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R0042-21741 · q-20261014-advance-20261109-r06-mobius-function-0066
Use positive integer n=16. Give (n, divisor_mu_rows, sum, target, equal).
G001 — Use positive integer n=16. Give (n, divisor_mu_rows, sum, target, equal).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.
Before
source operation=mobius-divisor-sum;original snapshot and mathematics retained
After
explicit ordered source output fields=[n, divisor_mu_rows, sum, target, equal]
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.
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
Before
ORIGINAL n=16;complete positive candidate range1..n
After
ALL actual positive divisors INCREASING=[1, 2, 4, 8, 16]
Conditions: none
Keep divisors over the complete positive range,including1 and n;every subsequent mu row needs a complete factorization.
Rule: mobius_ALL_positive_divisors_not_sample · Uses: G001, S001
Before
mobius divisor1 of original n=16: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.
Rule: mobius_complete_UPF_input_role · Uses: G001, S002
Before
mobius divisor1 of original n=16: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.
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S003
Before
mobius divisor2 of original n=16: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.
Rule: mobius_complete_UPF_input_role · Uses: G001, S004
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.
Rule: upf_factorization_remaining_product_invariant · Uses: G001, S005
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.
Rule: exact_rational_arithmetic · Uses: G001, S006
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.
Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S007
Before
mobius divisor2 of original n=16: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.
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S008
Before
mobius divisor4 of original n=16: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.
Rule: mobius_complete_UPF_input_role · Uses: G001, S009
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.
Rule: upf_factorization_remaining_product_invariant · Uses: G001, S010
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.
Rule: exact_rational_arithmetic · Uses: G001, S011
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.
Rule: exact_rational_arithmetic · Uses: G001, S012
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.
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S013
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.
Rule: exact_rational_arithmetic · Uses: G001, S014
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.
Rule: exact_rational_arithmetic · Uses: G001, S015
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
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.
Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S016
Before
mobius divisor4 of original n=16: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.
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S017
Before
mobius divisor8 of original n=16:actual positive input for mu=8
After
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
Conditions: none
Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.
Rule: mobius_complete_UPF_input_role · Uses: G001, S018
Before
ORIGINAL positive integer absolute magnitude=8>1
After
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
Conditions: none
Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime.
Rule: upf_factorization_remaining_product_invariant · Uses: G001, S019
Before
upf extract divisor2 occurrence0 := div(8, 2)
After
upf extract divisor2 occurrence0 = 4
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
upf extract divisor2 occurrence1 := div(4, 2)
After
upf extract divisor2 occurrence1 = 2
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
upf extract divisor2 occurrence2 := div(2, 2)
After
upf extract divisor2 occurrence2 = 1
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
next divisor=2;previous remaining=8;all smaller divisors exhausted
After
extracted multiplicity=3;next remaining=1;remaining modulo2=1;extracted multiset=[2, 2, 2]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S023
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.
Rule: exact_rational_arithmetic · Uses: G001, S024
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.
Rule: exact_rational_arithmetic · Uses: G001, S025
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.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=8
After
complete increasing prime multiset=[2, 2, 2];full product=8;a composite terminal remainder would have an already tested smaller divisor
Conditions: none
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.
Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S027
Before
mobius divisor8 of original n=16:input=8;COMPLETE prime multiset=[2, 2, 2];distinct primes=[2]
After
mu=0;empty product case=No;repeated prime=Yes
Conditions: none
The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S028
Before
mobius divisor16 of original n=16:actual positive input for mu=16
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.
Rule: mobius_complete_UPF_input_role · Uses: G001, S029
Before
ORIGINAL positive integer absolute magnitude=16>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.
Rule: upf_factorization_remaining_product_invariant · Uses: G001, S030
Before
upf extract divisor2 occurrence0 := div(16, 2)
After
upf extract divisor2 occurrence0 = 8
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
upf extract divisor2 occurrence1 := div(8, 2)
After
upf extract divisor2 occurrence1 = 4
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
upf extract divisor2 occurrence2 := div(4, 2)
After
upf extract divisor2 occurrence2 = 2
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
upf extract divisor2 occurrence3 := div(2, 2)
After
upf extract divisor2 occurrence3 = 1
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
next divisor=2;previous remaining=16;all smaller divisors exhausted
After
extracted multiplicity=4;next remaining=1;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.
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S035
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.
Rule: exact_rational_arithmetic · Uses: G001, S036
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.
Rule: exact_rational_arithmetic · Uses: G001, S037
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.
Rule: exact_rational_arithmetic · Uses: G001, S038
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.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=16
After
complete increasing prime multiset=[2, 2, 2, 2];full product=16;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.
Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S040
Before
mobius divisor16 of original n=16:input=16;COMPLETE prime multiset=[2, 2, 2, 2];distinct primes=[2]
After
mu=0;empty product case=No;repeated prime=Yes
Conditions: none
The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S041
Before
Mobius complete divisor mu sum := sum(1, -1, 0, 0, 0)
After
Mobius complete divisor mu sum = 0
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
ORIGINAL n=16;ALL increasing [divisor,mu] rows=[[1, 1], [2, -1], [4, 0], [8, 0], [16, 0]]
After
exact sum=0;target=0;squarefree divisors correspond to subsets of DISTINCT primes,so sum=(-1)^subset_size summed over all subsets=(1-1)^r;for n1,r0 gives1
Conditions: none
Nonsquarefree divisors contribute0;all others correspond to subsets of distinct primes.For r>0,pair subsets by one fixed prime to cancel signs.For n1,the sole empty subset contributes1.This supplies the general argument rather than extrapolating one equality.
Rule: mobius_divisor_sum_general_subset_cancellation · Uses: G001, S043
S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S043 · premise_and_result_audit — Recompute Mobius complete divisor mu sum with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S040 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals.S039 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.S038 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.S037 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S035 · premise_and_result_audit — Recompute upf extract divisor2 occurrence3 with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute upf extract divisor2 occurrence2 with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S027 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S023 · premise_and_result_audit — Recompute upf extract divisor2 occurrence2 with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S016 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S013 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Recompute upf canonical complete product factor0 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.(16, [[1, 1], [2, -1], [4, 0], [8, 0], [16, 0]], 0, 0, Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 2074da1a2eebb6f3389484a5af34e6108bf0e97ccdd5d0e3e7108ed639f0e1e2candidate record SHA-256: ab260458c8f5f0379240a18aea749fd4f09c192544fae4589535742498a2d778semantic review receipt SHA-256: dd44555041dfbb6fe9936aa00b9a5f8c5982c2784ef61dd0cbd5d1a303a49c1aquestion-first witness SHA-256: 9a6011263f6ac53b9a1fd57addf2f3e2f6088b54802c539f332d689480dc2d17Lean: NOT_RUN · human review: NOT_PERFORMED