{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261113-r02-real-exponential-function-0050", "pilot_number": "R0047-24275", "number": 24275, "family_id": "qf-mod44-exp-infinite-radius-certificate", "title": { "zh-Hant": "第 24275 題：實指數函數", "en": "Question 24275: The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261113-r02-real-exponential-function-0050", "batch_id": "batch-20261015-advance-20261113-r02-real-exponential-function-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-real-exponential-function" ], "title_zh": "實指數函數", "stem_zh": "E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={\"epsilon_denominator\":26,\"start_index\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(26, 26, epsilon=1/26, |a_(n+1)/a_n|=1/(n+1), 1/27, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)", "explanation_zh": "Wave6第七層核對=(26, 26, epsilon=1/26, |a_(n+1)/a_n|=1/(n+1), 1/27, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "實指數函數" ] }, "variant": { "family_id": "qf-mod44-exp-infinite-radius-certificate", "seed": 202611132050, "parameters": { "epsilon_denominator": 26, "start_index": 26, "domain": "Exact_real_exponential_series", "operation": "exp-infinite-radius-certificate", "semantic_template": "qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":26,\"start_index\":26}" } }, "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 normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r02-real-exponential-function-0050", "batch_id": "batch-20261015-advance-20261113-r02-real-exponential-function-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "e44253f5a491fcb1914edde090610a5d369b87deb49bad156c4829b0794c2bc7", "family_id": "qf-mod44-exp-infinite-radius-certificate", "seed": 202611132050, "semantic_template": "qf-mod44-exp-infinite-radius-certificate/{\"epsilon_denominator\":26,\"start_index\":26}" }, "fields": { "title": "The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe", "stem": "Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/26, N=26, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).", "answer": { "kind": "expression", "value": "(26, 26, epsilon=1/26, |a_(n+1)/a_n|=1/(n+1), 1/27, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)", "explanation": "For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={\"epsilon_denominator\":26,\"start_index\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/26, N=26, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [exp-infinite-radius-certificate]", "after": "derived = [[epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "本頁列完整來源欄序，不需JavaScript；gradient boundary依原mode選6／8／7欄，不補欄也不由答案猜mode。", "en": "List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode." }, "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": "Exp tail coefficient ratio bound := div(1, 27)", "after": "Exp tail coefficient ratio bound = 1/27", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp tail coefficient ratio bound，不以小數近似替代。", "en": "Recompute Exp tail coefficient ratio bound with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "27" ], "output": "1/27", "label": "Exp tail coefficient ratio bound" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Exp declared positive epsilon := div(1, 26)", "after": "Exp declared positive epsilon = 1/26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp declared positive epsilon，不以小數近似替代。", "en": "Recompute Exp declared positive epsilon with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "26" ], "output": "1/26", "label": "Exp declared positive epsilon" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [26, 26, 1/27, 1/26]", "after": "derived = [EVERY n>=N:1/(n+1)<=1/(N+1)<1/d, for ANY epsilon>0 choose integer N with N+1>1/epsilon, coefficient ratio limit0,radius infinity]", "relation": "implies", "rule_id": "Exp_ALL_n_tail_and_arbitrary_epsilon_recipe", "conditions": [], "explanation": { "zh-Hant": "N>=d給嚴格1/(N+1)<1/d；另對任意正epsilon的recipe才證極限，不只一個epsilon。代入固定x的項多|x|仍趨0，故所有實x絕對收斂。", "en": "N>=d gives the strict bound1/(N+1)<1/d.The arbitrary-positive-epsilon recipe proves a limit,not just one epsilon.Term ratios at fixed x have an extra |x| and still tend0,so all real x give absolute convergence." }, "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": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [[26, 26], a_n=1/n! for EVERY n>=0]", "after": "derived = [coefficient ratio=1/(n+1), for EVERY fixed real z,term absolute ratio=|z|/(n+1)->0, absolute convergence and real-valued E(z)]", "relation": "implies", "rule_id": "Exp_WHOLE_factorial_law_ALL_fixed_real_arguments", "conditions": [], "explanation": { "zh-Hant": "對任意固定實z選N+1>2|z|，之後項的絕對比值小於1/2，由幾何尾界保絕對收斂。不是由有限prefix或把係數比值漏乘|z|。", "en": "For any fixed real z,choose N+1>2|z|;later absolute term ratios are below1/2 and a geometric tail proves absolute convergence.This is not finite-prefix extrapolation or omission of |z|." }, "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": [ 26, 26, "epsilon=1/26", "|a_(n+1)/a_n|=1/(n+1)", "1/27", true, "for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity" ] }, "conclusion": { "canonical_value": "(26, 26, epsilon=1/26, |a_(n+1)/a_n|=1/(n+1), 1/27, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)", "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-20261113-r02-real-exponential-function-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261113-r02-real-exponential-function-foundations/questions.jsonl", "batch_sha256": "e937e416efd79ec1e73c82a290ba61f2b60a0d2fc00f91f29ef00d18b9473f95", "original_line": 50, "original_line_sha256": "5cc49f0bb958b07108b4ea57a3356047a4598cc511648e03ad09012a409277a5", "question_compact_sha256": "e44253f5a491fcb1914edde090610a5d369b87deb49bad156c4829b0794c2bc7", "english_row_compact_sha256": "ad625be3cbf0d83d2ca04822d8b039131aaf71fd4ea0d735b5b72de9ac434fff", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261113-r02-real-exponential-function-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261113-r02-real-exponential-function-foundations.en.jsonl" }, "mko_bindings": { "mko-real-exponential-function": { "sha256": "7b0736c512a2b1e27562551d8a9de463446e4b54c5998868cfe7573aee0b4f17", "english_sha256": "1fafde29fc55b9df97ad047683d8c6f01be2efa1ac103b831e7d23ec97eafb33", "original_path": "public/data/mko/real-exponential-function.json", "english_path": "public/data/i18n/editions/mko-real-exponential-function.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "b3aad195dad743751e4c8a4a6ce44f091c3824da1e52c21e9266d4ccd1dc1b68", "original_candidate_sha256": "95a6c8eb760123e488b1a3a8dcbabda791ac94235eafce85655b4896beec4024", "original_candidate_line_sha256": "95a6c8eb760123e488b1a3a8dcbabda791ac94235eafce85655b4896beec4024" }, "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-p48", "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": "9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386", "question_first_witness_sha256": "eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 5, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "de979634b04f1e89d3c18dfa72fee06349bd1744961f3cd1346a115d40feb447", "arithmetic_steps": 2 }, "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." } }