{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261007-advance-20261009-r05-antiderivative-0077", "pilot_number": "R0058-29952", "number": 29952, "family_id": "qf-density-primitive-candidate-derivative", "title": { "zh-Hant": "第 29952 題：有理候選的導數係數", "en": "Question 29952: Derivative Coefficients of a Rational Candidate" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261007-advance-20261009-r05-antiderivative-0077", "batch_id": "batch-20261007-advance-20261009-r05-antiderivative-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-antiderivative" ], "title_zh": "有理候選的導數係數", "stem_zh": "全實數論域；升冪係數的受限多項式模型。候選F的升冪有理係數為 [[3,1],[2,3]]，每對為[分子,正分母]。列F′的完整升冪最簡有理係數，零式列[0/1]。", "answer": { "kind": "expression", "value": "[2/3]", "explanation_zh": "候選導數升冪係數=[2/3]。以係數乘次方求導並去除末端零；候選常數項不影響導數，也不以代入抽樣作一般證明。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "有理候選的導數係數" ] }, "variant": { "family_id": "qf-density-primitive-candidate-derivative", "seed": 202610095077, "parameters": { "F": [ [ 3, 1 ], [ 2, 3 ] ], "operation": "density_primitive_candidate_derivative", "domain": "reals", "representation": "ascending_powers", "semantic_template": "qf-density-primitive-candidate-derivative/[\"2/3\"]" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-07", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261007-advance-20261009-r05-antiderivative-0077", "batch_id": "batch-20261007-advance-20261009-r05-antiderivative-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "544e978db9d8d39f6148b05930042fe8a11b91965c46e1b2bfcfc1a2441d2ee7", "family_id": "qf-density-primitive-candidate-derivative", "seed": 202610095077, "semantic_template": "qf-density-primitive-candidate-derivative/[\"2/3\"]" }, "fields": { "title": "Derivative Coefficients of a Rational Candidate", "stem": "On all R, F has ascending rational coefficients [3/1, 2/3]. Give its complete normalized ascending derivative coefficients as reduced fractions.", "answer": { "kind": "expression", "value": "[2/3]", "explanation": "Differentiate coefficientwise to obtain [2/3]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check." }, "skills": [ "Derivative Coefficients of a Rational Candidate" ], "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": "全實數論域；升冪係數的受限多項式模型。候選F的升冪有理係數為 [[3,1],[2,3]]，每對為[分子,正分母]。列F′的完整升冪最簡有理係數，零式列[0/1]。", "en": "On all R, F has ascending rational coefficients [3/1, 2/3]. Give its complete normalized ascending derivative coefficients as reduced fractions." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-density-primitive-candidate-derivative, density_primitive_candidate_derivative]", "after": "derived = [[ALL_normalized_ascending_derivative_coefficients_string_retaining1]]", "relation": "implies", "rule_id": "R58_EXACT_source_interface_and_answer_ROLE", "conditions": [], "explanation": { "zh-Hant": "依本題銀行的keys、domain或frame、operation與輸出角色。STR22/App20 tuple與legacy scalar不同；分數是否保/1按原契約，不能用共同外觀改題。", "en": "Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原題資料、同一索引／方向／代表式與本步前提；不只核答案。精確候選核對、獨立審查、形式證明與發布資格仍分開。", "en": "Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [candidate F, [[3, 1], [2, 3]], 是]", "after": "derived = [[3/1, 2/3]]", "relation": "implies", "rule_id": "Primitive_RAW_arity_SAFE_inputs_POSITIVE_rational_den_trim_KEEPzero", "conditions": [], "explanation": { "zh-Hant": "整數f原長1..4、rational候選F原長1..5；每個raw數SAFE，rational den嚴格正（不是先normalize負den）。升冪只trim尾端0但zero polynomial保一項；衍生運算不套1e6或degree cap。", "en": "Raw integer f has length1..4 and rational candidate F length1..5.Every raw number is SAFE;rational denominators are STRICTLY positive,not repaired from negatives.Trim only trailing zeros,keeping a singleton zero polynomial.Derived algebra has no1e6 or raw-degree cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原題資料、同一索引／方向／代表式與本步前提；不只核答案。精確候選核對、獨立審查、形式證明與發布資格仍分開。", "en": "Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Primitive candidate ORIGINAL derivative degree0 := mul(1, 2/3)", "after": "Primitive candidate ORIGINAL derivative degree0 = 2/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive candidate ORIGINAL derivative degree0，不以小數近似替代。", "en": "Recompute Primitive candidate ORIGINAL derivative degree0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2/3" ], "output": "2/3", "label": "Primitive candidate ORIGINAL derivative degree0" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [[3/1, 2/3]]", "after": "derived = [[2/3], [2/3]]", "relation": "implies", "rule_id": "Primitive_candidate_COMPLETE_derivative_coefficients_not_eligibility", "conditions": [], "explanation": { "zh-Hant": "逐每個非常數項i*c_i*x^(i-1)列完整升冪係數，再trim尾0。constant微分0，常數候選回[0/1]非[]；題目未給input f，不捏造F_prime=f的Yes/No資格判定。", "en": "Differentiate EVERY nonconstant term into i*c_i*x^(i-1),list the COMPLETE ascending coefficients,and trim trailing zeros.Constants differentiate to0,so a constant candidate returns[0/1],not[].No input f was provided;do not invent a Yes/No eligibility test F_prime=f." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原題資料、同一索引／方向／代表式與本步前提；不只核答案。精確候選核對、獨立審查、形式證明與發布資格仍分開。", "en": "Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null } ], "math_model": { "result": "[2/3]" }, "conclusion": { "canonical_value": "[2/3]", "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-20261007-advance-20261009-r05-antiderivative-foundations", "batch_path": "public/data/questions/batches/batch-20261007-advance-20261009-r05-antiderivative-foundations/questions.jsonl", "batch_sha256": "085ade26180bd6f2bb25667f7334717d6fd8127fddfdf06e74e67bdbe897b44e", "original_line": 77, "original_line_sha256": "25cc755b0bfd5ccb5ed4b08ec6d648be4bfa864f2e2c2ef15ded5cd45bc15d4f", "question_compact_sha256": "544e978db9d8d39f6148b05930042fe8a11b91965c46e1b2bfcfc1a2441d2ee7", "english_row_compact_sha256": "5c2a33a194669c800c8e12df682d7280aec6629dac9ee01215241c18ea4df655", "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261009-r05-antiderivative-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261009-r05-antiderivative-foundations.en.jsonl" }, "mko_bindings": { "mko-antiderivative": { "sha256": "5f2ecff6866e471d1f8da7fea1fa5e121364ae2d24b2fea405d2300fe9ff1cae", "english_sha256": "f983dd8b898bc9eac9d5237c7895e5f7dc182ac46efb5f9b58b925fc6c2ecb9d", "original_path": "public/data/mko/antiderivative.json", "english_path": "public/data/i18n/editions/mko-antiderivative.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "623c23f45586b08629c5a347ffbc994cf8dbff2133c4f4cdb227b851d74ea6b6", "original_candidate_sha256": "4e67445c3d9b37190e86022b9f67c642f7652e65e27a5eba012314238f8f52c9", "original_candidate_line_sha256": "4e67445c3d9b37190e86022b9f67c642f7652e65e27a5eba012314238f8f52c9" }, "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-20261026-011-p59", "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": "66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05df", "question_first_witness_sha256": "b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 4, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "bf9085933dc607c6020aa506b9b545a099708adb88c3f778910ede25152f5643", "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." } }