第 23098 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":[23,46,69,92,115,138],"order":7,"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":[23,46,69,92,115,138],"order":7,"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=[23, 46, 69, 92, 115, 138];declared order=7

declared degree BOUND=5=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list

Higher derivative level1 ORIGINAL power1 coefficient product := mul(46, 1)

Higher derivative level1 ORIGINAL power1 coefficient product = 46

Higher derivative level1 ORIGINAL power2 coefficient product := mul(69, 2)

Higher derivative level1 ORIGINAL power2 coefficient product = 138

Higher derivative level1 ORIGINAL power3 coefficient product := mul(92, 3)

Higher derivative level1 ORIGINAL power3 coefficient product = 276

Higher derivative level1 ORIGINAL power4 coefficient product := mul(115, 4)

Higher derivative level1 ORIGINAL power4 coefficient product = 460

Higher derivative level1 ORIGINAL power5 coefficient product := mul(138, 5)

Higher derivative level1 ORIGINAL power5 coefficient product = 690

previous derivative order=0;actual low-to-high list=[23, 46, 69, 92, 115, 138];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 46, 1, 46], [2, 69, 2, 138], [3, 92, 3, 276], [4, 115, 4, 460], [5, 138, 5, 690]];next raw list=[46, 138, 276, 460, 690];all intermediate products SAFE

Higher derivative level2 ORIGINAL power1 coefficient product := mul(138, 1)

Higher derivative level2 ORIGINAL power1 coefficient product = 138

Higher derivative level2 ORIGINAL power2 coefficient product := mul(276, 2)

Higher derivative level2 ORIGINAL power2 coefficient product = 552

Higher derivative level2 ORIGINAL power3 coefficient product := mul(460, 3)

Higher derivative level2 ORIGINAL power3 coefficient product = 1380

Higher derivative level2 ORIGINAL power4 coefficient product := mul(690, 4)

Higher derivative level2 ORIGINAL power4 coefficient product = 2760

previous derivative order=1;actual low-to-high list=[46, 138, 276, 460, 690];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 138, 1, 138], [2, 276, 2, 552], [3, 460, 3, 1380], [4, 690, 4, 2760]];next raw list=[138, 552, 1380, 2760];all intermediate products SAFE

Higher derivative level3 ORIGINAL power1 coefficient product := mul(552, 1)

Higher derivative level3 ORIGINAL power1 coefficient product = 552

Higher derivative level3 ORIGINAL power2 coefficient product := mul(1380, 2)

Higher derivative level3 ORIGINAL power2 coefficient product = 2760

Higher derivative level3 ORIGINAL power3 coefficient product := mul(2760, 3)

Higher derivative level3 ORIGINAL power3 coefficient product = 8280

previous derivative order=2;actual low-to-high list=[138, 552, 1380, 2760];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 552, 1, 552], [2, 1380, 2, 2760], [3, 2760, 3, 8280]];next raw list=[552, 2760, 8280];all intermediate products SAFE

Higher derivative level4 ORIGINAL power1 coefficient product := mul(2760, 1)

Higher derivative level4 ORIGINAL power1 coefficient product = 2760

Higher derivative level4 ORIGINAL power2 coefficient product := mul(8280, 2)

Higher derivative level4 ORIGINAL power2 coefficient product = 16560

previous derivative order=3;actual low-to-high list=[552, 2760, 8280];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 2760, 1, 2760], [2, 8280, 2, 16560]];next raw list=[2760, 16560];all intermediate products SAFE

Higher derivative level5 ORIGINAL power1 coefficient product := mul(16560, 1)

Higher derivative level5 ORIGINAL power1 coefficient product = 16560

previous derivative order=4;actual low-to-high list=[2760, 16560];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 16560, 1, 16560]];next raw list=[16560];all intermediate products SAFE

previous derivative order=5;actual low-to-high list=[16560];drop ONLY its constant term

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

previous derivative order=6;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=[23, 46, 69, 92, 115, 138];order=7;declared degree bound=5

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

• S027 · 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。
• S026 · 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。
• S025 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S022 · 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。
• S021 · 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。
• 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 level4 ORIGINAL power2 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 level4 ORIGINAL power1 coefficient product，不以小數近似替代。
• S017 · 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。
• S016 · 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。
• S015 · 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。
• S014 · 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。
• S013 · 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。
• S012 · 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。
• S011 · 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。
• S010 · 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。
• S009 · 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。
• S008 · 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。
• 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

(5, 7, [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。