R0043-22135 · q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0085
positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={"n":132,"a":10,"b":1,"omitted_divisor":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-incomplete-divisor-boundary；資料={"n":132,"a":10,"b":1,"omitted_divisor":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=mobinv-incomplete-divisor-boundary;original snapshot and mathematics retained
操作後
explicit ordered source output fields=[n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary]
條件：無
原題歷史介面標記保留，直接列來源有序欄位，不需JavaScript；MVT由原mode選欄序，不補空欄或看答案形狀猜。tag與固定No保原聲稱範圍。
規則：source_ordered_interface_explicit_without_javascript · 依賴：G001
操作前
Mobius inversion ORIGINAL inverse:actual positive input=132;check complete integer range1..132
操作後
ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132];not only primes/proper divisors or all integers<=n
條件：無
逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。
規則：mobinv_ALL_positive_divisors_complete_indexing · 依賴：G001, S001
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1
操作後
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
條件：無
先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。
規則：mobius_complete_UPF_input_role · 依賴：G001, S002
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]
操作後
mu=1;empty product case=是;repeated prime=否
條件：無
1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。
規則：mobius_value_one_repeated_prime_or_squarefree_parity · 依賴：G001, S003
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 := div(132, 1)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 132
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S004
操作前
Mobius inversion ORIGINAL inverse F(n/1):actual positive input=132;check complete integer range1..132
操作後
ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132];not only primes/proper divisors or all integers<=n
條件：無
逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。
規則：mobinv_ALL_positive_divisors_complete_indexing · 依賴：G001, S005
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor := mul(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 10
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S006
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = 11
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S007
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(10, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 20
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S008
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b := sum(20, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 21
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S009
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor := mul(10, 3)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor = 30
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S010
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b := sum(30, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b = 31
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S011
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor := mul(10, 4)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor = 40
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S012
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b := sum(40, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b = 41
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S013
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor := mul(10, 6)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor = 60
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S014
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b := sum(60, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b = 61
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S015
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(11) ORIGINAL a times divisor := mul(10, 11)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(11) ORIGINAL a times divisor = 110
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S016
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(11) plus ORIGINAL b := sum(110, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(11) plus ORIGINAL b = 111
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S017
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(12) ORIGINAL a times divisor := mul(10, 12)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(12) ORIGINAL a times divisor = 120
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S018
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(12) plus ORIGINAL b := sum(120, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(12) plus ORIGINAL b = 121
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S019
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(22) ORIGINAL a times divisor := mul(10, 22)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(22) ORIGINAL a times divisor = 220
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S020
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(22) plus ORIGINAL b := sum(220, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(22) plus ORIGINAL b = 221
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S021
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(33) ORIGINAL a times divisor := mul(10, 33)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(33) ORIGINAL a times divisor = 330
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S022
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(33) plus ORIGINAL b := sum(330, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(33) plus ORIGINAL b = 331
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S023
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(44) ORIGINAL a times divisor := mul(10, 44)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(44) ORIGINAL a times divisor = 440
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S024
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(44) plus ORIGINAL b := sum(440, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(44) plus ORIGINAL b = 441
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S025
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(66) ORIGINAL a times divisor := mul(10, 66)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(66) ORIGINAL a times divisor = 660
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S026
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(66) plus ORIGINAL b := sum(660, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(66) plus ORIGINAL b = 661
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S027
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(132) ORIGINAL a times divisor := mul(10, 132)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(132) ORIGINAL a times divisor = 1320
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S028
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(132) plus ORIGINAL b := sum(1320, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(132) plus ORIGINAL b = 1321
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S029
操作前
Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(11, 21, 31, 41, 61, 111, 121, 221, 331, 441, 661, 1321)
操作後
Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 3372
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S030
操作前
Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 3372)
操作後
Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 3372
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S031
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor2:actual positive input for mu=2
操作後
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
條件：無
先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。
規則：mobius_complete_UPF_input_role · 依賴：G001, S032
操作前
ORIGINAL positive integer absolute magnitude=2>1
操作後
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
條件：無
每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。
規則：upf_factorization_remaining_product_invariant · 依賴：G001, S033
操作前
upf canonical complete product factor0 := mul(1, 2)
操作後
upf canonical complete product factor0 = 2
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S034
操作前
exhausted trial divisors below2;terminal remaining=2;d*d>remaining;ORIGINAL n=2
操作後
complete increasing prime multiset=[2];full product=2;a composite terminal remainder would have an already tested smaller divisor
條件：無
若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。
規則：upf_final_prime_remainder_complete_existence_instance · 依賴：G001, S035
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor2:input=2;COMPLETE prime multiset=[2];distinct primes=[2]
操作後
mu=-1;empty product case=否;repeated prime=否
條件：無
1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。
規則：mobius_value_one_repeated_prime_or_squarefree_parity · 依賴：G001, S036
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 := div(132, 2)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 66
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S037
操作前
Mobius inversion ORIGINAL inverse F(n/2):actual positive input=66;check complete integer range1..66
操作後
ALL positive divisors INCREASING=[1, 2, 3, 6, 11, 22, 33, 66];not only primes/proper divisors or all integers<=n
條件：無
逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。
規則：mobinv_ALL_positive_divisors_complete_indexing · 依賴：G001, S038
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 10
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S039
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 11
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S040
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor := mul(10, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor = 20
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S041
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b := sum(20, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b = 21
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S042
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor := mul(10, 3)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor = 30
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S043
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b := sum(30, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b = 31
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S044
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(6) ORIGINAL a times divisor := mul(10, 6)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(6) ORIGINAL a times divisor = 60
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S045
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(6) plus ORIGINAL b := sum(60, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(6) plus ORIGINAL b = 61
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S046
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(11) ORIGINAL a times divisor := mul(10, 11)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(11) ORIGINAL a times divisor = 110
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S047
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(11) plus ORIGINAL b := sum(110, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(11) plus ORIGINAL b = 111
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S048
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(22) ORIGINAL a times divisor := mul(10, 22)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(22) ORIGINAL a times divisor = 220
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S049
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(22) plus ORIGINAL b := sum(220, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(22) plus ORIGINAL b = 221
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S050
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(33) ORIGINAL a times divisor := mul(10, 33)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(33) ORIGINAL a times divisor = 330
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S051
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(33) plus ORIGINAL b := sum(330, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(33) plus ORIGINAL b = 331
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S052
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(66) ORIGINAL a times divisor := mul(10, 66)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(66) ORIGINAL a times divisor = 660
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S053
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(66) plus ORIGINAL b := sum(660, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(66) plus ORIGINAL b = 661
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S054
操作前
Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(11, 21, 31, 61, 111, 221, 331, 661)
操作後
Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 1448
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S055
操作前
Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 1448)
操作後
Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -1448
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S056
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor3:actual positive input for mu=3
操作後
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
條件：無
先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。
規則：mobius_complete_UPF_input_role · 依賴：G001, S057
操作前
ORIGINAL positive integer absolute magnitude=3>1
操作後
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
條件：無
每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。
規則：upf_factorization_remaining_product_invariant · 依賴：G001, S058
操作前
upf canonical complete product factor0 := mul(1, 3)
操作後
upf canonical complete product factor0 = 3
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S059
操作前
exhausted trial divisors below2;terminal remaining=3;d*d>remaining;ORIGINAL n=3
操作後
complete increasing prime multiset=[3];full product=3;a composite terminal remainder would have an already tested smaller divisor
條件：無
若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。
規則：upf_final_prime_remainder_complete_existence_instance · 依賴：G001, S060
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor3:input=3;COMPLETE prime multiset=[3];distinct primes=[3]
操作後
mu=-1;empty product case=否;repeated prime=否
條件：無
1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。
規則：mobius_value_one_repeated_prime_or_squarefree_parity · 依賴：G001, S061
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 := div(132, 3)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 = 44
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S062
操作前
Mobius inversion ORIGINAL inverse F(n/3):actual positive input=44;check complete integer range1..44
操作後
ALL positive divisors INCREASING=[1, 2, 4, 11, 22, 44];not only primes/proper divisors or all integers<=n
條件：無
逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。
規則：mobinv_ALL_positive_divisors_complete_indexing · 依賴：G001, S063
操作前
Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor := mul(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor = 10
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S064
操作前
Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b := sum(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b = 11
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S065
操作前
Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor := mul(10, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor = 20
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S066
操作前
Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b := sum(20, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b = 21
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S067
操作前
Mobius inversion ORIGINAL inverse F(n/3) f(4) ORIGINAL a times divisor := mul(10, 4)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(4) ORIGINAL a times divisor = 40
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S068
操作前
Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b := sum(40, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b = 41
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S069
操作前
Mobius inversion ORIGINAL inverse F(n/3) f(11) ORIGINAL a times divisor := mul(10, 11)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(11) ORIGINAL a times divisor = 110
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S070
操作前
Mobius inversion ORIGINAL inverse F(n/3) f(11) plus ORIGINAL b := sum(110, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(11) plus ORIGINAL b = 111
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S071
操作前
Mobius inversion ORIGINAL inverse F(n/3) f(22) ORIGINAL a times divisor := mul(10, 22)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(22) ORIGINAL a times divisor = 220
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S072
操作前
Mobius inversion ORIGINAL inverse F(n/3) f(22) plus ORIGINAL b := sum(220, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(22) plus ORIGINAL b = 221
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S073
操作前
Mobius inversion ORIGINAL inverse F(n/3) f(44) ORIGINAL a times divisor := mul(10, 44)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(44) ORIGINAL a times divisor = 440
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S074
操作前
Mobius inversion ORIGINAL inverse F(n/3) f(44) plus ORIGINAL b := sum(440, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(44) plus ORIGINAL b = 441
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S075
操作前
Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum := sum(11, 21, 41, 111, 221, 441)
操作後
Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum = 846
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S076
操作前
Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) := mul(-1, 846)
操作後
Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) = -846
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S077
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor4:actual positive input for mu=4
操作後
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
條件：無
先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。
規則：mobius_complete_UPF_input_role · 依賴：G001, S078
操作前
ORIGINAL positive integer absolute magnitude=4>1
操作後
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
條件：無
每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。
規則：upf_factorization_remaining_product_invariant · 依賴：G001, S079
操作前
upf extract divisor2 occurrence0 := div(4, 2)
操作後
upf extract divisor2 occurrence0 = 2
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S080
操作前
upf extract divisor2 occurrence1 := div(2, 2)
操作後
upf extract divisor2 occurrence1 = 1
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S081
操作前
next divisor=2;previous remaining=4;all smaller divisors exhausted
操作後
extracted multiplicity=2;next remaining=1;remaining modulo2=1;extracted multiset=[2, 2]
條件：無
除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。
規則：upf_complete_trial_division_remaining_invariant · 依賴：G001, S082
操作前
upf canonical complete product factor0 := mul(1, 2)
操作後
upf canonical complete product factor0 = 2
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S083
操作前
upf canonical complete product factor1 := mul(2, 2)
操作後
upf canonical complete product factor1 = 4
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S084
操作前
exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=4
操作後
complete increasing prime multiset=[2, 2];full product=4;a composite terminal remainder would have an already tested smaller divisor
條件：無
若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。
規則：upf_final_prime_remainder_complete_existence_instance · 依賴：G001, S085
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor4:input=4;COMPLETE prime multiset=[2, 2];distinct primes=[2]
操作後
mu=0;empty product case=否;repeated prime=是
條件：無
1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。
規則：mobius_value_one_repeated_prime_or_squarefree_parity · 依賴：G001, S086
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 := div(132, 4)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 = 33
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S087
操作前
Mobius inversion ORIGINAL inverse F(n/4):actual positive input=33;check complete integer range1..33
操作後
ALL positive divisors INCREASING=[1, 3, 11, 33];not only primes/proper divisors or all integers<=n
條件：無
逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。
規則：mobinv_ALL_positive_divisors_complete_indexing · 依賴：G001, S088
操作前
Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor := mul(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor = 10
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S089
操作前
Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b := sum(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b = 11
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S090
操作前
Mobius inversion ORIGINAL inverse F(n/4) f(3) ORIGINAL a times divisor := mul(10, 3)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(3) ORIGINAL a times divisor = 30
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S091
操作前
Mobius inversion ORIGINAL inverse F(n/4) f(3) plus ORIGINAL b := sum(30, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(3) plus ORIGINAL b = 31
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S092
操作前
Mobius inversion ORIGINAL inverse F(n/4) f(11) ORIGINAL a times divisor := mul(10, 11)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(11) ORIGINAL a times divisor = 110
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S093
操作前
Mobius inversion ORIGINAL inverse F(n/4) f(11) plus ORIGINAL b := sum(110, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(11) plus ORIGINAL b = 111
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S094
操作前
Mobius inversion ORIGINAL inverse F(n/4) f(33) ORIGINAL a times divisor := mul(10, 33)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(33) ORIGINAL a times divisor = 330
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S095
操作前
Mobius inversion ORIGINAL inverse F(n/4) f(33) plus ORIGINAL b := sum(330, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(33) plus ORIGINAL b = 331
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S096
操作前
Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum := sum(11, 31, 111, 331)
操作後
Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum = 484
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S097
操作前
Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) := mul(0, 484)
操作後
Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) = 0
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S098
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor6:actual positive input for mu=6
操作後
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
條件：無
先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。
規則：mobius_complete_UPF_input_role · 依賴：G001, S099
操作前
ORIGINAL positive integer absolute magnitude=6>1
操作後
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
條件：無
每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。
規則：upf_factorization_remaining_product_invariant · 依賴：G001, S100
操作前
upf extract divisor2 occurrence0 := div(6, 2)
操作後
upf extract divisor2 occurrence0 = 3
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S101
操作前
next divisor=2;previous remaining=6;all smaller divisors exhausted
操作後
extracted multiplicity=1;next remaining=3;remaining modulo2=1;extracted multiset=[2]
條件：無
除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。
規則：upf_complete_trial_division_remaining_invariant · 依賴：G001, S102
操作前
upf canonical complete product factor0 := mul(1, 2)
操作後
upf canonical complete product factor0 = 2
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S103
操作前
upf canonical complete product factor1 := mul(2, 3)
操作後
upf canonical complete product factor1 = 6
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S104
操作前
exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=6
操作後
complete increasing prime multiset=[2, 3];full product=6;a composite terminal remainder would have an already tested smaller divisor
條件：無
若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。
規則：upf_final_prime_remainder_complete_existence_instance · 依賴：G001, S105
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor6:input=6;COMPLETE prime multiset=[2, 3];distinct primes=[2, 3]
操作後
mu=1;empty product case=否;repeated prime=否
條件：無
1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。
規則：mobius_value_one_repeated_prime_or_squarefree_parity · 依賴：G001, S106
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 := div(132, 6)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 = 22
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S107
操作前
Mobius inversion ORIGINAL inverse F(n/6):actual positive input=22;check complete integer range1..22
操作後
ALL positive divisors INCREASING=[1, 2, 11, 22];not only primes/proper divisors or all integers<=n
條件：無
逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。
規則：mobinv_ALL_positive_divisors_complete_indexing · 依賴：G001, S108
操作前
Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor := mul(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor = 10
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S109
操作前
Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b := sum(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b = 11
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S110
操作前
Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor := mul(10, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor = 20
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S111
操作前
Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b := sum(20, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b = 21
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S112
操作前
Mobius inversion ORIGINAL inverse F(n/6) f(11) ORIGINAL a times divisor := mul(10, 11)
操作後
Mobius inversion ORIGINAL inverse F(n/6) f(11) ORIGINAL a times divisor = 110
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S113
操作前
Mobius inversion ORIGINAL inverse F(n/6) f(11) plus ORIGINAL b := sum(110, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/6) f(11) plus ORIGINAL b = 111
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S114
操作前
Mobius inversion ORIGINAL inverse F(n/6) f(22) ORIGINAL a times divisor := mul(10, 22)
操作後
Mobius inversion ORIGINAL inverse F(n/6) f(22) ORIGINAL a times divisor = 220
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S115
操作前
Mobius inversion ORIGINAL inverse F(n/6) f(22) plus ORIGINAL b := sum(220, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/6) f(22) plus ORIGINAL b = 221
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S116
操作前
Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum := sum(11, 21, 111, 221)
操作後
Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum = 364
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S117
操作前
Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) := mul(1, 364)
操作後
Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) = 364
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S118
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor11:actual positive input for mu=11
操作後
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
條件：無
先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。
規則：mobius_complete_UPF_input_role · 依賴：G001, S119
操作前
ORIGINAL positive integer absolute magnitude=11>1
操作後
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
條件：無
每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。
規則：upf_factorization_remaining_product_invariant · 依賴：G001, S120
操作前
next divisor=2;previous remaining=11;all smaller divisors exhausted
操作後
extracted multiplicity=0;next remaining=11;remaining modulo2=1;extracted multiset=[]
條件：無
除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。
規則：upf_complete_trial_division_remaining_invariant · 依賴：G001, S121
操作前
next divisor=3;previous remaining=11;all smaller divisors exhausted
操作後
extracted multiplicity=0;next remaining=11;remaining modulo3=2;extracted multiset=[]
條件：無
除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。
規則：upf_complete_trial_division_remaining_invariant · 依賴：G001, S122
操作前
upf canonical complete product factor0 := mul(1, 11)
操作後
upf canonical complete product factor0 = 11
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S123
操作前
exhausted trial divisors below4;terminal remaining=11;d*d>remaining;ORIGINAL n=11
操作後
complete increasing prime multiset=[11];full product=11;a composite terminal remainder would have an already tested smaller divisor
條件：無
若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。
規則：upf_final_prime_remainder_complete_existence_instance · 依賴：G001, S124
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor11:input=11;COMPLETE prime multiset=[11];distinct primes=[11]
操作後
mu=-1;empty product case=否;repeated prime=否
條件：無
1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。
規則：mobius_value_one_repeated_prime_or_squarefree_parity · 依賴：G001, S125
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor11 := div(132, 11)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor11 = 12
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S126
操作前
Mobius inversion ORIGINAL inverse F(n/11):actual positive input=12;check complete integer range1..12
操作後
ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 12];not only primes/proper divisors or all integers<=n
條件：無
逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。
規則：mobinv_ALL_positive_divisors_complete_indexing · 依賴：G001, S127
操作前
Mobius inversion ORIGINAL inverse F(n/11) f(1) ORIGINAL a times divisor := mul(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/11) f(1) ORIGINAL a times divisor = 10
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S128
操作前
Mobius inversion ORIGINAL inverse F(n/11) f(1) plus ORIGINAL b := sum(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/11) f(1) plus ORIGINAL b = 11
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S129
操作前
Mobius inversion ORIGINAL inverse F(n/11) f(2) ORIGINAL a times divisor := mul(10, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/11) f(2) ORIGINAL a times divisor = 20
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S130
操作前
Mobius inversion ORIGINAL inverse F(n/11) f(2) plus ORIGINAL b := sum(20, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/11) f(2) plus ORIGINAL b = 21
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S131
操作前
Mobius inversion ORIGINAL inverse F(n/11) f(3) ORIGINAL a times divisor := mul(10, 3)
操作後
Mobius inversion ORIGINAL inverse F(n/11) f(3) ORIGINAL a times divisor = 30
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S132
操作前
Mobius inversion ORIGINAL inverse F(n/11) f(3) plus ORIGINAL b := sum(30, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/11) f(3) plus ORIGINAL b = 31
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S133
操作前
Mobius inversion ORIGINAL inverse F(n/11) f(4) ORIGINAL a times divisor := mul(10, 4)
操作後
Mobius inversion ORIGINAL inverse F(n/11) f(4) ORIGINAL a times divisor = 40
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S134
操作前
Mobius inversion ORIGINAL inverse F(n/11) f(4) plus ORIGINAL b := sum(40, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/11) f(4) plus ORIGINAL b = 41
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S135
操作前
Mobius inversion ORIGINAL inverse F(n/11) f(6) ORIGINAL a times divisor := mul(10, 6)
操作後
Mobius inversion ORIGINAL inverse F(n/11) f(6) ORIGINAL a times divisor = 60
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S136
操作前
Mobius inversion ORIGINAL inverse F(n/11) f(6) plus ORIGINAL b := sum(60, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/11) f(6) plus ORIGINAL b = 61
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S137
操作前
Mobius inversion ORIGINAL inverse F(n/11) f(12) ORIGINAL a times divisor := mul(10, 12)
操作後
Mobius inversion ORIGINAL inverse F(n/11) f(12) ORIGINAL a times divisor = 120
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S138
操作前
Mobius inversion ORIGINAL inverse F(n/11) f(12) plus ORIGINAL b := sum(120, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/11) f(12) plus ORIGINAL b = 121
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S139
操作前
Mobius inversion ORIGINAL inverse F(n/11) full forward divisor sum := sum(11, 21, 31, 41, 61, 121)
操作後
Mobius inversion ORIGINAL inverse F(n/11) full forward divisor sum = 286
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S140
操作前
Mobius inversion ORIGINAL inverse mu(11) times SAME F(n/11) := mul(-1, 286)
操作後
Mobius inversion ORIGINAL inverse mu(11) times SAME F(n/11) = -286
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S141
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor12:actual positive input for mu=12
操作後
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
條件：無
先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。
規則：mobius_complete_UPF_input_role · 依賴：G001, S142
操作前
ORIGINAL positive integer absolute magnitude=12>1
操作後
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
條件：無
每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。
規則：upf_factorization_remaining_product_invariant · 依賴：G001, S143
操作前
upf extract divisor2 occurrence0 := div(12, 2)
操作後
upf extract divisor2 occurrence0 = 6
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S144
操作前
upf extract divisor2 occurrence1 := div(6, 2)
操作後
upf extract divisor2 occurrence1 = 3
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S145
操作前
next divisor=2;previous remaining=12;all smaller divisors exhausted
操作後
extracted multiplicity=2;next remaining=3;remaining modulo2=1;extracted multiset=[2, 2]
條件：無
除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。
規則：upf_complete_trial_division_remaining_invariant · 依賴：G001, S146
操作前
upf canonical complete product factor0 := mul(1, 2)
操作後
upf canonical complete product factor0 = 2
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S147
操作前
upf canonical complete product factor1 := mul(2, 2)
操作後
upf canonical complete product factor1 = 4
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S148
操作前
upf canonical complete product factor2 := mul(4, 3)
操作後
upf canonical complete product factor2 = 12
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S149
操作前
exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=12
操作後
complete increasing prime multiset=[2, 2, 3];full product=12;a composite terminal remainder would have an already tested smaller divisor
條件：無
若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。
規則：upf_final_prime_remainder_complete_existence_instance · 依賴：G001, S150
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor12:input=12;COMPLETE prime multiset=[2, 2, 3];distinct primes=[2, 3]
操作後
mu=0;empty product case=否;repeated prime=是
條件：無
1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。
規則：mobius_value_one_repeated_prime_or_squarefree_parity · 依賴：G001, S151
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12 := div(132, 12)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12 = 11
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S152
操作前
Mobius inversion ORIGINAL inverse F(n/12):actual positive input=11;check complete integer range1..11
操作後
ALL positive divisors INCREASING=[1, 11];not only primes/proper divisors or all integers<=n
條件：無
逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。
規則：mobinv_ALL_positive_divisors_complete_indexing · 依賴：G001, S153
操作前
Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor := mul(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor = 10
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S154
操作前
Mobius inversion ORIGINAL inverse F(n/12) f(1) plus ORIGINAL b := sum(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/12) f(1) plus ORIGINAL b = 11
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S155
操作前
Mobius inversion ORIGINAL inverse F(n/12) f(11) ORIGINAL a times divisor := mul(10, 11)
操作後
Mobius inversion ORIGINAL inverse F(n/12) f(11) ORIGINAL a times divisor = 110
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S156
操作前
Mobius inversion ORIGINAL inverse F(n/12) f(11) plus ORIGINAL b := sum(110, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/12) f(11) plus ORIGINAL b = 111
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S157
操作前
Mobius inversion ORIGINAL inverse F(n/12) full forward divisor sum := sum(11, 111)
操作後
Mobius inversion ORIGINAL inverse F(n/12) full forward divisor sum = 122
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S158
操作前
Mobius inversion ORIGINAL inverse mu(12) times SAME F(n/12) := mul(0, 122)
操作後
Mobius inversion ORIGINAL inverse mu(12) times SAME F(n/12) = 0
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S159
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor22:actual positive input for mu=22
操作後
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
條件：無
先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。
規則：mobius_complete_UPF_input_role · 依賴：G001, S160
操作前
ORIGINAL positive integer absolute magnitude=22>1
操作後
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
條件：無
每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。
規則：upf_factorization_remaining_product_invariant · 依賴：G001, S161
操作前
upf extract divisor2 occurrence0 := div(22, 2)
操作後
upf extract divisor2 occurrence0 = 11
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S162
操作前
next divisor=2;previous remaining=22;all smaller divisors exhausted
操作後
extracted multiplicity=1;next remaining=11;remaining modulo2=1;extracted multiset=[2]
條件：無
除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。
規則：upf_complete_trial_division_remaining_invariant · 依賴：G001, S163
操作前
next divisor=3;previous remaining=11;all smaller divisors exhausted
操作後
extracted multiplicity=0;next remaining=11;remaining modulo3=2;extracted multiset=[2]
條件：無
除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。
規則：upf_complete_trial_division_remaining_invariant · 依賴：G001, S164
操作前
upf canonical complete product factor0 := mul(1, 2)
操作後
upf canonical complete product factor0 = 2
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S165
操作前
upf canonical complete product factor1 := mul(2, 11)
操作後
upf canonical complete product factor1 = 22
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S166
操作前
exhausted trial divisors below4;terminal remaining=11;d*d>remaining;ORIGINAL n=22
操作後
complete increasing prime multiset=[2, 11];full product=22;a composite terminal remainder would have an already tested smaller divisor
條件：無
若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。
規則：upf_final_prime_remainder_complete_existence_instance · 依賴：G001, S167
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor22:input=22;COMPLETE prime multiset=[2, 11];distinct primes=[2, 11]
操作後
mu=1;empty product case=否;repeated prime=否
條件：無
1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。
規則：mobius_value_one_repeated_prime_or_squarefree_parity · 依賴：G001, S168
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor22 := div(132, 22)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor22 = 6
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S169
操作前
Mobius inversion ORIGINAL inverse F(n/22):actual positive input=6;check complete integer range1..6
操作後
ALL positive divisors INCREASING=[1, 2, 3, 6];not only primes/proper divisors or all integers<=n
條件：無
逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。
規則：mobinv_ALL_positive_divisors_complete_indexing · 依賴：G001, S170
操作前
Mobius inversion ORIGINAL inverse F(n/22) f(1) ORIGINAL a times divisor := mul(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/22) f(1) ORIGINAL a times divisor = 10
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S171
操作前
Mobius inversion ORIGINAL inverse F(n/22) f(1) plus ORIGINAL b := sum(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/22) f(1) plus ORIGINAL b = 11
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S172
操作前
Mobius inversion ORIGINAL inverse F(n/22) f(2) ORIGINAL a times divisor := mul(10, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/22) f(2) ORIGINAL a times divisor = 20
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S173
操作前
Mobius inversion ORIGINAL inverse F(n/22) f(2) plus ORIGINAL b := sum(20, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/22) f(2) plus ORIGINAL b = 21
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S174
操作前
Mobius inversion ORIGINAL inverse F(n/22) f(3) ORIGINAL a times divisor := mul(10, 3)
操作後
Mobius inversion ORIGINAL inverse F(n/22) f(3) ORIGINAL a times divisor = 30
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S175
操作前
Mobius inversion ORIGINAL inverse F(n/22) f(3) plus ORIGINAL b := sum(30, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/22) f(3) plus ORIGINAL b = 31
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S176
操作前
Mobius inversion ORIGINAL inverse F(n/22) f(6) ORIGINAL a times divisor := mul(10, 6)
操作後
Mobius inversion ORIGINAL inverse F(n/22) f(6) ORIGINAL a times divisor = 60
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S177
操作前
Mobius inversion ORIGINAL inverse F(n/22) f(6) plus ORIGINAL b := sum(60, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/22) f(6) plus ORIGINAL b = 61
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S178
操作前
Mobius inversion ORIGINAL inverse F(n/22) full forward divisor sum := sum(11, 21, 31, 61)
操作後
Mobius inversion ORIGINAL inverse F(n/22) full forward divisor sum = 124
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S179
操作前
Mobius inversion ORIGINAL inverse mu(22) times SAME F(n/22) := mul(1, 124)
操作後
Mobius inversion ORIGINAL inverse mu(22) times SAME F(n/22) = 124
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S180
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor33:actual positive input for mu=33
操作後
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
條件：無
先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。
規則：mobius_complete_UPF_input_role · 依賴：G001, S181
操作前
ORIGINAL positive integer absolute magnitude=33>1
操作後
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
條件：無
每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。
規則：upf_factorization_remaining_product_invariant · 依賴：G001, S182
操作前
next divisor=2;previous remaining=33;all smaller divisors exhausted
操作後
extracted multiplicity=0;next remaining=33;remaining modulo2=1;extracted multiset=[]
條件：無
除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。
規則：upf_complete_trial_division_remaining_invariant · 依賴：G001, S183
操作前
upf extract divisor3 occurrence0 := div(33, 3)
操作後
upf extract divisor3 occurrence0 = 11
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S184
操作前
next divisor=3;previous remaining=33;all smaller divisors exhausted
操作後
extracted multiplicity=1;next remaining=11;remaining modulo3=2;extracted multiset=[3]
條件：無
除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。
規則：upf_complete_trial_division_remaining_invariant · 依賴：G001, S185
操作前
upf canonical complete product factor0 := mul(1, 3)
操作後
upf canonical complete product factor0 = 3
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S186
操作前
upf canonical complete product factor1 := mul(3, 11)
操作後
upf canonical complete product factor1 = 33
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S187
操作前
exhausted trial divisors below4;terminal remaining=11;d*d>remaining;ORIGINAL n=33
操作後
complete increasing prime multiset=[3, 11];full product=33;a composite terminal remainder would have an already tested smaller divisor
條件：無
若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。
規則：upf_final_prime_remainder_complete_existence_instance · 依賴：G001, S188
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor33:input=33;COMPLETE prime multiset=[3, 11];distinct primes=[3, 11]
操作後
mu=1;empty product case=否;repeated prime=否
條件：無
1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。
規則：mobius_value_one_repeated_prime_or_squarefree_parity · 依賴：G001, S189
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor33 := div(132, 33)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor33 = 4
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S190
操作前
Mobius inversion ORIGINAL inverse F(n/33):actual positive input=4;check complete integer range1..4
操作後
ALL positive divisors INCREASING=[1, 2, 4];not only primes/proper divisors or all integers<=n
條件：無
逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。
規則：mobinv_ALL_positive_divisors_complete_indexing · 依賴：G001, S191
操作前
Mobius inversion ORIGINAL inverse F(n/33) f(1) ORIGINAL a times divisor := mul(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/33) f(1) ORIGINAL a times divisor = 10
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S192
操作前
Mobius inversion ORIGINAL inverse F(n/33) f(1) plus ORIGINAL b := sum(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/33) f(1) plus ORIGINAL b = 11
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S193
操作前
Mobius inversion ORIGINAL inverse F(n/33) f(2) ORIGINAL a times divisor := mul(10, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/33) f(2) ORIGINAL a times divisor = 20
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S194
操作前
Mobius inversion ORIGINAL inverse F(n/33) f(2) plus ORIGINAL b := sum(20, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/33) f(2) plus ORIGINAL b = 21
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S195
操作前
Mobius inversion ORIGINAL inverse F(n/33) f(4) ORIGINAL a times divisor := mul(10, 4)
操作後
Mobius inversion ORIGINAL inverse F(n/33) f(4) ORIGINAL a times divisor = 40
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S196
操作前
Mobius inversion ORIGINAL inverse F(n/33) f(4) plus ORIGINAL b := sum(40, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/33) f(4) plus ORIGINAL b = 41
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S197
操作前
Mobius inversion ORIGINAL inverse F(n/33) full forward divisor sum := sum(11, 21, 41)
操作後
Mobius inversion ORIGINAL inverse F(n/33) full forward divisor sum = 73
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S198
操作前
Mobius inversion ORIGINAL inverse mu(33) times SAME F(n/33) := mul(1, 73)
操作後
Mobius inversion ORIGINAL inverse mu(33) times SAME F(n/33) = 73
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S199
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor44:actual positive input for mu=44
操作後
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
條件：無
先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。
規則：mobius_complete_UPF_input_role · 依賴：G001, S200
操作前
ORIGINAL positive integer absolute magnitude=44>1
操作後
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
條件：無
每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。
規則：upf_factorization_remaining_product_invariant · 依賴：G001, S201
操作前
upf extract divisor2 occurrence0 := div(44, 2)
操作後
upf extract divisor2 occurrence0 = 22
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S202
操作前
upf extract divisor2 occurrence1 := div(22, 2)
操作後
upf extract divisor2 occurrence1 = 11
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S203
操作前
next divisor=2;previous remaining=44;all smaller divisors exhausted
操作後
extracted multiplicity=2;next remaining=11;remaining modulo2=1;extracted multiset=[2, 2]
條件：無
除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。
規則：upf_complete_trial_division_remaining_invariant · 依賴：G001, S204
操作前
next divisor=3;previous remaining=11;all smaller divisors exhausted
操作後
extracted multiplicity=0;next remaining=11;remaining modulo3=2;extracted multiset=[2, 2]
條件：無
除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。
規則：upf_complete_trial_division_remaining_invariant · 依賴：G001, S205
操作前
upf canonical complete product factor0 := mul(1, 2)
操作後
upf canonical complete product factor0 = 2
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S206
操作前
upf canonical complete product factor1 := mul(2, 2)
操作後
upf canonical complete product factor1 = 4
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S207
操作前
upf canonical complete product factor2 := mul(4, 11)
操作後
upf canonical complete product factor2 = 44
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S208
操作前
exhausted trial divisors below4;terminal remaining=11;d*d>remaining;ORIGINAL n=44
操作後
complete increasing prime multiset=[2, 2, 11];full product=44;a composite terminal remainder would have an already tested smaller divisor
條件：無
若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。
規則：upf_final_prime_remainder_complete_existence_instance · 依賴：G001, S209
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor44:input=44;COMPLETE prime multiset=[2, 2, 11];distinct primes=[2, 11]
操作後
mu=0;empty product case=否;repeated prime=是
條件：無
1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。
規則：mobius_value_one_repeated_prime_or_squarefree_parity · 依賴：G001, S210
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor44 := div(132, 44)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor44 = 3
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S211
操作前
Mobius inversion ORIGINAL inverse F(n/44):actual positive input=3;check complete integer range1..3
操作後
ALL positive divisors INCREASING=[1, 3];not only primes/proper divisors or all integers<=n
條件：無
逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。
規則：mobinv_ALL_positive_divisors_complete_indexing · 依賴：G001, S212
操作前
Mobius inversion ORIGINAL inverse F(n/44) f(1) ORIGINAL a times divisor := mul(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/44) f(1) ORIGINAL a times divisor = 10
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S213
操作前
Mobius inversion ORIGINAL inverse F(n/44) f(1) plus ORIGINAL b := sum(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/44) f(1) plus ORIGINAL b = 11
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S214
操作前
Mobius inversion ORIGINAL inverse F(n/44) f(3) ORIGINAL a times divisor := mul(10, 3)
操作後
Mobius inversion ORIGINAL inverse F(n/44) f(3) ORIGINAL a times divisor = 30
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S215
操作前
Mobius inversion ORIGINAL inverse F(n/44) f(3) plus ORIGINAL b := sum(30, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/44) f(3) plus ORIGINAL b = 31
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S216
操作前
Mobius inversion ORIGINAL inverse F(n/44) full forward divisor sum := sum(11, 31)
操作後
Mobius inversion ORIGINAL inverse F(n/44) full forward divisor sum = 42
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S217
操作前
Mobius inversion ORIGINAL inverse mu(44) times SAME F(n/44) := mul(0, 42)
操作後
Mobius inversion ORIGINAL inverse mu(44) times SAME F(n/44) = 0
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S218
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor66:actual positive input for mu=66
操作後
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
條件：無
先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。
規則：mobius_complete_UPF_input_role · 依賴：G001, S219
操作前
ORIGINAL positive integer absolute magnitude=66>1
操作後
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
條件：無
每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。
規則：upf_factorization_remaining_product_invariant · 依賴：G001, S220
操作前
upf extract divisor2 occurrence0 := div(66, 2)
操作後
upf extract divisor2 occurrence0 = 33
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S221
操作前
next divisor=2;previous remaining=66;all smaller divisors exhausted
操作後
extracted multiplicity=1;next remaining=33;remaining modulo2=1;extracted multiset=[2]
條件：無
除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。
規則：upf_complete_trial_division_remaining_invariant · 依賴：G001, S222
操作前
upf extract divisor3 occurrence1 := div(33, 3)
操作後
upf extract divisor3 occurrence1 = 11
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S223
操作前
next divisor=3;previous remaining=33;all smaller divisors exhausted
操作後
extracted multiplicity=1;next remaining=11;remaining modulo3=2;extracted multiset=[2, 3]
條件：無
除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。
規則：upf_complete_trial_division_remaining_invariant · 依賴：G001, S224
操作前
upf canonical complete product factor0 := mul(1, 2)
操作後
upf canonical complete product factor0 = 2
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S225
操作前
upf canonical complete product factor1 := mul(2, 3)
操作後
upf canonical complete product factor1 = 6
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S226
操作前
upf canonical complete product factor2 := mul(6, 11)
操作後
upf canonical complete product factor2 = 66
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S227
操作前
exhausted trial divisors below4;terminal remaining=11;d*d>remaining;ORIGINAL n=66
操作後
complete increasing prime multiset=[2, 3, 11];full product=66;a composite terminal remainder would have an already tested smaller divisor
條件：無
若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。
規則：upf_final_prime_remainder_complete_existence_instance · 依賴：G001, S228
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor66:input=66;COMPLETE prime multiset=[2, 3, 11];distinct primes=[2, 3, 11]
操作後
mu=-1;empty product case=否;repeated prime=否
條件：無
1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。
規則：mobius_value_one_repeated_prime_or_squarefree_parity · 依賴：G001, S229
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor66 := div(132, 66)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor66 = 2
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S230
操作前
Mobius inversion ORIGINAL inverse F(n/66):actual positive input=2;check complete integer range1..2
操作後
ALL positive divisors INCREASING=[1, 2];not only primes/proper divisors or all integers<=n
條件：無
逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。
規則：mobinv_ALL_positive_divisors_complete_indexing · 依賴：G001, S231
操作前
Mobius inversion ORIGINAL inverse F(n/66) f(1) ORIGINAL a times divisor := mul(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/66) f(1) ORIGINAL a times divisor = 10
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S232
操作前
Mobius inversion ORIGINAL inverse F(n/66) f(1) plus ORIGINAL b := sum(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/66) f(1) plus ORIGINAL b = 11
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S233
操作前
Mobius inversion ORIGINAL inverse F(n/66) f(2) ORIGINAL a times divisor := mul(10, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/66) f(2) ORIGINAL a times divisor = 20
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S234
操作前
Mobius inversion ORIGINAL inverse F(n/66) f(2) plus ORIGINAL b := sum(20, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/66) f(2) plus ORIGINAL b = 21
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S235
操作前
Mobius inversion ORIGINAL inverse F(n/66) full forward divisor sum := sum(11, 21)
操作後
Mobius inversion ORIGINAL inverse F(n/66) full forward divisor sum = 32
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S236
操作前
Mobius inversion ORIGINAL inverse mu(66) times SAME F(n/66) := mul(-1, 32)
操作後
Mobius inversion ORIGINAL inverse mu(66) times SAME F(n/66) = -32
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S237
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor132:actual positive input for mu=132
操作後
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
條件：無
先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。
規則：mobius_complete_UPF_input_role · 依賴：G001, S238
操作前
ORIGINAL positive integer absolute magnitude=132>1
操作後
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
條件：無
每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。
規則：upf_factorization_remaining_product_invariant · 依賴：G001, S239
操作前
upf extract divisor2 occurrence0 := div(132, 2)
操作後
upf extract divisor2 occurrence0 = 66
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S240
操作前
upf extract divisor2 occurrence1 := div(66, 2)
操作後
upf extract divisor2 occurrence1 = 33
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S241
操作前
next divisor=2;previous remaining=132;all smaller divisors exhausted
操作後
extracted multiplicity=2;next remaining=33;remaining modulo2=1;extracted multiset=[2, 2]
條件：無
除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。
規則：upf_complete_trial_division_remaining_invariant · 依賴：G001, S242
操作前
upf extract divisor3 occurrence2 := div(33, 3)
操作後
upf extract divisor3 occurrence2 = 11
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S243
操作前
next divisor=3;previous remaining=33;all smaller divisors exhausted
操作後
extracted multiplicity=1;next remaining=11;remaining modulo3=2;extracted multiset=[2, 2, 3]
條件：無
除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。
規則：upf_complete_trial_division_remaining_invariant · 依賴：G001, S244
操作前
upf canonical complete product factor0 := mul(1, 2)
操作後
upf canonical complete product factor0 = 2
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S245
操作前
upf canonical complete product factor1 := mul(2, 2)
操作後
upf canonical complete product factor1 = 4
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S246
操作前
upf canonical complete product factor2 := mul(4, 3)
操作後
upf canonical complete product factor2 = 12
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S247
操作前
upf canonical complete product factor3 := mul(12, 11)
操作後
upf canonical complete product factor3 = 132
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S248
操作前
exhausted trial divisors below4;terminal remaining=11;d*d>remaining;ORIGINAL n=132
操作後
complete increasing prime multiset=[2, 2, 3, 11];full product=132;a composite terminal remainder would have an already tested smaller divisor
條件：無
若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。
規則：upf_final_prime_remainder_complete_existence_instance · 依賴：G001, S249
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor132:input=132;COMPLETE prime multiset=[2, 2, 3, 11];distinct primes=[2, 3, 11]
操作後
mu=0;empty product case=否;repeated prime=是
條件：無
1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。
規則：mobius_value_one_repeated_prime_or_squarefree_parity · 依賴：G001, S250
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor132 := div(132, 132)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor132 = 1
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S251
操作前
Mobius inversion ORIGINAL inverse F(n/132):actual positive input=1;check complete integer range1..1
操作後
ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n
條件：無
逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。
規則：mobinv_ALL_positive_divisors_complete_indexing · 依賴：G001, S252
操作前
Mobius inversion ORIGINAL inverse F(n/132) f(1) ORIGINAL a times divisor := mul(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/132) f(1) ORIGINAL a times divisor = 10
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S253
操作前
Mobius inversion ORIGINAL inverse F(n/132) f(1) plus ORIGINAL b := sum(10, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/132) f(1) plus ORIGINAL b = 11
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S254
操作前
Mobius inversion ORIGINAL inverse F(n/132) full forward divisor sum := sum(11)
操作後
Mobius inversion ORIGINAL inverse F(n/132) full forward divisor sum = 11
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S255
操作前
Mobius inversion ORIGINAL inverse mu(132) times SAME F(n/132) := mul(0, 11)
操作後
Mobius inversion ORIGINAL inverse mu(132) times SAME F(n/132) = 0
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S256
操作前
Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(3372, -1448, -846, 0, 364, -286, 0, 124, 73, 0, -32, 0)
操作後
Mobius inversion ORIGINAL inverse complete inverse-term sum = 1321
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S257
操作前
Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(10, 132)
操作後
Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 1320
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S258
操作前
Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(1320, 1)
操作後
Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 1321
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S259
操作前
Mobius inversion ORIGINAL inverse:actual n=132;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 132, 3372, 3372], [2, -1, 66, 1448, -1448], [3, -1, 44, 846, -846], [4, 0, 33, 484, 0], [6, 1, 22, 364, 364], [11, -1, 12, 286, -286], [12, 0, 11, 122, 0], [22, 1, 6, 124, 124], [33, 1, 4, 73, 73], [44, 0, 3, 42, 0], [66, -1, 2, 32, -32], [132, 0, 1, 11, 0]]
操作後
complete reconstructed=1321;direct f(n)=1321;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)
條件：無
有限雙和可換序，d*e|n；內層mu和在n/e=1時為1，否則由不同prime子集合正負配對為0，故只留e=n。n=1也有d=1、mu1=1，不混另一等價公式的index。
規則：mobinv_general_finite_double_sum_and_Mobius_cancellation · 依賴：G001, S260
操作前
Mobius inversion ORIGINAL omitted-divisor partial sum := sum(-1448, -846, 0, 364, -286, 0, 124, 73, 0, -32, 0)
操作後
Mobius inversion ORIGINAL omitted-divisor partial sum = -2051
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S261
操作前
Mobius inversion complete minus partial := sub(1321, -2051)
操作後
Mobius inversion complete minus partial = 3372
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S262
操作前
ORIGINAL full divisors=[1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132]
操作後
complete=1321=f(n)=1321;partial=-2051;lost term=3372!=0;incomplete_certificate fixed No
條件：無
必須真漏一個divisor且其貢獻非零，使部分結果與目標不同；漏零項而恰巧相等不給本boundary證書。No拒絕不完整證書，不否認完整反演正確。
規則：mobinv_omission_actual_nonzero_discriminative_term · 依賴：G001, S263
S264 · 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。S263 · premise_and_result_audit — 以精確有理數重算 Mobius inversion complete minus partial，不以小數近似替代。S262 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL omitted-divisor partial sum，不以小數近似替代。S261 · 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。S260 · 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。S259 · 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。S258 · 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。S257 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(132) times SAME F(n/132)，不以小數近似替代。S256 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/132) full forward divisor sum，不以小數近似替代。S255 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/132) f(1) plus ORIGINAL b，不以小數近似替代。S254 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/132) f(1) ORIGINAL a times divisor，不以小數近似替代。S253 · 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。S252 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor132，不以小數近似替代。S251 · 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。S250 · 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。S249 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor3，不以小數近似替代。S248 · 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。S247 · 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。S246 · 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。S245 · 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。S244 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence2，不以小數近似替代。S243 · 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。S242 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。S241 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。S240 · 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。S239 · 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。S238 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(66) times SAME F(n/66)，不以小數近似替代。S237 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/66) full forward divisor sum，不以小數近似替代。S236 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/66) f(2) plus ORIGINAL b，不以小數近似替代。S235 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/66) f(2) ORIGINAL a times divisor，不以小數近似替代。S234 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/66) f(1) plus ORIGINAL b，不以小數近似替代。S233 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/66) f(1) ORIGINAL a times divisor，不以小數近似替代。S232 · 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。S231 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor66，不以小數近似替代。S230 · 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。S229 · 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。S228 · 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。S227 · 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。S226 · 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。S225 · 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。S224 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence1，不以小數近似替代。S223 · 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。S222 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。S221 · 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。S220 · 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。S219 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(44) times SAME F(n/44)，不以小數近似替代。S218 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/44) full forward divisor sum，不以小數近似替代。S217 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/44) f(3) plus ORIGINAL b，不以小數近似替代。S216 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/44) f(3) ORIGINAL a times divisor，不以小數近似替代。S215 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/44) f(1) plus ORIGINAL b，不以小數近似替代。S214 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/44) f(1) ORIGINAL a times divisor，不以小數近似替代。S213 · 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。S212 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor44，不以小數近似替代。S211 · 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。S210 · 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。S209 · 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。S208 · 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。S207 · 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。S206 · 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。S205 · 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。S204 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。S203 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。S202 · 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。S201 · 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。S200 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(33) times SAME F(n/33)，不以小數近似替代。S199 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/33) full forward divisor sum，不以小數近似替代。S198 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/33) f(4) plus ORIGINAL b，不以小數近似替代。S197 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/33) f(4) ORIGINAL a times divisor，不以小數近似替代。S196 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/33) f(2) plus ORIGINAL b，不以小數近似替代。S195 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/33) f(2) ORIGINAL a times divisor，不以小數近似替代。S194 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/33) f(1) plus ORIGINAL b，不以小數近似替代。S193 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/33) f(1) ORIGINAL a times divisor，不以小數近似替代。S192 · 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。S191 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor33，不以小數近似替代。S190 · 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。S189 · 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。S188 · 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。S187 · 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。S186 · 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。S185 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence0，不以小數近似替代。S184 · 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。S183 · 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。S182 · 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。S181 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(22) times SAME F(n/22)，不以小數近似替代。S180 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) full forward divisor sum，不以小數近似替代。S179 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(6) plus ORIGINAL b，不以小數近似替代。S178 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(6) ORIGINAL a times divisor，不以小數近似替代。S177 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(3) plus ORIGINAL b，不以小數近似替代。S176 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(3) ORIGINAL a times divisor，不以小數近似替代。S175 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(2) plus ORIGINAL b，不以小數近似替代。S174 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(2) ORIGINAL a times divisor，不以小數近似替代。S173 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(1) plus ORIGINAL b，不以小數近似替代。S172 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(1) ORIGINAL a times divisor，不以小數近似替代。S171 · 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。S170 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor22，不以小數近似替代。S169 · 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。S168 · 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。S167 · 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。S166 · 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。S165 · 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。S164 · 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。S163 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。S162 · 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。S161 · 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。S160 · 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。S159 · 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。S158 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/12) f(11) plus ORIGINAL b，不以小數近似替代。S157 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/12) f(11) ORIGINAL a times divisor，不以小數近似替代。S156 · 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。S155 · 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。S154 · 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。S153 · 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。S152 · 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。S151 · 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。S150 · 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。S149 · 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。S148 · 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。S147 · 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。S146 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。S145 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。S144 · 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。S143 · 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。S142 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(11) times SAME F(n/11)，不以小數近似替代。S141 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) full forward divisor sum，不以小數近似替代。S140 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(12) plus ORIGINAL b，不以小數近似替代。S139 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(12) ORIGINAL a times divisor，不以小數近似替代。S138 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(6) plus ORIGINAL b，不以小數近似替代。S137 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(6) ORIGINAL a times divisor，不以小數近似替代。S136 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(4) plus ORIGINAL b，不以小數近似替代。S135 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(4) ORIGINAL a times divisor，不以小數近似替代。S134 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(3) plus ORIGINAL b，不以小數近似替代。S133 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(3) ORIGINAL a times divisor，不以小數近似替代。S132 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(2) plus ORIGINAL b，不以小數近似替代。S131 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(2) ORIGINAL a times divisor，不以小數近似替代。S130 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(1) plus ORIGINAL b，不以小數近似替代。S129 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(1) ORIGINAL a times divisor，不以小數近似替代。S128 · 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。S127 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor11，不以小數近似替代。S126 · 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。S125 · 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。S124 · 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。S123 · 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。S122 · 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。S121 · 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。S120 · 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。S119 · 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。S118 · 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。S117 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(22) plus ORIGINAL b，不以小數近似替代。S116 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(22) ORIGINAL a times divisor，不以小數近似替代。S115 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(11) plus ORIGINAL b，不以小數近似替代。S114 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(11) ORIGINAL a times divisor，不以小數近似替代。S113 · 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。S112 · 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。S111 · 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。S110 · 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。S109 · 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。S108 · 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。S107 · 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。S106 · 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。S105 · 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。S104 · 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。S103 · 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。S102 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。S101 · 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。S100 · 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。S099 · 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。S098 · 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。S097 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(33) plus ORIGINAL b，不以小數近似替代。S096 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(33) ORIGINAL a times divisor，不以小數近似替代。S095 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(11) plus ORIGINAL b，不以小數近似替代。S094 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(11) ORIGINAL a times divisor，不以小數近似替代。S093 · 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。S092 · 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。S091 · 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。S090 · 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。S089 · 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。S088 · 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。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 upf canonical complete product factor1，不以小數近似替代。S084 · 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。S083 · 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。S082 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。S081 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。S080 · 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。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 mu(3) times SAME F(n/3)，不以小數近似替代。S077 · 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。S076 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(44) plus ORIGINAL b，不以小數近似替代。S075 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(44) ORIGINAL a times divisor，不以小數近似替代。S074 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(22) plus ORIGINAL b，不以小數近似替代。S073 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(22) ORIGINAL a times divisor，不以小數近似替代。S072 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(11) plus ORIGINAL b，不以小數近似替代。S071 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(11) ORIGINAL a times divisor，不以小數近似替代。S070 · 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。S069 · 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。S068 · 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。S067 · 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。S066 · 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。S065 · 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。S064 · 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。S063 · 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。S062 · 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。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 — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。S059 · 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。S058 · 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。S057 · 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。S056 · 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。S055 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(66) plus ORIGINAL b，不以小數近似替代。S054 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(66) ORIGINAL a times divisor，不以小數近似替代。S053 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(33) plus ORIGINAL b，不以小數近似替代。S052 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(33) ORIGINAL a times divisor，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/2) f(22) plus ORIGINAL b，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/2) f(22) ORIGINAL a times divisor，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/2) f(11) plus ORIGINAL b，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/2) f(11) ORIGINAL a times divisor，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/2) f(6) plus ORIGINAL b，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/2) f(6) ORIGINAL a times divisor，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/2) f(3) plus ORIGINAL b，不以小數近似替代。S044 · 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。S043 · 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。S042 · 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。S041 · 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。S040 · 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。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 — 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。S037 · 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。S036 · 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。S035 · 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。S034 · 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。S033 · 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。S032 · 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。S031 · 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。S030 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(132) plus ORIGINAL b，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/1) f(132) ORIGINAL a times divisor，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/1) f(66) plus ORIGINAL b，不以小數近似替代。S027 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(66) ORIGINAL a times divisor，不以小數近似替代。S026 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(44) plus ORIGINAL b，不以小數近似替代。S025 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(44) ORIGINAL a times divisor，不以小數近似替代。S024 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(33) plus ORIGINAL b，不以小數近似替代。S023 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(33) ORIGINAL a times divisor，不以小數近似替代。S022 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(22) plus ORIGINAL b，不以小數近似替代。S021 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(22) ORIGINAL a times divisor，不以小數近似替代。S020 · 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。S019 · 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。S018 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(11) 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(11) 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。(132, [1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132], 1, [2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132], 1321, -2051, 1321, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)
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: b829f82283a4fd24d0aa9a546fc9a109e5eda62ed55f708954993f12d090324dcandidate record SHA-256: f8bb695d735fd8001957dc16f3ece2e1ee37c7a0c09747b83d0cd73c80556262semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5eLean: NOT_RUN · human review: NOT_PERFORMED