第 22162 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-inverse-reconstruction；資料={"n":12,"a":2,"b":-1}。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-inverse-reconstruction；資料={"n":12,"a":2,"b":-1}。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-inverse-reconstruction;original snapshot and mathematics retained

explicit ordered source output fields=[n, inverse_terms, reconstructed, f_n, equal]

Mobius inversion ORIGINAL inverse:actual positive input=12;check complete integer range1..12

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 12];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]

mu=1;empty product case=是;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 := div(12, 1)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 12

Mobius inversion ORIGINAL inverse F(n/1):actual positive input=12;check complete integer range1..12

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 12];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor := mul(2, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 2

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(2, -1)

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = 1

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(2, 2)

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 4

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b := sum(4, -1)

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 3

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor := mul(2, 3)

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor = 6

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b := sum(6, -1)

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b = 5

Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor := mul(2, 4)

Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor = 8

Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b := sum(8, -1)

Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b = 7

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor := mul(2, 6)

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor = 12

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b := sum(12, -1)

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b = 11

Mobius inversion ORIGINAL inverse F(n/1) f(12) ORIGINAL a times divisor := mul(2, 12)

Mobius inversion ORIGINAL inverse F(n/1) f(12) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/1) f(12) plus ORIGINAL b := sum(24, -1)

Mobius inversion ORIGINAL inverse F(n/1) f(12) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(1, 3, 5, 7, 11, 23)

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 50

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 50)

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 50

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor2:actual positive input for mu=2

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=2>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

exhausted trial divisors below2;terminal remaining=2;d*d>remaining;ORIGINAL n=2

complete increasing prime multiset=[2];full product=2;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor2:input=2;COMPLETE prime multiset=[2];distinct primes=[2]

mu=-1;empty product case=否;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 := div(12, 2)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 6

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=6;check complete integer range1..6

ALL positive divisors INCREASING=[1, 2, 3, 6];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(2, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 2

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(2, -1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 1

Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor := mul(2, 2)

Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor = 4

Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b := sum(4, -1)

Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b = 3

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor := mul(2, 3)

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor = 6

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b := sum(6, -1)

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b = 5

Mobius inversion ORIGINAL inverse F(n/2) f(6) ORIGINAL a times divisor := mul(2, 6)

Mobius inversion ORIGINAL inverse F(n/2) f(6) ORIGINAL a times divisor = 12

Mobius inversion ORIGINAL inverse F(n/2) f(6) plus ORIGINAL b := sum(12, -1)

Mobius inversion ORIGINAL inverse F(n/2) f(6) plus ORIGINAL b = 11

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(1, 3, 5, 11)

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 20

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 20)

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -20

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor3:actual positive input for mu=3

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=3>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf canonical complete product factor0 := mul(1, 3)

upf canonical complete product factor0 = 3

exhausted trial divisors below2;terminal remaining=3;d*d>remaining;ORIGINAL n=3

complete increasing prime multiset=[3];full product=3;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor3:input=3;COMPLETE prime multiset=[3];distinct primes=[3]

mu=-1;empty product case=否;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 := div(12, 3)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 = 4

Mobius inversion ORIGINAL inverse F(n/3):actual positive input=4;check complete integer range1..4

ALL positive divisors INCREASING=[1, 2, 4];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor := mul(2, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor = 2

Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b := sum(2, -1)

Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b = 1

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor := mul(2, 2)

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor = 4

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b := sum(4, -1)

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b = 3

Mobius inversion ORIGINAL inverse F(n/3) f(4) ORIGINAL a times divisor := mul(2, 4)

Mobius inversion ORIGINAL inverse F(n/3) f(4) ORIGINAL a times divisor = 8

Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b := sum(8, -1)

Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b = 7

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum := sum(1, 3, 7)

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum = 11

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) := mul(-1, 11)

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) = -11

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor4:actual positive input for mu=4

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=4>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(4, 2)

upf extract divisor2 occurrence0 = 2

upf extract divisor2 occurrence1 := div(2, 2)

upf extract divisor2 occurrence1 = 1

next divisor=2;previous remaining=4;all smaller divisors exhausted

extracted multiplicity=2;next remaining=1;remaining modulo2=1;extracted multiset=[2, 2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=4

complete increasing prime multiset=[2, 2];full product=4;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor4:input=4;COMPLETE prime multiset=[2, 2];distinct primes=[2]

mu=0;empty product case=否;repeated prime=是

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 := div(12, 4)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 = 3

Mobius inversion ORIGINAL inverse F(n/4):actual positive input=3;check complete integer range1..3

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

Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor := mul(2, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor = 2

Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b := sum(2, -1)

Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b = 1

Mobius inversion ORIGINAL inverse F(n/4) f(3) ORIGINAL a times divisor := mul(2, 3)

Mobius inversion ORIGINAL inverse F(n/4) f(3) ORIGINAL a times divisor = 6

Mobius inversion ORIGINAL inverse F(n/4) f(3) plus ORIGINAL b := sum(6, -1)

Mobius inversion ORIGINAL inverse F(n/4) f(3) plus ORIGINAL b = 5

Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum := sum(1, 5)

Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum = 6

Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) := mul(0, 6)

Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor6:actual positive input for mu=6

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=6>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(6, 2)

upf extract divisor2 occurrence0 = 3

next divisor=2;previous remaining=6;all smaller divisors exhausted

extracted multiplicity=1;next remaining=3;remaining modulo2=1;extracted multiset=[2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 3)

upf canonical complete product factor1 = 6

exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=6

complete increasing prime multiset=[2, 3];full product=6;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor6:input=6;COMPLETE prime multiset=[2, 3];distinct primes=[2, 3]

mu=1;empty product case=否;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 := div(12, 6)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 = 2

Mobius inversion ORIGINAL inverse F(n/6):actual positive input=2;check complete integer range1..2

ALL positive divisors INCREASING=[1, 2];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor := mul(2, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor = 2

Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b := sum(2, -1)

Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b = 1

Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor := mul(2, 2)

Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor = 4

Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b := sum(4, -1)

Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b = 3

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum := sum(1, 3)

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum = 4

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) := mul(1, 4)

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) = 4

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor12:actual positive input for mu=12

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=12>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(12, 2)

upf extract divisor2 occurrence0 = 6

upf extract divisor2 occurrence1 := div(6, 2)

upf extract divisor2 occurrence1 = 3

next divisor=2;previous remaining=12;all smaller divisors exhausted

extracted multiplicity=2;next remaining=3;remaining modulo2=1;extracted multiset=[2, 2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 3)

upf canonical complete product factor2 = 12

exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=12

complete increasing prime multiset=[2, 2, 3];full product=12;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor12:input=12;COMPLETE prime multiset=[2, 2, 3];distinct primes=[2, 3]

mu=0;empty product case=否;repeated prime=是

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12 := div(12, 12)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12 = 1

Mobius inversion ORIGINAL inverse F(n/12):actual positive input=1;check complete integer range1..1

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

Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor := mul(2, 1)

Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor = 2

Mobius inversion ORIGINAL inverse F(n/12) f(1) plus ORIGINAL b := sum(2, -1)

Mobius inversion ORIGINAL inverse F(n/12) f(1) plus ORIGINAL b = 1

Mobius inversion ORIGINAL inverse F(n/12) full forward divisor sum := sum(1)

Mobius inversion ORIGINAL inverse F(n/12) full forward divisor sum = 1

Mobius inversion ORIGINAL inverse mu(12) times SAME F(n/12) := mul(0, 1)

Mobius inversion ORIGINAL inverse mu(12) times SAME F(n/12) = 0

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(50, -20, -11, 0, 4, 0)

Mobius inversion ORIGINAL inverse complete inverse-term sum = 23

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(2, 12)

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 24

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(24, -1)

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse:actual n=12;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 12, 50, 50], [2, -1, 6, 20, -20], [3, -1, 4, 11, -11], [4, 0, 3, 6, 0], [6, 1, 2, 4, 4], [12, 0, 1, 1, 0]]

complete reconstructed=23;direct f(n)=23;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

逐步反向核對

• S105 · 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。
• S104 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse direct plus ORIGINAL b，不以小數近似替代。
• S103 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse direct ORIGINAL a times n，不以小數近似替代。
• S102 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse complete inverse-term sum，不以小數近似替代。
• S101 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(12) times SAME F(n/12)，不以小數近似替代。
• S100 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/12) full forward divisor sum，不以小數近似替代。
• S099 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/12) f(1) plus ORIGINAL b，不以小數近似替代。
• S098 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S097 · 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。
• S096 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12，不以小數近似替代。
• S095 · 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。
• S094 · 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。
• S093 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。
• S092 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S091 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S090 · 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。
• S089 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S088 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S087 · 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。
• S086 · 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。
• S085 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6)，不以小數近似替代。
• S084 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum，不以小數近似替代。
• S083 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b，不以小數近似替代。
• S082 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S081 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b，不以小數近似替代。
• S080 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S079 · 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。
• S078 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6，不以小數近似替代。
• S077 · 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。
• S076 · 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。
• S075 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S074 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S073 · 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。
• S072 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S071 · 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。
• S070 · 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。
• S069 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4)，不以小數近似替代。
• S068 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum，不以小數近似替代。
• S067 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(3) plus ORIGINAL b，不以小數近似替代。
• S066 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S065 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S063 · 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。
• S062 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4，不以小數近似替代。
• S061 · 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。
• S060 · 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。
• S059 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S057 · 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。
• S056 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S054 · 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。
• S053 · 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。
• S052 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3)，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S044 · 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。
• S043 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3，不以小數近似替代。
• S042 · 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。
• S041 · 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。
• S040 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S039 · 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。
• S038 · 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。
• S037 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2)，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(6) plus ORIGINAL b，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor，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 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2，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 — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S020 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1)，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(12) plus ORIGINAL b，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(12) ORIGINAL a times divisor，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(2) 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 inverse F(n/1) f(2) 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 inverse F(n/1) f(1) 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 inverse F(n/1) f(1) ORIGINAL a times divisor，不以小數近似替代。
• 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 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1，不以小數近似替代。
• S004 · 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。
• 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。

結論

(12, [[1, 1, 12, 50, 50], [2, -1, 6, 20, -20], [3, -1, 4, 11, -11], [4, 0, 3, 6, 0], [6, 1, 2, 4, 4], [12, 0, 1, 1, 0]], 23, 23, 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。

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。