R0045-23042 · q-20261014-advance-20261111-r02-higher-order-derivative-0017
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":[17,34,51,68],"order":1}。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-order-qualification；資料={"coefficients":[17,34,51,68],"order":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。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。
操作前
source operation=higher-order-qualification;original snapshot and mathematics retained
操作後
explicit ordered source output fields=[declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate]
條件：無
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。
規則：source_ordered_interface_explicit_without_javascript · 依賴：G001
操作前
ORIGINAL low-to-high coefficients=[17, 34, 51, 68];declared order=1
操作後
declared degree BOUND=3=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list
條件：無
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。
規則：higher_ORIGINAL_low_to_high_coefficients_degree_BOUND_order0 · 依賴：G001, S001
操作前
Higher derivative level1 ORIGINAL power1 coefficient product := mul(34, 1)
操作後
Higher derivative level1 ORIGINAL power1 coefficient product = 34
條件：無
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。
規則：exact_rational_arithmetic · 依賴：G001, S002
操作前
Higher derivative level1 ORIGINAL power2 coefficient product := mul(51, 2)
操作後
Higher derivative level1 ORIGINAL power2 coefficient product = 102
條件：無
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。
規則：exact_rational_arithmetic · 依賴：G001, S003
操作前
Higher derivative level1 ORIGINAL power3 coefficient product := mul(68, 3)
操作後
Higher derivative level1 ORIGINAL power3 coefficient product = 204
條件：無
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。
規則：exact_rational_arithmetic · 依賴：G001, S004
操作前
previous derivative order=0;actual low-to-high list=[17, 34, 51, 68];drop ONLY its constant term
操作後
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 34, 1, 34], [2, 51, 2, 102], [3, 68, 3, 204]];next raw list=[34, 102, 204];all intermediate products SAFE
條件：無
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。
規則：higher_EVERY_recursive_derivative_level_not_reordered · 依賴：G001, S005
操作前
ORIGINAL coefficients=[17, 34, 51, 68];order=1;declared degree bound=3
操作後
final derivative coefficients=[34, 102, 204];for each original power k>=order multiplier=k*(k-1)*...*(k-order+1);lower powers vanish
條件：無
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。
規則：higher_falling_factorial_rule_all_orders_polynomial_only · 依賴：G001, S006
S007 · 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。S006 · 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。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。(3, 1, 是, recursive_derivative_chain_required, 是)
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。
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。
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: eab6b1c444c6f698b9667d5dc75e198d001532302da5bd71ed5e191bec3d114dcandidate record SHA-256: 52905e51675be563893cba01852e5baa146d135db2450b9cc6fdd9b217c1dc81semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2aLean: NOT_RUN · human review: NOT_PERFORMED