第 23083 題：高階導數

原題與已知條件

f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={"coefficients":[8,16,24,32,40,48,56],"order":8,"point":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={"coefficients":[8,16,24,32,40,48,56],"order":8,"point":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

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。

• mko-higher-order-derivative

逐步推導

source operation=higher-zero-beyond-degree;original snapshot and mathematics retained

explicit ordered source output fields=[declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate]

ORIGINAL low-to-high coefficients=[8, 16, 24, 32, 40, 48, 56];declared order=8

declared degree BOUND=6=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list

Higher derivative level1 ORIGINAL power1 coefficient product := mul(16, 1)

Higher derivative level1 ORIGINAL power1 coefficient product = 16

Higher derivative level1 ORIGINAL power2 coefficient product := mul(24, 2)

Higher derivative level1 ORIGINAL power2 coefficient product = 48

Higher derivative level1 ORIGINAL power3 coefficient product := mul(32, 3)

Higher derivative level1 ORIGINAL power3 coefficient product = 96

Higher derivative level1 ORIGINAL power4 coefficient product := mul(40, 4)

Higher derivative level1 ORIGINAL power4 coefficient product = 160

Higher derivative level1 ORIGINAL power5 coefficient product := mul(48, 5)

Higher derivative level1 ORIGINAL power5 coefficient product = 240

Higher derivative level1 ORIGINAL power6 coefficient product := mul(56, 6)

Higher derivative level1 ORIGINAL power6 coefficient product = 336

previous derivative order=0;actual low-to-high list=[8, 16, 24, 32, 40, 48, 56];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 16, 1, 16], [2, 24, 2, 48], [3, 32, 3, 96], [4, 40, 4, 160], [5, 48, 5, 240], [6, 56, 6, 336]];next raw list=[16, 48, 96, 160, 240, 336];all intermediate products SAFE

Higher derivative level2 ORIGINAL power1 coefficient product := mul(48, 1)

Higher derivative level2 ORIGINAL power1 coefficient product = 48

Higher derivative level2 ORIGINAL power2 coefficient product := mul(96, 2)

Higher derivative level2 ORIGINAL power2 coefficient product = 192

Higher derivative level2 ORIGINAL power3 coefficient product := mul(160, 3)

Higher derivative level2 ORIGINAL power3 coefficient product = 480

Higher derivative level2 ORIGINAL power4 coefficient product := mul(240, 4)

Higher derivative level2 ORIGINAL power4 coefficient product = 960

Higher derivative level2 ORIGINAL power5 coefficient product := mul(336, 5)

Higher derivative level2 ORIGINAL power5 coefficient product = 1680

previous derivative order=1;actual low-to-high list=[16, 48, 96, 160, 240, 336];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 48, 1, 48], [2, 96, 2, 192], [3, 160, 3, 480], [4, 240, 4, 960], [5, 336, 5, 1680]];next raw list=[48, 192, 480, 960, 1680];all intermediate products SAFE

Higher derivative level3 ORIGINAL power1 coefficient product := mul(192, 1)

Higher derivative level3 ORIGINAL power1 coefficient product = 192

Higher derivative level3 ORIGINAL power2 coefficient product := mul(480, 2)

Higher derivative level3 ORIGINAL power2 coefficient product = 960

Higher derivative level3 ORIGINAL power3 coefficient product := mul(960, 3)

Higher derivative level3 ORIGINAL power3 coefficient product = 2880

Higher derivative level3 ORIGINAL power4 coefficient product := mul(1680, 4)

Higher derivative level3 ORIGINAL power4 coefficient product = 6720

previous derivative order=2;actual low-to-high list=[48, 192, 480, 960, 1680];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 192, 1, 192], [2, 480, 2, 960], [3, 960, 3, 2880], [4, 1680, 4, 6720]];next raw list=[192, 960, 2880, 6720];all intermediate products SAFE

Higher derivative level4 ORIGINAL power1 coefficient product := mul(960, 1)

Higher derivative level4 ORIGINAL power1 coefficient product = 960

Higher derivative level4 ORIGINAL power2 coefficient product := mul(2880, 2)

Higher derivative level4 ORIGINAL power2 coefficient product = 5760

Higher derivative level4 ORIGINAL power3 coefficient product := mul(6720, 3)

Higher derivative level4 ORIGINAL power3 coefficient product = 20160

previous derivative order=3;actual low-to-high list=[192, 960, 2880, 6720];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 960, 1, 960], [2, 2880, 2, 5760], [3, 6720, 3, 20160]];next raw list=[960, 5760, 20160];all intermediate products SAFE

Higher derivative level5 ORIGINAL power1 coefficient product := mul(5760, 1)

Higher derivative level5 ORIGINAL power1 coefficient product = 5760

Higher derivative level5 ORIGINAL power2 coefficient product := mul(20160, 2)

Higher derivative level5 ORIGINAL power2 coefficient product = 40320

previous derivative order=4;actual low-to-high list=[960, 5760, 20160];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 5760, 1, 5760], [2, 20160, 2, 40320]];next raw list=[5760, 40320];all intermediate products SAFE

Higher derivative level6 ORIGINAL power1 coefficient product := mul(40320, 1)

Higher derivative level6 ORIGINAL power1 coefficient product = 40320

previous derivative order=5;actual low-to-high list=[5760, 40320];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 40320, 1, 40320]];next raw list=[40320];all intermediate products SAFE

previous derivative order=6;actual low-to-high list=[40320];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[];next raw list=[];all intermediate products SAFE

previous derivative order=7;actual low-to-high list=[];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[];next raw list=[];all intermediate products SAFE

ORIGINAL coefficients=[8, 16, 24, 32, 40, 48, 56];order=8;declared degree bound=6

final derivative coefficients=[0];for each original power k>=order multiplier=k*(k-1)*...*(k-order+1);lower powers vanish

Higher derivative Horner power0 previous value times ORIGINAL point := mul(0, 0)

Higher derivative Horner power0 previous value times ORIGINAL point = 0

Higher derivative Horner power0 plus ORIGINAL derivative coefficient := sum(0, 0)

Higher derivative Horner power0 plus ORIGINAL derivative coefficient = 0

逐步反向核對

• S034 · premise_and_result_audit — 以精確有理數重算 Higher derivative Horner power0 plus ORIGINAL derivative coefficient，不以小數近似替代。
• S033 · premise_and_result_audit — 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。
• S032 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S031 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S030 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S029 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S028 · premise_and_result_audit — 以精確有理數重算 Higher derivative level6 ORIGINAL power1 coefficient product，不以小數近似替代。
• S027 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S026 · premise_and_result_audit — 以精確有理數重算 Higher derivative level5 ORIGINAL power2 coefficient product，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Higher derivative level5 ORIGINAL power1 coefficient product，不以小數近似替代。
• S024 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S023 · premise_and_result_audit — 以精確有理數重算 Higher derivative level4 ORIGINAL power3 coefficient product，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Higher derivative level4 ORIGINAL power2 coefficient product，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Higher derivative level4 ORIGINAL power1 coefficient product，不以小數近似替代。
• S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S019 · premise_and_result_audit — 以精確有理數重算 Higher derivative level3 ORIGINAL power4 coefficient product，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Higher derivative level3 ORIGINAL power3 coefficient product，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Higher derivative level3 ORIGINAL power2 coefficient product，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Higher derivative level3 ORIGINAL power1 coefficient product，不以小數近似替代。
• S015 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S014 · premise_and_result_audit — 以精確有理數重算 Higher derivative level2 ORIGINAL power5 coefficient product，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Higher derivative level2 ORIGINAL power4 coefficient product，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Higher derivative level2 ORIGINAL power3 coefficient product，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Higher derivative level2 ORIGINAL power2 coefficient product，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Higher derivative level2 ORIGINAL power1 coefficient product，不以小數近似替代。
• S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S008 · premise_and_result_audit — 以精確有理數重算 Higher derivative level1 ORIGINAL power6 coefficient product，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Higher derivative level1 ORIGINAL power5 coefficient product，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Higher derivative level1 ORIGINAL power4 coefficient product，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Higher derivative level1 ORIGINAL power3 coefficient product，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Higher derivative level1 ORIGINAL power2 coefficient product，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Higher derivative level1 ORIGINAL power1 coefficient product，不以小數近似替代。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(6, 8, [0], 0, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

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。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。