Question 22144: An Omitted Divisor Must Be a Discriminative Missing-Term Witness

On this page

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

← Question derivation index · Original question batch

R0043-22144 · q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0094

Original question and givens

Use positive integer n=420, exact integer arithmetic function f(k)=a*k+b with a=19, b=1. 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. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

G001 — Use positive integer n=420, exact integer arithmetic function f(k)=a*k+b with a=19, b=1. 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. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

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-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=420;check complete integer range1..420

After

ALL positive divisors INCREASING=[1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420];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(420, 1)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 420

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=420;check complete integer range1..420

After

ALL positive divisors INCREASING=[1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420];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(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 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, S006

S008 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = 20

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(19, 2)

After

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 38

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(38, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 39

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(3) ORIGINAL a times divisor := mul(19, 3)

After

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor = 57

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(3) plus ORIGINAL b := sum(57, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b = 58

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(4) ORIGINAL a times divisor := mul(19, 4)

After

Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor = 76

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(4) plus ORIGINAL b := sum(76, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b = 77

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(5) ORIGINAL a times divisor := mul(19, 5)

After

Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor = 95

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(5) plus ORIGINAL b := sum(95, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b = 96

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(6) ORIGINAL a times divisor := mul(19, 6)

After

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor = 114

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(6) plus ORIGINAL b := sum(114, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b = 115

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(7) ORIGINAL a times divisor := mul(19, 7)

After

Mobius inversion ORIGINAL inverse F(n/1) f(7) ORIGINAL a times divisor = 133

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(7) plus ORIGINAL b := sum(133, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b = 134

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(10) ORIGINAL a times divisor := mul(19, 10)

After

Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor = 190

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(10) plus ORIGINAL b := sum(190, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b = 191

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) f(12) ORIGINAL a times divisor := mul(19, 12)

After

Mobius inversion ORIGINAL inverse F(n/1) f(12) ORIGINAL a times divisor = 228

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 F(n/1) f(12) plus ORIGINAL b := sum(228, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(12) plus ORIGINAL b = 229

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/1) f(14) ORIGINAL a times divisor := mul(19, 14)

After

Mobius inversion ORIGINAL inverse F(n/1) f(14) ORIGINAL a times divisor = 266

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/1) f(14) plus ORIGINAL b := sum(266, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(14) plus ORIGINAL b = 267

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/1) f(15) ORIGINAL a times divisor := mul(19, 15)

After

Mobius inversion ORIGINAL inverse F(n/1) f(15) ORIGINAL a times divisor = 285

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/1) f(15) plus ORIGINAL b := sum(285, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(15) plus ORIGINAL b = 286

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/1) f(20) ORIGINAL a times divisor := mul(19, 20)

After

Mobius inversion ORIGINAL inverse F(n/1) f(20) ORIGINAL a times divisor = 380

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/1) f(20) plus ORIGINAL b := sum(380, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(20) plus ORIGINAL b = 381

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 F(n/1) f(21) ORIGINAL a times divisor := mul(19, 21)

After

Mobius inversion ORIGINAL inverse F(n/1) f(21) ORIGINAL a times divisor = 399

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 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(21) plus ORIGINAL b := sum(399, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(21) plus ORIGINAL b = 400

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/1) f(28) ORIGINAL a times divisor := mul(19, 28)

After

Mobius inversion ORIGINAL inverse F(n/1) f(28) ORIGINAL a times divisor = 532

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/1) f(28) plus ORIGINAL b := sum(532, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(28) plus ORIGINAL b = 533

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/1) f(30) ORIGINAL a times divisor := mul(19, 30)

After

Mobius inversion ORIGINAL inverse F(n/1) f(30) ORIGINAL a times divisor = 570

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/1) f(30) plus ORIGINAL b := sum(570, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(30) plus ORIGINAL b = 571

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/1) f(35) ORIGINAL a times divisor := mul(19, 35)

After

Mobius inversion ORIGINAL inverse F(n/1) f(35) ORIGINAL a times divisor = 665

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/1) f(35) plus ORIGINAL b := sum(665, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(35) plus ORIGINAL b = 666

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/1) f(42) ORIGINAL a times divisor := mul(19, 42)

After

Mobius inversion ORIGINAL inverse F(n/1) f(42) ORIGINAL a times divisor = 798

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/1) f(42) plus ORIGINAL b := sum(798, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(42) plus ORIGINAL b = 799

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 F(n/1) f(60) ORIGINAL a times divisor := mul(19, 60)

After

Mobius inversion ORIGINAL inverse F(n/1) f(60) ORIGINAL a times divisor = 1140

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 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(60) plus ORIGINAL b := sum(1140, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(60) plus ORIGINAL b = 1141

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, S041

S043 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(70) ORIGINAL a times divisor := mul(19, 70)

After

Mobius inversion ORIGINAL inverse F(n/1) f(70) ORIGINAL a times divisor = 1330

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/1) f(70) plus ORIGINAL b := sum(1330, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(70) plus ORIGINAL b = 1331

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/1) f(84) ORIGINAL a times divisor := mul(19, 84)

After

Mobius inversion ORIGINAL inverse F(n/1) f(84) ORIGINAL a times divisor = 1596

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/1) f(84) plus ORIGINAL b := sum(1596, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(84) plus ORIGINAL b = 1597

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/1) f(105) ORIGINAL a times divisor := mul(19, 105)

After

Mobius inversion ORIGINAL inverse F(n/1) f(105) ORIGINAL a times divisor = 1995

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 F(n/1) f(105) plus ORIGINAL b := sum(1995, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(105) plus ORIGINAL b = 1996

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 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(140) ORIGINAL a times divisor := mul(19, 140)

After

Mobius inversion ORIGINAL inverse F(n/1) f(140) ORIGINAL a times divisor = 2660

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 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(140) plus ORIGINAL b := sum(2660, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(140) plus ORIGINAL b = 2661

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, S049

S051 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(210) ORIGINAL a times divisor := mul(19, 210)

After

Mobius inversion ORIGINAL inverse F(n/1) f(210) ORIGINAL a times divisor = 3990

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/1) f(210) plus ORIGINAL b := sum(3990, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(210) plus ORIGINAL b = 3991

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/1) f(420) ORIGINAL a times divisor := mul(19, 420)

After

Mobius inversion ORIGINAL inverse F(n/1) f(420) ORIGINAL a times divisor = 7980

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/1) f(420) plus ORIGINAL b := sum(7980, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(420) plus ORIGINAL b = 7981

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/1) full forward divisor sum := sum(20, 39, 58, 77, 96, 115, 134, 191, 229, 267, 286, 381, 400, 533, 571, 666, 799, 1141, 1331, 1597, 1996, 2661, 3991, 7981)

After

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 25560

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 mu(1) times SAME F(n/1) := mul(1, 25560)

After

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 25560

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 · 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, S056

S058 · 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, S057

S059 · 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, S058

S060 · 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, S059

S061 · 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, S060

S062 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 := div(420, 2)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 210

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 · implies

Before

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=210;check complete integer range1..210

After

ALL positive divisors INCREASING=[1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210];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, S062

S064 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 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, S063

S065 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 20

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 F(n/2) f(2) ORIGINAL a times divisor := mul(19, 2)

After

Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor = 38

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 F(n/2) f(2) plus ORIGINAL b := sum(38, 1)

After

Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b = 39

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 F(n/2) f(3) ORIGINAL a times divisor := mul(19, 3)

After

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor = 57

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 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b := sum(57, 1)

After

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b = 58

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, S068

S070 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor := mul(19, 5)

After

Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor = 95

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 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b := sum(95, 1)

After

Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b = 96

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, S070

S072 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(6) ORIGINAL a times divisor := mul(19, 6)

After

Mobius inversion ORIGINAL inverse F(n/2) f(6) ORIGINAL a times divisor = 114

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/2) f(6) plus ORIGINAL b := sum(114, 1)

After

Mobius inversion ORIGINAL inverse F(n/2) f(6) plus ORIGINAL b = 115

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/2) f(7) ORIGINAL a times divisor := mul(19, 7)

After

Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor = 133

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/2) f(7) plus ORIGINAL b := sum(133, 1)

After

Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b = 134

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/2) f(10) ORIGINAL a times divisor := mul(19, 10)

After

Mobius inversion ORIGINAL inverse F(n/2) f(10) ORIGINAL a times divisor = 190

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 F(n/2) f(10) plus ORIGINAL b := sum(190, 1)

After

Mobius inversion ORIGINAL inverse F(n/2) f(10) plus ORIGINAL b = 191

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 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(14) ORIGINAL a times divisor := mul(19, 14)

After

Mobius inversion ORIGINAL inverse F(n/2) f(14) ORIGINAL a times divisor = 266

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, S077

S079 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(14) plus ORIGINAL b := sum(266, 1)

After

Mobius inversion ORIGINAL inverse F(n/2) f(14) plus ORIGINAL b = 267

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, S078

S080 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(15) ORIGINAL a times divisor := mul(19, 15)

After

Mobius inversion ORIGINAL inverse F(n/2) f(15) ORIGINAL a times divisor = 285

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, S079

S081 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(15) plus ORIGINAL b := sum(285, 1)

After

Mobius inversion ORIGINAL inverse F(n/2) f(15) plus ORIGINAL b = 286

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, S080

S082 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(21) ORIGINAL a times divisor := mul(19, 21)

After

Mobius inversion ORIGINAL inverse F(n/2) f(21) ORIGINAL a times divisor = 399

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, S081

S083 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(21) plus ORIGINAL b := sum(399, 1)

After

Mobius inversion ORIGINAL inverse F(n/2) f(21) plus ORIGINAL b = 400

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, S082

S084 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(30) ORIGINAL a times divisor := mul(19, 30)

After

Mobius inversion ORIGINAL inverse F(n/2) f(30) ORIGINAL a times divisor = 570

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, S083

S085 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(30) plus ORIGINAL b := sum(570, 1)

After

Mobius inversion ORIGINAL inverse F(n/2) f(30) plus ORIGINAL b = 571

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, S084

S086 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(35) ORIGINAL a times divisor := mul(19, 35)

After

Mobius inversion ORIGINAL inverse F(n/2) f(35) ORIGINAL a times divisor = 665

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 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(35) plus ORIGINAL b := sum(665, 1)

After

Mobius inversion ORIGINAL inverse F(n/2) f(35) plus ORIGINAL b = 666

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, S086

S088 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(42) ORIGINAL a times divisor := mul(19, 42)

After

Mobius inversion ORIGINAL inverse F(n/2) f(42) ORIGINAL a times divisor = 798

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/2) f(42) plus ORIGINAL b := sum(798, 1)

After

Mobius inversion ORIGINAL inverse F(n/2) f(42) plus ORIGINAL b = 799

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/2) f(70) ORIGINAL a times divisor := mul(19, 70)

After

Mobius inversion ORIGINAL inverse F(n/2) f(70) ORIGINAL a times divisor = 1330

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/2) f(70) plus ORIGINAL b := sum(1330, 1)

After

Mobius inversion ORIGINAL inverse F(n/2) f(70) plus ORIGINAL b = 1331

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/2) f(105) ORIGINAL a times divisor := mul(19, 105)

After

Mobius inversion ORIGINAL inverse F(n/2) f(105) ORIGINAL a times divisor = 1995

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/2) f(105) plus ORIGINAL b := sum(1995, 1)

After

Mobius inversion ORIGINAL inverse F(n/2) f(105) plus ORIGINAL b = 1996

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/2) f(210) ORIGINAL a times divisor := mul(19, 210)

After

Mobius inversion ORIGINAL inverse F(n/2) f(210) ORIGINAL a times divisor = 3990

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/2) f(210) plus ORIGINAL b := sum(3990, 1)

After

Mobius inversion ORIGINAL inverse F(n/2) f(210) plus ORIGINAL b = 3991

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/2) full forward divisor sum := sum(20, 39, 58, 96, 115, 134, 191, 267, 286, 400, 571, 666, 799, 1331, 1996, 3991)

After

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 10960

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(2) times SAME F(n/2) := mul(-1, 10960)

After

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -10960

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 divisor3:actual positive input for mu=3

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=3>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 canonical complete product factor0 := mul(1, 3)

After

upf canonical complete product factor0 = 3

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

exhausted trial divisors below2;terminal remaining=3;d*d>remaining;ORIGINAL n=3

After

complete increasing prime multiset=[3];full product=3;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, S100

S102 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor3:input=3;COMPLETE prime multiset=[3];distinct primes=[3]

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, S101

S103 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 := div(420, 3)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 = 140

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, S102

S104 · implies

Before

Mobius inversion ORIGINAL inverse F(n/3):actual positive input=140;check complete integer range1..140

After

ALL positive divisors INCREASING=[1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140];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, S103

S105 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor = 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, S104

S106 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b = 20

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, S105

S107 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor := mul(19, 2)

After

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor = 38

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, S106

S108 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b := sum(38, 1)

After

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b = 39

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 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(4) ORIGINAL a times divisor := mul(19, 4)

After

Mobius inversion ORIGINAL inverse F(n/3) f(4) ORIGINAL a times divisor = 76

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, S108

S110 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b := sum(76, 1)

After

Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b = 77

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/3) f(5) ORIGINAL a times divisor := mul(19, 5)

After

Mobius inversion ORIGINAL inverse F(n/3) f(5) ORIGINAL a times divisor = 95

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/3) f(5) plus ORIGINAL b := sum(95, 1)

After

Mobius inversion ORIGINAL inverse F(n/3) f(5) plus ORIGINAL b = 96

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/3) f(7) ORIGINAL a times divisor := mul(19, 7)

After

Mobius inversion ORIGINAL inverse F(n/3) f(7) ORIGINAL a times divisor = 133

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/3) f(7) plus ORIGINAL b := sum(133, 1)

After

Mobius inversion ORIGINAL inverse F(n/3) f(7) plus ORIGINAL b = 134

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 F(n/3) f(10) ORIGINAL a times divisor := mul(19, 10)

After

Mobius inversion ORIGINAL inverse F(n/3) f(10) ORIGINAL a times divisor = 190

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 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(10) plus ORIGINAL b := sum(190, 1)

After

Mobius inversion ORIGINAL inverse F(n/3) f(10) plus ORIGINAL b = 191

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, S115

S117 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(14) ORIGINAL a times divisor := mul(19, 14)

After

Mobius inversion ORIGINAL inverse F(n/3) f(14) ORIGINAL a times divisor = 266

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, S116

S118 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(14) plus ORIGINAL b := sum(266, 1)

After

Mobius inversion ORIGINAL inverse F(n/3) f(14) plus ORIGINAL b = 267

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, S117

S119 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(20) ORIGINAL a times divisor := mul(19, 20)

After

Mobius inversion ORIGINAL inverse F(n/3) f(20) ORIGINAL a times divisor = 380

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, S118

S120 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(20) plus ORIGINAL b := sum(380, 1)

After

Mobius inversion ORIGINAL inverse F(n/3) f(20) plus ORIGINAL b = 381

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, S119

S121 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(28) ORIGINAL a times divisor := mul(19, 28)

After

Mobius inversion ORIGINAL inverse F(n/3) f(28) ORIGINAL a times divisor = 532

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, S120

S122 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(28) plus ORIGINAL b := sum(532, 1)

After

Mobius inversion ORIGINAL inverse F(n/3) f(28) plus ORIGINAL b = 533

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, S121

S123 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(35) ORIGINAL a times divisor := mul(19, 35)

After

Mobius inversion ORIGINAL inverse F(n/3) f(35) ORIGINAL a times divisor = 665

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, S122

S124 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(35) plus ORIGINAL b := sum(665, 1)

After

Mobius inversion ORIGINAL inverse F(n/3) f(35) plus ORIGINAL b = 666

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, S123

S125 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(70) ORIGINAL a times divisor := mul(19, 70)

After

Mobius inversion ORIGINAL inverse F(n/3) f(70) ORIGINAL a times divisor = 1330

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, S124

S126 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(70) plus ORIGINAL b := sum(1330, 1)

After

Mobius inversion ORIGINAL inverse F(n/3) f(70) plus ORIGINAL b = 1331

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, S125

S127 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(140) ORIGINAL a times divisor := mul(19, 140)

After

Mobius inversion ORIGINAL inverse F(n/3) f(140) ORIGINAL a times divisor = 2660

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, S126

S128 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) f(140) plus ORIGINAL b := sum(2660, 1)

After

Mobius inversion ORIGINAL inverse F(n/3) f(140) plus ORIGINAL b = 2661

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, S127

S129 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum := sum(20, 39, 77, 96, 134, 191, 267, 381, 533, 666, 1331, 2661)

After

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum = 6396

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, S128

S130 · calculation

Before

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) := mul(-1, 6396)

After

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) = -6396

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, S129

S131 · 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, S130

S132 · 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, S131

S133 · 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, S132

S134 · 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, S133

S135 · 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, S134

S136 · 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, S135

S137 · 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, S136

S138 · 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, S137

S139 · 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, S138

S140 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 := div(420, 4)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 = 105

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 F(n/4):actual positive input=105;check complete integer range1..105

After

ALL positive divisors INCREASING=[1, 3, 5, 7, 15, 21, 35, 105];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, S140

S142 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor = 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, S141

S143 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b = 20

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, S142

S144 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(3) ORIGINAL a times divisor := mul(19, 3)

After

Mobius inversion ORIGINAL inverse F(n/4) f(3) ORIGINAL a times divisor = 57

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, S143

S145 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(3) plus ORIGINAL b := sum(57, 1)

After

Mobius inversion ORIGINAL inverse F(n/4) f(3) plus ORIGINAL b = 58

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, S144

S146 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(5) ORIGINAL a times divisor := mul(19, 5)

After

Mobius inversion ORIGINAL inverse F(n/4) f(5) ORIGINAL a times divisor = 95

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, S145

S147 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(5) plus ORIGINAL b := sum(95, 1)

After

Mobius inversion ORIGINAL inverse F(n/4) f(5) plus ORIGINAL b = 96

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, S146

S148 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(7) ORIGINAL a times divisor := mul(19, 7)

After

Mobius inversion ORIGINAL inverse F(n/4) f(7) ORIGINAL a times divisor = 133

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, S147

S149 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(7) plus ORIGINAL b := sum(133, 1)

After

Mobius inversion ORIGINAL inverse F(n/4) f(7) plus ORIGINAL b = 134

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, S148

S150 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(15) ORIGINAL a times divisor := mul(19, 15)

After

Mobius inversion ORIGINAL inverse F(n/4) f(15) ORIGINAL a times divisor = 285

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, S149

S151 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(15) plus ORIGINAL b := sum(285, 1)

After

Mobius inversion ORIGINAL inverse F(n/4) f(15) plus ORIGINAL b = 286

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, S150

S152 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(21) ORIGINAL a times divisor := mul(19, 21)

After

Mobius inversion ORIGINAL inverse F(n/4) f(21) ORIGINAL a times divisor = 399

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, S151

S153 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(21) plus ORIGINAL b := sum(399, 1)

After

Mobius inversion ORIGINAL inverse F(n/4) f(21) plus ORIGINAL b = 400

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, S152

S154 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(35) ORIGINAL a times divisor := mul(19, 35)

After

Mobius inversion ORIGINAL inverse F(n/4) f(35) ORIGINAL a times divisor = 665

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, S153

S155 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(35) plus ORIGINAL b := sum(665, 1)

After

Mobius inversion ORIGINAL inverse F(n/4) f(35) plus ORIGINAL b = 666

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, S154

S156 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(105) ORIGINAL a times divisor := mul(19, 105)

After

Mobius inversion ORIGINAL inverse F(n/4) f(105) ORIGINAL a times divisor = 1995

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, S155

S157 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(105) plus ORIGINAL b := sum(1995, 1)

After

Mobius inversion ORIGINAL inverse F(n/4) f(105) plus ORIGINAL b = 1996

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, S156

S158 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum := sum(20, 58, 96, 134, 286, 400, 666, 1996)

After

Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum = 3656

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, S157

S159 · calculation

Before

Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) := mul(0, 3656)

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, S158

S160 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor5:actual positive input for mu=5

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, S159

S161 · implies

Before

ORIGINAL positive integer absolute magnitude=5>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, S160

S162 · implies

Before

next divisor=2;previous remaining=5;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=5;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, S161

S163 · calculation

Before

upf canonical complete product factor0 := mul(1, 5)

After

upf canonical complete product factor0 = 5

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, S162

S164 · implies

Before

exhausted trial divisors below3;terminal remaining=5;d*d>remaining;ORIGINAL n=5

After

complete increasing prime multiset=[5];full product=5;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, S163

S165 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor5:input=5;COMPLETE prime multiset=[5];distinct primes=[5]

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, S164

S166 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 := div(420, 5)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 = 84

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 · implies

Before

Mobius inversion ORIGINAL inverse F(n/5):actual positive input=84;check complete integer range1..84

After

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84];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, S166

S168 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor = 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, S167

S169 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b = 20

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, S168

S170 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor := mul(19, 2)

After

Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor = 38

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 ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b := sum(38, 1)

After

Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b = 39

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 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(3) ORIGINAL a times divisor := mul(19, 3)

After

Mobius inversion ORIGINAL inverse F(n/5) f(3) ORIGINAL a times divisor = 57

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, S171

S173 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(3) plus ORIGINAL b := sum(57, 1)

After

Mobius inversion ORIGINAL inverse F(n/5) f(3) plus ORIGINAL b = 58

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, S172

S174 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(4) ORIGINAL a times divisor := mul(19, 4)

After

Mobius inversion ORIGINAL inverse F(n/5) f(4) ORIGINAL a times divisor = 76

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, S173

S175 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(4) plus ORIGINAL b := sum(76, 1)

After

Mobius inversion ORIGINAL inverse F(n/5) f(4) plus ORIGINAL b = 77

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, S174

S176 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(6) ORIGINAL a times divisor := mul(19, 6)

After

Mobius inversion ORIGINAL inverse F(n/5) f(6) ORIGINAL a times divisor = 114

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, S175

S177 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(6) plus ORIGINAL b := sum(114, 1)

After

Mobius inversion ORIGINAL inverse F(n/5) f(6) plus ORIGINAL b = 115

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, S176

S178 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(7) ORIGINAL a times divisor := mul(19, 7)

After

Mobius inversion ORIGINAL inverse F(n/5) f(7) ORIGINAL a times divisor = 133

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, S177

S179 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(7) plus ORIGINAL b := sum(133, 1)

After

Mobius inversion ORIGINAL inverse F(n/5) f(7) plus ORIGINAL b = 134

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, S178

S180 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(12) ORIGINAL a times divisor := mul(19, 12)

After

Mobius inversion ORIGINAL inverse F(n/5) f(12) ORIGINAL a times divisor = 228

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, S179

S181 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(12) plus ORIGINAL b := sum(228, 1)

After

Mobius inversion ORIGINAL inverse F(n/5) f(12) plus ORIGINAL b = 229

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, S180

S182 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(14) ORIGINAL a times divisor := mul(19, 14)

After

Mobius inversion ORIGINAL inverse F(n/5) f(14) ORIGINAL a times divisor = 266

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, S181

S183 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(14) plus ORIGINAL b := sum(266, 1)

After

Mobius inversion ORIGINAL inverse F(n/5) f(14) plus ORIGINAL b = 267

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, S182

S184 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(21) ORIGINAL a times divisor := mul(19, 21)

After

Mobius inversion ORIGINAL inverse F(n/5) f(21) ORIGINAL a times divisor = 399

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, S183

S185 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(21) plus ORIGINAL b := sum(399, 1)

After

Mobius inversion ORIGINAL inverse F(n/5) f(21) plus ORIGINAL b = 400

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, S184

S186 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(28) ORIGINAL a times divisor := mul(19, 28)

After

Mobius inversion ORIGINAL inverse F(n/5) f(28) ORIGINAL a times divisor = 532

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, S185

S187 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(28) plus ORIGINAL b := sum(532, 1)

After

Mobius inversion ORIGINAL inverse F(n/5) f(28) plus ORIGINAL b = 533

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, S186

S188 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(42) ORIGINAL a times divisor := mul(19, 42)

After

Mobius inversion ORIGINAL inverse F(n/5) f(42) ORIGINAL a times divisor = 798

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, S187

S189 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(42) plus ORIGINAL b := sum(798, 1)

After

Mobius inversion ORIGINAL inverse F(n/5) f(42) plus ORIGINAL b = 799

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, S188

S190 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(84) ORIGINAL a times divisor := mul(19, 84)

After

Mobius inversion ORIGINAL inverse F(n/5) f(84) ORIGINAL a times divisor = 1596

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, S189

S191 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(84) plus ORIGINAL b := sum(1596, 1)

After

Mobius inversion ORIGINAL inverse F(n/5) f(84) plus ORIGINAL b = 1597

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, S190

S192 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum := sum(20, 39, 58, 77, 115, 134, 229, 267, 400, 533, 799, 1597)

After

Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum = 4268

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, S191

S193 · calculation

Before

Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) := mul(-1, 4268)

After

Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) = -4268

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, S192

S194 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor6:actual positive input for mu=6

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, S193

S195 · implies

Before

ORIGINAL positive integer absolute magnitude=6>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, S194

S196 · calculation

Before

upf extract divisor2 occurrence0 := div(6, 2)

After

upf extract divisor2 occurrence0 = 3

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, S195

S197 · implies

Before

next divisor=2;previous remaining=6;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=3;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, S196

S198 · 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, S197

S199 · calculation

Before

upf canonical complete product factor1 := mul(2, 3)

After

upf canonical complete product factor1 = 6

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, S198

S200 · implies

Before

exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=6

After

complete increasing prime multiset=[2, 3];full product=6;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, S199

S201 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor6:input=6;COMPLETE prime multiset=[2, 3];distinct primes=[2, 3]

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, S200

S202 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 := div(420, 6)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 = 70

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, S201

S203 · implies

Before

Mobius inversion ORIGINAL inverse F(n/6):actual positive input=70;check complete integer range1..70

After

ALL positive divisors INCREASING=[1, 2, 5, 7, 10, 14, 35, 70];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, S202

S204 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor = 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, S203

S205 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b = 20

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, S204

S206 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor := mul(19, 2)

After

Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor = 38

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, S205

S207 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b := sum(38, 1)

After

Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b = 39

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, S206

S208 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/6) f(5) ORIGINAL a times divisor := mul(19, 5)

After

Mobius inversion ORIGINAL inverse F(n/6) f(5) ORIGINAL a times divisor = 95

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, S207

S209 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/6) f(5) plus ORIGINAL b := sum(95, 1)

After

Mobius inversion ORIGINAL inverse F(n/6) f(5) plus ORIGINAL b = 96

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, S208

S210 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor := mul(19, 7)

After

Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor = 133

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, S209

S211 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b := sum(133, 1)

After

Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b = 134

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, S210

S212 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/6) f(10) ORIGINAL a times divisor := mul(19, 10)

After

Mobius inversion ORIGINAL inverse F(n/6) f(10) ORIGINAL a times divisor = 190

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, S211

S213 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/6) f(10) plus ORIGINAL b := sum(190, 1)

After

Mobius inversion ORIGINAL inverse F(n/6) f(10) plus ORIGINAL b = 191

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, S212

S214 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/6) f(14) ORIGINAL a times divisor := mul(19, 14)

After

Mobius inversion ORIGINAL inverse F(n/6) f(14) ORIGINAL a times divisor = 266

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, S213

S215 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/6) f(14) plus ORIGINAL b := sum(266, 1)

After

Mobius inversion ORIGINAL inverse F(n/6) f(14) plus ORIGINAL b = 267

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, S214

S216 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/6) f(35) ORIGINAL a times divisor := mul(19, 35)

After

Mobius inversion ORIGINAL inverse F(n/6) f(35) ORIGINAL a times divisor = 665

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, S215

S217 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/6) f(35) plus ORIGINAL b := sum(665, 1)

After

Mobius inversion ORIGINAL inverse F(n/6) f(35) plus ORIGINAL b = 666

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, S216

S218 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/6) f(70) ORIGINAL a times divisor := mul(19, 70)

After

Mobius inversion ORIGINAL inverse F(n/6) f(70) ORIGINAL a times divisor = 1330

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, S217

S219 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/6) f(70) plus ORIGINAL b := sum(1330, 1)

After

Mobius inversion ORIGINAL inverse F(n/6) f(70) plus ORIGINAL b = 1331

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, S218

S220 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum := sum(20, 39, 96, 134, 191, 267, 666, 1331)

After

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum = 2744

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, S219

S221 · calculation

Before

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) := mul(1, 2744)

After

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) = 2744

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, S220

S222 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor7:actual positive input for mu=7

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, S221

S223 · implies

Before

ORIGINAL positive integer absolute magnitude=7>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, S222

S224 · implies

Before

next divisor=2;previous remaining=7;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=7;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, S223

S225 · calculation

Before

upf canonical complete product factor0 := mul(1, 7)

After

upf canonical complete product factor0 = 7

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, S224

S226 · implies

Before

exhausted trial divisors below3;terminal remaining=7;d*d>remaining;ORIGINAL n=7

After

complete increasing prime multiset=[7];full product=7;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, S225

S227 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor7:input=7;COMPLETE prime multiset=[7];distinct primes=[7]

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, S226

S228 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 := div(420, 7)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 = 60

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, S227

S229 · implies

Before

Mobius inversion ORIGINAL inverse F(n/7):actual positive input=60;check complete integer range1..60

After

ALL positive divisors INCREASING=[1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60];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, S228

S230 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor = 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, S229

S231 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b = 20

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, S230

S232 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor := mul(19, 2)

After

Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor = 38

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, S231

S233 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b := sum(38, 1)

After

Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b = 39

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, S232

S234 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(3) ORIGINAL a times divisor := mul(19, 3)

After

Mobius inversion ORIGINAL inverse F(n/7) f(3) ORIGINAL a times divisor = 57

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, S233

S235 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b := sum(57, 1)

After

Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b = 58

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, S234

S236 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(4) ORIGINAL a times divisor := mul(19, 4)

After

Mobius inversion ORIGINAL inverse F(n/7) f(4) ORIGINAL a times divisor = 76

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, S235

S237 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(4) plus ORIGINAL b := sum(76, 1)

After

Mobius inversion ORIGINAL inverse F(n/7) f(4) plus ORIGINAL b = 77

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, S236

S238 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(5) ORIGINAL a times divisor := mul(19, 5)

After

Mobius inversion ORIGINAL inverse F(n/7) f(5) ORIGINAL a times divisor = 95

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, S237

S239 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(5) plus ORIGINAL b := sum(95, 1)

After

Mobius inversion ORIGINAL inverse F(n/7) f(5) plus ORIGINAL b = 96

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, S238

S240 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(6) ORIGINAL a times divisor := mul(19, 6)

After

Mobius inversion ORIGINAL inverse F(n/7) f(6) ORIGINAL a times divisor = 114

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, S239

S241 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(6) plus ORIGINAL b := sum(114, 1)

After

Mobius inversion ORIGINAL inverse F(n/7) f(6) plus ORIGINAL b = 115

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, S240

S242 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(10) ORIGINAL a times divisor := mul(19, 10)

After

Mobius inversion ORIGINAL inverse F(n/7) f(10) ORIGINAL a times divisor = 190

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, S241

S243 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(10) plus ORIGINAL b := sum(190, 1)

After

Mobius inversion ORIGINAL inverse F(n/7) f(10) plus ORIGINAL b = 191

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, S242

S244 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(12) ORIGINAL a times divisor := mul(19, 12)

After

Mobius inversion ORIGINAL inverse F(n/7) f(12) ORIGINAL a times divisor = 228

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, S243

S245 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(12) plus ORIGINAL b := sum(228, 1)

After

Mobius inversion ORIGINAL inverse F(n/7) f(12) plus ORIGINAL b = 229

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, S244

S246 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(15) ORIGINAL a times divisor := mul(19, 15)

After

Mobius inversion ORIGINAL inverse F(n/7) f(15) ORIGINAL a times divisor = 285

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, S245

S247 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(15) plus ORIGINAL b := sum(285, 1)

After

Mobius inversion ORIGINAL inverse F(n/7) f(15) plus ORIGINAL b = 286

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, S246

S248 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(20) ORIGINAL a times divisor := mul(19, 20)

After

Mobius inversion ORIGINAL inverse F(n/7) f(20) ORIGINAL a times divisor = 380

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, S247

S249 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(20) plus ORIGINAL b := sum(380, 1)

After

Mobius inversion ORIGINAL inverse F(n/7) f(20) plus ORIGINAL b = 381

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, S248

S250 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(30) ORIGINAL a times divisor := mul(19, 30)

After

Mobius inversion ORIGINAL inverse F(n/7) f(30) ORIGINAL a times divisor = 570

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, S249

S251 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(30) plus ORIGINAL b := sum(570, 1)

After

Mobius inversion ORIGINAL inverse F(n/7) f(30) plus ORIGINAL b = 571

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, S250

S252 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(60) ORIGINAL a times divisor := mul(19, 60)

After

Mobius inversion ORIGINAL inverse F(n/7) f(60) ORIGINAL a times divisor = 1140

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, S251

S253 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) f(60) plus ORIGINAL b := sum(1140, 1)

After

Mobius inversion ORIGINAL inverse F(n/7) f(60) plus ORIGINAL b = 1141

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, S252

S254 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum := sum(20, 39, 58, 77, 96, 115, 191, 229, 286, 381, 571, 1141)

After

Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum = 3204

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, S253

S255 · calculation

Before

Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) := mul(-1, 3204)

After

Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) = -3204

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, S254

S256 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor10:actual positive input for mu=10

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, S255

S257 · implies

Before

ORIGINAL positive integer absolute magnitude=10>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, S256

S258 · calculation

Before

upf extract divisor2 occurrence0 := div(10, 2)

After

upf extract divisor2 occurrence0 = 5

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, S257

S259 · implies

Before

next divisor=2;previous remaining=10;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=5;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, S258

S260 · 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, S259

S261 · calculation

Before

upf canonical complete product factor1 := mul(2, 5)

After

upf canonical complete product factor1 = 10

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, S260

S262 · implies

Before

exhausted trial divisors below3;terminal remaining=5;d*d>remaining;ORIGINAL n=10

After

complete increasing prime multiset=[2, 5];full product=10;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, S261

S263 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor10:input=10;COMPLETE prime multiset=[2, 5];distinct primes=[2, 5]

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, S262

S264 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 := div(420, 10)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 = 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, S263

S265 · implies

Before

Mobius inversion ORIGINAL inverse F(n/10):actual positive input=42;check complete integer range1..42

After

ALL positive divisors INCREASING=[1, 2, 3, 6, 7, 14, 21, 42];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, S264

S266 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor = 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, S265

S267 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b = 20

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, S266

S268 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(2) ORIGINAL a times divisor := mul(19, 2)

After

Mobius inversion ORIGINAL inverse F(n/10) f(2) ORIGINAL a times divisor = 38

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, S267

S269 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(2) plus ORIGINAL b := sum(38, 1)

After

Mobius inversion ORIGINAL inverse F(n/10) f(2) plus ORIGINAL b = 39

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, S268

S270 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(3) ORIGINAL a times divisor := mul(19, 3)

After

Mobius inversion ORIGINAL inverse F(n/10) f(3) ORIGINAL a times divisor = 57

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, S269

S271 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(3) plus ORIGINAL b := sum(57, 1)

After

Mobius inversion ORIGINAL inverse F(n/10) f(3) plus ORIGINAL b = 58

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, S270

S272 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(6) ORIGINAL a times divisor := mul(19, 6)

After

Mobius inversion ORIGINAL inverse F(n/10) f(6) ORIGINAL a times divisor = 114

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, S271

S273 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(6) plus ORIGINAL b := sum(114, 1)

After

Mobius inversion ORIGINAL inverse F(n/10) f(6) plus ORIGINAL b = 115

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, S272

S274 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(7) ORIGINAL a times divisor := mul(19, 7)

After

Mobius inversion ORIGINAL inverse F(n/10) f(7) ORIGINAL a times divisor = 133

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, S273

S275 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(7) plus ORIGINAL b := sum(133, 1)

After

Mobius inversion ORIGINAL inverse F(n/10) f(7) plus ORIGINAL b = 134

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, S274

S276 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(14) ORIGINAL a times divisor := mul(19, 14)

After

Mobius inversion ORIGINAL inverse F(n/10) f(14) ORIGINAL a times divisor = 266

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, S275

S277 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(14) plus ORIGINAL b := sum(266, 1)

After

Mobius inversion ORIGINAL inverse F(n/10) f(14) plus ORIGINAL b = 267

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, S276

S278 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(21) ORIGINAL a times divisor := mul(19, 21)

After

Mobius inversion ORIGINAL inverse F(n/10) f(21) ORIGINAL a times divisor = 399

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, S277

S279 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(21) plus ORIGINAL b := sum(399, 1)

After

Mobius inversion ORIGINAL inverse F(n/10) f(21) plus ORIGINAL b = 400

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, S278

S280 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(42) ORIGINAL a times divisor := mul(19, 42)

After

Mobius inversion ORIGINAL inverse F(n/10) f(42) ORIGINAL a times divisor = 798

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, S279

S281 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(42) plus ORIGINAL b := sum(798, 1)

After

Mobius inversion ORIGINAL inverse F(n/10) f(42) plus ORIGINAL b = 799

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, S280

S282 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum := sum(20, 39, 58, 115, 134, 267, 400, 799)

After

Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum = 1832

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, S281

S283 · calculation

Before

Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) := mul(1, 1832)

After

Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) = 1832

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, S282

S284 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor12:actual positive input for mu=12

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, S283

S285 · implies

Before

ORIGINAL positive integer absolute magnitude=12>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, S284

S286 · calculation

Before

upf extract divisor2 occurrence0 := div(12, 2)

After

upf extract divisor2 occurrence0 = 6

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, S285

S287 · calculation

Before

upf extract divisor2 occurrence1 := div(6, 2)

After

upf extract divisor2 occurrence1 = 3

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, S286

S288 · implies

Before

next divisor=2;previous remaining=12;all smaller divisors exhausted

After

extracted multiplicity=2;next remaining=3;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, S287

S289 · 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, S288

S290 · 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, S289

S291 · calculation

Before

upf canonical complete product factor2 := mul(4, 3)

After

upf canonical complete product factor2 = 12

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, S290

S292 · implies

Before

exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=12

After

complete increasing prime multiset=[2, 2, 3];full product=12;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, S291

S293 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor12:input=12;COMPLETE prime multiset=[2, 2, 3];distinct primes=[2, 3]

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, S292

S294 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12 := div(420, 12)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12 = 35

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, S293

S295 · implies

Before

Mobius inversion ORIGINAL inverse F(n/12):actual positive input=35;check complete integer range1..35

After

ALL positive divisors INCREASING=[1, 5, 7, 35];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, S294

S296 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor = 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, S295

S297 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/12) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/12) f(1) plus ORIGINAL b = 20

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, S296

S298 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/12) f(5) ORIGINAL a times divisor := mul(19, 5)

After

Mobius inversion ORIGINAL inverse F(n/12) f(5) ORIGINAL a times divisor = 95

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, S297

S299 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/12) f(5) plus ORIGINAL b := sum(95, 1)

After

Mobius inversion ORIGINAL inverse F(n/12) f(5) plus ORIGINAL b = 96

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, S298

S300 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/12) f(7) ORIGINAL a times divisor := mul(19, 7)

After

Mobius inversion ORIGINAL inverse F(n/12) f(7) ORIGINAL a times divisor = 133

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, S299

S301 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/12) f(7) plus ORIGINAL b := sum(133, 1)

After

Mobius inversion ORIGINAL inverse F(n/12) f(7) plus ORIGINAL b = 134

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, S300

S302 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/12) f(35) ORIGINAL a times divisor := mul(19, 35)

After

Mobius inversion ORIGINAL inverse F(n/12) f(35) ORIGINAL a times divisor = 665

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, S301

S303 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/12) f(35) plus ORIGINAL b := sum(665, 1)

After

Mobius inversion ORIGINAL inverse F(n/12) f(35) plus ORIGINAL b = 666

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, S302

S304 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/12) full forward divisor sum := sum(20, 96, 134, 666)

After

Mobius inversion ORIGINAL inverse F(n/12) full forward divisor sum = 916

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, S303

S305 · calculation

Before

Mobius inversion ORIGINAL inverse mu(12) times SAME F(n/12) := mul(0, 916)

After

Mobius inversion ORIGINAL inverse mu(12) times SAME F(n/12) = 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, S304

S306 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor14:actual positive input for mu=14

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, S305

S307 · implies

Before

ORIGINAL positive integer absolute magnitude=14>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, S306

S308 · calculation

Before

upf extract divisor2 occurrence0 := div(14, 2)

After

upf extract divisor2 occurrence0 = 7

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, S307

S309 · implies

Before

next divisor=2;previous remaining=14;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=7;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, S308

S310 · 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, S309

S311 · calculation

Before

upf canonical complete product factor1 := mul(2, 7)

After

upf canonical complete product factor1 = 14

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, S310

S312 · implies

Before

exhausted trial divisors below3;terminal remaining=7;d*d>remaining;ORIGINAL n=14

After

complete increasing prime multiset=[2, 7];full product=14;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, S311

S313 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor14:input=14;COMPLETE prime multiset=[2, 7];distinct primes=[2, 7]

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, S312

S314 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 := div(420, 14)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 = 30

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, S313

S315 · implies

Before

Mobius inversion ORIGINAL inverse F(n/14):actual positive input=30;check complete integer range1..30

After

ALL positive divisors INCREASING=[1, 2, 3, 5, 6, 10, 15, 30];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, S314

S316 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor = 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, S315

S317 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b = 20

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, S316

S318 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/14) f(2) ORIGINAL a times divisor := mul(19, 2)

After

Mobius inversion ORIGINAL inverse F(n/14) f(2) ORIGINAL a times divisor = 38

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, S317

S319 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/14) f(2) plus ORIGINAL b := sum(38, 1)

After

Mobius inversion ORIGINAL inverse F(n/14) f(2) plus ORIGINAL b = 39

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, S318

S320 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor := mul(19, 3)

After

Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor = 57

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, S319

S321 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b := sum(57, 1)

After

Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b = 58

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, S320

S322 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/14) f(5) ORIGINAL a times divisor := mul(19, 5)

After

Mobius inversion ORIGINAL inverse F(n/14) f(5) ORIGINAL a times divisor = 95

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, S321

S323 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/14) f(5) plus ORIGINAL b := sum(95, 1)

After

Mobius inversion ORIGINAL inverse F(n/14) f(5) plus ORIGINAL b = 96

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, S322

S324 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/14) f(6) ORIGINAL a times divisor := mul(19, 6)

After

Mobius inversion ORIGINAL inverse F(n/14) f(6) ORIGINAL a times divisor = 114

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, S323

S325 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/14) f(6) plus ORIGINAL b := sum(114, 1)

After

Mobius inversion ORIGINAL inverse F(n/14) f(6) plus ORIGINAL b = 115

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, S324

S326 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/14) f(10) ORIGINAL a times divisor := mul(19, 10)

After

Mobius inversion ORIGINAL inverse F(n/14) f(10) ORIGINAL a times divisor = 190

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, S325

S327 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/14) f(10) plus ORIGINAL b := sum(190, 1)

After

Mobius inversion ORIGINAL inverse F(n/14) f(10) plus ORIGINAL b = 191

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, S326

S328 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/14) f(15) ORIGINAL a times divisor := mul(19, 15)

After

Mobius inversion ORIGINAL inverse F(n/14) f(15) ORIGINAL a times divisor = 285

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, S327

S329 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/14) f(15) plus ORIGINAL b := sum(285, 1)

After

Mobius inversion ORIGINAL inverse F(n/14) f(15) plus ORIGINAL b = 286

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, S328

S330 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/14) f(30) ORIGINAL a times divisor := mul(19, 30)

After

Mobius inversion ORIGINAL inverse F(n/14) f(30) ORIGINAL a times divisor = 570

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, S329

S331 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/14) f(30) plus ORIGINAL b := sum(570, 1)

After

Mobius inversion ORIGINAL inverse F(n/14) f(30) plus ORIGINAL b = 571

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, S330

S332 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum := sum(20, 39, 58, 96, 115, 191, 286, 571)

After

Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum = 1376

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, S331

S333 · calculation

Before

Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) := mul(1, 1376)

After

Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) = 1376

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, S332

S334 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor15:actual positive input for mu=15

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, S333

S335 · implies

Before

ORIGINAL positive integer absolute magnitude=15>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, S334

S336 · implies

Before

next divisor=2;previous remaining=15;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=15;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, S335

S337 · calculation

Before

upf extract divisor3 occurrence0 := div(15, 3)

After

upf extract divisor3 occurrence0 = 5

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, S336

S338 · implies

Before

next divisor=3;previous remaining=15;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=5;remaining modulo3=2;extracted multiset=[3]

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, S337

S339 · calculation

Before

upf canonical complete product factor0 := mul(1, 3)

After

upf canonical complete product factor0 = 3

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, S338

S340 · calculation

Before

upf canonical complete product factor1 := mul(3, 5)

After

upf canonical complete product factor1 = 15

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, S339

S341 · implies

Before

exhausted trial divisors below4;terminal remaining=5;d*d>remaining;ORIGINAL n=15

After

complete increasing prime multiset=[3, 5];full product=15;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, S340

S342 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor15:input=15;COMPLETE prime multiset=[3, 5];distinct primes=[3, 5]

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, S341

S343 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor15 := div(420, 15)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor15 = 28

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, S342

S344 · implies

Before

Mobius inversion ORIGINAL inverse F(n/15):actual positive input=28;check complete integer range1..28

After

ALL positive divisors INCREASING=[1, 2, 4, 7, 14, 28];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, S343

S345 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/15) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/15) f(1) ORIGINAL a times divisor = 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, S344

S346 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/15) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/15) f(1) plus ORIGINAL b = 20

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, S345

S347 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/15) f(2) ORIGINAL a times divisor := mul(19, 2)

After

Mobius inversion ORIGINAL inverse F(n/15) f(2) ORIGINAL a times divisor = 38

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, S346

S348 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/15) f(2) plus ORIGINAL b := sum(38, 1)

After

Mobius inversion ORIGINAL inverse F(n/15) f(2) plus ORIGINAL b = 39

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, S347

S349 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/15) f(4) ORIGINAL a times divisor := mul(19, 4)

After

Mobius inversion ORIGINAL inverse F(n/15) f(4) ORIGINAL a times divisor = 76

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, S348

S350 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/15) f(4) plus ORIGINAL b := sum(76, 1)

After

Mobius inversion ORIGINAL inverse F(n/15) f(4) plus ORIGINAL b = 77

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, S349

S351 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/15) f(7) ORIGINAL a times divisor := mul(19, 7)

After

Mobius inversion ORIGINAL inverse F(n/15) f(7) ORIGINAL a times divisor = 133

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, S350

S352 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/15) f(7) plus ORIGINAL b := sum(133, 1)

After

Mobius inversion ORIGINAL inverse F(n/15) f(7) plus ORIGINAL b = 134

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, S351

S353 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/15) f(14) ORIGINAL a times divisor := mul(19, 14)

After

Mobius inversion ORIGINAL inverse F(n/15) f(14) ORIGINAL a times divisor = 266

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, S352

S354 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/15) f(14) plus ORIGINAL b := sum(266, 1)

After

Mobius inversion ORIGINAL inverse F(n/15) f(14) plus ORIGINAL b = 267

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, S353

S355 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/15) f(28) ORIGINAL a times divisor := mul(19, 28)

After

Mobius inversion ORIGINAL inverse F(n/15) f(28) ORIGINAL a times divisor = 532

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, S354

S356 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/15) f(28) plus ORIGINAL b := sum(532, 1)

After

Mobius inversion ORIGINAL inverse F(n/15) f(28) plus ORIGINAL b = 533

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, S355

S357 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/15) full forward divisor sum := sum(20, 39, 77, 134, 267, 533)

After

Mobius inversion ORIGINAL inverse F(n/15) full forward divisor sum = 1070

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, S356

S358 · calculation

Before

Mobius inversion ORIGINAL inverse mu(15) times SAME F(n/15) := mul(1, 1070)

After

Mobius inversion ORIGINAL inverse mu(15) times SAME F(n/15) = 1070

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, S357

S359 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor20:actual positive input for mu=20

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, S358

S360 · implies

Before

ORIGINAL positive integer absolute magnitude=20>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, S359

S361 · calculation

Before

upf extract divisor2 occurrence0 := div(20, 2)

After

upf extract divisor2 occurrence0 = 10

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, S360

S362 · calculation

Before

upf extract divisor2 occurrence1 := div(10, 2)

After

upf extract divisor2 occurrence1 = 5

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, S361

S363 · implies

Before

next divisor=2;previous remaining=20;all smaller divisors exhausted

After

extracted multiplicity=2;next remaining=5;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, S362

S364 · 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, S363

S365 · 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, S364

S366 · calculation

Before

upf canonical complete product factor2 := mul(4, 5)

After

upf canonical complete product factor2 = 20

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, S365

S367 · implies

Before

exhausted trial divisors below3;terminal remaining=5;d*d>remaining;ORIGINAL n=20

After

complete increasing prime multiset=[2, 2, 5];full product=20;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, S366

S368 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor20:input=20;COMPLETE prime multiset=[2, 2, 5];distinct primes=[2, 5]

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, S367

S369 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor20 := div(420, 20)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor20 = 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, S368

S370 · implies

Before

Mobius inversion ORIGINAL inverse F(n/20):actual positive input=21;check complete integer range1..21

After

ALL positive divisors INCREASING=[1, 3, 7, 21];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, S369

S371 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/20) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/20) f(1) ORIGINAL a times divisor = 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, S370

S372 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/20) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/20) f(1) plus ORIGINAL b = 20

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, S371

S373 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/20) f(3) ORIGINAL a times divisor := mul(19, 3)

After

Mobius inversion ORIGINAL inverse F(n/20) f(3) ORIGINAL a times divisor = 57

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, S372

S374 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/20) f(3) plus ORIGINAL b := sum(57, 1)

After

Mobius inversion ORIGINAL inverse F(n/20) f(3) plus ORIGINAL b = 58

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, S373

S375 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/20) f(7) ORIGINAL a times divisor := mul(19, 7)

After

Mobius inversion ORIGINAL inverse F(n/20) f(7) ORIGINAL a times divisor = 133

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, S374

S376 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/20) f(7) plus ORIGINAL b := sum(133, 1)

After

Mobius inversion ORIGINAL inverse F(n/20) f(7) plus ORIGINAL b = 134

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, S375

S377 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/20) f(21) ORIGINAL a times divisor := mul(19, 21)

After

Mobius inversion ORIGINAL inverse F(n/20) f(21) ORIGINAL a times divisor = 399

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, S376

S378 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/20) f(21) plus ORIGINAL b := sum(399, 1)

After

Mobius inversion ORIGINAL inverse F(n/20) f(21) plus ORIGINAL b = 400

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, S377

S379 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/20) full forward divisor sum := sum(20, 58, 134, 400)

After

Mobius inversion ORIGINAL inverse F(n/20) full forward divisor sum = 612

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, S378

S380 · calculation

Before

Mobius inversion ORIGINAL inverse mu(20) times SAME F(n/20) := mul(0, 612)

After

Mobius inversion ORIGINAL inverse mu(20) times SAME F(n/20) = 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, S379

S381 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor21:actual positive input for mu=21

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, S380

S382 · implies

Before

ORIGINAL positive integer absolute magnitude=21>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, S381

S383 · implies

Before

next divisor=2;previous remaining=21;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=21;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, S382

S384 · calculation

Before

upf extract divisor3 occurrence0 := div(21, 3)

After

upf extract divisor3 occurrence0 = 7

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, S383

S385 · implies

Before

next divisor=3;previous remaining=21;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=7;remaining modulo3=1;extracted multiset=[3]

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, S384

S386 · calculation

Before

upf canonical complete product factor0 := mul(1, 3)

After

upf canonical complete product factor0 = 3

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, S385

S387 · calculation

Before

upf canonical complete product factor1 := mul(3, 7)

After

upf canonical complete product factor1 = 21

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, S386

S388 · implies

Before

exhausted trial divisors below4;terminal remaining=7;d*d>remaining;ORIGINAL n=21

After

complete increasing prime multiset=[3, 7];full product=21;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, S387

S389 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor21:input=21;COMPLETE prime multiset=[3, 7];distinct primes=[3, 7]

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, S388

S390 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21 := div(420, 21)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21 = 20

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, S389

S391 · implies

Before

Mobius inversion ORIGINAL inverse F(n/21):actual positive input=20;check complete integer range1..20

After

ALL positive divisors INCREASING=[1, 2, 4, 5, 10, 20];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, S390

S392 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor = 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, S391

S393 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/21) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/21) f(1) plus ORIGINAL b = 20

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, S392

S394 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor := mul(19, 2)

After

Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor = 38

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, S393

S395 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b := sum(38, 1)

After

Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b = 39

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, S394

S396 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/21) f(4) ORIGINAL a times divisor := mul(19, 4)

After

Mobius inversion ORIGINAL inverse F(n/21) f(4) ORIGINAL a times divisor = 76

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, S395

S397 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/21) f(4) plus ORIGINAL b := sum(76, 1)

After

Mobius inversion ORIGINAL inverse F(n/21) f(4) plus ORIGINAL b = 77

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, S396

S398 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/21) f(5) ORIGINAL a times divisor := mul(19, 5)

After

Mobius inversion ORIGINAL inverse F(n/21) f(5) ORIGINAL a times divisor = 95

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, S397

S399 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/21) f(5) plus ORIGINAL b := sum(95, 1)

After

Mobius inversion ORIGINAL inverse F(n/21) f(5) plus ORIGINAL b = 96

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, S398

S400 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/21) f(10) ORIGINAL a times divisor := mul(19, 10)

After

Mobius inversion ORIGINAL inverse F(n/21) f(10) ORIGINAL a times divisor = 190

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, S399

S401 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/21) f(10) plus ORIGINAL b := sum(190, 1)

After

Mobius inversion ORIGINAL inverse F(n/21) f(10) plus ORIGINAL b = 191

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, S400

S402 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/21) f(20) ORIGINAL a times divisor := mul(19, 20)

After

Mobius inversion ORIGINAL inverse F(n/21) f(20) ORIGINAL a times divisor = 380

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, S401

S403 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/21) f(20) plus ORIGINAL b := sum(380, 1)

After

Mobius inversion ORIGINAL inverse F(n/21) f(20) plus ORIGINAL b = 381

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, S402

S404 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum := sum(20, 39, 77, 96, 191, 381)

After

Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum = 804

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, S403

S405 · calculation

Before

Mobius inversion ORIGINAL inverse mu(21) times SAME F(n/21) := mul(1, 804)

After

Mobius inversion ORIGINAL inverse mu(21) times SAME F(n/21) = 804

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, S404

S406 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor28:actual positive input for mu=28

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, S405

S407 · implies

Before

ORIGINAL positive integer absolute magnitude=28>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, S406

S408 · calculation

Before

upf extract divisor2 occurrence0 := div(28, 2)

After

upf extract divisor2 occurrence0 = 14

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, S407

S409 · calculation

Before

upf extract divisor2 occurrence1 := div(14, 2)

After

upf extract divisor2 occurrence1 = 7

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, S408

S410 · implies

Before

next divisor=2;previous remaining=28;all smaller divisors exhausted

After

extracted multiplicity=2;next remaining=7;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, S409

S411 · 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, S410

S412 · 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, S411

S413 · calculation

Before

upf canonical complete product factor2 := mul(4, 7)

After

upf canonical complete product factor2 = 28

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, S412

S414 · implies

Before

exhausted trial divisors below3;terminal remaining=7;d*d>remaining;ORIGINAL n=28

After

complete increasing prime multiset=[2, 2, 7];full product=28;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, S413

S415 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor28:input=28;COMPLETE prime multiset=[2, 2, 7];distinct primes=[2, 7]

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, S414

S416 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor28 := div(420, 28)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor28 = 15

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, S415

S417 · implies

Before

Mobius inversion ORIGINAL inverse F(n/28):actual positive input=15;check complete integer range1..15

After

ALL positive divisors INCREASING=[1, 3, 5, 15];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, S416

S418 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/28) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/28) f(1) ORIGINAL a times divisor = 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, S417

S419 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/28) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/28) f(1) plus ORIGINAL b = 20

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, S418

S420 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/28) f(3) ORIGINAL a times divisor := mul(19, 3)

After

Mobius inversion ORIGINAL inverse F(n/28) f(3) ORIGINAL a times divisor = 57

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, S419

S421 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/28) f(3) plus ORIGINAL b := sum(57, 1)

After

Mobius inversion ORIGINAL inverse F(n/28) f(3) plus ORIGINAL b = 58

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, S420

S422 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/28) f(5) ORIGINAL a times divisor := mul(19, 5)

After

Mobius inversion ORIGINAL inverse F(n/28) f(5) ORIGINAL a times divisor = 95

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, S421

S423 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/28) f(5) plus ORIGINAL b := sum(95, 1)

After

Mobius inversion ORIGINAL inverse F(n/28) f(5) plus ORIGINAL b = 96

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, S422

S424 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/28) f(15) ORIGINAL a times divisor := mul(19, 15)

After

Mobius inversion ORIGINAL inverse F(n/28) f(15) ORIGINAL a times divisor = 285

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, S423

S425 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/28) f(15) plus ORIGINAL b := sum(285, 1)

After

Mobius inversion ORIGINAL inverse F(n/28) f(15) plus ORIGINAL b = 286

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, S424

S426 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/28) full forward divisor sum := sum(20, 58, 96, 286)

After

Mobius inversion ORIGINAL inverse F(n/28) full forward divisor sum = 460

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, S425

S427 · calculation

Before

Mobius inversion ORIGINAL inverse mu(28) times SAME F(n/28) := mul(0, 460)

After

Mobius inversion ORIGINAL inverse mu(28) times SAME F(n/28) = 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, S426

S428 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor30:actual positive input for mu=30

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, S427

S429 · implies

Before

ORIGINAL positive integer absolute magnitude=30>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, S428

S430 · calculation

Before

upf extract divisor2 occurrence0 := div(30, 2)

After

upf extract divisor2 occurrence0 = 15

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, S429

S431 · implies

Before

next divisor=2;previous remaining=30;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=15;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, S430

S432 · calculation

Before

upf extract divisor3 occurrence1 := div(15, 3)

After

upf extract divisor3 occurrence1 = 5

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, S431

S433 · implies

Before

next divisor=3;previous remaining=15;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=5;remaining modulo3=2;extracted multiset=[2, 3]

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, S432

S434 · 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, S433

S435 · calculation

Before

upf canonical complete product factor1 := mul(2, 3)

After

upf canonical complete product factor1 = 6

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, S434

S436 · calculation

Before

upf canonical complete product factor2 := mul(6, 5)

After

upf canonical complete product factor2 = 30

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, S435

S437 · implies

Before

exhausted trial divisors below4;terminal remaining=5;d*d>remaining;ORIGINAL n=30

After

complete increasing prime multiset=[2, 3, 5];full product=30;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, S436

S438 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor30:input=30;COMPLETE prime multiset=[2, 3, 5];distinct primes=[2, 3, 5]

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, S437

S439 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor30 := div(420, 30)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor30 = 14

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, S438

S440 · implies

Before

Mobius inversion ORIGINAL inverse F(n/30):actual positive input=14;check complete integer range1..14

After

ALL positive divisors INCREASING=[1, 2, 7, 14];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, S439

S441 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/30) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/30) f(1) ORIGINAL a times divisor = 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, S440

S442 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/30) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/30) f(1) plus ORIGINAL b = 20

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, S441

S443 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/30) f(2) ORIGINAL a times divisor := mul(19, 2)

After

Mobius inversion ORIGINAL inverse F(n/30) f(2) ORIGINAL a times divisor = 38

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, S442

S444 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/30) f(2) plus ORIGINAL b := sum(38, 1)

After

Mobius inversion ORIGINAL inverse F(n/30) f(2) plus ORIGINAL b = 39

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, S443

S445 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/30) f(7) ORIGINAL a times divisor := mul(19, 7)

After

Mobius inversion ORIGINAL inverse F(n/30) f(7) ORIGINAL a times divisor = 133

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, S444

S446 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/30) f(7) plus ORIGINAL b := sum(133, 1)

After

Mobius inversion ORIGINAL inverse F(n/30) f(7) plus ORIGINAL b = 134

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, S445

S447 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/30) f(14) ORIGINAL a times divisor := mul(19, 14)

After

Mobius inversion ORIGINAL inverse F(n/30) f(14) ORIGINAL a times divisor = 266

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, S446

S448 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/30) f(14) plus ORIGINAL b := sum(266, 1)

After

Mobius inversion ORIGINAL inverse F(n/30) f(14) plus ORIGINAL b = 267

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, S447

S449 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/30) full forward divisor sum := sum(20, 39, 134, 267)

After

Mobius inversion ORIGINAL inverse F(n/30) full forward divisor sum = 460

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, S448

S450 · calculation

Before

Mobius inversion ORIGINAL inverse mu(30) times SAME F(n/30) := mul(-1, 460)

After

Mobius inversion ORIGINAL inverse mu(30) times SAME F(n/30) = -460

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, S449

S451 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor35:actual positive input for mu=35

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, S450

S452 · implies

Before

ORIGINAL positive integer absolute magnitude=35>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, S451

S453 · implies

Before

next divisor=2;previous remaining=35;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=35;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, S452

S454 · implies

Before

next divisor=3;previous remaining=35;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=35;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, S453

S455 · implies

Before

next divisor=4;previous remaining=35;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=35;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, S454

S456 · calculation

Before

upf extract divisor5 occurrence0 := div(35, 5)

After

upf extract divisor5 occurrence0 = 7

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, S455

S457 · implies

Before

next divisor=5;previous remaining=35;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=7;remaining modulo5=2;extracted multiset=[5]

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, S456

S458 · calculation

Before

upf canonical complete product factor0 := mul(1, 5)

After

upf canonical complete product factor0 = 5

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, S457

S459 · calculation

Before

upf canonical complete product factor1 := mul(5, 7)

After

upf canonical complete product factor1 = 35

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, S458

S460 · implies

Before

exhausted trial divisors below6;terminal remaining=7;d*d>remaining;ORIGINAL n=35

After

complete increasing prime multiset=[5, 7];full product=35;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, S459

S461 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor35:input=35;COMPLETE prime multiset=[5, 7];distinct primes=[5, 7]

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, S460

S462 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor35 := div(420, 35)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor35 = 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, S461

S463 · implies

Before

Mobius inversion ORIGINAL inverse F(n/35):actual positive input=12;check complete integer range1..12

After

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 12];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, S462

S464 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/35) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/35) f(1) ORIGINAL a times divisor = 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, S463

S465 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/35) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/35) f(1) plus ORIGINAL b = 20

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, S464

S466 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/35) f(2) ORIGINAL a times divisor := mul(19, 2)

After

Mobius inversion ORIGINAL inverse F(n/35) f(2) ORIGINAL a times divisor = 38

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, S465

S467 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/35) f(2) plus ORIGINAL b := sum(38, 1)

After

Mobius inversion ORIGINAL inverse F(n/35) f(2) plus ORIGINAL b = 39

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, S466

S468 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/35) f(3) ORIGINAL a times divisor := mul(19, 3)

After

Mobius inversion ORIGINAL inverse F(n/35) f(3) ORIGINAL a times divisor = 57

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, S467

S469 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/35) f(3) plus ORIGINAL b := sum(57, 1)

After

Mobius inversion ORIGINAL inverse F(n/35) f(3) plus ORIGINAL b = 58

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, S468

S470 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/35) f(4) ORIGINAL a times divisor := mul(19, 4)

After

Mobius inversion ORIGINAL inverse F(n/35) f(4) ORIGINAL a times divisor = 76

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, S469

S471 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/35) f(4) plus ORIGINAL b := sum(76, 1)

After

Mobius inversion ORIGINAL inverse F(n/35) f(4) plus ORIGINAL b = 77

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, S470

S472 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/35) f(6) ORIGINAL a times divisor := mul(19, 6)

After

Mobius inversion ORIGINAL inverse F(n/35) f(6) ORIGINAL a times divisor = 114

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, S471

S473 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/35) f(6) plus ORIGINAL b := sum(114, 1)

After

Mobius inversion ORIGINAL inverse F(n/35) f(6) plus ORIGINAL b = 115

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, S472

S474 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/35) f(12) ORIGINAL a times divisor := mul(19, 12)

After

Mobius inversion ORIGINAL inverse F(n/35) f(12) ORIGINAL a times divisor = 228

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, S473

S475 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/35) f(12) plus ORIGINAL b := sum(228, 1)

After

Mobius inversion ORIGINAL inverse F(n/35) f(12) plus ORIGINAL b = 229

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, S474

S476 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/35) full forward divisor sum := sum(20, 39, 58, 77, 115, 229)

After

Mobius inversion ORIGINAL inverse F(n/35) full forward divisor sum = 538

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, S475

S477 · calculation

Before

Mobius inversion ORIGINAL inverse mu(35) times SAME F(n/35) := mul(1, 538)

After

Mobius inversion ORIGINAL inverse mu(35) times SAME F(n/35) = 538

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, S476

S478 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor42:actual positive input for mu=42

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, S477

S479 · implies

Before

ORIGINAL positive integer absolute magnitude=42>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, S478

S480 · calculation

Before

upf extract divisor2 occurrence0 := div(42, 2)

After

upf extract divisor2 occurrence0 = 21

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, S479

S481 · implies

Before

next divisor=2;previous remaining=42;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=21;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, S480

S482 · calculation

Before

upf extract divisor3 occurrence1 := div(21, 3)

After

upf extract divisor3 occurrence1 = 7

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, S481

S483 · implies

Before

next divisor=3;previous remaining=21;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=7;remaining modulo3=1;extracted multiset=[2, 3]

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, S482

S484 · 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, S483

S485 · calculation

Before

upf canonical complete product factor1 := mul(2, 3)

After

upf canonical complete product factor1 = 6

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, S484

S486 · calculation

Before

upf canonical complete product factor2 := mul(6, 7)

After

upf canonical complete product factor2 = 42

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, S485

S487 · implies

Before

exhausted trial divisors below4;terminal remaining=7;d*d>remaining;ORIGINAL n=42

After

complete increasing prime multiset=[2, 3, 7];full product=42;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, S486

S488 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor42:input=42;COMPLETE prime multiset=[2, 3, 7];distinct primes=[2, 3, 7]

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, S487

S489 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor42 := div(420, 42)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor42 = 10

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, S488

S490 · implies

Before

Mobius inversion ORIGINAL inverse F(n/42):actual positive input=10;check complete integer range1..10

After

ALL positive divisors INCREASING=[1, 2, 5, 10];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, S489

S491 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor = 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, S490

S492 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b = 20

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, S491

S493 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/42) f(2) ORIGINAL a times divisor := mul(19, 2)

After

Mobius inversion ORIGINAL inverse F(n/42) f(2) ORIGINAL a times divisor = 38

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, S492

S494 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/42) f(2) plus ORIGINAL b := sum(38, 1)

After

Mobius inversion ORIGINAL inverse F(n/42) f(2) plus ORIGINAL b = 39

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, S493

S495 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/42) f(5) ORIGINAL a times divisor := mul(19, 5)

After

Mobius inversion ORIGINAL inverse F(n/42) f(5) ORIGINAL a times divisor = 95

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, S494

S496 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/42) f(5) plus ORIGINAL b := sum(95, 1)

After

Mobius inversion ORIGINAL inverse F(n/42) f(5) plus ORIGINAL b = 96

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, S495

S497 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/42) f(10) ORIGINAL a times divisor := mul(19, 10)

After

Mobius inversion ORIGINAL inverse F(n/42) f(10) ORIGINAL a times divisor = 190

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, S496

S498 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/42) f(10) plus ORIGINAL b := sum(190, 1)

After

Mobius inversion ORIGINAL inverse F(n/42) f(10) plus ORIGINAL b = 191

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, S497

S499 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum := sum(20, 39, 96, 191)

After

Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum = 346

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, S498

S500 · calculation

Before

Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42) := mul(-1, 346)

After

Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42) = -346

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, S499

S501 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor60:actual positive input for mu=60

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, S500

S502 · implies

Before

ORIGINAL positive integer absolute magnitude=60>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, S501

S503 · calculation

Before

upf extract divisor2 occurrence0 := div(60, 2)

After

upf extract divisor2 occurrence0 = 30

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, S502

S504 · calculation

Before

upf extract divisor2 occurrence1 := div(30, 2)

After

upf extract divisor2 occurrence1 = 15

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, S503

S505 · implies

Before

next divisor=2;previous remaining=60;all smaller divisors exhausted

After

extracted multiplicity=2;next remaining=15;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, S504

S506 · calculation

Before

upf extract divisor3 occurrence2 := div(15, 3)

After

upf extract divisor3 occurrence2 = 5

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, S505

S507 · implies

Before

next divisor=3;previous remaining=15;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=5;remaining modulo3=2;extracted multiset=[2, 2, 3]

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, S506

S508 · 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, S507

S509 · 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, S508

S510 · calculation

Before

upf canonical complete product factor2 := mul(4, 3)

After

upf canonical complete product factor2 = 12

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, S509

S511 · calculation

Before

upf canonical complete product factor3 := mul(12, 5)

After

upf canonical complete product factor3 = 60

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, S510

S512 · implies

Before

exhausted trial divisors below4;terminal remaining=5;d*d>remaining;ORIGINAL n=60

After

complete increasing prime multiset=[2, 2, 3, 5];full product=60;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, S511

S513 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor60:input=60;COMPLETE prime multiset=[2, 2, 3, 5];distinct primes=[2, 3, 5]

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, S512

S514 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor60 := div(420, 60)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor60 = 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, S513

S515 · implies

Before

Mobius inversion ORIGINAL inverse F(n/60):actual positive input=7;check complete integer range1..7

After

ALL positive divisors INCREASING=[1, 7];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, S514

S516 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/60) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/60) f(1) ORIGINAL a times divisor = 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, S515

S517 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/60) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/60) f(1) plus ORIGINAL b = 20

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, S516

S518 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/60) f(7) ORIGINAL a times divisor := mul(19, 7)

After

Mobius inversion ORIGINAL inverse F(n/60) f(7) ORIGINAL a times divisor = 133

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, S517

S519 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/60) f(7) plus ORIGINAL b := sum(133, 1)

After

Mobius inversion ORIGINAL inverse F(n/60) f(7) plus ORIGINAL b = 134

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, S518

S520 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/60) full forward divisor sum := sum(20, 134)

After

Mobius inversion ORIGINAL inverse F(n/60) full forward divisor sum = 154

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, S519

S521 · calculation

Before

Mobius inversion ORIGINAL inverse mu(60) times SAME F(n/60) := mul(0, 154)

After

Mobius inversion ORIGINAL inverse mu(60) times SAME F(n/60) = 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, S520

S522 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor70:actual positive input for mu=70

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, S521

S523 · implies

Before

ORIGINAL positive integer absolute magnitude=70>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, S522

S524 · calculation

Before

upf extract divisor2 occurrence0 := div(70, 2)

After

upf extract divisor2 occurrence0 = 35

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, S523

S525 · implies

Before

next divisor=2;previous remaining=70;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=35;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, S524

S526 · implies

Before

next divisor=3;previous remaining=35;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=35;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, S525

S527 · implies

Before

next divisor=4;previous remaining=35;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=35;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, S526

S528 · calculation

Before

upf extract divisor5 occurrence1 := div(35, 5)

After

upf extract divisor5 occurrence1 = 7

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, S527

S529 · implies

Before

next divisor=5;previous remaining=35;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=7;remaining modulo5=2;extracted multiset=[2, 5]

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, S528

S530 · 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, S529

S531 · calculation

Before

upf canonical complete product factor1 := mul(2, 5)

After

upf canonical complete product factor1 = 10

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, S530

S532 · calculation

Before

upf canonical complete product factor2 := mul(10, 7)

After

upf canonical complete product factor2 = 70

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, S531

S533 · implies

Before

exhausted trial divisors below6;terminal remaining=7;d*d>remaining;ORIGINAL n=70

After

complete increasing prime multiset=[2, 5, 7];full product=70;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, S532

S534 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor70:input=70;COMPLETE prime multiset=[2, 5, 7];distinct primes=[2, 5, 7]

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, S533

S535 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor70 := div(420, 70)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor70 = 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, S534

S536 · implies

Before

Mobius inversion ORIGINAL inverse F(n/70):actual positive input=6;check complete integer range1..6

After

ALL positive divisors INCREASING=[1, 2, 3, 6];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, S535

S537 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/70) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/70) f(1) ORIGINAL a times divisor = 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, S536

S538 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/70) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/70) f(1) plus ORIGINAL b = 20

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, S537

S539 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/70) f(2) ORIGINAL a times divisor := mul(19, 2)

After

Mobius inversion ORIGINAL inverse F(n/70) f(2) ORIGINAL a times divisor = 38

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, S538

S540 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/70) f(2) plus ORIGINAL b := sum(38, 1)

After

Mobius inversion ORIGINAL inverse F(n/70) f(2) plus ORIGINAL b = 39

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, S539

S541 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/70) f(3) ORIGINAL a times divisor := mul(19, 3)

After

Mobius inversion ORIGINAL inverse F(n/70) f(3) ORIGINAL a times divisor = 57

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, S540

S542 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/70) f(3) plus ORIGINAL b := sum(57, 1)

After

Mobius inversion ORIGINAL inverse F(n/70) f(3) plus ORIGINAL b = 58

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, S541

S543 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/70) f(6) ORIGINAL a times divisor := mul(19, 6)

After

Mobius inversion ORIGINAL inverse F(n/70) f(6) ORIGINAL a times divisor = 114

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, S542

S544 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/70) f(6) plus ORIGINAL b := sum(114, 1)

After

Mobius inversion ORIGINAL inverse F(n/70) f(6) plus ORIGINAL b = 115

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, S543

S545 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/70) full forward divisor sum := sum(20, 39, 58, 115)

After

Mobius inversion ORIGINAL inverse F(n/70) full forward divisor sum = 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, S544

S546 · calculation

Before

Mobius inversion ORIGINAL inverse mu(70) times SAME F(n/70) := mul(-1, 232)

After

Mobius inversion ORIGINAL inverse mu(70) times SAME F(n/70) = -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, S545

S547 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor84:actual positive input for mu=84

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, S546

S548 · implies

Before

ORIGINAL positive integer absolute magnitude=84>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, S547

S549 · calculation

Before

upf extract divisor2 occurrence0 := div(84, 2)

After

upf extract divisor2 occurrence0 = 42

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, S548

S550 · calculation

Before

upf extract divisor2 occurrence1 := div(42, 2)

After

upf extract divisor2 occurrence1 = 21

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, S549

S551 · implies

Before

next divisor=2;previous remaining=84;all smaller divisors exhausted

After

extracted multiplicity=2;next remaining=21;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, S550

S552 · calculation

Before

upf extract divisor3 occurrence2 := div(21, 3)

After

upf extract divisor3 occurrence2 = 7

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, S551

S553 · implies

Before

next divisor=3;previous remaining=21;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=7;remaining modulo3=1;extracted multiset=[2, 2, 3]

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, S552

S554 · 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, S553

S555 · 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, S554

S556 · calculation

Before

upf canonical complete product factor2 := mul(4, 3)

After

upf canonical complete product factor2 = 12

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, S555

S557 · calculation

Before

upf canonical complete product factor3 := mul(12, 7)

After

upf canonical complete product factor3 = 84

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, S556

S558 · implies

Before

exhausted trial divisors below4;terminal remaining=7;d*d>remaining;ORIGINAL n=84

After

complete increasing prime multiset=[2, 2, 3, 7];full product=84;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, S557

S559 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor84:input=84;COMPLETE prime multiset=[2, 2, 3, 7];distinct primes=[2, 3, 7]

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, S558

S560 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor84 := div(420, 84)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor84 = 5

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, S559

S561 · implies

Before

Mobius inversion ORIGINAL inverse F(n/84):actual positive input=5;check complete integer range1..5

After

ALL positive divisors INCREASING=[1, 5];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, S560

S562 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/84) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/84) f(1) ORIGINAL a times divisor = 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, S561

S563 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/84) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/84) f(1) plus ORIGINAL b = 20

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, S562

S564 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/84) f(5) ORIGINAL a times divisor := mul(19, 5)

After

Mobius inversion ORIGINAL inverse F(n/84) f(5) ORIGINAL a times divisor = 95

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, S563

S565 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/84) f(5) plus ORIGINAL b := sum(95, 1)

After

Mobius inversion ORIGINAL inverse F(n/84) f(5) plus ORIGINAL b = 96

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, S564

S566 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/84) full forward divisor sum := sum(20, 96)

After

Mobius inversion ORIGINAL inverse F(n/84) full forward divisor sum = 116

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, S565

S567 · calculation

Before

Mobius inversion ORIGINAL inverse mu(84) times SAME F(n/84) := mul(0, 116)

After

Mobius inversion ORIGINAL inverse mu(84) times SAME F(n/84) = 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, S566

S568 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor105:actual positive input for mu=105

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, S567

S569 · implies

Before

ORIGINAL positive integer absolute magnitude=105>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, S568

S570 · implies

Before

next divisor=2;previous remaining=105;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=105;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, S569

S571 · calculation

Before

upf extract divisor3 occurrence0 := div(105, 3)

After

upf extract divisor3 occurrence0 = 35

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, S570

S572 · implies

Before

next divisor=3;previous remaining=105;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=35;remaining modulo3=2;extracted multiset=[3]

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, S571

S573 · implies

Before

next divisor=4;previous remaining=35;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=35;remaining modulo4=3;extracted multiset=[3]

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, S572

S574 · calculation

Before

upf extract divisor5 occurrence1 := div(35, 5)

After

upf extract divisor5 occurrence1 = 7

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, S573

S575 · implies

Before

next divisor=5;previous remaining=35;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=7;remaining modulo5=2;extracted multiset=[3, 5]

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, S574

S576 · calculation

Before

upf canonical complete product factor0 := mul(1, 3)

After

upf canonical complete product factor0 = 3

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, S575

S577 · calculation

Before

upf canonical complete product factor1 := mul(3, 5)

After

upf canonical complete product factor1 = 15

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, S576

S578 · calculation

Before

upf canonical complete product factor2 := mul(15, 7)

After

upf canonical complete product factor2 = 105

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, S577

S579 · implies

Before

exhausted trial divisors below6;terminal remaining=7;d*d>remaining;ORIGINAL n=105

After

complete increasing prime multiset=[3, 5, 7];full product=105;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, S578

S580 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor105:input=105;COMPLETE prime multiset=[3, 5, 7];distinct primes=[3, 5, 7]

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, S579

S581 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor105 := div(420, 105)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor105 = 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, S580

S582 · implies

Before

Mobius inversion ORIGINAL inverse F(n/105):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, S581

S583 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/105) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/105) f(1) ORIGINAL a times divisor = 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, S582

S584 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/105) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/105) f(1) plus ORIGINAL b = 20

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, S583

S585 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/105) f(2) ORIGINAL a times divisor := mul(19, 2)

After

Mobius inversion ORIGINAL inverse F(n/105) f(2) ORIGINAL a times divisor = 38

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, S584

S586 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/105) f(2) plus ORIGINAL b := sum(38, 1)

After

Mobius inversion ORIGINAL inverse F(n/105) f(2) plus ORIGINAL b = 39

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, S585

S587 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/105) f(4) ORIGINAL a times divisor := mul(19, 4)

After

Mobius inversion ORIGINAL inverse F(n/105) f(4) ORIGINAL a times divisor = 76

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, S586

S588 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/105) f(4) plus ORIGINAL b := sum(76, 1)

After

Mobius inversion ORIGINAL inverse F(n/105) f(4) plus ORIGINAL b = 77

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, S587

S589 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/105) full forward divisor sum := sum(20, 39, 77)

After

Mobius inversion ORIGINAL inverse F(n/105) full forward divisor sum = 136

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, S588

S590 · calculation

Before

Mobius inversion ORIGINAL inverse mu(105) times SAME F(n/105) := mul(-1, 136)

After

Mobius inversion ORIGINAL inverse mu(105) times SAME F(n/105) = -136

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, S589

S591 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor140:actual positive input for mu=140

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, S590

S592 · implies

Before

ORIGINAL positive integer absolute magnitude=140>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, S591

S593 · calculation

Before

upf extract divisor2 occurrence0 := div(140, 2)

After

upf extract divisor2 occurrence0 = 70

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, S592

S594 · calculation

Before

upf extract divisor2 occurrence1 := div(70, 2)

After

upf extract divisor2 occurrence1 = 35

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, S593

S595 · implies

Before

next divisor=2;previous remaining=140;all smaller divisors exhausted

After

extracted multiplicity=2;next remaining=35;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, S594

S596 · implies

Before

next divisor=3;previous remaining=35;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=35;remaining modulo3=2;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, S595

S597 · implies

Before

next divisor=4;previous remaining=35;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=35;remaining modulo4=3;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, S596

S598 · calculation

Before

upf extract divisor5 occurrence2 := div(35, 5)

After

upf extract divisor5 occurrence2 = 7

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, S597

S599 · implies

Before

next divisor=5;previous remaining=35;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=7;remaining modulo5=2;extracted multiset=[2, 2, 5]

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, S598

S600 · 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, S599

S601 · 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, S600

S602 · calculation

Before

upf canonical complete product factor2 := mul(4, 5)

After

upf canonical complete product factor2 = 20

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, S601

S603 · calculation

Before

upf canonical complete product factor3 := mul(20, 7)

After

upf canonical complete product factor3 = 140

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, S602

S604 · implies

Before

exhausted trial divisors below6;terminal remaining=7;d*d>remaining;ORIGINAL n=140

After

complete increasing prime multiset=[2, 2, 5, 7];full product=140;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, S603

S605 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor140:input=140;COMPLETE prime multiset=[2, 2, 5, 7];distinct primes=[2, 5, 7]

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, S604

S606 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor140 := div(420, 140)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor140 = 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, S605

S607 · implies

Before

Mobius inversion ORIGINAL inverse F(n/140):actual positive input=3;check complete integer range1..3

After

ALL positive divisors INCREASING=[1, 3];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, S606

S608 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/140) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/140) f(1) ORIGINAL a times divisor = 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, S607

S609 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/140) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/140) f(1) plus ORIGINAL b = 20

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, S608

S610 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/140) f(3) ORIGINAL a times divisor := mul(19, 3)

After

Mobius inversion ORIGINAL inverse F(n/140) f(3) ORIGINAL a times divisor = 57

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, S609

S611 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/140) f(3) plus ORIGINAL b := sum(57, 1)

After

Mobius inversion ORIGINAL inverse F(n/140) f(3) plus ORIGINAL b = 58

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, S610

S612 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/140) full forward divisor sum := sum(20, 58)

After

Mobius inversion ORIGINAL inverse F(n/140) full forward divisor sum = 78

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, S611

S613 · calculation

Before

Mobius inversion ORIGINAL inverse mu(140) times SAME F(n/140) := mul(0, 78)

After

Mobius inversion ORIGINAL inverse mu(140) times SAME F(n/140) = 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, S612

S614 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor210:actual positive input for mu=210

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, S613

S615 · implies

Before

ORIGINAL positive integer absolute magnitude=210>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, S614

S616 · calculation

Before

upf extract divisor2 occurrence0 := div(210, 2)

After

upf extract divisor2 occurrence0 = 105

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, S615

S617 · implies

Before

next divisor=2;previous remaining=210;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=105;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, S616

S618 · calculation

Before

upf extract divisor3 occurrence1 := div(105, 3)

After

upf extract divisor3 occurrence1 = 35

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, S617

S619 · implies

Before

next divisor=3;previous remaining=105;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=35;remaining modulo3=2;extracted multiset=[2, 3]

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, S618

S620 · implies

Before

next divisor=4;previous remaining=35;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=35;remaining modulo4=3;extracted multiset=[2, 3]

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, S619

S621 · calculation

Before

upf extract divisor5 occurrence2 := div(35, 5)

After

upf extract divisor5 occurrence2 = 7

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, S620

S622 · implies

Before

next divisor=5;previous remaining=35;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=7;remaining modulo5=2;extracted multiset=[2, 3, 5]

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, S621

S623 · 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, S622

S624 · calculation

Before

upf canonical complete product factor1 := mul(2, 3)

After

upf canonical complete product factor1 = 6

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, S623

S625 · calculation

Before

upf canonical complete product factor2 := mul(6, 5)

After

upf canonical complete product factor2 = 30

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, S624

S626 · calculation

Before

upf canonical complete product factor3 := mul(30, 7)

After

upf canonical complete product factor3 = 210

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, S625

S627 · implies

Before

exhausted trial divisors below6;terminal remaining=7;d*d>remaining;ORIGINAL n=210

After

complete increasing prime multiset=[2, 3, 5, 7];full product=210;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, S626

S628 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor210:input=210;COMPLETE prime multiset=[2, 3, 5, 7];distinct primes=[2, 3, 5, 7]

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, S627

S629 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor210 := div(420, 210)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor210 = 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, S628

S630 · implies

Before

Mobius inversion ORIGINAL inverse F(n/210):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, S629

S631 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/210) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/210) f(1) ORIGINAL a times divisor = 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, S630

S632 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/210) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/210) f(1) plus ORIGINAL b = 20

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, S631

S633 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/210) f(2) ORIGINAL a times divisor := mul(19, 2)

After

Mobius inversion ORIGINAL inverse F(n/210) f(2) ORIGINAL a times divisor = 38

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, S632

S634 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/210) f(2) plus ORIGINAL b := sum(38, 1)

After

Mobius inversion ORIGINAL inverse F(n/210) f(2) plus ORIGINAL b = 39

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, S633

S635 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/210) full forward divisor sum := sum(20, 39)

After

Mobius inversion ORIGINAL inverse F(n/210) full forward divisor sum = 59

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, S634

S636 · calculation

Before

Mobius inversion ORIGINAL inverse mu(210) times SAME F(n/210) := mul(1, 59)

After

Mobius inversion ORIGINAL inverse mu(210) times SAME F(n/210) = 59

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, S635

S637 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor420:actual positive input for mu=420

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, S636

S638 · implies

Before

ORIGINAL positive integer absolute magnitude=420>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, S637

S639 · calculation

Before

upf extract divisor2 occurrence0 := div(420, 2)

After

upf extract divisor2 occurrence0 = 210

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, S638

S640 · calculation

Before

upf extract divisor2 occurrence1 := div(210, 2)

After

upf extract divisor2 occurrence1 = 105

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, S639

S641 · implies

Before

next divisor=2;previous remaining=420;all smaller divisors exhausted

After

extracted multiplicity=2;next remaining=105;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, S640

S642 · calculation

Before

upf extract divisor3 occurrence2 := div(105, 3)

After

upf extract divisor3 occurrence2 = 35

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, S641

S643 · implies

Before

next divisor=3;previous remaining=105;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=35;remaining modulo3=2;extracted multiset=[2, 2, 3]

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, S642

S644 · implies

Before

next divisor=4;previous remaining=35;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=35;remaining modulo4=3;extracted multiset=[2, 2, 3]

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, S643

S645 · calculation

Before

upf extract divisor5 occurrence3 := div(35, 5)

After

upf extract divisor5 occurrence3 = 7

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, S644

S646 · implies

Before

next divisor=5;previous remaining=35;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=7;remaining modulo5=2;extracted multiset=[2, 2, 3, 5]

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, S645

S647 · 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, S646

S648 · 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, S647

S649 · calculation

Before

upf canonical complete product factor2 := mul(4, 3)

After

upf canonical complete product factor2 = 12

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, S648

S650 · calculation

Before

upf canonical complete product factor3 := mul(12, 5)

After

upf canonical complete product factor3 = 60

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, S649

S651 · calculation

Before

upf canonical complete product factor4 := mul(60, 7)

After

upf canonical complete product factor4 = 420

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, S650

S652 · implies

Before

exhausted trial divisors below6;terminal remaining=7;d*d>remaining;ORIGINAL n=420

After

complete increasing prime multiset=[2, 2, 3, 5, 7];full product=420;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, S651

S653 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor420:input=420;COMPLETE prime multiset=[2, 2, 3, 5, 7];distinct primes=[2, 3, 5, 7]

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, S652

S654 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor420 := div(420, 420)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor420 = 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, S653

S655 · implies

Before

Mobius inversion ORIGINAL inverse F(n/420):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, S654

S656 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/420) f(1) ORIGINAL a times divisor := mul(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/420) f(1) ORIGINAL a times divisor = 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, S655

S657 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/420) f(1) plus ORIGINAL b := sum(19, 1)

After

Mobius inversion ORIGINAL inverse F(n/420) f(1) plus ORIGINAL b = 20

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, S656

S658 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/420) full forward divisor sum := sum(20)

After

Mobius inversion ORIGINAL inverse F(n/420) full forward divisor sum = 20

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, S657

S659 · calculation

Before

Mobius inversion ORIGINAL inverse mu(420) times SAME F(n/420) := mul(0, 20)

After

Mobius inversion ORIGINAL inverse mu(420) times SAME F(n/420) = 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, S658

S660 · calculation

Before

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(25560, -10960, -6396, 0, -4268, 2744, -3204, 1832, 0, 1376, 1070, 0, 804, 0, -460, 538, -346, 0, -232, 0, -136, 0, 59, 0)

After

Mobius inversion ORIGINAL inverse complete inverse-term sum = 7981

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, S659

S661 · calculation

Before

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(19, 420)

After

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 7980

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, S660

S662 · calculation

Before

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(7980, 1)

After

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 7981

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, S661

S663 · implies

Before

Mobius inversion ORIGINAL inverse:actual n=420;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 420, 25560, 25560], [2, -1, 210, 10960, -10960], [3, -1, 140, 6396, -6396], [4, 0, 105, 3656, 0], [5, -1, 84, 4268, -4268], [6, 1, 70, 2744, 2744], [7, -1, 60, 3204, -3204], [10, 1, 42, 1832, 1832], [12, 0, 35, 916, 0], [14, 1, 30, 1376, 1376], [15, 1, 28, 1070, 1070], [20, 0, 21, 612, 0], [21, 1, 20, 804, 804], [28, 0, 15, 460, 0], [30, -1, 14, 460, -460], [35, 1, 12, 538, 538], [42, -1, 10, 346, -346], [60, 0, 7, 154, 0], [70, -1, 6, 232, -232], [84, 0, 5, 116, 0], [105, -1, 4, 136, -136], [140, 0, 3, 78, 0], [210, 1, 2, 59, 59], [420, 0, 1, 20, 0]]

After

complete reconstructed=7981;direct f(n)=7981;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, S662

S664 · calculation

Before

Mobius inversion ORIGINAL omitted-divisor partial sum := sum(-10960, -6396, 0, -4268, 2744, -3204, 1832, 0, 1376, 1070, 0, 804, 0, -460, 538, -346, 0, -232, 0, -136, 0, 59, 0)

After

Mobius inversion ORIGINAL omitted-divisor partial sum = -17579

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, S663

S665 · calculation

Before

Mobius inversion complete minus partial := sub(7981, -17579)

After

Mobius inversion complete minus partial = 25560

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, S664

S666 · implies

Before

ORIGINAL full divisors=[1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420]

After

complete=7981=f(n)=7981;partial=-17579;lost term=25560!=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, S665

Reverse checks

S666 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S665 · premise_and_result_audit — Recompute Mobius inversion complete minus partial with exact rational arithmetic, not rounded decimals.
S664 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL omitted-divisor partial sum with exact rational arithmetic, not rounded decimals.
S663 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S662 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S661 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
S660 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
S659 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(420) times SAME F(n/420) with exact rational arithmetic, not rounded decimals.
S658 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/420) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S657 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/420) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S656 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/420) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S655 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S654 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor420 with exact rational arithmetic, not rounded decimals.
S653 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S652 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S651 · premise_and_result_audit — Recompute upf canonical complete product factor4 with exact rational arithmetic, not rounded decimals.
S650 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals.
S649 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
S648 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S647 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S646 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S645 · premise_and_result_audit — Recompute upf extract divisor5 occurrence3 with exact rational arithmetic, not rounded decimals.
S644 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S643 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S642 · premise_and_result_audit — Recompute upf extract divisor3 occurrence2 with exact rational arithmetic, not rounded decimals.
S641 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S640 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
S639 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
S638 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S637 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S636 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(210) times SAME F(n/210) with exact rational arithmetic, not rounded decimals.
S635 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/210) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S634 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/210) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S633 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/210) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S632 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/210) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S631 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/210) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S630 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S629 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor210 with exact rational arithmetic, not rounded decimals.
S628 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S627 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S626 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals.
S625 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
S624 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S623 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S622 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S621 · premise_and_result_audit — Recompute upf extract divisor5 occurrence2 with exact rational arithmetic, not rounded decimals.
S620 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S619 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S618 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
S617 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S616 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
S615 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S614 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S613 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(140) times SAME F(n/140) with exact rational arithmetic, not rounded decimals.
S612 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/140) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S611 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/140) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S610 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/140) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S609 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/140) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S608 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/140) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S607 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S606 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor140 with exact rational arithmetic, not rounded decimals.
S605 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S604 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S603 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals.
S602 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
S601 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S600 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S599 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S598 · premise_and_result_audit — Recompute upf extract divisor5 occurrence2 with exact rational arithmetic, not rounded decimals.
S597 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S596 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S595 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S594 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
S593 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
S592 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S591 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S590 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(105) times SAME F(n/105) with exact rational arithmetic, not rounded decimals.
S589 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S588 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S587 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S586 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S585 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S584 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S583 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S582 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S581 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor105 with exact rational arithmetic, not rounded decimals.
S580 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S579 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S578 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
S577 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S576 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S575 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S574 · premise_and_result_audit — Recompute upf extract divisor5 occurrence1 with exact rational arithmetic, not rounded decimals.
S573 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S572 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S571 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals.
S570 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S569 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S568 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S567 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(84) times SAME F(n/84) with exact rational arithmetic, not rounded decimals.
S566 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/84) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S565 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/84) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S564 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/84) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S563 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/84) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S562 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/84) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S561 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S560 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor84 with exact rational arithmetic, not rounded decimals.
S559 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S558 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S557 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals.
S556 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
S555 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S554 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S553 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S552 · premise_and_result_audit — Recompute upf extract divisor3 occurrence2 with exact rational arithmetic, not rounded decimals.
S551 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S550 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
S549 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
S548 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S547 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S546 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(70) times SAME F(n/70) with exact rational arithmetic, not rounded decimals.
S545 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S544 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S543 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S542 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S541 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S540 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S539 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S538 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S537 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S536 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S535 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor70 with exact rational arithmetic, not rounded decimals.
S534 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S533 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S532 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
S531 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S530 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S529 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S528 · premise_and_result_audit — Recompute upf extract divisor5 occurrence1 with exact rational arithmetic, not rounded decimals.
S527 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S526 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S525 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S524 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
S523 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S522 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S521 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(60) times SAME F(n/60) with exact rational arithmetic, not rounded decimals.
S520 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/60) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S519 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/60) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S518 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/60) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S517 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/60) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S516 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/60) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S515 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S514 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor60 with exact rational arithmetic, not rounded decimals.
S513 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S512 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S511 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals.
S510 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
S509 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S508 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S507 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S506 · premise_and_result_audit — Recompute upf extract divisor3 occurrence2 with exact rational arithmetic, not rounded decimals.
S505 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S504 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
S503 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
S502 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S501 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S500 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42) with exact rational arithmetic, not rounded decimals.
S499 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S498 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S497 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S496 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S495 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S494 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S493 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S492 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S491 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S490 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S489 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor42 with exact rational arithmetic, not rounded decimals.
S488 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S487 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S486 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
S485 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S484 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S483 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S482 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
S481 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S480 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
S479 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S478 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S477 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(35) times SAME F(n/35) with exact rational arithmetic, not rounded decimals.
S476 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S475 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(12) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S474 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(12) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S473 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S472 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S471 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S470 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S469 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S468 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S467 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S466 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S465 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S464 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S463 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S462 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor35 with exact rational arithmetic, not rounded decimals.
S461 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S460 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S459 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S458 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S457 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S456 · premise_and_result_audit — Recompute upf extract divisor5 occurrence0 with exact rational arithmetic, not rounded decimals.
S455 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S454 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S453 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S452 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S451 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S450 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(30) times SAME F(n/30) with exact rational arithmetic, not rounded decimals.
S449 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S448 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S447 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S446 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S445 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S444 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S443 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S442 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S441 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S440 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S439 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor30 with exact rational arithmetic, not rounded decimals.
S438 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S437 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S436 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
S435 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S434 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S433 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S432 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
S431 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S430 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
S429 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S428 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S427 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(28) times SAME F(n/28) with exact rational arithmetic, not rounded decimals.
S426 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/28) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S425 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/28) f(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S424 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/28) f(15) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S423 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/28) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S422 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/28) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S421 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/28) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S420 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/28) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S419 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/28) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S418 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/28) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S417 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S416 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor28 with exact rational arithmetic, not rounded decimals.
S415 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S414 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S413 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
S412 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S411 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S410 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S409 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
S408 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
S407 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S406 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S405 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(21) times SAME F(n/21) with exact rational arithmetic, not rounded decimals.
S404 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S403 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(20) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S402 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(20) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S401 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S400 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S399 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S398 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S397 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S396 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S395 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S394 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S393 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S392 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S391 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S390 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21 with exact rational arithmetic, not rounded decimals.
S389 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S388 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S387 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S386 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S385 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S384 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals.
S383 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S382 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S381 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S380 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(20) times SAME F(n/20) with exact rational arithmetic, not rounded decimals.
S379 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S378 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S377 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S376 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S375 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S374 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S373 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S372 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S371 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S370 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S369 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor20 with exact rational arithmetic, not rounded decimals.
S368 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S367 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S366 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
S365 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S364 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S363 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S362 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
S361 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
S360 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S359 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S358 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(15) times SAME F(n/15) with exact rational arithmetic, not rounded decimals.
S357 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S356 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(28) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S355 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(28) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S354 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S353 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S352 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S351 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S350 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S349 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S348 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S347 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S346 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S345 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S344 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S343 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor15 with exact rational arithmetic, not rounded decimals.
S342 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S341 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S340 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S339 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S338 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S337 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals.
S336 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S335 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S334 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S333 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) with exact rational arithmetic, not rounded decimals.
S332 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S331 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(30) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S330 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(30) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S329 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S328 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(15) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S327 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S326 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S325 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S324 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S323 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S322 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S321 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S320 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S319 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S318 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S317 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S316 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S315 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S314 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 with exact rational arithmetic, not rounded decimals.
S313 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S312 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S311 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S310 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S309 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S308 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
S307 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S306 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S305 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(12) times SAME F(n/12) with exact rational arithmetic, not rounded decimals.
S304 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S303 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(35) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S302 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(35) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S301 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S300 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S299 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S298 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S297 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S296 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S295 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S294 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12 with exact rational arithmetic, not rounded decimals.
S293 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S292 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S291 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
S290 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S289 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S288 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S287 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
S286 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
S285 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S284 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S283 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) with exact rational arithmetic, not rounded decimals.
S282 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S281 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(42) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S280 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(42) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S279 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S278 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S277 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S276 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S275 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S274 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S273 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S272 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S271 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S270 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S269 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S268 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S267 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S266 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S265 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S264 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 with exact rational arithmetic, not rounded decimals.
S263 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S262 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S261 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S260 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S259 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S258 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
S257 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S256 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S255 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) with exact rational arithmetic, not rounded decimals.
S254 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S253 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(60) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S252 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(60) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S251 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(30) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S250 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(30) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S249 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(20) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S248 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(20) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S247 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S246 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(15) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S245 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(12) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S244 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(12) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S243 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S242 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S241 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S240 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S239 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S238 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S237 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S236 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S235 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S234 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S233 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S232 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S231 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S230 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S229 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S228 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 with exact rational arithmetic, not rounded decimals.
S227 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S226 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S225 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S224 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S223 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S222 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S221 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) with exact rational arithmetic, not rounded decimals.
S220 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S219 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(70) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S218 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(70) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S217 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(35) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S216 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(35) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S215 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S214 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S213 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S212 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S211 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S210 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S209 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S208 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S207 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S206 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S205 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S204 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S203 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S202 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 with exact rational arithmetic, not rounded decimals.
S201 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S200 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S199 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S198 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S197 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S196 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
S195 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S194 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S193 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) with exact rational arithmetic, not rounded decimals.
S192 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S191 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(84) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S190 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(84) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S189 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(42) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S188 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(42) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S187 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(28) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S186 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(28) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S185 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S184 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S183 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S182 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S181 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(12) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S180 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(12) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S179 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S178 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S177 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S176 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S175 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S174 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S173 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S172 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S171 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S170 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S169 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S168 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S167 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S166 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 with exact rational arithmetic, not rounded decimals.
S165 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S164 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S163 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S162 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S161 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S160 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S159 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) with exact rational arithmetic, not rounded decimals.
S158 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S157 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(105) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S156 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(105) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S155 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(35) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S154 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(35) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S153 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S152 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S151 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S150 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(15) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S149 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S148 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S147 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S146 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S145 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S144 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S143 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S142 · 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.
S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S140 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 with exact rational arithmetic, not rounded decimals.
S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S137 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
S136 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S135 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S134 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
S133 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S131 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S130 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) with exact rational arithmetic, not rounded decimals.
S129 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S128 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(140) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S127 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(140) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S126 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(70) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S125 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(70) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S124 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(35) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S123 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(35) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S122 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(28) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S121 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(28) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S120 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(20) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S119 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(20) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S118 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S117 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S116 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S115 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S114 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S113 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S112 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S111 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S110 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S109 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S108 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S107 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S106 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S105 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S103 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 with exact rational arithmetic, not rounded decimals.
S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S100 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S097 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals.
S096 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S095 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(210) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S094 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(210) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S093 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(105) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S092 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(105) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S091 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(70) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S090 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(70) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S089 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(42) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S088 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(42) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S087 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(35) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S086 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(35) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S085 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(30) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S084 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(30) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S083 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S082 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S081 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S080 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(15) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S079 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S078 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S077 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S076 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S075 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S074 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S073 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S072 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S071 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S070 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S069 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S068 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S067 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S066 · 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.
S065 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S064 · 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.
S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S062 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S059 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S056 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
S055 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
S054 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(420) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S053 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(420) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S052 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(210) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S051 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(210) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S050 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(140) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S049 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(140) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S048 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(105) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S047 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(105) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S046 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(84) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S045 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(84) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S044 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(70) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S043 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(70) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S042 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(60) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S041 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(60) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S040 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(42) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S039 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(42) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S038 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(35) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S037 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(35) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S036 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(30) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S035 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(30) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(28) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(28) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S032 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S031 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S030 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(20) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S029 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(20) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S028 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(15) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(12) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(12) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S019 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S013 · 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.
S012 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(3) 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

(420, [1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420], 1, [2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420], 7981, -17579, 7981, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

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: 7806e4ff3d8277bd94ac33a0f0e8730891b188bfb72572ca8aafed07af72349e
candidate record SHA-256: 946c79b18fc168292d79eea689bbb5d3f12350e923c73cf96dc22efa7aa05921
semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5
question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON