{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0093", "pilot_number": "R0053-27068", "number": 27068, "family_id": "qf-mod48-uniform-limit-boundary", "title": { "zh-Hant": "第 27068 題：一致極限保持連續", "en": "Question 27068: Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0093", "batch_id": "batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-uniform-limit-continuity-theorem" ], "title_zh": "一致極限保持連續", "stem_zh": "共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"discontinuous_terms\",\"scale\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(discontinuous_terms, 19, f_n(x)=1_if_x>=1/20_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, 否)", "explanation_zh": "Wave6第十一層核對=(discontinuous_terms, 19, f_n(x)=1_if_x>=1/20_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "一致極限保持連續" ] }, "variant": { "family_id": "qf-mod48-uniform-limit-boundary", "seed": 202611173093, "parameters": { "mode": "discontinuous_terms", "scale": 19, "domain": "Exact_uniform_limit_continuity", "operation": "uniform-limit-boundary", "semantic_template": "uniform-limit-boundary|{mode:\"discontinuous_terms\",scale:19}" } }, "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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0093", "batch_id": "batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "bf59462125915d3785b32082a32dabad12168bf5a184b17163d2b32554377938", "family_id": "qf-mod48-uniform-limit-boundary", "seed": 202611173093, "semantic_template": "uniform-limit-boundary|{mode:\"discontinuous_terms\",scale:19}" }, "fields": { "title": "Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms", "stem": "Use mode=discontinuous_terms, positive integer scale=19, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).", "answer": { "kind": "expression", "value": "(discontinuous_terms, 19, f_n(x)=1_if_x>=1/20_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)", "explanation": "x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (discontinuous_terms, 19, f_n(x)=1_if_x>=1/20_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)." }, "skills": [ "Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms" ], "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": "共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"discontinuous_terms\",\"scale\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use mode=discontinuous_terms, positive integer scale=19, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [uniform-limit-boundary]", "after": "derived = [[mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "此頁直接列完整來源欄序與mode特定角色，不靠JavaScript或由答案外形猜介面；No、undefined、infinity各保原意。", "en": "List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [discontinuous_terms, 19]", "after": "derived = [discontinuous_terms, 19, f_n(x)=1_if_x>=1/20_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, 否]", "relation": "implies", "rule_id": "Uniform_original_WHOLE_boundary_premise_obstruction", "conditions": [], "explanation": { "zh-Hant": "x的冪在x<1趨0、在1為1，只有逐點收斂；固定step sequence雖uniform但每項不連續。有限grid可漏掉點間unit bump。moving spike每項連續，在移動peak取1，固定x最終取0，所以sup誤差仍1。No拒定理前提／方法推論，不把所有underlying convergence都否定，也不冒稱已做形式證明。", "en": "Powers tend to0 for x<1 but equal1 at x=1,so convergence is only pointwise.A constant step sequence is uniform but its terms are discontinuous.A finite grid can miss a unit bump between points.A moving spike is continuous,attains1 at its moving peak and eventually vanishes at each fixed x,so its sup error stays1.No rejects the theorem premises/method,not every underlying convergence claim or a performed formal proof." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null } ], "math_model": { "result": [ "discontinuous_terms", 19, "f_n(x)=1_if_x>=1/20_else_0_for_every_n", "uniform_but_every_term_and_limit_are_discontinuous", false ] }, "conclusion": { "canonical_value": "(discontinuous_terms, 19, f_n(x)=1_if_x>=1/20_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, 否)", "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-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations/questions.jsonl", "batch_sha256": "44b65c723885d425fc0e5b2e6a3b0313fdbda2cfdd29d739ed78d03875c75af0", "original_line": 93, "original_line_sha256": "c28a46a5408724b91767e90070a587abc76c959602fe68482f7698a24ab1586d", "question_compact_sha256": "bf59462125915d3785b32082a32dabad12168bf5a184b17163d2b32554377938", "english_row_compact_sha256": "8a87b86033c279159aa8ab00dc69bae64bb9a8ab8382c0b3735c028a2c3c3427", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations.en.jsonl" }, "mko_bindings": { "mko-uniform-limit-continuity-theorem": { "sha256": "7dd17546858c30c475f0f2eead9893cb35e9e8ed4969e65933c7a8fd7a621632", "english_sha256": "6c9744f93c7afdccbc56b487121dbdd22b7d603cb95927b0a7147b9456c1cd5d", "original_path": "public/data/mko/uniform-limit-continuity-theorem.json", "english_path": "public/data/i18n/editions/mko-uniform-limit-continuity-theorem.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "a54409c0bc3f68730e1072c6d3d772970462c1bc7ffc203b4030a6136a610283", "original_candidate_sha256": "65f3a19924631690e376766a8adc945d854f65483974e48dbe1fe13b8b736550", "original_candidate_line_sha256": "65f3a19924631690e376766a8adc945d854f65483974e48dbe1fe13b8b736550" }, "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-20261026-010-p54", "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": "1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62", "question_first_witness_sha256": "79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 2, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "2ca3acdbdf1c195b500295a7b86444f3932ce4544f9878f9762c1d2a8fc4aa54", "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." } }