Question 22158: Keep mu(d) Paired with F(n Divided by d)

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0043-22158 · q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0033

Original question and givens

Use positive integer n=8, exact integer arithmetic function f(k)=a*k+b with a=3, b=-5. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

G001 — Use positive integer n=8, exact integer arithmetic function f(k)=a*k+b with a=3, b=-5. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

mko-mobius-inversion-finite-divisors

Step-by-step derivation

S001 · implies

Before

source operation=mobinv-inverse-reconstruction;original snapshot and mathematics retained

After

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

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=8;check complete integer range1..8

After

ALL positive divisors INCREASING=[1, 2, 4, 8];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(8, 1)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 8

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=8;check complete integer range1..8

After

ALL positive divisors INCREASING=[1, 2, 4, 8];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(3, 1)

After

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

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(3, -5)

After

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = -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, S007

S009 · calculation

Before

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

After

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

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(6, -5)

After

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 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, S009

S011 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor := mul(3, 4)

After

Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor = 12

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(4) plus ORIGINAL b := sum(12, -5)

After

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

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(8) ORIGINAL a times divisor := mul(3, 8)

After

Mobius inversion ORIGINAL inverse F(n/1) f(8) ORIGINAL a times divisor = 24

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(8) plus ORIGINAL b := sum(24, -5)

After

Mobius inversion ORIGINAL inverse F(n/1) f(8) plus ORIGINAL b = 19

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) full forward divisor sum := sum(-2, 1, 7, 19)

After

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 25

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 mu(1) times SAME F(n/1) := mul(1, 25)

After

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 25

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 · 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, S016

S018 · 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, S017

S019 · 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, S018

S020 · 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, S019

S021 · 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, S020

S022 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 := div(8, 2)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 4

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 · implies

Before

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=4;check complete integer range1..4

After

ALL positive divisors INCREASING=[1, 2, 4];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, S022

S024 · calculation

Before

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

After

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

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 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(3, -5)

After

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = -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, S024

S026 · calculation

Before

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

After

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

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, S025

S027 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b := sum(6, -5)

After

Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b = 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, S026

S028 · calculation

Before

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

After

Mobius inversion ORIGINAL inverse F(n/2) f(4) ORIGINAL a times divisor = 12

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, S027

S029 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(4) plus ORIGINAL b := sum(12, -5)

After

Mobius inversion ORIGINAL inverse F(n/2) f(4) plus ORIGINAL b = 7

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, S028

S030 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(-2, 1, 7)

After

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 6

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 · calculation

Before

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 6)

After

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -6

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, S030

S032 · implies

Before

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

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, S031

S033 · implies

Before

ORIGINAL positive integer absolute magnitude=4>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, S032

S034 · calculation

Before

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

After

upf extract divisor2 occurrence0 = 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, S033

S035 · calculation

Before

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

After

upf extract divisor2 occurrence1 = 1

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, S034

S036 · implies

Before

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

After

extracted multiplicity=2;next remaining=1;remaining modulo2=1;extracted multiset=[2, 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, S035

S037 · 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, S036

S038 · calculation

Before

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

After

upf canonical complete product factor1 = 4

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, S037

S039 · implies

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S038

S040 · implies

Before

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

After

mu=0;empty product case=No;repeated prime=Yes

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, S039

S041 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 := div(8, 4)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 = 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, S040

S042 · implies

Before

Mobius inversion ORIGINAL inverse F(n/4):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, S041

S043 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor := mul(3, 1)

After

Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor = 3

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, S042

S044 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b := sum(3, -5)

After

Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b = -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, S043

S045 · calculation

Before

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

After

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

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, S044

S046 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(2) plus ORIGINAL b := sum(6, -5)

After

Mobius inversion ORIGINAL inverse F(n/4) f(2) plus ORIGINAL b = 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, S045

S047 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum := sum(-2, 1)

After

Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum = -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, S046

S048 · calculation

Before

Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) := mul(0, -1)

After

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

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, S047

S049 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor8:actual positive input for mu=8

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, S048

S050 · implies

Before

ORIGINAL positive integer absolute magnitude=8>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, S049

S051 · calculation

Before

upf extract divisor2 occurrence0 := div(8, 2)

After

upf extract divisor2 occurrence0 = 4

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, S050

S052 · calculation

Before

upf extract divisor2 occurrence1 := div(4, 2)

After

upf extract divisor2 occurrence1 = 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, S051

S053 · calculation

Before

upf extract divisor2 occurrence2 := div(2, 2)

After

upf extract divisor2 occurrence2 = 1

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, S052

S054 · implies

Before

next divisor=2;previous remaining=8;all smaller divisors exhausted

After

extracted multiplicity=3;next remaining=1;remaining modulo2=1;extracted multiset=[2, 2, 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, S053

S055 · 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, S054

S056 · calculation

Before

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

After

upf canonical complete product factor1 = 4

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, S055

S057 · calculation

Before

upf canonical complete product factor2 := mul(4, 2)

After

upf canonical complete product factor2 = 8

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, S056

S058 · implies

Before

exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=8

After

complete increasing prime multiset=[2, 2, 2];full product=8;a composite terminal remainder would have an already tested smaller divisor

Conditions: none

Step identifiers and dependencies

Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S057

S059 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor8:input=8;COMPLETE prime multiset=[2, 2, 2];distinct primes=[2]

After

mu=0;empty product case=No;repeated prime=Yes

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, S058

S060 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor8 := div(8, 8)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor8 = 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, S059

S061 · implies

Before

Mobius inversion ORIGINAL inverse F(n/8):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, S060

S062 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/8) f(1) ORIGINAL a times divisor := mul(3, 1)

After

Mobius inversion ORIGINAL inverse F(n/8) f(1) ORIGINAL a times divisor = 3

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, S061

S063 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/8) f(1) plus ORIGINAL b := sum(3, -5)

After

Mobius inversion ORIGINAL inverse F(n/8) f(1) plus ORIGINAL b = -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, S062

S064 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/8) full forward divisor sum := sum(-2)

After

Mobius inversion ORIGINAL inverse F(n/8) full forward divisor sum = -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, S063

S065 · calculation

Before

Mobius inversion ORIGINAL inverse mu(8) times SAME F(n/8) := mul(0, -2)

After

Mobius inversion ORIGINAL inverse mu(8) times SAME F(n/8) = 0

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, S064

S066 · calculation

Before

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(25, -6, 0, 0)

After

Mobius inversion ORIGINAL inverse complete inverse-term sum = 19

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, S065

S067 · calculation

Before

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(3, 8)

After

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 24

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, S066

S068 · calculation

Before

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(24, -5)

After

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 19

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, S067

S069 · implies

Before

Mobius inversion ORIGINAL inverse:actual n=8;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 8, 25, 25], [2, -1, 4, 6, -6], [4, 0, 2, -1, 0], [8, 0, 1, -2, 0]]

After

complete reconstructed=19;direct f(n)=19;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, S068

Reverse checks

S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S068 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S067 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
S066 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
S065 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(8) times SAME F(n/8) with exact rational arithmetic, not rounded decimals.
S064 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/8) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S063 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/8) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S062 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/8) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S060 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor8 with exact rational arithmetic, not rounded decimals.
S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S057 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
S056 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S055 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S053 · premise_and_result_audit — Recompute upf extract divisor2 occurrence2 with exact rational arithmetic, not rounded decimals.
S052 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
S051 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S048 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) with exact rational arithmetic, not rounded decimals.
S047 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S046 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S045 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S044 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S043 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S041 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 with exact rational arithmetic, not rounded decimals.
S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S038 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S037 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S035 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S031 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals.
S030 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S029 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S028 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S022 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S019 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S016 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(8) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(8) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S010 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S005 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(8, [[1, 1, 8, 25, 25], [2, -1, 4, 6, -6], [4, 0, 2, -1, 0], [8, 0, 1, -2, 0]], 19, 19, Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts

input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: a461d75417144d2838f82561eb3848c789de5870e6050fc2a87db7866feff777
candidate record SHA-256: 94baac4e5a4b06e9901fe43fe74aa46aafb9d3761f434aa4a8bea15c7a9b02c7
semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5
question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON