounded, [0, 1, 2], 是, 否, 否, monotone_without_bound_not_covered)", "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-20261108-r02-monotone-sequence-convergence-theorem-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations/questions.jsonl", "batch_sha256": "89dab9b3bab56cf32df54dfb1052a43027c5e475f1fe7279fd2a6f3e5b292774", "original_line": 76, "original_line_sha256": "62a89587f8b4bcf3ec7fec2842879ae8a04ecb19dd1cb2ba774904592d35c49a", "question_compact_sha256": "7c756ff17beb78c3aed7d75a4dd74ee1e762b91b585c4dc19dc57e1d65eca3ef", "english_row_compact_sha256": "a4d5f2abef55f9b63457112fe2b3e456d09935b37ce61f3cc47d7012d826f772", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations.en.jsonl" }, "mko_bindings": { "mko-monotone-sequence-convergence-theorem": { "sha256": "7347d54ee6a5bad0a10d31d2740a6f5af3d2539f9dd91cfc6e85d51ea7c966dc", "english_sha256": "04ac52a1866dfc205ff11c701b766a16b5e2bfa894396f50a16b4f6eb495d448", "original_path": "public/data/mko/monotone-sequence-convergence-theorem.json", "english_path": "public/data/i18n/editions/mko-monotone-sequence-convergence-theorem.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "2bda8f45e99ed8658b697be77ed3e5ec90ceff1b4f72fa7aa6a47d1a0c6254cf", "original_candidate_sha256": "47c88bfcfa00f3de276bf005a60c1d873e5a209717014f332e820d3da2c3b53c", "original_candidate_line_sha256": "47c88bfcfa00f3de276bf005a60c1d873e5a209717014f332e820d3da2c3b53c", "mko_binding_revision": { "id": "edition-symbol-bindings-20261101", "path": "content/edition-symbol-bindings-20261101/revision.json", "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c", "previous_english_sha256": { "mko-monotone-sequence-convergence-theorem": "c3634e802309cc74e6545f1ff6fbce10c51061adddafe94aa8151785626efdc9" } } }, "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. 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