R0043-22146 · q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0096
Step-by-step derivation
S001 · implies
Before
source operation=mobinv-incomplete-divisor-boundary;original snapshot and mathematics retained
After
explicit ordered source output fields=[n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary]
Conditions: none
Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
Mobius inversion ORIGINAL inverse:actual positive input=506;check complete integer range1..506
After
ALL positive divisors INCREASING=[1, 2, 11, 22, 23, 46, 253, 506];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S001
S003 · implies
Before
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1
After
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
Conditions: none
Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.
Step identifiers and dependencies
Rule: mobius_complete_UPF_input_role · Uses: G001, S002
S004 · implies
Before
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]
After
mu=1;empty product case=Yes;repeated prime=No
Conditions: none
The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.
Step identifiers and dependencies
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S003
S005 · calculation
Before
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 := div(506, 1)
After
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 506
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · implies
Before
Mobius inversion ORIGINAL inverse F(n/1):actual positive input=506;check complete integer range1..506
After
ALL positive divisors INCREASING=[1, 2, 11, 22, 23, 46, 253, 506];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S005
S007 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor := mul(21, 1)
After
Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 21
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(21, 1)
After
Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = 22
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(21, 2)
After
Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 42
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b := sum(42, 1)
After
Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 43
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/1) f(11) ORIGINAL a times divisor := mul(21, 11)
After
Mobius inversion ORIGINAL inverse F(n/1) f(11) ORIGINAL a times divisor = 231
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/1) f(11) plus ORIGINAL b := sum(231, 1)
After
Mobius inversion ORIGINAL inverse F(n/1) f(11) plus ORIGINAL b = 232
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S011
S013 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/1) f(22) ORIGINAL a times divisor := mul(21, 22)
After
Mobius inversion ORIGINAL inverse F(n/1) f(22) ORIGINAL a times divisor = 462
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/1) f(22) plus ORIGINAL b := sum(462, 1)
After
Mobius inversion ORIGINAL inverse F(n/1) f(22) plus ORIGINAL b = 463
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S013
S015 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/1) f(23) ORIGINAL a times divisor := mul(21, 23)
After
Mobius inversion ORIGINAL inverse F(n/1) f(23) ORIGINAL a times divisor = 483
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S014
S016 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/1) f(23) plus ORIGINAL b := sum(483, 1)
After
Mobius inversion ORIGINAL inverse F(n/1) f(23) plus ORIGINAL b = 484
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S015
S017 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/1) f(46) ORIGINAL a times divisor := mul(21, 46)
After
Mobius inversion ORIGINAL inverse F(n/1) f(46) ORIGINAL a times divisor = 966
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S016
S018 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/1) f(46) plus ORIGINAL b := sum(966, 1)
After
Mobius inversion ORIGINAL inverse F(n/1) f(46) plus ORIGINAL b = 967
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S017
S019 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/1) f(253) ORIGINAL a times divisor := mul(21, 253)
After
Mobius inversion ORIGINAL inverse F(n/1) f(253) ORIGINAL a times divisor = 5313
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S018
S020 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/1) f(253) plus ORIGINAL b := sum(5313, 1)
After
Mobius inversion ORIGINAL inverse F(n/1) f(253) plus ORIGINAL b = 5314
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S019
S021 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/1) f(506) ORIGINAL a times divisor := mul(21, 506)
After
Mobius inversion ORIGINAL inverse F(n/1) f(506) ORIGINAL a times divisor = 10626
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S020
S022 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/1) f(506) plus ORIGINAL b := sum(10626, 1)
After
Mobius inversion ORIGINAL inverse F(n/1) f(506) plus ORIGINAL b = 10627
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S021
S023 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(22, 43, 232, 463, 484, 967, 5314, 10627)
After
Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 18152
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S022
S024 · calculation
Before
Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 18152)
After
Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 18152
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S023
S025 · implies
Before
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor2:actual positive input for mu=2
After
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
Conditions: none
Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.
Step identifiers and dependencies
Rule: mobius_complete_UPF_input_role · Uses: G001, S024
S026 · implies
Before
ORIGINAL positive integer absolute magnitude=2>1
After
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
Conditions: none
Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime.
Step identifiers and dependencies
Rule: upf_factorization_remaining_product_invariant · Uses: G001, S025
S027 · calculation
Before
upf canonical complete product factor0 := mul(1, 2)
After
upf canonical complete product factor0 = 2
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S026
S028 · implies
Before
exhausted trial divisors below2;terminal remaining=2;d*d>remaining;ORIGINAL n=2
After
complete increasing prime multiset=[2];full product=2;a composite terminal remainder would have an already tested smaller divisor
Conditions: none
If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors.
Step identifiers and dependencies
Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S027
S029 · implies
Before
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor2:input=2;COMPLETE prime multiset=[2];distinct primes=[2]
After
mu=-1;empty product case=No;repeated prime=No
Conditions: none
The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.
Step identifiers and dependencies
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S028
S030 · calculation
Before
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 := div(506, 2)
After
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 253
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S029
S031 · implies
Before
Mobius inversion ORIGINAL inverse F(n/2):actual positive input=253;check complete integer range1..253
After
ALL positive divisors INCREASING=[1, 11, 23, 253];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S030
S032 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(21, 1)
After
Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 21
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S031
S033 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(21, 1)
After
Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 22
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S032
S034 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/2) f(11) ORIGINAL a times divisor := mul(21, 11)
After
Mobius inversion ORIGINAL inverse F(n/2) f(11) ORIGINAL a times divisor = 231
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S033
S035 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/2) f(11) plus ORIGINAL b := sum(231, 1)
After
Mobius inversion ORIGINAL inverse F(n/2) f(11) plus ORIGINAL b = 232
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S034
S036 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/2) f(23) ORIGINAL a times divisor := mul(21, 23)
After
Mobius inversion ORIGINAL inverse F(n/2) f(23) ORIGINAL a times divisor = 483
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S035
S037 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/2) f(23) plus ORIGINAL b := sum(483, 1)
After
Mobius inversion ORIGINAL inverse F(n/2) f(23) plus ORIGINAL b = 484
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S036
S038 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/2) f(253) ORIGINAL a times divisor := mul(21, 253)
After
Mobius inversion ORIGINAL inverse F(n/2) f(253) ORIGINAL a times divisor = 5313
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S037
S039 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/2) f(253) plus ORIGINAL b := sum(5313, 1)
After
Mobius inversion ORIGINAL inverse F(n/2) f(253) plus ORIGINAL b = 5314
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S038
S040 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(22, 232, 484, 5314)
After
Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 6052
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S039
S041 · calculation
Before
Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 6052)
After
Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -6052
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S040
S042 · implies
Before
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor11:actual positive input for mu=11
After
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
Conditions: none
Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.
Step identifiers and dependencies
Rule: mobius_complete_UPF_input_role · Uses: G001, S041
S043 · implies
Before
ORIGINAL positive integer absolute magnitude=11>1
After
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
Conditions: none
Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime.
Step identifiers and dependencies
Rule: upf_factorization_remaining_product_invariant · Uses: G001, S042
S044 · implies
Before
next divisor=2;previous remaining=11;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=11;remaining modulo2=1;extracted multiset=[]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S043
S045 · implies
Before
next divisor=3;previous remaining=11;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=11;remaining modulo3=2;extracted multiset=[]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S044
S046 · calculation
Before
upf canonical complete product factor0 := mul(1, 11)
After
upf canonical complete product factor0 = 11
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S045
S047 · implies
Before
exhausted trial divisors below4;terminal remaining=11;d*d>remaining;ORIGINAL n=11
After
complete increasing prime multiset=[11];full product=11;a composite terminal remainder would have an already tested smaller divisor
Conditions: none
If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors.
Step identifiers and dependencies
Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S046
S048 · implies
Before
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor11:input=11;COMPLETE prime multiset=[11];distinct primes=[11]
After
mu=-1;empty product case=No;repeated prime=No
Conditions: none
The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.
Step identifiers and dependencies
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S047
S049 · calculation
Before
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor11 := div(506, 11)
After
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor11 = 46
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S048
S050 · implies
Before
Mobius inversion ORIGINAL inverse F(n/11):actual positive input=46;check complete integer range1..46
After
ALL positive divisors INCREASING=[1, 2, 23, 46];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S049
S051 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/11) f(1) ORIGINAL a times divisor := mul(21, 1)
After
Mobius inversion ORIGINAL inverse F(n/11) f(1) ORIGINAL a times divisor = 21
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S050
S052 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/11) f(1) plus ORIGINAL b := sum(21, 1)
After
Mobius inversion ORIGINAL inverse F(n/11) f(1) plus ORIGINAL b = 22
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S051
S053 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/11) f(2) ORIGINAL a times divisor := mul(21, 2)
After
Mobius inversion ORIGINAL inverse F(n/11) f(2) ORIGINAL a times divisor = 42
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S052
S054 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/11) f(2) plus ORIGINAL b := sum(42, 1)
After
Mobius inversion ORIGINAL inverse F(n/11) f(2) plus ORIGINAL b = 43
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S053
S055 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/11) f(23) ORIGINAL a times divisor := mul(21, 23)
After
Mobius inversion ORIGINAL inverse F(n/11) f(23) ORIGINAL a times divisor = 483
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S054
S056 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/11) f(23) plus ORIGINAL b := sum(483, 1)
After
Mobius inversion ORIGINAL inverse F(n/11) f(23) plus ORIGINAL b = 484
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S055
S057 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/11) f(46) ORIGINAL a times divisor := mul(21, 46)
After
Mobius inversion ORIGINAL inverse F(n/11) f(46) ORIGINAL a times divisor = 966
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S056
S058 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/11) f(46) plus ORIGINAL b := sum(966, 1)
After
Mobius inversion ORIGINAL inverse F(n/11) f(46) plus ORIGINAL b = 967
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S057
S059 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/11) full forward divisor sum := sum(22, 43, 484, 967)
After
Mobius inversion ORIGINAL inverse F(n/11) full forward divisor sum = 1516
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S058
S060 · calculation
Before
Mobius inversion ORIGINAL inverse mu(11) times SAME F(n/11) := mul(-1, 1516)
After
Mobius inversion ORIGINAL inverse mu(11) times SAME F(n/11) = -1516
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S059
S061 · implies
Before
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor22:actual positive input for mu=22
After
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
Conditions: none
Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.
Step identifiers and dependencies
Rule: mobius_complete_UPF_input_role · Uses: G001, S060
S062 · implies
Before
ORIGINAL positive integer absolute magnitude=22>1
After
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
Conditions: none
Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime.
Step identifiers and dependencies
Rule: upf_factorization_remaining_product_invariant · Uses: G001, S061
S063 · calculation
Before
upf extract divisor2 occurrence0 := div(22, 2)
After
upf extract divisor2 occurrence0 = 11
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S062
S064 · implies
Before
next divisor=2;previous remaining=22;all smaller divisors exhausted
After
extracted multiplicity=1;next remaining=11;remaining modulo2=1;extracted multiset=[2]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S063
S065 · implies
Before
next divisor=3;previous remaining=11;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=11;remaining modulo3=2;extracted multiset=[2]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S064
S066 · calculation
Before
upf canonical complete product factor0 := mul(1, 2)
After
upf canonical complete product factor0 = 2
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S065
S067 · calculation
Before
upf canonical complete product factor1 := mul(2, 11)
After
upf canonical complete product factor1 = 22
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S066
S068 · implies
Before
exhausted trial divisors below4;terminal remaining=11;d*d>remaining;ORIGINAL n=22
After
complete increasing prime multiset=[2, 11];full product=22;a composite terminal remainder would have an already tested smaller divisor
Conditions: none
If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors.
Step identifiers and dependencies
Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S067
S069 · implies
Before
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor22:input=22;COMPLETE prime multiset=[2, 11];distinct primes=[2, 11]
After
mu=1;empty product case=No;repeated prime=No
Conditions: none
The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.
Step identifiers and dependencies
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S068
S070 · calculation
Before
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor22 := div(506, 22)
After
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor22 = 23
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S069
S071 · implies
Before
Mobius inversion ORIGINAL inverse F(n/22):actual positive input=23;check complete integer range1..23
After
ALL positive divisors INCREASING=[1, 23];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S070
S072 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/22) f(1) ORIGINAL a times divisor := mul(21, 1)
After
Mobius inversion ORIGINAL inverse F(n/22) f(1) ORIGINAL a times divisor = 21
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S071
S073 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/22) f(1) plus ORIGINAL b := sum(21, 1)
After
Mobius inversion ORIGINAL inverse F(n/22) f(1) plus ORIGINAL b = 22
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S072
S074 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/22) f(23) ORIGINAL a times divisor := mul(21, 23)
After
Mobius inversion ORIGINAL inverse F(n/22) f(23) ORIGINAL a times divisor = 483
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S073
S075 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/22) f(23) plus ORIGINAL b := sum(483, 1)
After
Mobius inversion ORIGINAL inverse F(n/22) f(23) plus ORIGINAL b = 484
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S074
S076 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/22) full forward divisor sum := sum(22, 484)
After
Mobius inversion ORIGINAL inverse F(n/22) full forward divisor sum = 506
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S075
S077 · calculation
Before
Mobius inversion ORIGINAL inverse mu(22) times SAME F(n/22) := mul(1, 506)
After
Mobius inversion ORIGINAL inverse mu(22) times SAME F(n/22) = 506
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S076
S078 · implies
Before
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor23:actual positive input for mu=23
After
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
Conditions: none
Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.
Step identifiers and dependencies
Rule: mobius_complete_UPF_input_role · Uses: G001, S077
S079 · implies
Before
ORIGINAL positive integer absolute magnitude=23>1
After
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
Conditions: none
Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime.
Step identifiers and dependencies
Rule: upf_factorization_remaining_product_invariant · Uses: G001, S078
S080 · implies
Before
next divisor=2;previous remaining=23;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=23;remaining modulo2=1;extracted multiset=[]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S079
S081 · implies
Before
next divisor=3;previous remaining=23;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=23;remaining modulo3=2;extracted multiset=[]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S080
S082 · implies
Before
next divisor=4;previous remaining=23;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=23;remaining modulo4=3;extracted multiset=[]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S081
S083 · calculation
Before
upf canonical complete product factor0 := mul(1, 23)
After
upf canonical complete product factor0 = 23
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S082
S084 · implies
Before
exhausted trial divisors below5;terminal remaining=23;d*d>remaining;ORIGINAL n=23
After
complete increasing prime multiset=[23];full product=23;a composite terminal remainder would have an already tested smaller divisor
Conditions: none
If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors.
Step identifiers and dependencies
Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S083
S085 · implies
Before
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor23:input=23;COMPLETE prime multiset=[23];distinct primes=[23]
After
mu=-1;empty product case=No;repeated prime=No
Conditions: none
The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.
Step identifiers and dependencies
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S084
S086 · calculation
Before
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor23 := div(506, 23)
After
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor23 = 22
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S085
S087 · implies
Before
Mobius inversion ORIGINAL inverse F(n/23):actual positive input=22;check complete integer range1..22
After
ALL positive divisors INCREASING=[1, 2, 11, 22];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S086
S088 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/23) f(1) ORIGINAL a times divisor := mul(21, 1)
After
Mobius inversion ORIGINAL inverse F(n/23) f(1) ORIGINAL a times divisor = 21
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S087
S089 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/23) f(1) plus ORIGINAL b := sum(21, 1)
After
Mobius inversion ORIGINAL inverse F(n/23) f(1) plus ORIGINAL b = 22
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S088
S090 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/23) f(2) ORIGINAL a times divisor := mul(21, 2)
After
Mobius inversion ORIGINAL inverse F(n/23) f(2) ORIGINAL a times divisor = 42
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S089
S091 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/23) f(2) plus ORIGINAL b := sum(42, 1)
After
Mobius inversion ORIGINAL inverse F(n/23) f(2) plus ORIGINAL b = 43
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S090
S092 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/23) f(11) ORIGINAL a times divisor := mul(21, 11)
After
Mobius inversion ORIGINAL inverse F(n/23) f(11) ORIGINAL a times divisor = 231
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S091
S093 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/23) f(11) plus ORIGINAL b := sum(231, 1)
After
Mobius inversion ORIGINAL inverse F(n/23) f(11) plus ORIGINAL b = 232
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S092
S094 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/23) f(22) ORIGINAL a times divisor := mul(21, 22)
After
Mobius inversion ORIGINAL inverse F(n/23) f(22) ORIGINAL a times divisor = 462
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S093
S095 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/23) f(22) plus ORIGINAL b := sum(462, 1)
After
Mobius inversion ORIGINAL inverse F(n/23) f(22) plus ORIGINAL b = 463
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S094
S096 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/23) full forward divisor sum := sum(22, 43, 232, 463)
After
Mobius inversion ORIGINAL inverse F(n/23) full forward divisor sum = 760
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S095
S097 · calculation
Before
Mobius inversion ORIGINAL inverse mu(23) times SAME F(n/23) := mul(-1, 760)
After
Mobius inversion ORIGINAL inverse mu(23) times SAME F(n/23) = -760
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S096
S098 · implies
Before
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor46:actual positive input for mu=46
After
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
Conditions: none
Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.
Step identifiers and dependencies
Rule: mobius_complete_UPF_input_role · Uses: G001, S097
S099 · implies
Before
ORIGINAL positive integer absolute magnitude=46>1
After
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
Conditions: none
Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime.
Step identifiers and dependencies
Rule: upf_factorization_remaining_product_invariant · Uses: G001, S098
S100 · calculation
Before
upf extract divisor2 occurrence0 := div(46, 2)
After
upf extract divisor2 occurrence0 = 23
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S099
S101 · implies
Before
next divisor=2;previous remaining=46;all smaller divisors exhausted
After
extracted multiplicity=1;next remaining=23;remaining modulo2=1;extracted multiset=[2]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S100
S102 · implies
Before
next divisor=3;previous remaining=23;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=23;remaining modulo3=2;extracted multiset=[2]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S101
S103 · implies
Before
next divisor=4;previous remaining=23;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=23;remaining modulo4=3;extracted multiset=[2]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S102
S104 · calculation
Before
upf canonical complete product factor0 := mul(1, 2)
After
upf canonical complete product factor0 = 2
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S103
S105 · calculation
Before
upf canonical complete product factor1 := mul(2, 23)
After
upf canonical complete product factor1 = 46
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S104
S106 · implies
Before
exhausted trial divisors below5;terminal remaining=23;d*d>remaining;ORIGINAL n=46
After
complete increasing prime multiset=[2, 23];full product=46;a composite terminal remainder would have an already tested smaller divisor
Conditions: none
If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors.
Step identifiers and dependencies
Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S105
S107 · implies
Before
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor46:input=46;COMPLETE prime multiset=[2, 23];distinct primes=[2, 23]
After
mu=1;empty product case=No;repeated prime=No
Conditions: none
The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.
Step identifiers and dependencies
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S106
S108 · calculation
Before
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor46 := div(506, 46)
After
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor46 = 11
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S107
S109 · implies
Before
Mobius inversion ORIGINAL inverse F(n/46):actual positive input=11;check complete integer range1..11
After
ALL positive divisors INCREASING=[1, 11];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S108
S110 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/46) f(1) ORIGINAL a times divisor := mul(21, 1)
After
Mobius inversion ORIGINAL inverse F(n/46) f(1) ORIGINAL a times divisor = 21
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S109
S111 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/46) f(1) plus ORIGINAL b := sum(21, 1)
After
Mobius inversion ORIGINAL inverse F(n/46) f(1) plus ORIGINAL b = 22
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S110
S112 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/46) f(11) ORIGINAL a times divisor := mul(21, 11)
After
Mobius inversion ORIGINAL inverse F(n/46) f(11) ORIGINAL a times divisor = 231
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S111
S113 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/46) f(11) plus ORIGINAL b := sum(231, 1)
After
Mobius inversion ORIGINAL inverse F(n/46) f(11) plus ORIGINAL b = 232
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S112
S114 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/46) full forward divisor sum := sum(22, 232)
After
Mobius inversion ORIGINAL inverse F(n/46) full forward divisor sum = 254
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S113
S115 · calculation
Before
Mobius inversion ORIGINAL inverse mu(46) times SAME F(n/46) := mul(1, 254)
After
Mobius inversion ORIGINAL inverse mu(46) times SAME F(n/46) = 254
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S114
S116 · implies
Before
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor253:actual positive input for mu=253
After
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
Conditions: none
Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.
Step identifiers and dependencies
Rule: mobius_complete_UPF_input_role · Uses: G001, S115
S117 · implies
Before
ORIGINAL positive integer absolute magnitude=253>1
After
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
Conditions: none
Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime.
Step identifiers and dependencies
Rule: upf_factorization_remaining_product_invariant · Uses: G001, S116
S118 · implies
Before
next divisor=2;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=253;remaining modulo2=1;extracted multiset=[]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S117
S119 · implies
Before
next divisor=3;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=253;remaining modulo3=1;extracted multiset=[]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S118
S120 · implies
Before
next divisor=4;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=253;remaining modulo4=1;extracted multiset=[]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S119
S121 · implies
Before
next divisor=5;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=253;remaining modulo5=3;extracted multiset=[]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S120
S122 · implies
Before
next divisor=6;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=253;remaining modulo6=1;extracted multiset=[]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S121
S123 · implies
Before
next divisor=7;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=253;remaining modulo7=1;extracted multiset=[]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S122
S124 · implies
Before
next divisor=8;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=253;remaining modulo8=5;extracted multiset=[]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S123
S125 · implies
Before
next divisor=9;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=253;remaining modulo9=1;extracted multiset=[]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S124
S126 · implies
Before
next divisor=10;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=253;remaining modulo10=3;extracted multiset=[]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S125
S127 · calculation
Before
upf extract divisor11 occurrence0 := div(253, 11)
After
upf extract divisor11 occurrence0 = 23
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S126
S128 · implies
Before
next divisor=11;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=1;next remaining=23;remaining modulo11=1;extracted multiset=[11]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S127
S129 · calculation
Before
upf canonical complete product factor0 := mul(1, 11)
After
upf canonical complete product factor0 = 11
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S128
S130 · calculation
Before
upf canonical complete product factor1 := mul(11, 23)
After
upf canonical complete product factor1 = 253
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S129
S131 · implies
Before
exhausted trial divisors below12;terminal remaining=23;d*d>remaining;ORIGINAL n=253
After
complete increasing prime multiset=[11, 23];full product=253;a composite terminal remainder would have an already tested smaller divisor
Conditions: none
If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors.
Step identifiers and dependencies
Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S130
S132 · implies
Before
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor253:input=253;COMPLETE prime multiset=[11, 23];distinct primes=[11, 23]
After
mu=1;empty product case=No;repeated prime=No
Conditions: none
The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.
Step identifiers and dependencies
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S131
S133 · calculation
Before
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor253 := div(506, 253)
After
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor253 = 2
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S132
S134 · implies
Before
Mobius inversion ORIGINAL inverse F(n/253):actual positive input=2;check complete integer range1..2
After
ALL positive divisors INCREASING=[1, 2];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S133
S135 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/253) f(1) ORIGINAL a times divisor := mul(21, 1)
After
Mobius inversion ORIGINAL inverse F(n/253) f(1) ORIGINAL a times divisor = 21
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S134
S136 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/253) f(1) plus ORIGINAL b := sum(21, 1)
After
Mobius inversion ORIGINAL inverse F(n/253) f(1) plus ORIGINAL b = 22
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S135
S137 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/253) f(2) ORIGINAL a times divisor := mul(21, 2)
After
Mobius inversion ORIGINAL inverse F(n/253) f(2) ORIGINAL a times divisor = 42
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S136
S138 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/253) f(2) plus ORIGINAL b := sum(42, 1)
After
Mobius inversion ORIGINAL inverse F(n/253) f(2) plus ORIGINAL b = 43
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S137
S139 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/253) full forward divisor sum := sum(22, 43)
After
Mobius inversion ORIGINAL inverse F(n/253) full forward divisor sum = 65
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S138
S140 · calculation
Before
Mobius inversion ORIGINAL inverse mu(253) times SAME F(n/253) := mul(1, 65)
After
Mobius inversion ORIGINAL inverse mu(253) times SAME F(n/253) = 65
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S139
S141 · implies
Before
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor506:actual positive input for mu=506
After
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
Conditions: none
Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.
Step identifiers and dependencies
Rule: mobius_complete_UPF_input_role · Uses: G001, S140
S142 · implies
Before
ORIGINAL positive integer absolute magnitude=506>1
After
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
Conditions: none
Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime.
Step identifiers and dependencies
Rule: upf_factorization_remaining_product_invariant · Uses: G001, S141
S143 · calculation
Before
upf extract divisor2 occurrence0 := div(506, 2)
After
upf extract divisor2 occurrence0 = 253
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S142
S144 · implies
Before
next divisor=2;previous remaining=506;all smaller divisors exhausted
After
extracted multiplicity=1;next remaining=253;remaining modulo2=1;extracted multiset=[2]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S143
S145 · implies
Before
next divisor=3;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=253;remaining modulo3=1;extracted multiset=[2]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S144
S146 · implies
Before
next divisor=4;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=253;remaining modulo4=1;extracted multiset=[2]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S145
S147 · implies
Before
next divisor=5;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=253;remaining modulo5=3;extracted multiset=[2]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S146
S148 · implies
Before
next divisor=6;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=253;remaining modulo6=1;extracted multiset=[2]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S147
S149 · implies
Before
next divisor=7;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=253;remaining modulo7=1;extracted multiset=[2]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S148
S150 · implies
Before
next divisor=8;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=253;remaining modulo8=5;extracted multiset=[2]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S149
S151 · implies
Before
next divisor=9;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=253;remaining modulo9=1;extracted multiset=[2]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S150
S152 · implies
Before
next divisor=10;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=253;remaining modulo10=3;extracted multiset=[2]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S151
S153 · calculation
Before
upf extract divisor11 occurrence1 := div(253, 11)
After
upf extract divisor11 occurrence1 = 23
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S152
S154 · implies
Before
next divisor=11;previous remaining=253;all smaller divisors exhausted
After
extracted multiplicity=1;next remaining=23;remaining modulo11=1;extracted multiset=[2, 11]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S153
S155 · calculation
Before
upf canonical complete product factor0 := mul(1, 2)
After
upf canonical complete product factor0 = 2
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S154
S156 · calculation
Before
upf canonical complete product factor1 := mul(2, 11)
After
upf canonical complete product factor1 = 22
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S155
S157 · calculation
Before
upf canonical complete product factor2 := mul(22, 23)
After
upf canonical complete product factor2 = 506
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S156
S158 · implies
Before
exhausted trial divisors below12;terminal remaining=23;d*d>remaining;ORIGINAL n=506
After
complete increasing prime multiset=[2, 11, 23];full product=506;a composite terminal remainder would have an already tested smaller divisor
Conditions: none
If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors.
Step identifiers and dependencies
Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S157
S159 · implies
Before
Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor506:input=506;COMPLETE prime multiset=[2, 11, 23];distinct primes=[2, 11, 23]
After
mu=-1;empty product case=No;repeated prime=No
Conditions: none
The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.
Step identifiers and dependencies
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S158
S160 · calculation
Before
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor506 := div(506, 506)
After
Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor506 = 1
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S159
S161 · implies
Before
Mobius inversion ORIGINAL inverse F(n/506):actual positive input=1;check complete integer range1..1
After
ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S160
S162 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/506) f(1) ORIGINAL a times divisor := mul(21, 1)
After
Mobius inversion ORIGINAL inverse F(n/506) f(1) ORIGINAL a times divisor = 21
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S161
S163 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/506) f(1) plus ORIGINAL b := sum(21, 1)
After
Mobius inversion ORIGINAL inverse F(n/506) f(1) plus ORIGINAL b = 22
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S162
S164 · calculation
Before
Mobius inversion ORIGINAL inverse F(n/506) full forward divisor sum := sum(22)
After
Mobius inversion ORIGINAL inverse F(n/506) full forward divisor sum = 22
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S163
S165 · calculation
Before
Mobius inversion ORIGINAL inverse mu(506) times SAME F(n/506) := mul(-1, 22)
After
Mobius inversion ORIGINAL inverse mu(506) times SAME F(n/506) = -22
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S164
S166 · calculation
Before
Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(18152, -6052, -1516, 506, -760, 254, 65, -22)
After
Mobius inversion ORIGINAL inverse complete inverse-term sum = 10627
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S165
S167 · calculation
Before
Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(21, 506)
After
Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 10626
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S166
S168 · calculation
Before
Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(10626, 1)
After
Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 10627
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S167
S169 · implies
Before
Mobius inversion ORIGINAL inverse:actual n=506;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 506, 18152, 18152], [2, -1, 253, 6052, -6052], [11, -1, 46, 1516, -1516], [22, 1, 23, 506, 506], [23, -1, 22, 760, -760], [46, 1, 11, 254, 254], [253, 1, 2, 65, 65], [506, -1, 1, 22, -22]]
After
complete reconstructed=10627;direct f(n)=10627;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)
Conditions: none
Swap the finite sums over d*e|n.The inner Mobius sum is1 for n/e=1 and0 otherwise by pairing distinct-prime subsets,leaving only e=n.For n1,the d1 term and mu1=1 remain;do not mix indices from the alternate equivalent convention.
Step identifiers and dependencies
Rule: mobinv_general_finite_double_sum_and_Mobius_cancellation · Uses: G001, S168
S170 · calculation
Before
Mobius inversion ORIGINAL omitted-divisor partial sum := sum(-6052, -1516, 506, -760, 254, 65, -22)
After
Mobius inversion ORIGINAL omitted-divisor partial sum = -7525
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S169
S171 · calculation
Before
Mobius inversion complete minus partial := sub(10627, -7525)
After
Mobius inversion complete minus partial = 18152
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S170
S172 · implies
Before
ORIGINAL full divisors=[1, 2, 11, 22, 23, 46, 253, 506];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 11, 22, 23, 46, 253, 506]
After
complete=10627=f(n)=10627;partial=-7525;lost term=18152!=0;incomplete_certificate fixed No
Conditions: none
The omitted value must be an actual divisor with nonzero contribution,so the partial result differs from the target.Omitting a zero term does not qualify this discriminative boundary.The No rejects the incomplete certificate,not the complete inversion.
Step identifiers and dependencies
Rule: mobinv_omission_actual_nonzero_discriminative_term · Uses: G001, S171