Step-by-step derivation
S001 · implies
Before
source operation=mobius-canonical-value;original snapshot and mathematics retained
After
explicit ordered source output fields=[n, prime_multiset, distinct_primes, mu, certificate]
Conditions: none
Keep the original historical interface marker while listing ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their source claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
mobius original n:actual positive input for mu=240
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, S001
S003 · implies
Before
ORIGINAL positive integer absolute magnitude=240>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, S002
S004 · calculation
Before
upf extract divisor2 occurrence0 := div(240, 2)
After
upf extract divisor2 occurrence0 = 120
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, S003
S005 · calculation
Before
upf extract divisor2 occurrence1 := div(120, 2)
After
upf extract divisor2 occurrence1 = 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, S004
S006 · calculation
Before
upf extract divisor2 occurrence2 := div(60, 2)
After
upf extract divisor2 occurrence2 = 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, S005
S007 · calculation
Before
upf extract divisor2 occurrence3 := div(30, 2)
After
upf extract divisor2 occurrence3 = 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, S006
S008 · implies
Before
next divisor=2;previous remaining=240;all smaller divisors exhausted
After
extracted multiplicity=4;next remaining=15;remaining modulo2=1;extracted multiset=[2, 2, 2, 2]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S007
S009 · calculation
Before
upf extract divisor3 occurrence4 := div(15, 3)
After
upf extract divisor3 occurrence4 = 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, S008
S010 · 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, 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, S009
S011 · 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, S010
S012 · 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, S011
S013 · calculation
Before
upf canonical complete product factor2 := mul(4, 2)
After
upf canonical complete product factor2 = 8
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · calculation
Before
upf canonical complete product factor3 := mul(8, 2)
After
upf canonical complete product factor3 = 16
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, S013
S015 · calculation
Before
upf canonical complete product factor4 := mul(16, 3)
After
upf canonical complete product factor4 = 48
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, S014
S016 · calculation
Before
upf canonical complete product factor5 := mul(48, 5)
After
upf canonical complete product factor5 = 240
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, S015
S017 · implies
Before
exhausted trial divisors below4;terminal remaining=5;d*d>remaining;ORIGINAL n=240
After
complete increasing prime multiset=[2, 2, 2, 2, 3, 5];full product=240;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, S016
S018 · implies
Before
mobius original n:input=240;COMPLETE prime multiset=[2, 2, 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, S017