R0045-23046 · q-20261014-advance-20261111-r02-higher-order-derivative-0021
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":[21,42,63,84],"order":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-order-qualification；資料={"coefficients":[21,42,63,84],"order":0}。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=[21, 42, 63, 84];declared order=0
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
ORIGINAL coefficients=[21, 42, 63, 84];order=0;declared degree bound=3
操作後
final derivative coefficients=[21, 42, 63, 84];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, S002
S003 · 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。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, 0, 是, f_zero_equals_f, 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: 9d942ab97e384ea8a9b9a9597e31e77c9d10ed541204c2c93b139984832dadaacandidate record SHA-256: 1206fb3951fe370f573f96260cec952682cdf3b7f99ed2ac60d1772fbf978ee4semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2aLean: NOT_RUN · human review: NOT_PERFORMED