{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0023", "pilot_number": "R0053-27098", "number": 27098, "family_id": "qf-mod48-uniform-sup-certificate", "title": { "zh-Hant": "第 27098 題：一致極限保持連續", "en": "Question 27098: An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0023", "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-sup-certificate；資料={\"base_slope\":-2,\"base_intercept\":23,\"error_scale\":23,\"n\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(-2, 23, 23, 23, f_n(x)=-2x+23+23x/24, 23/24, sup_on_[0,1]_attained_at_x=1, 是)", "explanation_zh": "Wave6第十一層核對=(-2, 23, 23, 23, f_n(x)=-2x+23+23x/24, 23/24, sup_on_[0,1]_attained_at_x=1, 是)；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-sup-certificate", "seed": 202611173023, "parameters": { "base_slope": -2, "base_intercept": 23, "error_scale": 23, "n": 23, "domain": "Exact_uniform_limit_continuity", "operation": "uniform-sup-certificate", "semantic_template": "uniform-sup-certificate|{base_intercept:[23,1],base_slope:[-2,1],error_scale:[23,1],n:23}" } }, "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-0023", "batch_id": "batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "da3daf7ccc02da0a382ba8e5f754f752930c2436746a3dddc0c4b621ccb32837", "family_id": "qf-mod48-uniform-sup-certificate", "seed": 202611173023, "semantic_template": "uniform-sup-certificate|{base_intercept:[23,1],base_slope:[-2,1],error_scale:[23,1],n:23}" }, "fields": { "title": "An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum", "stem": "On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-2,b=23,E=23>0,n=23≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).", "answer": { "kind": "expression", "value": "(-2, 23, 23, 23, f_n(x)=-2x+23+23x/24, 23/24, sup_on_[0,1]_attained_at_x=1, Yes)", "explanation": "All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-2, 23, 23, 23, f_n(x)=-2x+23+23x/24, 23/24, sup_on_[0,1]_attained_at_x=1, Yes)." }, "skills": [ "An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum" ], "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-sup-certificate；資料={\"base_slope\":-2,\"base_intercept\":23,\"error_scale\":23,\"n\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-2,b=23,E=23>0,n=23≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [uniform-sup-certificate]", "after": "derived = [[base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate]]", "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": "Uniform ORIGINAL error over n plus one := div(23, 24)", "after": "Uniform ORIGINAL error over n plus one = 23/24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Uniform ORIGINAL error over n plus one，不以小數近似替代。", "en": "Recompute Uniform ORIGINAL error over n plus one with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "23", "24" ], "output": "23/24", "label": "Uniform ORIGINAL error over n plus one" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [-2, 23, 23, 23]", "after": "derived = [f(x)=base_slope*x+base_intercept, f_n(x)-f(x)=(23)*x/24, 23/24, EVERY x in[0,1]:0<=x<=1;attain at x=1, 是]", "relation": "implies", "rule_id": "Uniform_WHOLE_affine_difference_absolute_bound_and_attainment", "conditions": [], "explanation": { "zh-Hant": "原題給的是整個[0,1]上的affine law，不是grid samples。相減消去同一base，因E>0，所有x的絕對誤差=E*x/(n+1)<=E/(n+1)，且x=1達到，故這是確切sup；affine項與limit都連續。", "en": "The given affine law covers the WHOLE[0,1],not sampled points.Subtraction cancels the same base.With E>0,the absolute error is E*x/(n+1)<=E/(n+1) for every x,and x=1 attains it.Hence this is the exact supremum;the affine terms and limit are continuous." }, "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": [ "-2", "23", "23", 23, "f_n(x)=-2x+23+23x/24", "23/24", "sup_on_[0,1]_attained_at_x=1", true ] }, "conclusion": { "canonical_value": "(-2, 23, 23, 23, f_n(x)=-2x+23+23x/24, 23/24, sup_on_[0,1]_attained_at_x=1, 是)", "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": 23, "original_line_sha256": "545b92df423fc663a91a174397d0248de2766e4922db513be1204fbadbcfe7cc", "question_compact_sha256": "da3daf7ccc02da0a382ba8e5f754f752930c2436746a3dddc0c4b621ccb32837", "english_row_compact_sha256": "6ad8bea8e85e24d49199c86d710990c653488fd5be5055fc297673016d2673bd", "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": "36ca3f24564d33654260f71252f1c60a525335ba745835ab15d0405e7bc748be", "original_candidate_line_sha256": "36ca3f24564d33654260f71252f1c60a525335ba745835ab15d0405e7bc748be" }, "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": 3, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "39b6f1f4b239b31b9f21829cb5658495ef465e458f2e629aa1b1854d5e5c1454", "arithmetic_steps": 1 }, "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." } }