{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0080", "pilot_number": "R0036-18530", "number": 18530, "family_id": "qf-mod35-uniform-finite-prefix-boundary", "title": { "zh-Hant": "第 18530 題：一致收斂", "en": "Question 18530: A Finite Prefix Does Not Supply the Uniform All-n Obligation" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0080", "batch_id": "batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-uniform-convergence-of-functions" ], "title_zh": "一致收斂", "stem_zh": "共同有限domain的symbolic tail與同一N(epsilon)。操作=uniform-finite-prefix-boundary；資料={\"limit\":[4,5],\"coefficients\":[5,-6],\"shift\":2,\"prefix_length\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(5, 否, finite_prefix_not_uniform_convergence_certificate)", "explanation_zh": "Wave5第四層核對=(5, 否, finite_prefix_not_uniform_convergence_certificate)；依完整明示資料重算；同一N控制所有x且有限domain可取pointwise witnesses最大值；不可外推無限domain且finite prefix不足。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "一致收斂" ] }, "variant": { "family_id": "qf-mod35-uniform-finite-prefix-boundary", "seed": 202611042080, "parameters": { "limit": [ 4, 5 ], "coefficients": [ 5, -6 ], "shift": 2, "prefix_length": 5, "domain": "Finite_domain_symbolic_uniform_tail", "operation": "uniform-finite-prefix-boundary", "semantic_template": "qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[5,-6],\"limit\":[4,5],\"prefix_length\":5,\"shift\":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、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、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-20261104-r02-uniform-convergence-of-functions-0080", "batch_id": "batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "5d66025658fb2370877d5a2528dfdf1d43e95eca3d86691c30b11441cf2670ef", "family_id": "qf-mod35-uniform-finite-prefix-boundary", "seed": 202611042080, "semantic_template": "qf-mod35-uniform-finite-prefix-boundary/{\"coefficients\":[5,-6],\"limit\":[4,5],\"prefix_length\":5,\"shift\":2}" }, "fields": { "title": "A Finite Prefix Does Not Supply the Uniform All-n Obligation", "stem": "Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[4, 5], coefficients=[5, -6], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=5. Give (prefix_length, sufficient, boundary).", "answer": { "kind": "expression", "value": "(5, No, finite_prefix_not_uniform_convergence_certificate)", "explanation": "The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (5, No, finite_prefix_not_uniform_convergence_certificate)." }, "skills": [ "A Finite Prefix Does Not Supply the Uniform All-n Obligation" ], "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": "共同有限domain的symbolic tail與同一N(epsilon)。操作=uniform-finite-prefix-boundary；資料={\"limit\":[4,5],\"coefficients\":[5,-6],\"shift\":2,\"prefix_length\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[4, 5], coefficients=[5, -6], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=5. Give (prefix_length, sufficient, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=uniform-finite-prefix-boundary;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[prefix_length, sufficient, boundary]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題歷史介面標記仍在快照；這裡直接給原始欄位順序，主文不需JavaScript或外部表。描述欄位不自動證成任何數學聲稱。", "en": "Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence." }, "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 complete domain Fin(2);aligned limit=[4, 5];signed coefficients=[5, -6];shift=2", "after": "for EVERY n>=0 and EVERY original point x,f_n(x)-limit[x]=coefficient[x]/(n+shift);same positive denominator on the whole domain", "relation": "implies", "rule_id": "uniform_declared_full_family_error_cancellation", "conditions": [], "explanation": { "zh-Hant": "先從明示的全序列定義消去同一limit[x]，不是只看已觀察數值；limit與coefficient保原點序，係數與limit不是機率值。", "en": "Cancel the same limit[x] from the declared whole-sequence formula,not merely observed samples.Retain original point order;these coefficients and limits are not probabilities." }, "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": "finite inspected prefix length=5;SEPARATE explicit full family coefficients=[5, -6];shift=2", "after": "prefix-alone sufficiency=否;finite_prefix_not_uniform_convergence_certificate", "relation": "implies", "rule_id": "uniform_prefix_no_not_denial_of_separate_full_tail", "conditions": [], "explanation": { "zh-Hant": "No只限制有限prefix自身的證明力，不否定另外明示的完整尾模型。一致收斂需要同一N同時控制所有點及每個後續n。", "en": "The No concerns evidence from the prefix alone,not failure of the separately declared whole-tail model.Uniform convergence needs one N for all points and every later 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 } ], "math_model": { "result": [ 5, false, "finite_prefix_not_uniform_convergence_certificate" ] }, "conclusion": { "canonical_value": "(5, 否, finite_prefix_not_uniform_convergence_certificate)", "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-20261104-r02-uniform-convergence-of-functions-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations/questions.jsonl", "batch_sha256": "c5e99643b18779ea8210be153283f9fac0853c289c3f16e39f8c065d3178ecea", "original_line": 80, "original_line_sha256": "4d6055e6971cdd116b16ef93b51868d40d6fd5a3318b654b18c6b1f336fff133", "question_compact_sha256": "5d66025658fb2370877d5a2528dfdf1d43e95eca3d86691c30b11441cf2670ef", "english_row_compact_sha256": "53c6701b61f163c17dbcd7bb1b3cc4661921c32960526f4491bb8b24f3de3d27", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations.en.jsonl" }, "mko_bindings": { "mko-uniform-convergence-of-functions": { "sha256": "c28f03cdaf36d8b06ca1718e4253a1d705745496ed7c655d09f55c2a82148ac0", "english_sha256": "f3588cebfd521dd72a1db18c1ae10af42c4b9398d451eb7680dac158b73a4462", "original_path": "public/data/mko/uniform-convergence-of-functions.json", "english_path": "public/data/i18n/editions/mko-uniform-convergence-of-functions.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "49e5d7674cefa01298c2386543f777d2bdfc99446b4027156b80142f10031994", "original_candidate_sha256": "4d5436b545a71186531154c958dc600591dc75f9e5267d5555424032c62c5374", "original_candidate_line_sha256": "4d5436b545a71186531154c958dc600591dc75f9e5267d5555424032c62c5374" }, "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-20261026-008-p37", "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": "efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301", "question_first_witness_sha256": "acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 3, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "2a75afa420ecef4f5a452282a5f51ae92fdffe7ba6dc387b034e745d0da114ee", "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." } }