Step-by-step derivation
S001 · implies
Before
source operation=upf-canonical-factorization;original snapshot and mathematics retained
After
explicit ordered source output fields=[n, prime_multiset, exponents, product, complete_factorization_certificate, certificate]
Conditions: none
Retain the historical interface marker while listing ordered output fields without JavaScript.Fields and tags are not extra mathematical evidence;fixed No markers retain their claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ORIGINAL positive integer absolute magnitude=402>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, S001
S003 · calculation
Before
upf extract divisor2 occurrence0 := div(402, 2)
After
upf extract divisor2 occurrence0 = 201
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, S002
S004 · implies
Before
next divisor=2;previous remaining=402;all smaller divisors exhausted
After
extracted multiplicity=1;next remaining=201;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, S003
S005 · calculation
Before
upf extract divisor3 occurrence1 := div(201, 3)
After
upf extract divisor3 occurrence1 = 67
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 · implies
Before
next divisor=3;previous remaining=201;all smaller divisors exhausted
After
extracted multiplicity=1;next remaining=67;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, S005
S007 · implies
Before
next divisor=4;previous remaining=67;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=67;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, S006
S008 · implies
Before
next divisor=5;previous remaining=67;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=67;remaining modulo5=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, S007
S009 · implies
Before
next divisor=6;previous remaining=67;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=67;remaining modulo6=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, S008
S010 · implies
Before
next divisor=7;previous remaining=67;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=67;remaining modulo7=4;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, S009
S011 · implies
Before
next divisor=8;previous remaining=67;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=67;remaining modulo8=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, S010
S012 · 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, S011
S013 · 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, S012
S014 · calculation
Before
upf canonical complete product factor2 := mul(6, 67)
After
upf canonical complete product factor2 = 402
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 · implies
Before
exhausted trial divisors below9;terminal remaining=67;d*d>remaining;ORIGINAL n=402
After
complete increasing prime multiset=[2, 3, 67];full product=402;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, S014
S016 · implies
Before
ORIGINAL n=402;complete prime multiset=[2, 3, 67]
After
ALL [prime,exponent] rows=[[2, 1], [3, 1], [67, 1]];product=402;no multiplicities discarded
Conditions: none
Retain every multiplicity,not just distinct prime support.Uniqueness compares all exponents,not only which prime species occur.
Step identifiers and dependencies
Rule: upf_exponents_keep_every_prime_multiplicity · Uses: G001, S015