{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261111-r02-higher-order-derivative-0041", "pilot_number": "R0045-23066", "number": 23066, "family_id": "qf-mod42-higher-polynomial-nth-derivative", "title": { "zh-Hant": "第 23066 題：高階導數", "en": "Question 23066: Exact Falling Factorials Preserve Low-to-High Derivative Coefficients" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261111-r02-higher-order-derivative-0041", "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-polynomial-nth-derivative；資料={\"coefficients\":[16,32,48,64,80,96,112],\"order\":7,\"point\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([16, 32, 48, 64, 80, 96, 112], 7, [0], -2, 0, falling_factorial_coefficients_exact)", "explanation_zh": "Wave6第五層核對=([16, 32, 48, 64, 80, 96, 112], 7, [0], -2, 0, falling_factorial_coefficients_exact)；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": 2, "skills": [ "高階導數" ] }, "variant": { "family_id": "qf-mod42-higher-polynomial-nth-derivative", "seed": 202611112041, "parameters": { "coefficients": [ 16, 32, 48, 64, 80, 96, 112 ], "order": 7, "point": -2, "domain": "Exact_polynomial_higher_derivative", "operation": "higher-polynomial-nth-derivative", "semantic_template": "qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[16,32,48,64,80,96,112],\"order\":7,\"point\":-2}" } }, "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-0041", "batch_id": "batch-20261014-advance-20261111-r02-higher-order-derivative-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "4de15f92aa3c0b467ff9bfc96224a3134c99cc2b47071e84434a67bdea72b106", "family_id": "qf-mod42-higher-polynomial-nth-derivative", "seed": 202611112041, "semantic_template": "qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[16,32,48,64,80,96,112],\"order\":7,\"point\":-2}" }, "fields": { "title": "Exact Falling Factorials Preserve Low-to-High Derivative Coefficients", "stem": "Use original low-to-high coefficients=[16, 32, 48, 64, 80, 96, 112] for f(t)=sum coefficients[k]*t^k and derivative order=7≥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. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).", "answer": { "kind": "expression", "value": "([16, 32, 48, 64, 80, 96, 112], 7, [0], -2, 0, falling_factorial_coefficients_exact)", "explanation": "Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([16, 32, 48, 64, 80, 96, 112], 7, [0], -2, 0, falling_factorial_coefficients_exact)." }, "skills": [ "Exact Falling Factorials Preserve Low-to-High Derivative Coefficients" ], "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-polynomial-nth-derivative；資料={\"coefficients\":[16,32,48,64,80,96,112],\"order\":7,\"point\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use original low-to-high coefficients=[16, 32, 48, 64, 80, 96, 112] for f(t)=sum coefficients[k]*t^k and derivative order=7≥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. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=higher-polynomial-nth-derivative;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[coefficients, order, derivative_coefficients, point, value, 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=[16, 32, 48, 64, 80, 96, 112];declared order=7", "after": "declared degree BOUND=6=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": "Higher derivative level1 ORIGINAL power1 coefficient product := mul(32, 1)", "after": "Higher derivative level1 ORIGINAL power1 coefficient product = 32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level1 ORIGINAL power1 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level1 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "32", "1" ], "output": "32", "label": "Higher derivative level1 ORIGINAL power1 coefficient product" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Higher derivative level1 ORIGINAL power2 coefficient product := mul(48, 2)", "after": "Higher derivative level1 ORIGINAL power2 coefficient product = 96", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level1 ORIGINAL power2 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level1 ORIGINAL power2 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "48", "2" ], "output": "96", "label": "Higher derivative level1 ORIGINAL power2 coefficient product" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Higher derivative level1 ORIGINAL power3 coefficient product := mul(64, 3)", "after": "Higher derivative level1 ORIGINAL power3 coefficient product = 192", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level1 ORIGINAL power3 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level1 ORIGINAL power3 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "64", "3" ], "output": "192", "label": "Higher derivative level1 ORIGINAL power3 coefficient product" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Higher derivative level1 ORIGINAL power4 coefficient product := mul(80, 4)", "after": "Higher derivative level1 ORIGINAL power4 coefficient product = 320", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level1 ORIGINAL power4 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level1 ORIGINAL power4 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "80", "4" ], "output": "320", "label": "Higher derivative level1 ORIGINAL power4 coefficient product" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Higher derivative level1 ORIGINAL power5 coefficient product := mul(96, 5)", "after": "Higher derivative level1 ORIGINAL power5 coefficient product = 480", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level1 ORIGINAL power5 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level1 ORIGINAL power5 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "96", "5" ], "output": "480", "label": "Higher derivative level1 ORIGINAL power5 coefficient product" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Higher derivative level1 ORIGINAL power6 coefficient product := mul(112, 6)", "after": "Higher derivative level1 ORIGINAL power6 coefficient product = 672", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level1 ORIGINAL power6 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level1 ORIGINAL power6 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "112", "6" ], "output": "672", "label": "Higher derivative level1 ORIGINAL power6 coefficient product" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "previous derivative order=0;actual low-to-high list=[16, 32, 48, 64, 80, 96, 112];drop ONLY its constant term", "after": "ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 32, 1, 32], [2, 48, 2, 96], [3, 64, 3, 192], [4, 80, 4, 320], [5, 96, 5, 480], [6, 112, 6, 672]];next raw list=[32, 96, 192, 320, 480, 672];all intermediate products SAFE", "relation": "implies", "rule_id": "higher_EVERY_recursive_derivative_level_not_reordered", "conditions": [], "explanation": { "zh-Hant": "每一層把舊power k乘原係數，再按低到高放到power k−1；原raw list耗盡後維持空，最後才以[0]表示零導數。英文來源要求所有中間乘積安全，不只最後結果。", "en": "At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value." }, "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": "S010", "uses": [ "G001", "S009" ], "before": "Higher derivative level2 ORIGINAL power1 coefficient product := mul(96, 1)", "after": "Higher derivative level2 ORIGINAL power1 coefficient product = 96", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level2 ORIGINAL power1 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level2 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "96", "1" ], "output": "96", "label": "Higher derivative level2 ORIGINAL power1 coefficient product" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Higher derivative level2 ORIGINAL power2 coefficient product := mul(192, 2)", "after": "Higher derivative level2 ORIGINAL power2 coefficient product = 384", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level2 ORIGINAL power2 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level2 ORIGINAL power2 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "192", "2" ], "output": "384", "label": "Higher derivative level2 ORIGINAL power2 coefficient product" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Higher derivative level2 ORIGINAL power3 coefficient product := mul(320, 3)", "after": "Higher derivative level2 ORIGINAL power3 coefficient product = 960", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level2 ORIGINAL power3 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level2 ORIGINAL power3 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "320", "3" ], "output": "960", "label": "Higher derivative level2 ORIGINAL power3 coefficient product" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Higher derivative level2 ORIGINAL power4 coefficient product := mul(480, 4)", "after": "Higher derivative level2 ORIGINAL power4 coefficient product = 1920", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level2 ORIGINAL power4 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level2 ORIGINAL power4 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "480", "4" ], "output": "1920", "label": "Higher derivative level2 ORIGINAL power4 coefficient product" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "Higher derivative level2 ORIGINAL power5 coefficient product := mul(672, 5)", "after": "Higher derivative level2 ORIGINAL power5 coefficient product = 3360", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level2 ORIGINAL power5 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level2 ORIGINAL power5 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "672", "5" ], "output": "3360", "label": "Higher derivative level2 ORIGINAL power5 coefficient product" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "previous derivative order=1;actual low-to-high list=[32, 96, 192, 320, 480, 672];drop ONLY its constant term", "after": "ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 96, 1, 96], [2, 192, 2, 384], [3, 320, 3, 960], [4, 480, 4, 1920], [5, 672, 5, 3360]];next raw list=[96, 384, 960, 1920, 3360];all intermediate products SAFE", "relation": "implies", "rule_id": "higher_EVERY_recursive_derivative_level_not_reordered", "conditions": [], "explanation": { "zh-Hant": "每一層把舊power k乘原係數，再按低到高放到power k−1；原raw list耗盡後維持空，最後才以[0]表示零導數。英文來源要求所有中間乘積安全，不只最後結果。", "en": "At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value." }, "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": "S016", "uses": [ "G001", "S015" ], "before": "Higher derivative level3 ORIGINAL power1 coefficient product := mul(384, 1)", "after": "Higher derivative level3 ORIGINAL power1 coefficient product = 384", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level3 ORIGINAL power1 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level3 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "384", "1" ], "output": "384", "label": "Higher derivative level3 ORIGINAL power1 coefficient product" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "Higher derivative level3 ORIGINAL power2 coefficient product := mul(960, 2)", "after": "Higher derivative level3 ORIGINAL power2 coefficient product = 1920", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level3 ORIGINAL power2 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level3 ORIGINAL power2 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "960", "2" ], "output": "1920", "label": "Higher derivative level3 ORIGINAL power2 coefficient product" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "Higher derivative level3 ORIGINAL power3 coefficient product := mul(1920, 3)", "after": "Higher derivative level3 ORIGINAL power3 coefficient product = 5760", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level3 ORIGINAL power3 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level3 ORIGINAL power3 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1920", "3" ], "output": "5760", "label": "Higher derivative level3 ORIGINAL power3 coefficient product" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "Higher derivative level3 ORIGINAL power4 coefficient product := mul(3360, 4)", "after": "Higher derivative level3 ORIGINAL power4 coefficient product = 13440", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level3 ORIGINAL power4 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level3 ORIGINAL power4 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3360", "4" ], "output": "13440", "label": "Higher derivative level3 ORIGINAL power4 coefficient product" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "previous derivative order=2;actual low-to-high list=[96, 384, 960, 1920, 3360];drop ONLY its constant term", "after": "ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 384, 1, 384], [2, 960, 2, 1920], [3, 1920, 3, 5760], [4, 3360, 4, 13440]];next raw list=[384, 1920, 5760, 13440];all intermediate products SAFE", "relation": "implies", "rule_id": "higher_EVERY_recursive_derivative_level_not_reordered", "conditions": [], "explanation": { "zh-Hant": "每一層把舊power k乘原係數，再按低到高放到power k−1；原raw list耗盡後維持空，最後才以[0]表示零導數。英文來源要求所有中間乘積安全，不只最後結果。", "en": "At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value." }, "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": "S021", "uses": [ "G001", "S020" ], "before": "Higher derivative level4 ORIGINAL power1 coefficient product := mul(1920, 1)", "after": "Higher derivative level4 ORIGINAL power1 coefficient product = 1920", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level4 ORIGINAL power1 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level4 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1920", "1" ], "output": "1920", "label": "Higher derivative level4 ORIGINAL power1 coefficient product" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "Higher derivative level4 ORIGINAL power2 coefficient product := mul(5760, 2)", "after": "Higher derivative level4 ORIGINAL power2 coefficient product = 11520", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level4 ORIGINAL power2 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level4 ORIGINAL power2 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "5760", "2" ], "output": "11520", "label": "Higher derivative level4 ORIGINAL power2 coefficient product" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "Higher derivative level4 ORIGINAL power3 coefficient product := mul(13440, 3)", "after": "Higher derivative level4 ORIGINAL power3 coefficient product = 40320", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level4 ORIGINAL power3 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level4 ORIGINAL power3 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13440", "3" ], "output": "40320", "label": "Higher derivative level4 ORIGINAL power3 coefficient product" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "previous derivative order=3;actual low-to-high list=[384, 1920, 5760, 13440];drop ONLY its constant term", "after": "ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 1920, 1, 1920], [2, 5760, 2, 11520], [3, 13440, 3, 40320]];next raw list=[1920, 11520, 40320];all intermediate products SAFE", "relation": "implies", "rule_id": "higher_EVERY_recursive_derivative_level_not_reordered", "conditions": [], "explanation": { "zh-Hant": "每一層把舊power k乘原係數，再按低到高放到power k−1；原raw list耗盡後維持空，最後才以[0]表示零導數。英文來源要求所有中間乘積安全，不只最後結果。", "en": "At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value." }, "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": "S025", "uses": [ "G001", "S024" ], "before": "Higher derivative level5 ORIGINAL power1 coefficient product := mul(11520, 1)", "after": "Higher derivative level5 ORIGINAL power1 coefficient product = 11520", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level5 ORIGINAL power1 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level5 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "11520", "1" ], "output": "11520", "label": "Higher derivative level5 ORIGINAL power1 coefficient product" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "Higher derivative level5 ORIGINAL power2 coefficient product := mul(40320, 2)", "after": "Higher derivative level5 ORIGINAL power2 coefficient product = 80640", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level5 ORIGINAL power2 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level5 ORIGINAL power2 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "40320", "2" ], "output": "80640", "label": "Higher derivative level5 ORIGINAL power2 coefficient product" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "previous derivative order=4;actual low-to-high list=[1920, 11520, 40320];drop ONLY its constant term", "after": "ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 11520, 1, 11520], [2, 40320, 2, 80640]];next raw list=[11520, 80640];all intermediate products SAFE", "relation": "implies", "rule_id": "higher_EVERY_recursive_derivative_level_not_reordered", "conditions": [], "explanation": { "zh-Hant": "每一層把舊power k乘原係數，再按低到高放到power k−1；原raw list耗盡後維持空，最後才以[0]表示零導數。英文來源要求所有中間乘積安全，不只最後結果。", "en": "At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value." }, "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": "S028", "uses": [ "G001", "S027" ], "before": "Higher derivative level6 ORIGINAL power1 coefficient product := mul(80640, 1)", "after": "Higher derivative level6 ORIGINAL power1 coefficient product = 80640", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative level6 ORIGINAL power1 coefficient product，不以小數近似替代。", "en": "Recompute Higher derivative level6 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "80640", "1" ], "output": "80640", "label": "Higher derivative level6 ORIGINAL power1 coefficient product" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "previous derivative order=5;actual low-to-high list=[11520, 80640];drop ONLY its constant term", "after": "ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 80640, 1, 80640]];next raw list=[80640];all intermediate products SAFE", "relation": "implies", "rule_id": "higher_EVERY_recursive_derivative_level_not_reordered", "conditions": [], "explanation": { "zh-Hant": "每一層把舊power k乘原係數，再按低到高放到power k−1；原raw list耗盡後維持空，最後才以[0]表示零導數。英文來源要求所有中間乘積安全，不只最後結果。", "en": "At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value." }, "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": "S030", "uses": [ "G001", "S029" ], "before": "previous derivative order=6;actual low-to-high list=[80640];drop ONLY its constant term", "after": "ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[];next raw list=[];all intermediate products SAFE", "relation": "implies", "rule_id": "higher_EVERY_recursive_derivative_level_not_reordered", "conditions": [], "explanation": { "zh-Hant": "每一層把舊power k乘原係數，再按低到高放到power k−1；原raw list耗盡後維持空，最後才以[0]表示零導數。英文來源要求所有中間乘積安全，不只最後結果。", "en": "At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value." }, "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": "S031", "uses": [ "G001", "S030" ], "before": "ORIGINAL coefficients=[16, 32, 48, 64, 80, 96, 112];order=7;declared degree bound=6", "after": "final derivative coefficients=[0];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 }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "Higher derivative Horner power0 previous value times ORIGINAL point := mul(0, -2)", "after": "Higher derivative Horner power0 previous value times ORIGINAL point = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative Horner power0 previous value times ORIGINAL point，不以小數近似替代。", "en": "Recompute Higher derivative Horner power0 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "-2" ], "output": "0", "label": "Higher derivative Horner power0 previous value times ORIGINAL point" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "Higher derivative Horner power0 plus ORIGINAL derivative coefficient := sum(0, 0)", "after": "Higher derivative Horner power0 plus ORIGINAL derivative coefficient = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本題原始角色、次序與已核中間量精確代值，不用浮點近似或另一題的資料。", "en": "Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Higher derivative Horner power0 plus ORIGINAL derivative coefficient，不以小數近似替代。", "en": "Recompute Higher derivative Horner power0 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "0" ], "output": "0", "label": "Higher derivative Horner power0 plus ORIGINAL derivative coefficient" } } ], "math_model": { "result": [ [ 16, 32, 48, 64, 80, 96, 112 ], 7, [ 0 ], -2, 0, "falling_factorial_coefficients_exact" ] }, "conclusion": { "canonical_value": "([16, 32, 48, 64, 80, 96, 112], 7, [0], -2, 0, falling_factorial_coefficients_exact)", "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": 41, "original_line_sha256": "75fe824c0c1cc393cc6b9b29e268a2fe80749a7d7a875b03d3d3c3eb070c598a", "question_compact_sha256": "4de15f92aa3c0b467ff9bfc96224a3134c99cc2b47071e84434a67bdea72b106", "english_row_compact_sha256": "2f5c9d46cf1cbe0e3cc1c279048eaa91194f4febb03fbfa45b2998828cf2b7a6", "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": "d177509b25dd7d3cfb4cb33361a95eba2fb3971e89a52bd7cc56ad354f8c2a4b", "original_candidate_line_sha256": "d177509b25dd7d3cfb4cb33361a95eba2fb3971e89a52bd7cc56ad354f8c2a4b" }, "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": 33, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "bccbde67c7bb125968c4508ac0963464859509bbf813de2ebdd1ba571a6aea51", "arithmetic_steps": 23 }, "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." } }