R0043-22136 · q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0086
positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={"n":156,"a":11,"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":156,"a":11,"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=156;check complete integer range1..156
操作後
ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156];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(156, 1)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 156
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=156;check complete integer range1..156
操作後
ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156];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(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 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, S006
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = 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, S007
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(11, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 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, S008
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b := sum(22, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 23
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(11, 3)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor = 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, S010
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b := sum(33, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b = 34
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(11, 4)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor = 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, S012
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b := sum(44, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b = 45
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(11, 6)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor = 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, S014
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b := sum(66, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b = 67
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(12) ORIGINAL a times divisor := mul(11, 12)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(12) ORIGINAL a times divisor = 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, S016
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(12) plus ORIGINAL b := sum(132, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(12) plus ORIGINAL b = 133
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(13) ORIGINAL a times divisor := mul(11, 13)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(13) ORIGINAL a times divisor = 143
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(13) plus ORIGINAL b := sum(143, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(13) plus ORIGINAL b = 144
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(26) ORIGINAL a times divisor := mul(11, 26)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(26) ORIGINAL a times divisor = 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, S020
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(26) plus ORIGINAL b := sum(286, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(26) plus ORIGINAL b = 287
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(39) ORIGINAL a times divisor := mul(11, 39)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(39) ORIGINAL a times divisor = 429
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(39) plus ORIGINAL b := sum(429, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(39) plus ORIGINAL b = 430
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(52) ORIGINAL a times divisor := mul(11, 52)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(52) ORIGINAL a times divisor = 572
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(52) plus ORIGINAL b := sum(572, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(52) plus ORIGINAL b = 573
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(78) ORIGINAL a times divisor := mul(11, 78)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(78) ORIGINAL a times divisor = 858
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(78) plus ORIGINAL b := sum(858, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(78) plus ORIGINAL b = 859
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(156) ORIGINAL a times divisor := mul(11, 156)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(156) ORIGINAL a times divisor = 1716
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(156) plus ORIGINAL b := sum(1716, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(156) plus ORIGINAL b = 1717
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(12, 23, 34, 45, 67, 133, 144, 287, 430, 573, 859, 1717)
操作後
Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 4324
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, 4324)
操作後
Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 4324
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(156, 2)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 78
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=78;check complete integer range1..78
操作後
ALL positive divisors INCREASING=[1, 2, 3, 6, 13, 26, 39, 78];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(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 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, S039
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 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, S040
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor := mul(11, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor = 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, S041
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b := sum(22, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b = 23
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(11, 3)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor = 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, S043
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b := sum(33, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b = 34
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(11, 6)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(6) ORIGINAL a times divisor = 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, S045
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(6) plus ORIGINAL b := sum(66, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(6) plus ORIGINAL b = 67
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(13) ORIGINAL a times divisor := mul(11, 13)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(13) ORIGINAL a times divisor = 143
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(13) plus ORIGINAL b := sum(143, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(13) plus ORIGINAL b = 144
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(26) ORIGINAL a times divisor := mul(11, 26)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(26) ORIGINAL a times divisor = 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, S049
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(26) plus ORIGINAL b := sum(286, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(26) plus ORIGINAL b = 287
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(39) ORIGINAL a times divisor := mul(11, 39)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(39) ORIGINAL a times divisor = 429
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(39) plus ORIGINAL b := sum(429, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(39) plus ORIGINAL b = 430
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(78) ORIGINAL a times divisor := mul(11, 78)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(78) ORIGINAL a times divisor = 858
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(78) plus ORIGINAL b := sum(858, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(78) plus ORIGINAL b = 859
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(12, 23, 34, 67, 144, 287, 430, 859)
操作後
Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 1856
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, 1856)
操作後
Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -1856
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(156, 3)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 = 52
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=52;check complete integer range1..52
操作後
ALL positive divisors INCREASING=[1, 2, 4, 13, 26, 52];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(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor = 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, S064
操作前
Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b := sum(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b = 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, S065
操作前
Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor := mul(11, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor = 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, S066
操作前
Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b := sum(22, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b = 23
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(11, 4)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(4) ORIGINAL a times divisor = 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, S068
操作前
Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b := sum(44, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b = 45
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(13) ORIGINAL a times divisor := mul(11, 13)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(13) ORIGINAL a times divisor = 143
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(13) plus ORIGINAL b := sum(143, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(13) plus ORIGINAL b = 144
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(26) ORIGINAL a times divisor := mul(11, 26)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(26) ORIGINAL a times divisor = 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, S072
操作前
Mobius inversion ORIGINAL inverse F(n/3) f(26) plus ORIGINAL b := sum(286, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(26) plus ORIGINAL b = 287
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(52) ORIGINAL a times divisor := mul(11, 52)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(52) ORIGINAL a times divisor = 572
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(52) plus ORIGINAL b := sum(572, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/3) f(52) plus ORIGINAL b = 573
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(12, 23, 45, 144, 287, 573)
操作後
Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum = 1084
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, 1084)
操作後
Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) = -1084
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(156, 4)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 = 39
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=39;check complete integer range1..39
操作後
ALL positive divisors INCREASING=[1, 3, 13, 39];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(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor = 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, S089
操作前
Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b := sum(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b = 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, S090
操作前
Mobius inversion ORIGINAL inverse F(n/4) f(3) ORIGINAL a times divisor := mul(11, 3)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(3) ORIGINAL a times divisor = 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, S091
操作前
Mobius inversion ORIGINAL inverse F(n/4) f(3) plus ORIGINAL b := sum(33, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(3) plus ORIGINAL b = 34
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(13) ORIGINAL a times divisor := mul(11, 13)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(13) ORIGINAL a times divisor = 143
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(13) plus ORIGINAL b := sum(143, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(13) plus ORIGINAL b = 144
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(39) ORIGINAL a times divisor := mul(11, 39)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(39) ORIGINAL a times divisor = 429
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(39) plus ORIGINAL b := sum(429, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(39) plus ORIGINAL b = 430
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(12, 34, 144, 430)
操作後
Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum = 620
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, 620)
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(156, 6)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 = 26
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=26;check complete integer range1..26
操作後
ALL positive divisors INCREASING=[1, 2, 13, 26];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(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor = 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, S109
操作前
Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b := sum(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b = 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, S110
操作前
Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor := mul(11, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor = 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, S111
操作前
Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b := sum(22, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b = 23
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(13) ORIGINAL a times divisor := mul(11, 13)
操作後
Mobius inversion ORIGINAL inverse F(n/6) f(13) ORIGINAL a times divisor = 143
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(13) plus ORIGINAL b := sum(143, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/6) f(13) plus ORIGINAL b = 144
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(26) ORIGINAL a times divisor := mul(11, 26)
操作後
Mobius inversion ORIGINAL inverse F(n/6) f(26) ORIGINAL a times divisor = 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, S115
操作前
Mobius inversion ORIGINAL inverse F(n/6) f(26) plus ORIGINAL b := sum(286, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/6) f(26) plus ORIGINAL b = 287
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(12, 23, 144, 287)
操作後
Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum = 466
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, 466)
操作後
Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) = 466
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 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, S119
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, S120
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, S121
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, S122
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, S123
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, S124
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, S125
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, S126
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, S127
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, S128
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12 := div(156, 12)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12 = 13
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/12):actual positive input=13;check complete integer range1..13
操作後
ALL positive divisors INCREASING=[1, 13];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, S130
操作前
Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor := mul(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor = 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, S131
操作前
Mobius inversion ORIGINAL inverse F(n/12) f(1) plus ORIGINAL b := sum(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/12) f(1) plus ORIGINAL b = 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, S132
操作前
Mobius inversion ORIGINAL inverse F(n/12) f(13) ORIGINAL a times divisor := mul(11, 13)
操作後
Mobius inversion ORIGINAL inverse F(n/12) f(13) ORIGINAL a times divisor = 143
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/12) f(13) plus ORIGINAL b := sum(143, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/12) f(13) plus ORIGINAL b = 144
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/12) full forward divisor sum := sum(12, 144)
操作後
Mobius inversion ORIGINAL inverse F(n/12) full forward divisor sum = 156
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 mu(12) times SAME F(n/12) := mul(0, 156)
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, S136
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor13:actual positive input for mu=13
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, S137
操作前
ORIGINAL positive integer absolute magnitude=13>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, S138
操作前
next divisor=2;previous remaining=13;all smaller divisors exhausted
操作後
extracted multiplicity=0;next remaining=13;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, S139
操作前
next divisor=3;previous remaining=13;all smaller divisors exhausted
操作後
extracted multiplicity=0;next remaining=13;remaining modulo3=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, S140
操作前
upf canonical complete product factor0 := mul(1, 13)
操作後
upf canonical complete product factor0 = 13
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, S141
操作前
exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=13
操作後
complete increasing prime multiset=[13];full product=13;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, S142
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor13:input=13;COMPLETE prime multiset=[13];distinct primes=[13]
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, S143
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor13 := div(156, 13)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor13 = 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, S144
操作前
Mobius inversion ORIGINAL inverse F(n/13):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, S145
操作前
Mobius inversion ORIGINAL inverse F(n/13) f(1) ORIGINAL a times divisor := mul(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/13) f(1) ORIGINAL a times divisor = 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, S146
操作前
Mobius inversion ORIGINAL inverse F(n/13) f(1) plus ORIGINAL b := sum(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/13) f(1) plus ORIGINAL b = 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, S147
操作前
Mobius inversion ORIGINAL inverse F(n/13) f(2) ORIGINAL a times divisor := mul(11, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/13) f(2) ORIGINAL a times divisor = 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, S148
操作前
Mobius inversion ORIGINAL inverse F(n/13) f(2) plus ORIGINAL b := sum(22, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/13) f(2) plus ORIGINAL b = 23
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, S149
操作前
Mobius inversion ORIGINAL inverse F(n/13) f(3) ORIGINAL a times divisor := mul(11, 3)
操作後
Mobius inversion ORIGINAL inverse F(n/13) f(3) ORIGINAL a times divisor = 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, S150
操作前
Mobius inversion ORIGINAL inverse F(n/13) f(3) plus ORIGINAL b := sum(33, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/13) f(3) plus ORIGINAL b = 34
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, S151
操作前
Mobius inversion ORIGINAL inverse F(n/13) f(4) ORIGINAL a times divisor := mul(11, 4)
操作後
Mobius inversion ORIGINAL inverse F(n/13) f(4) ORIGINAL a times divisor = 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, S152
操作前
Mobius inversion ORIGINAL inverse F(n/13) f(4) plus ORIGINAL b := sum(44, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/13) f(4) plus ORIGINAL b = 45
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, S153
操作前
Mobius inversion ORIGINAL inverse F(n/13) f(6) ORIGINAL a times divisor := mul(11, 6)
操作後
Mobius inversion ORIGINAL inverse F(n/13) f(6) ORIGINAL a times divisor = 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, S154
操作前
Mobius inversion ORIGINAL inverse F(n/13) f(6) plus ORIGINAL b := sum(66, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/13) f(6) plus ORIGINAL b = 67
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/13) f(12) ORIGINAL a times divisor := mul(11, 12)
操作後
Mobius inversion ORIGINAL inverse F(n/13) f(12) ORIGINAL a times divisor = 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, S156
操作前
Mobius inversion ORIGINAL inverse F(n/13) f(12) plus ORIGINAL b := sum(132, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/13) f(12) plus ORIGINAL b = 133
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/13) full forward divisor sum := sum(12, 23, 34, 45, 67, 133)
操作後
Mobius inversion ORIGINAL inverse F(n/13) full forward divisor sum = 314
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(13) times SAME F(n/13) := mul(-1, 314)
操作後
Mobius inversion ORIGINAL inverse mu(13) times SAME F(n/13) = -314
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 divisor26:actual positive input for mu=26
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=26>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(26, 2)
操作後
upf extract divisor2 occurrence0 = 13
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=26;all smaller divisors exhausted
操作後
extracted multiplicity=1;next remaining=13;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=13;all smaller divisors exhausted
操作後
extracted multiplicity=0;next remaining=13;remaining modulo3=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, 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, 13)
操作後
upf canonical complete product factor1 = 26
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=13;d*d>remaining;ORIGINAL n=26
操作後
complete increasing prime multiset=[2, 13];full product=26;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 divisor26:input=26;COMPLETE prime multiset=[2, 13];distinct primes=[2, 13]
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 divisor26 := div(156, 26)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor26 = 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/26):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/26) f(1) ORIGINAL a times divisor := mul(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/26) f(1) ORIGINAL a times divisor = 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, S171
操作前
Mobius inversion ORIGINAL inverse F(n/26) f(1) plus ORIGINAL b := sum(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/26) f(1) plus ORIGINAL b = 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, S172
操作前
Mobius inversion ORIGINAL inverse F(n/26) f(2) ORIGINAL a times divisor := mul(11, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/26) f(2) ORIGINAL a times divisor = 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, S173
操作前
Mobius inversion ORIGINAL inverse F(n/26) f(2) plus ORIGINAL b := sum(22, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/26) f(2) plus ORIGINAL b = 23
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/26) f(3) ORIGINAL a times divisor := mul(11, 3)
操作後
Mobius inversion ORIGINAL inverse F(n/26) f(3) ORIGINAL a times divisor = 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, S175
操作前
Mobius inversion ORIGINAL inverse F(n/26) f(3) plus ORIGINAL b := sum(33, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/26) f(3) plus ORIGINAL b = 34
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/26) f(6) ORIGINAL a times divisor := mul(11, 6)
操作後
Mobius inversion ORIGINAL inverse F(n/26) f(6) ORIGINAL a times divisor = 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, S177
操作前
Mobius inversion ORIGINAL inverse F(n/26) f(6) plus ORIGINAL b := sum(66, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/26) f(6) plus ORIGINAL b = 67
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/26) full forward divisor sum := sum(12, 23, 34, 67)
操作後
Mobius inversion ORIGINAL inverse F(n/26) full forward divisor sum = 136
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(26) times SAME F(n/26) := mul(1, 136)
操作後
Mobius inversion ORIGINAL inverse mu(26) times SAME F(n/26) = 136
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 divisor39:actual positive input for mu=39
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=39>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=39;all smaller divisors exhausted
操作後
extracted multiplicity=0;next remaining=39;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(39, 3)
操作後
upf extract divisor3 occurrence0 = 13
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=39;all smaller divisors exhausted
操作後
extracted multiplicity=1;next remaining=13;remaining modulo3=1;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, 13)
操作後
upf canonical complete product factor1 = 39
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=13;d*d>remaining;ORIGINAL n=39
操作後
complete increasing prime multiset=[3, 13];full product=39;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 divisor39:input=39;COMPLETE prime multiset=[3, 13];distinct primes=[3, 13]
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 divisor39 := div(156, 39)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor39 = 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/39):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/39) f(1) ORIGINAL a times divisor := mul(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/39) f(1) ORIGINAL a times divisor = 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, S192
操作前
Mobius inversion ORIGINAL inverse F(n/39) f(1) plus ORIGINAL b := sum(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/39) f(1) plus ORIGINAL b = 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, S193
操作前
Mobius inversion ORIGINAL inverse F(n/39) f(2) ORIGINAL a times divisor := mul(11, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/39) f(2) ORIGINAL a times divisor = 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, S194
操作前
Mobius inversion ORIGINAL inverse F(n/39) f(2) plus ORIGINAL b := sum(22, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/39) f(2) plus ORIGINAL b = 23
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/39) f(4) ORIGINAL a times divisor := mul(11, 4)
操作後
Mobius inversion ORIGINAL inverse F(n/39) f(4) ORIGINAL a times divisor = 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, S196
操作前
Mobius inversion ORIGINAL inverse F(n/39) f(4) plus ORIGINAL b := sum(44, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/39) f(4) plus ORIGINAL b = 45
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/39) full forward divisor sum := sum(12, 23, 45)
操作後
Mobius inversion ORIGINAL inverse F(n/39) full forward divisor sum = 80
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(39) times SAME F(n/39) := mul(1, 80)
操作後
Mobius inversion ORIGINAL inverse mu(39) times SAME F(n/39) = 80
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 divisor52:actual positive input for mu=52
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=52>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(52, 2)
操作後
upf extract divisor2 occurrence0 = 26
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(26, 2)
操作後
upf extract divisor2 occurrence1 = 13
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=52;all smaller divisors exhausted
操作後
extracted multiplicity=2;next remaining=13;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=13;all smaller divisors exhausted
操作後
extracted multiplicity=0;next remaining=13;remaining modulo3=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, 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, 13)
操作後
upf canonical complete product factor2 = 52
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=13;d*d>remaining;ORIGINAL n=52
操作後
complete increasing prime multiset=[2, 2, 13];full product=52;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 divisor52:input=52;COMPLETE prime multiset=[2, 2, 13];distinct primes=[2, 13]
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 divisor52 := div(156, 52)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor52 = 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/52):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/52) f(1) ORIGINAL a times divisor := mul(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/52) f(1) ORIGINAL a times divisor = 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, S213
操作前
Mobius inversion ORIGINAL inverse F(n/52) f(1) plus ORIGINAL b := sum(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/52) f(1) plus ORIGINAL b = 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, S214
操作前
Mobius inversion ORIGINAL inverse F(n/52) f(3) ORIGINAL a times divisor := mul(11, 3)
操作後
Mobius inversion ORIGINAL inverse F(n/52) f(3) ORIGINAL a times divisor = 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, S215
操作前
Mobius inversion ORIGINAL inverse F(n/52) f(3) plus ORIGINAL b := sum(33, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/52) f(3) plus ORIGINAL b = 34
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/52) full forward divisor sum := sum(12, 34)
操作後
Mobius inversion ORIGINAL inverse F(n/52) full forward divisor sum = 46
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(52) times SAME F(n/52) := mul(0, 46)
操作後
Mobius inversion ORIGINAL inverse mu(52) times SAME F(n/52) = 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 divisor78:actual positive input for mu=78
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=78>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(78, 2)
操作後
upf extract divisor2 occurrence0 = 39
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=78;all smaller divisors exhausted
操作後
extracted multiplicity=1;next remaining=39;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(39, 3)
操作後
upf extract divisor3 occurrence1 = 13
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=39;all smaller divisors exhausted
操作後
extracted multiplicity=1;next remaining=13;remaining modulo3=1;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, 13)
操作後
upf canonical complete product factor2 = 78
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=13;d*d>remaining;ORIGINAL n=78
操作後
complete increasing prime multiset=[2, 3, 13];full product=78;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 divisor78:input=78;COMPLETE prime multiset=[2, 3, 13];distinct primes=[2, 3, 13]
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 divisor78 := div(156, 78)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor78 = 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/78):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/78) f(1) ORIGINAL a times divisor := mul(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/78) f(1) ORIGINAL a times divisor = 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, S232
操作前
Mobius inversion ORIGINAL inverse F(n/78) f(1) plus ORIGINAL b := sum(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/78) f(1) plus ORIGINAL b = 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, S233
操作前
Mobius inversion ORIGINAL inverse F(n/78) f(2) ORIGINAL a times divisor := mul(11, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/78) f(2) ORIGINAL a times divisor = 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, S234
操作前
Mobius inversion ORIGINAL inverse F(n/78) f(2) plus ORIGINAL b := sum(22, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/78) f(2) plus ORIGINAL b = 23
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/78) full forward divisor sum := sum(12, 23)
操作後
Mobius inversion ORIGINAL inverse F(n/78) full forward divisor sum = 35
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(78) times SAME F(n/78) := mul(-1, 35)
操作後
Mobius inversion ORIGINAL inverse mu(78) times SAME F(n/78) = -35
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 divisor156:actual positive input for mu=156
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=156>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(156, 2)
操作後
upf extract divisor2 occurrence0 = 78
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(78, 2)
操作後
upf extract divisor2 occurrence1 = 39
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=156;all smaller divisors exhausted
操作後
extracted multiplicity=2;next remaining=39;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(39, 3)
操作後
upf extract divisor3 occurrence2 = 13
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=39;all smaller divisors exhausted
操作後
extracted multiplicity=1;next remaining=13;remaining modulo3=1;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, 13)
操作後
upf canonical complete product factor3 = 156
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=13;d*d>remaining;ORIGINAL n=156
操作後
complete increasing prime multiset=[2, 2, 3, 13];full product=156;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 divisor156:input=156;COMPLETE prime multiset=[2, 2, 3, 13];distinct primes=[2, 3, 13]
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 divisor156 := div(156, 156)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor156 = 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/156):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/156) f(1) ORIGINAL a times divisor := mul(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/156) f(1) ORIGINAL a times divisor = 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, S253
操作前
Mobius inversion ORIGINAL inverse F(n/156) f(1) plus ORIGINAL b := sum(11, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/156) f(1) plus ORIGINAL b = 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, S254
操作前
Mobius inversion ORIGINAL inverse F(n/156) full forward divisor sum := sum(12)
操作後
Mobius inversion ORIGINAL inverse F(n/156) full forward divisor sum = 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, S255
操作前
Mobius inversion ORIGINAL inverse mu(156) times SAME F(n/156) := mul(0, 12)
操作後
Mobius inversion ORIGINAL inverse mu(156) times SAME F(n/156) = 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(4324, -1856, -1084, 0, 466, 0, -314, 136, 80, 0, -35, 0)
操作後
Mobius inversion ORIGINAL inverse complete inverse-term sum = 1717
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(11, 156)
操作後
Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 1716
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(1716, 1)
操作後
Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 1717
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=156;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 156, 4324, 4324], [2, -1, 78, 1856, -1856], [3, -1, 52, 1084, -1084], [4, 0, 39, 620, 0], [6, 1, 26, 466, 466], [12, 0, 13, 156, 0], [13, -1, 12, 314, -314], [26, 1, 6, 136, 136], [39, 1, 4, 80, 80], [52, 0, 3, 46, 0], [78, -1, 2, 35, -35], [156, 0, 1, 12, 0]]
操作後
complete reconstructed=1717;direct f(n)=1717;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(-1856, -1084, 0, 466, 0, -314, 136, 80, 0, -35, 0)
操作後
Mobius inversion ORIGINAL omitted-divisor partial sum = -2607
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(1717, -2607)
操作後
Mobius inversion complete minus partial = 4324
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, 12, 13, 26, 39, 52, 78, 156];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156]
操作後
complete=1717=f(n)=1717;partial=-2607;lost term=4324!=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(156) times SAME F(n/156)，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/156) 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/156) 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/156) 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 divisor156，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(78) times SAME F(n/78)，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/78) 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/78) 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/78) 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/78) 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/78) 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 divisor78，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(52) times SAME F(n/52)，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/52) 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/52) 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/52) 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/52) 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/52) 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 divisor52，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(39) times SAME F(n/39)，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/39) 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/39) 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/39) 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/39) 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/39) 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/39) 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/39) 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 divisor39，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(26) times SAME F(n/26)，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/26) 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/26) 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/26) 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/26) 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/26) 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/26) 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/26) 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/26) 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/26) 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 divisor26，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(13) times SAME F(n/13)，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/13) 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/13) f(12) 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/13) f(12) 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/13) f(6) 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/13) f(6) ORIGINAL a times divisor，不以小數近似替代。S154 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(4) plus ORIGINAL b，不以小數近似替代。S153 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(4) ORIGINAL a times divisor，不以小數近似替代。S152 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(3) plus ORIGINAL b，不以小數近似替代。S151 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(3) ORIGINAL a times divisor，不以小數近似替代。S150 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(2) plus ORIGINAL b，不以小數近似替代。S149 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(2) ORIGINAL a times divisor，不以小數近似替代。S148 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(1) plus ORIGINAL b，不以小數近似替代。S147 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(1) ORIGINAL a times divisor，不以小數近似替代。S146 · 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。S145 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor13，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 upf canonical complete product factor0，不以小數近似替代。S141 · 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。S140 · 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。S139 · 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。S138 · 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。S137 · 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。S136 · 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。S135 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/12) f(13) plus ORIGINAL b，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/12) f(13) ORIGINAL a times divisor，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/12) f(1) plus ORIGINAL b，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/12) f(1) ORIGINAL a times divisor，不以小數近似替代。S131 · 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。S130 · 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。S129 · 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。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 upf canonical complete product factor2，不以小數近似替代。S126 · 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。S125 · 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。S124 · 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。S123 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。S122 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。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(26) 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(26) 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(13) 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(13) 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(39) 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(39) 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(13) 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(13) 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(52) 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(52) 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(26) 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(26) 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(13) 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(13) 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(78) 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(78) 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(39) 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(39) 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(26) 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(26) 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(13) 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(13) 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(156) 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(156) 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(78) 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(78) 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(52) 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(52) 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(39) 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(39) 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(26) 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(26) 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(13) 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(13) 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(12) plus ORIGINAL b，不以小數近似替代。S017 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(12) ORIGINAL a times divisor，不以小數近似替代。S016 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b，不以小數近似替代。S015 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor，不以小數近似替代。S014 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b，不以小數近似替代。S013 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor，不以小數近似替代。S012 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b，不以小數近似替代。S011 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor，不以小數近似替代。S010 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b，不以小數近似替代。S009 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor，不以小數近似替代。S008 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b，不以小數近似替代。S007 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor，不以小數近似替代。S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S005 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1，不以小數近似替代。S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。(156, [1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156], 1, [2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156], 1717, -2607, 1717, 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: 22e9a31cde649814099c983e46ea8450f59cef468fe9be5f15c0624508c87356candidate record SHA-256: 9b1785cd9bbedaf970e26496670bf48deef37ec249c13c7aade552bc16f4f40bsemantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5eLean: NOT_RUN · human review: NOT_PERFORMED