{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261111-r02-higher-order-derivative-0001", "pilot_number": "R0045-23026", "number": 23026, "family_id": "qf-mod42-higher-order-qualification", "title": { "zh-Hant": "第 23026 題：高階導數", "en": "Question 23026: Order Zero Is the Original Function; Later Orders Need the Recursive Chain" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261111-r02-higher-order-derivative-0001", "batch_id": "batch-20261014-advance-20261111-r02-higher-order-derivative-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-higher-order-derivative" ], "title_zh": "高階導數", "stem_zh": "f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[1,2,3,4],\"order\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(3, 0, 是, f_zero_equals_f, 是)", "explanation_zh": "Wave6第五層核對=(3, 0, 是, f_zero_equals_f, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "高階導數" ] }, "variant": { "family_id": "qf-mod42-higher-order-qualification", "seed": 202611112001, "parameters": { "coefficients": [ 1, 2, 3, 4 ], "order": 0, "domain": "Exact_polynomial_higher_derivative", "operation": "higher-order-qualification", "semantic_template": "qf-mod42-higher-order-qualification/{\"coefficients\":[1,2,3,4],\"order\":0}" } }, "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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r02-higher-order-derivative-0001", "batch_id": "batch-20261014-advance-20261111-r02-higher-order-derivative-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "1b94351548db7e92bdce63983c9baabc1d3f03891f86c9353c225e7f25200872", "family_id": "qf-mod42-higher-order-qualification", "seed": 202611112001, "semantic_template": "qf-mod42-higher-order-qualification/{\"coefficients\":[1,2,3,4],\"order\":0}" }, "fields": { "title": "Order Zero Is the Original Function; Later Orders Need the Recursive Chain", "stem": "Use original low-to-high coefficients=[1, 2, 3, 4] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).", "answer": { "kind": "expression", "value": "(3, 0, Yes, f_zero_equals_f, Yes)", "explanation": "Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 0, Yes, f_zero_equals_f, Yes)." }, "skills": [ "Order Zero Is the Original Function; Later Orders Need the Recursive Chain" ], "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": "f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[1,2,3,4],\"order\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use original low-to-high coefficients=[1, 2, 3, 4] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=higher-order-qualification;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題快照保留歷史介面；此處直接列來源欄序，不依賴JavaScript。tag與No保聲稱範圍，固定來源排序不冒充數學證明。", "en": "Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof." }, "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 low-to-high coefficients=[1, 2, 3, 4];declared order=0", "after": "declared degree BOUND=3=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list", "relation": "implies", "rule_id": "higher_ORIGINAL_low_to_high_coefficients_degree_BOUND_order0", "conditions": [], "explanation": { "zh-Hant": "長度減1是宣告上界，trailing zeros不先刪掉；order0就是原函數。單項式導數規則與有限和給多項式逐階存在，不是任意函數的無限外推。", "en": "Length minus1 is the declared bound;do not trim trailing zeros.Order0 is the original function.The monomial rule and finite sums justify each polynomial derivative,not an infinite extrapolation for arbitrary functions." }, "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": "ORIGINAL coefficients=[1, 2, 3, 4];order=0;declared degree bound=3", "after": "final derivative coefficients=[1, 2, 3, 4];for each original power k>=order multiplier=k*(k-1)*...*(k-order+1);lower powers vanish", "relation": "implies", "rule_id": "higher_falling_factorial_rule_all_orders_polynomial_only", "conditions": [], "explanation": { "zh-Hant": "反覆微分給falling factorial，與原係數序一致；order超過宣告bound時必為0。這不聲稱一般可微函數analytic，也不把trailing-zero上界冒充最小degree。", "en": "Repeated differentiation gives the falling factorial in original coefficient order;orders above the declared bound vanish.This gives no analyticity claim for arbitrary differentiable functions and does not turn a trailing-zero bound into the minimal degree." }, "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": [ 3, 0, true, "f_zero_equals_f", true ] }, "conclusion": { "canonical_value": "(3, 0, 是, f_zero_equals_f, 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-20261014-advance-20261111-r02-higher-order-derivative-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261111-r02-higher-order-derivative-foundations/questions.jsonl", "batch_sha256": "53a54bdcff64574bbf5f352d4badfe20bbb98f5da23d3abdfdc2f3b20d16f3b1", "original_line": 1, "original_line_sha256": "508b751958144930d8ee803882a2bb8ba429740dbb19c322285fc786395968a4", "question_compact_sha256": "1b94351548db7e92bdce63983c9baabc1d3f03891f86c9353c225e7f25200872", "english_row_compact_sha256": "b2cd1936ac03e7b43559281cb2c224ff34c9ddde17d790d891a6233f766609e9", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261111-r02-higher-order-derivative-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261111-r02-higher-order-derivative-foundations.en.jsonl" }, "mko_bindings": { "mko-higher-order-derivative": { "sha256": "78b36c7cb74b3af2a4e2f3cbb7c42f790660d13aa1347fdd50826cc718717741", "english_sha256": "c1886e7e00a610edfc2e3a3fb75552d4afb78b49f57c24e7e1abd7d5ce5775d6", "original_path": "public/data/mko/higher-order-derivative.json", "english_path": "public/data/i18n/editions/mko-higher-order-derivative.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "9732ef02cb296d2e89f9b6cade5a76a83e9d37bdcf6a3de7f8613ebce02311e8", "original_candidate_sha256": "172cb1f07ee84deca0476d9d857d0eb71633ea35d1039e7ea867b3d8ad788421", "original_candidate_line_sha256": "172cb1f07ee84deca0476d9d857d0eb71633ea35d1039e7ea867b3d8ad788421" }, "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-p46", "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": "357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438", "question_first_witness_sha256": "99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a" }, "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": "a757a2dc5acffc864018eb5d310b1a6c1f464cb6c68a33a22ff6370f4f2f1231", "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." } }