{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0070", "pilot_number": "R0053-27020", "number": 27020, "family_id": "qf-mod48-uniform-continuity-transfer", "title": { "zh-Hant": "第 27020 題：一致極限保持連續", "en": "Question 27020: A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0070", "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-continuity-transfer；資料={\"n\":20,\"lipschitz\":5,\"error_bound\":[1,23],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(20, 5, 1/23, 1, 1/15, 29/69, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)", "explanation_zh": "Wave6第十一層核對=(20, 5, 1/23, 1, 1/15, 29/69, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；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-continuity-transfer", "seed": 202611173070, "parameters": { "n": 20, "lipschitz": 5, "error_bound": [ 1, 23 ], "epsilon": 1, "domain": "Exact_uniform_limit_continuity", "operation": "uniform-continuity-transfer", "semantic_template": "uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,23],lipschitz:[5,1],n:20}" } }, "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-0070", "batch_id": "batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "c8f58434933c9ae1276b486233e5427c385dd2ff0253a356304aa9051333b9b3", "family_id": "qf-mod48-uniform-continuity-transfer", "seed": 202611173070, "semantic_template": "uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,23],lipschitz:[5,1],n:20}" }, "fields": { "title": "A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer", "stem": "Fix N=20, a positive Lipschitz bound L=5, a whole-domain uniform error bound E=1/23≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).", "answer": { "kind": "expression", "value": "(20, 5, 1/23, 1, 1/15, 29/69, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)", "explanation": "Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (20, 5, 1/23, 1, 1/15, 29/69, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)." }, "skills": [ "A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer" ], "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-continuity-transfer；資料={\"n\":20,\"lipschitz\":5,\"error_bound\":[1,23],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Fix N=20, a positive Lipschitz bound L=5, a whole-domain uniform error bound E=1/23≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [uniform-continuity-transfer]", "after": "derived = [[N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, 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 three times uniform error := mul(3, 1/23)", "after": "Uniform ORIGINAL three times uniform error = 3/23", "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 three times uniform error，不以小數近似替代。", "en": "Recompute Uniform ORIGINAL three times uniform error with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "1/23" ], "output": "3/23", "label": "Uniform ORIGINAL three times uniform error" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Uniform ORIGINAL three times Lipschitz constant := mul(3, 5)", "after": "Uniform ORIGINAL three times Lipschitz constant = 15", "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 three times Lipschitz constant，不以小數近似替代。", "en": "Recompute Uniform ORIGINAL three times Lipschitz constant with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "5" ], "output": "15", "label": "Uniform ORIGINAL three times Lipschitz constant" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Uniform ORIGINAL epsilon over threeL := div(1, 15)", "after": "Uniform ORIGINAL epsilon over threeL = 1/15", "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 epsilon over threeL，不以小數近似替代。", "en": "Recompute Uniform ORIGINAL epsilon over threeL with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "15" ], "output": "1/15", "label": "Uniform ORIGINAL epsilon over threeL" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Uniform ORIGINAL twice uniform error := mul(2, 1/23)", "after": "Uniform ORIGINAL twice uniform error = 2/23", "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 twice uniform error，不以小數近似替代。", "en": "Recompute Uniform ORIGINAL twice uniform error with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1/23" ], "output": "2/23", "label": "Uniform ORIGINAL twice uniform error" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Uniform ORIGINAL epsilon over three := div(1, 3)", "after": "Uniform ORIGINAL epsilon over three = 1/3", "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 epsilon over three，不以小數近似替代。", "en": "Recompute Uniform ORIGINAL epsilon over three with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "3" ], "output": "1/3", "label": "Uniform ORIGINAL epsilon over three" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Uniform ORIGINAL three term triangle upper bound := sum(2/23, 1/3)", "after": "Uniform ORIGINAL three term triangle upper bound = 29/69", "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 three term triangle upper bound，不以小數近似替代。", "en": "Recompute Uniform ORIGINAL three term triangle upper bound with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2/23", "1/3" ], "output": "29/69", "label": "Uniform ORIGINAL three term triangle upper bound" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [20, 5, 1/23, 1]", "after": "derived = [ASSUME f_N is L-Lipschitz on the WHOLE domain, ASSUME EVERY x:abs(f_N(x)-f(x))<=error_bound, 1/15]", "relation": "implies", "rule_id": "Uniform_GIVEN_same_fN_WHOLE_error_and_Lipschitz_premises", "conditions": [], "explanation": { "zh-Hant": "固定同一N；whole uniform error與f_N的L-Lipschitz是題設前提，不是從N這個scalar、有限samples或grid算出。由L>0及epsilon>3error得delta>0。", "en": "Fix the same N.The WHOLE uniform error and L-Lipschitz property of f_N are GIVEN premises,not inferred from a scalar N,samples or grids.L>0 and epsilon>3error give delta>0." }, "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": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [20, 5, 1/23, 1, 1/15]", "after": "derived = [abs(f(x)-f(y))<=abs(f(x)-f_N(x))+abs(f_N(x)-f_N(y))+abs(f_N(y)-f(y)), abs(x-y)<1/15 => bound<29/69, 是]", "relation": "implies", "rule_id": "Uniform_arbitrary_xy_three_term_triangle_transfer", "conditions": [], "explanation": { "zh-Hant": "任意x,y若|x-y|