第 22121 題：有限因數格上的Möbius反演

原題與已知條件

positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={"n":21,"a":1,"b":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={"n":21,"a":1,"b":8}。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-mobius-inversion-finite-divisors

逐步推導

source operation=mobinv-forward-divisor-transform;original snapshot and mathematics retained

explicit ordered source output fields=[n, divisor_value_rows, F_n, certificate]

Mobius inversion ORIGINAL forward:actual positive input=21;check complete integer range1..21

ALL positive divisors INCREASING=[1, 3, 7, 21];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL forward f(1) ORIGINAL a times divisor := mul(1, 1)

Mobius inversion ORIGINAL forward f(1) ORIGINAL a times divisor = 1

Mobius inversion ORIGINAL forward f(1) plus ORIGINAL b := sum(1, 8)

Mobius inversion ORIGINAL forward f(1) plus ORIGINAL b = 9

Mobius inversion ORIGINAL forward f(3) ORIGINAL a times divisor := mul(1, 3)

Mobius inversion ORIGINAL forward f(3) ORIGINAL a times divisor = 3

Mobius inversion ORIGINAL forward f(3) plus ORIGINAL b := sum(3, 8)

Mobius inversion ORIGINAL forward f(3) plus ORIGINAL b = 11

Mobius inversion ORIGINAL forward f(7) ORIGINAL a times divisor := mul(1, 7)

Mobius inversion ORIGINAL forward f(7) ORIGINAL a times divisor = 7

Mobius inversion ORIGINAL forward f(7) plus ORIGINAL b := sum(7, 8)

Mobius inversion ORIGINAL forward f(7) plus ORIGINAL b = 15

Mobius inversion ORIGINAL forward f(21) ORIGINAL a times divisor := mul(1, 21)

Mobius inversion ORIGINAL forward f(21) ORIGINAL a times divisor = 21

Mobius inversion ORIGINAL forward f(21) plus ORIGINAL b := sum(21, 8)

Mobius inversion ORIGINAL forward f(21) plus ORIGINAL b = 29

Mobius inversion ORIGINAL forward full forward divisor sum := sum(9, 11, 15, 29)

Mobius inversion ORIGINAL forward full forward divisor sum = 64

逐步反向核對

• S011 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL forward full forward divisor sum，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL forward f(21) plus ORIGINAL b，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL forward f(21) ORIGINAL a times divisor，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL forward f(7) plus ORIGINAL b，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL forward f(7) ORIGINAL a times divisor，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL forward f(3) plus ORIGINAL b，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL forward f(3) ORIGINAL a times divisor，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL forward f(1) plus ORIGINAL b，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL forward f(1) ORIGINAL a times divisor，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

(21, [[1, 9], [3, 11], [7, 15], [21, 29]], 64, sum_over_all_positive_divisors)

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。