R0043-22131 · q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0081
positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={"n":56,"a":6,"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":56,"a":6,"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=56;check complete integer range1..56
操作後
ALL positive divisors INCREASING=[1, 2, 4, 7, 8, 14, 28, 56];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(56, 1)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 56
條件：無
依原始角色、次序與已核中間量精確代值；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=56;check complete integer range1..56
操作後
ALL positive divisors INCREASING=[1, 2, 4, 7, 8, 14, 28, 56];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(6, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 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, S006
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(6, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = 7
條件：無
依原始角色、次序與已核中間量精確代值；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(6, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 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, S008
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b := sum(12, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 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, S009
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor := mul(6, 4)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor = 24
條件：無
依原始角色、次序與已核中間量精確代值；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(4) plus ORIGINAL b := sum(24, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b = 25
條件：無
依原始角色、次序與已核中間量精確代值；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(7) ORIGINAL a times divisor := mul(6, 7)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(7) ORIGINAL a times divisor = 42
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S012
操作前
Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b := sum(42, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b = 43
條件：無
依原始角色、次序與已核中間量精確代值；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(8) ORIGINAL a times divisor := mul(6, 8)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(8) ORIGINAL a times divisor = 48
條件：無
依原始角色、次序與已核中間量精確代值；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(8) plus ORIGINAL b := sum(48, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(8) plus ORIGINAL b = 49
條件：無
依原始角色、次序與已核中間量精確代值；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(14) ORIGINAL a times divisor := mul(6, 14)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(14) ORIGINAL a times divisor = 84
條件：無
依原始角色、次序與已核中間量精確代值；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(14) plus ORIGINAL b := sum(84, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(14) plus ORIGINAL b = 85
條件：無
依原始角色、次序與已核中間量精確代值；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(28) ORIGINAL a times divisor := mul(6, 28)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(28) ORIGINAL a times divisor = 168
條件：無
依原始角色、次序與已核中間量精確代值；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(28) plus ORIGINAL b := sum(168, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(28) plus ORIGINAL b = 169
條件：無
依原始角色、次序與已核中間量精確代值；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(56) ORIGINAL a times divisor := mul(6, 56)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(56) ORIGINAL a times divisor = 336
條件：無
依原始角色、次序與已核中間量精確代值；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(56) plus ORIGINAL b := sum(336, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/1) f(56) plus ORIGINAL b = 337
條件：無
依原始角色、次序與已核中間量精確代值；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) full forward divisor sum := sum(7, 13, 25, 43, 49, 85, 169, 337)
操作後
Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 728
條件：無
依原始角色、次序與已核中間量精確代值；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 mu(1) times SAME F(n/1) := mul(1, 728)
操作後
Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 728
條件：無
依原始角色、次序與已核中間量精確代值；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 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, S024
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, S025
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, S026
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, S027
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, S028
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 := div(56, 2)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 28
條件：無
依原始角色、次序與已核中間量精確代值；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/2):actual positive input=28;check complete integer range1..28
操作後
ALL positive divisors INCREASING=[1, 2, 4, 7, 14, 28];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, S030
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(6, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 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, S031
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(6, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 7
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S032
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor := mul(6, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor = 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, S033
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b := sum(12, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b = 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, S034
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(4) ORIGINAL a times divisor := mul(6, 4)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(4) ORIGINAL a times divisor = 24
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S035
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(4) plus ORIGINAL b := sum(24, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(4) plus ORIGINAL b = 25
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S036
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor := mul(6, 7)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor = 42
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S037
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b := sum(42, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b = 43
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S038
操作前
Mobius inversion ORIGINAL inverse F(n/2) f(14) ORIGINAL a times divisor := mul(6, 14)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(14) ORIGINAL a times divisor = 84
條件：無
依原始角色、次序與已核中間量精確代值；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(14) plus ORIGINAL b := sum(84, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(14) plus ORIGINAL b = 85
條件：無
依原始角色、次序與已核中間量精確代值；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(28) ORIGINAL a times divisor := mul(6, 28)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(28) ORIGINAL a times divisor = 168
條件：無
依原始角色、次序與已核中間量精確代值；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(28) plus ORIGINAL b := sum(168, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/2) f(28) plus ORIGINAL b = 169
條件：無
依原始角色、次序與已核中間量精確代值；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) full forward divisor sum := sum(7, 13, 25, 43, 85, 169)
操作後
Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 342
條件：無
依原始角色、次序與已核中間量精確代值；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 mu(2) times SAME F(n/2) := mul(-1, 342)
操作後
Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -342
條件：無
依原始角色、次序與已核中間量精確代值；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 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, S045
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, S046
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, S047
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, S048
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, S049
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, S050
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, S051
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, S052
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, S053
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 := div(56, 4)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 = 14
條件：無
依原始角色、次序與已核中間量精確代值；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/4):actual positive input=14;check complete integer range1..14
操作後
ALL positive divisors INCREASING=[1, 2, 7, 14];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, S055
操作前
Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor := mul(6, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor = 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, S056
操作前
Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b := sum(6, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b = 7
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S057
操作前
Mobius inversion ORIGINAL inverse F(n/4) f(2) ORIGINAL a times divisor := mul(6, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(2) ORIGINAL a times divisor = 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, S058
操作前
Mobius inversion ORIGINAL inverse F(n/4) f(2) plus ORIGINAL b := sum(12, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(2) plus ORIGINAL b = 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, S059
操作前
Mobius inversion ORIGINAL inverse F(n/4) f(7) ORIGINAL a times divisor := mul(6, 7)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(7) ORIGINAL a times divisor = 42
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S060
操作前
Mobius inversion ORIGINAL inverse F(n/4) f(7) plus ORIGINAL b := sum(42, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(7) plus ORIGINAL b = 43
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S061
操作前
Mobius inversion ORIGINAL inverse F(n/4) f(14) ORIGINAL a times divisor := mul(6, 14)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(14) ORIGINAL a times divisor = 84
條件：無
依原始角色、次序與已核中間量精確代值；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/4) f(14) plus ORIGINAL b := sum(84, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/4) f(14) plus ORIGINAL b = 85
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S063
操作前
Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum := sum(7, 13, 43, 85)
操作後
Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum = 148
條件：無
依原始角色、次序與已核中間量精確代值；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 mu(4) times SAME F(n/4) := mul(0, 148)
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, S065
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor7:actual positive input for mu=7
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, S066
操作前
ORIGINAL positive integer absolute magnitude=7>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, S067
操作前
next divisor=2;previous remaining=7;all smaller divisors exhausted
操作後
extracted multiplicity=0;next remaining=7;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, S068
操作前
upf canonical complete product factor0 := mul(1, 7)
操作後
upf canonical complete product factor0 = 7
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, S069
操作前
exhausted trial divisors below3;terminal remaining=7;d*d>remaining;ORIGINAL n=7
操作後
complete increasing prime multiset=[7];full product=7;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, S070
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor7:input=7;COMPLETE prime multiset=[7];distinct primes=[7]
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, S071
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 := div(56, 7)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 = 8
條件：無
依原始角色、次序與已核中間量精確代值；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/7):actual positive input=8;check complete integer range1..8
操作後
ALL positive divisors INCREASING=[1, 2, 4, 8];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, S073
操作前
Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor := mul(6, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor = 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, S074
操作前
Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b := sum(6, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b = 7
條件：無
依原始角色、次序與已核中間量精確代值；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/7) f(2) ORIGINAL a times divisor := mul(6, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor = 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, S076
操作前
Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b := sum(12, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b = 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, S077
操作前
Mobius inversion ORIGINAL inverse F(n/7) f(4) ORIGINAL a times divisor := mul(6, 4)
操作後
Mobius inversion ORIGINAL inverse F(n/7) f(4) ORIGINAL a times divisor = 24
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S078
操作前
Mobius inversion ORIGINAL inverse F(n/7) f(4) plus ORIGINAL b := sum(24, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/7) f(4) plus ORIGINAL b = 25
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S079
操作前
Mobius inversion ORIGINAL inverse F(n/7) f(8) ORIGINAL a times divisor := mul(6, 8)
操作後
Mobius inversion ORIGINAL inverse F(n/7) f(8) ORIGINAL a times divisor = 48
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S080
操作前
Mobius inversion ORIGINAL inverse F(n/7) f(8) plus ORIGINAL b := sum(48, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/7) f(8) plus ORIGINAL b = 49
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S081
操作前
Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum := sum(7, 13, 25, 49)
操作後
Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum = 94
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S082
操作前
Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) := mul(-1, 94)
操作後
Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) = -94
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S083
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor8:actual positive input for mu=8
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, S084
操作前
ORIGINAL positive integer absolute magnitude=8>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, S085
操作前
upf extract divisor2 occurrence0 := div(8, 2)
操作後
upf extract divisor2 occurrence0 = 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, S086
操作前
upf extract divisor2 occurrence1 := div(4, 2)
操作後
upf extract divisor2 occurrence1 = 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, S087
操作前
upf extract divisor2 occurrence2 := div(2, 2)
操作後
upf extract divisor2 occurrence2 = 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, S088
操作前
next divisor=2;previous remaining=8;all smaller divisors exhausted
操作後
extracted multiplicity=3;next remaining=1;remaining modulo2=1;extracted multiset=[2, 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, S089
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, S090
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, S091
操作前
upf canonical complete product factor2 := mul(4, 2)
操作後
upf canonical complete product factor2 = 8
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, S092
操作前
exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=8
操作後
complete increasing prime multiset=[2, 2, 2];full product=8;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, S093
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor8:input=8;COMPLETE prime multiset=[2, 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, S094
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor8 := div(56, 8)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor8 = 7
條件：無
依原始角色、次序與已核中間量精確代值；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/8):actual positive input=7;check complete integer range1..7
操作後
ALL positive divisors INCREASING=[1, 7];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, S096
操作前
Mobius inversion ORIGINAL inverse F(n/8) f(1) ORIGINAL a times divisor := mul(6, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/8) f(1) ORIGINAL a times divisor = 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, S097
操作前
Mobius inversion ORIGINAL inverse F(n/8) f(1) plus ORIGINAL b := sum(6, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/8) f(1) plus ORIGINAL b = 7
條件：無
依原始角色、次序與已核中間量精確代值；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 F(n/8) f(7) ORIGINAL a times divisor := mul(6, 7)
操作後
Mobius inversion ORIGINAL inverse F(n/8) f(7) ORIGINAL a times divisor = 42
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S099
操作前
Mobius inversion ORIGINAL inverse F(n/8) f(7) plus ORIGINAL b := sum(42, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/8) f(7) plus ORIGINAL b = 43
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S100
操作前
Mobius inversion ORIGINAL inverse F(n/8) full forward divisor sum := sum(7, 43)
操作後
Mobius inversion ORIGINAL inverse F(n/8) full forward divisor sum = 50
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S101
操作前
Mobius inversion ORIGINAL inverse mu(8) times SAME F(n/8) := mul(0, 50)
操作後
Mobius inversion ORIGINAL inverse mu(8) times SAME F(n/8) = 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, S102
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor14:actual positive input for mu=14
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, S103
操作前
ORIGINAL positive integer absolute magnitude=14>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, S104
操作前
upf extract divisor2 occurrence0 := div(14, 2)
操作後
upf extract divisor2 occurrence0 = 7
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, S105
操作前
next divisor=2;previous remaining=14;all smaller divisors exhausted
操作後
extracted multiplicity=1;next remaining=7;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, S106
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, S107
操作前
upf canonical complete product factor1 := mul(2, 7)
操作後
upf canonical complete product factor1 = 14
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, S108
操作前
exhausted trial divisors below3;terminal remaining=7;d*d>remaining;ORIGINAL n=14
操作後
complete increasing prime multiset=[2, 7];full product=14;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, S109
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor14:input=14;COMPLETE prime multiset=[2, 7];distinct primes=[2, 7]
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, S110
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 := div(56, 14)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 = 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, S111
操作前
Mobius inversion ORIGINAL inverse F(n/14):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, S112
操作前
Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor := mul(6, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor = 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, S113
操作前
Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b := sum(6, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b = 7
條件：無
依原始角色、次序與已核中間量精確代值；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/14) f(2) ORIGINAL a times divisor := mul(6, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/14) f(2) ORIGINAL a times divisor = 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, S115
操作前
Mobius inversion ORIGINAL inverse F(n/14) f(2) plus ORIGINAL b := sum(12, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/14) f(2) plus ORIGINAL b = 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, S116
操作前
Mobius inversion ORIGINAL inverse F(n/14) f(4) ORIGINAL a times divisor := mul(6, 4)
操作後
Mobius inversion ORIGINAL inverse F(n/14) f(4) ORIGINAL a times divisor = 24
條件：無
依原始角色、次序與已核中間量精確代值；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 F(n/14) f(4) plus ORIGINAL b := sum(24, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/14) f(4) plus ORIGINAL b = 25
條件：無
依原始角色、次序與已核中間量精確代值；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 F(n/14) full forward divisor sum := sum(7, 13, 25)
操作後
Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum = 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, S119
操作前
Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) := mul(1, 45)
操作後
Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) = 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, S120
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor28:actual positive input for mu=28
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, S121
操作前
ORIGINAL positive integer absolute magnitude=28>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, S122
操作前
upf extract divisor2 occurrence0 := div(28, 2)
操作後
upf extract divisor2 occurrence0 = 14
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S123
操作前
upf extract divisor2 occurrence1 := div(14, 2)
操作後
upf extract divisor2 occurrence1 = 7
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
next divisor=2;previous remaining=28;all smaller divisors exhausted
操作後
extracted multiplicity=2;next remaining=7;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, S125
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, S126
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, S127
操作前
upf canonical complete product factor2 := mul(4, 7)
操作後
upf canonical complete product factor2 = 28
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, S128
操作前
exhausted trial divisors below3;terminal remaining=7;d*d>remaining;ORIGINAL n=28
操作後
complete increasing prime multiset=[2, 2, 7];full product=28;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, S129
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor28:input=28;COMPLETE prime multiset=[2, 2, 7];distinct primes=[2, 7]
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, S130
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor28 := div(56, 28)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor28 = 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, S131
操作前
Mobius inversion ORIGINAL inverse F(n/28):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, S132
操作前
Mobius inversion ORIGINAL inverse F(n/28) f(1) ORIGINAL a times divisor := mul(6, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/28) f(1) ORIGINAL a times divisor = 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, S133
操作前
Mobius inversion ORIGINAL inverse F(n/28) f(1) plus ORIGINAL b := sum(6, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/28) f(1) plus ORIGINAL b = 7
條件：無
依原始角色、次序與已核中間量精確代值；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/28) f(2) ORIGINAL a times divisor := mul(6, 2)
操作後
Mobius inversion ORIGINAL inverse F(n/28) f(2) ORIGINAL a times divisor = 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, S135
操作前
Mobius inversion ORIGINAL inverse F(n/28) f(2) plus ORIGINAL b := sum(12, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/28) f(2) plus ORIGINAL b = 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, S136
操作前
Mobius inversion ORIGINAL inverse F(n/28) full forward divisor sum := sum(7, 13)
操作後
Mobius inversion ORIGINAL inverse F(n/28) full forward divisor sum = 20
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S137
操作前
Mobius inversion ORIGINAL inverse mu(28) times SAME F(n/28) := mul(0, 20)
操作後
Mobius inversion ORIGINAL inverse mu(28) times SAME F(n/28) = 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, S138
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor56:actual positive input for mu=56
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, S139
操作前
ORIGINAL positive integer absolute magnitude=56>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, S140
操作前
upf extract divisor2 occurrence0 := div(56, 2)
操作後
upf extract divisor2 occurrence0 = 28
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
upf extract divisor2 occurrence1 := div(28, 2)
操作後
upf extract divisor2 occurrence1 = 14
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, S142
操作前
upf extract divisor2 occurrence2 := div(14, 2)
操作後
upf extract divisor2 occurrence2 = 7
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, S143
操作前
next divisor=2;previous remaining=56;all smaller divisors exhausted
操作後
extracted multiplicity=3;next remaining=7;remaining modulo2=1;extracted multiset=[2, 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, S144
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, S145
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, S146
操作前
upf canonical complete product factor2 := mul(4, 2)
操作後
upf canonical complete product factor2 = 8
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S147
操作前
upf canonical complete product factor3 := mul(8, 7)
操作後
upf canonical complete product factor3 = 56
條件：無
依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。
規則：exact_rational_arithmetic · 依賴：G001, S148
操作前
exhausted trial divisors below3;terminal remaining=7;d*d>remaining;ORIGINAL n=56
操作後
complete increasing prime multiset=[2, 2, 2, 7];full product=56;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, S149
操作前
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor56:input=56;COMPLETE prime multiset=[2, 2, 2, 7];distinct primes=[2, 7]
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, S150
操作前
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor56 := div(56, 56)
操作後
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor56 = 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, S151
操作前
Mobius inversion ORIGINAL inverse F(n/56):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, S152
操作前
Mobius inversion ORIGINAL inverse F(n/56) f(1) ORIGINAL a times divisor := mul(6, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/56) f(1) ORIGINAL a times divisor = 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, S153
操作前
Mobius inversion ORIGINAL inverse F(n/56) f(1) plus ORIGINAL b := sum(6, 1)
操作後
Mobius inversion ORIGINAL inverse F(n/56) f(1) plus ORIGINAL b = 7
條件：無
依原始角色、次序與已核中間量精確代值；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/56) full forward divisor sum := sum(7)
操作後
Mobius inversion ORIGINAL inverse F(n/56) full forward divisor sum = 7
條件：無
依原始角色、次序與已核中間量精確代值；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 mu(56) times SAME F(n/56) := mul(0, 7)
操作後
Mobius inversion ORIGINAL inverse mu(56) times SAME F(n/56) = 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, S156
操作前
Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(728, -342, 0, -94, 0, 45, 0, 0)
操作後
Mobius inversion ORIGINAL inverse complete inverse-term sum = 337
條件：無
依原始角色、次序與已核中間量精確代值；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 direct ORIGINAL a times n := mul(6, 56)
操作後
Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 336
條件：無
依原始角色、次序與已核中間量精確代值；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 direct plus ORIGINAL b := sum(336, 1)
操作後
Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 337
條件：無
依原始角色、次序與已核中間量精確代值；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:actual n=56;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 56, 728, 728], [2, -1, 28, 342, -342], [4, 0, 14, 148, 0], [7, -1, 8, 94, -94], [8, 0, 7, 50, 0], [14, 1, 4, 45, 45], [28, 0, 2, 20, 0], [56, 0, 1, 7, 0]]
操作後
complete reconstructed=337;direct f(n)=337;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, S160
操作前
Mobius inversion ORIGINAL omitted-divisor partial sum := sum(-342, 0, -94, 0, 45, 0, 0)
操作後
Mobius inversion ORIGINAL omitted-divisor partial sum = -391
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S161
操作前
Mobius inversion complete minus partial := sub(337, -391)
操作後
Mobius inversion complete minus partial = 728
條件：無
依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S162
操作前
ORIGINAL full divisors=[1, 2, 4, 7, 8, 14, 28, 56];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 4, 7, 8, 14, 28, 56]
操作後
complete=337=f(n)=337;partial=-391;lost term=728!=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, S163
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 Mobius inversion complete minus partial，不以小數近似替代。S162 · 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。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 direct plus ORIGINAL b，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 direct ORIGINAL a times n，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 complete inverse-term sum，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 mu(56) times SAME F(n/56)，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/56) full forward divisor sum，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/56) f(1) plus ORIGINAL b，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/56) f(1) ORIGINAL a times divisor，不以小數近似替代。S153 · 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。S152 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor56，不以小數近似替代。S151 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S150 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S149 · 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。S148 · 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。S147 · 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。S146 · 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。S145 · 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。S144 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence2，不以小數近似替代。S143 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。S142 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，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 — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(28) times SAME F(n/28)，不以小數近似替代。S138 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/28) full forward divisor sum，不以小數近似替代。S137 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/28) f(2) plus ORIGINAL b，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/28) f(2) ORIGINAL a times divisor，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/28) f(1) 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/28) f(1) ORIGINAL a times divisor，不以小數近似替代。S133 · 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。S132 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor28，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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S129 · 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。S128 · 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。S127 · 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。S126 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S125 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。S124 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。S123 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S122 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S121 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14)，不以小數近似替代。S120 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum，不以小數近似替代。S119 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/14) f(4) plus ORIGINAL b，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/14) f(4) ORIGINAL a times divisor，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/14) f(2) 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/14) f(2) 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/14) f(1) 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/14) f(1) ORIGINAL a times divisor，不以小數近似替代。S113 · 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。S112 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14，不以小數近似替代。S111 · 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。S110 · 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。S109 · 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。S108 · 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。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 — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。S105 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S104 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S103 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(8) times SAME F(n/8)，不以小數近似替代。S102 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/8) full forward divisor sum，不以小數近似替代。S101 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/8) f(7) plus ORIGINAL b，不以小數近似替代。S100 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/8) f(7) ORIGINAL a times divisor，不以小數近似替代。S099 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/8) f(1) plus ORIGINAL b，不以小數近似替代。S098 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/8) f(1) ORIGINAL a times divisor，不以小數近似替代。S097 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S096 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor8，不以小數近似替代。S095 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S094 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S093 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。S092 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。S091 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。S090 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S089 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence2，不以小數近似替代。S088 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。S087 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S084 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7)，不以小數近似替代。S083 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum，不以小數近似替代。S082 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(8) plus ORIGINAL b，不以小數近似替代。S081 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(8) ORIGINAL a times divisor，不以小數近似替代。S080 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(4) plus ORIGINAL b，不以小數近似替代。S079 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(4) ORIGINAL a times divisor，不以小數近似替代。S078 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b，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/7) f(2) ORIGINAL a times divisor，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/7) f(1) 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/7) f(1) ORIGINAL a times divisor，不以小數近似替代。S074 · 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。S073 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7，不以小數近似替代。S072 · 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。S071 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S070 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。S069 · 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。S068 · 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。S067 · 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。S066 · 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。S065 · 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。S064 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(14) plus ORIGINAL b，不以小數近似替代。S063 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(14) ORIGINAL a times divisor，不以小數近似替代。S062 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(7) plus ORIGINAL b，不以小數近似替代。S061 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(7) ORIGINAL a times divisor，不以小數近似替代。S060 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(2) plus ORIGINAL b，不以小數近似替代。S059 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/4) f(2) ORIGINAL a times divisor，不以小數近似替代。S058 · 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。S057 · 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。S056 · 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。S055 · 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。S054 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S053 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S052 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。S051 · 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。S050 · 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。S049 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。S048 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。S047 · 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。S046 · 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。S045 · 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。S044 · 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。S043 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(28) 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(28) 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(14) 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(14) ORIGINAL a times divisor，不以小數近似替代。S039 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b，不以小數近似替代。S038 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor，不以小數近似替代。S037 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(4) plus ORIGINAL b，不以小數近似替代。S036 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(4) ORIGINAL a times divisor，不以小數近似替代。S035 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b，不以小數近似替代。S034 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor，不以小數近似替代。S033 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b，不以小數近似替代。S032 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor，不以小數近似替代。S031 · 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。S030 · 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。S029 · 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。S028 · 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。S027 · 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。S026 · 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。S025 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S024 · premise_and_result_audit — 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。S023 · 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。S022 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(56) 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(56) 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(28) 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(28) 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(14) 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(14) 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(8) 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(8) 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(7) 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(7) 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(4) 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(4) 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。(56, [1, 2, 4, 7, 8, 14, 28, 56], 1, [2, 4, 7, 8, 14, 28, 56], 337, -391, 337, 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: 2bb61cf7a712f74b843df381a414927782c1a87b49fb5d3ec8a9122ed80d7ac2candidate record SHA-256: 6fece9b11e84d67c3a95fb088b7993024462501a6effce4bffe3d6cc63b742ddsemantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5eLean: NOT_RUN · human review: NOT_PERFORMED