# Question 22200: Every Divisor Row Recovers Its Original Arithmetic Value [Return to catalogue page 111](https://ocme.evemisslab.com/en/questions/page/111/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations/#q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0075) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0043-22200 · q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0075 ## Original question and givens Use positive integer n=25, exact integer arithmetic function f(k)=a*k+b with a=5, b=12. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary). - G001 — Use positive integer n=25, exact integer arithmetic function f(k)=a*k+b with a=5, b=12. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, 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](https://ocme.evemisslab.com/en/mko/mko-mobius-inversion-finite-divisors/) ## Step-by-step derivation S001 · impliesBefore source operation=mobinv-roundtrip-table;original snapshot and mathematics retained After explicit ordered source output fields=[n, roundtrip_rows, all_equal, 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 dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore Mobius inversion ORIGINAL roundtrip:actual positive input=25;check complete integer range1..25 After ALL positive divisors INCREASING=[1, 5, 25];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 dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S001 S003 · impliesBefore Mobius roundtrip m=1 forward: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 dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S002 S004 · calculationBefore Mobius roundtrip m=1 forward f(1) ORIGINAL a times divisor := mul(5, 1) After Mobius roundtrip m=1 forward f(1) ORIGINAL a times divisor = 5 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Mobius roundtrip m=1 forward f(1) plus ORIGINAL b := sum(5, 12) After Mobius roundtrip m=1 forward f(1) plus ORIGINAL b = 17 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Mobius roundtrip m=1 forward full forward divisor sum := sum(17) After Mobius roundtrip m=1 forward full forward divisor sum = 17 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore Mobius roundtrip m=1 inverse: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 dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S006 S008 · impliesBefore Mobius roundtrip m=1 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 dependenciesRule: mobius_complete_UPF_input_role · Uses: G001, S007 S009 · impliesBefore Mobius roundtrip m=1 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 dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S008 S010 · calculationBefore Mobius roundtrip m=1 inverse n divided by ORIGINAL divisor1 := div(1, 1) After Mobius roundtrip m=1 inverse n divided by ORIGINAL divisor1 = 1 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore Mobius roundtrip m=1 inverse F(n/1):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 dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S010 S012 · calculationBefore Mobius roundtrip m=1 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(5, 1) After Mobius roundtrip m=1 inverse F(n/1) f(1) ORIGINAL a times divisor = 5 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Mobius roundtrip m=1 inverse F(n/1) f(1) plus ORIGINAL b := sum(5, 12) After Mobius roundtrip m=1 inverse F(n/1) f(1) plus ORIGINAL b = 17 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Mobius roundtrip m=1 inverse F(n/1) full forward divisor sum := sum(17) After Mobius roundtrip m=1 inverse F(n/1) full forward divisor sum = 17 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Mobius roundtrip m=1 inverse mu(1) times SAME F(n/1) := mul(1, 17) After Mobius roundtrip m=1 inverse mu(1) times SAME F(n/1) = 17 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Mobius roundtrip m=1 inverse complete inverse-term sum := sum(17) After Mobius roundtrip m=1 inverse complete inverse-term sum = 17 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Mobius roundtrip m=1 inverse direct ORIGINAL a times n := mul(5, 1) After Mobius roundtrip m=1 inverse direct ORIGINAL a times n = 5 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Mobius roundtrip m=1 inverse direct plus ORIGINAL b := sum(5, 12) After Mobius roundtrip m=1 inverse direct plus ORIGINAL b = 17 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore Mobius roundtrip m=1 inverse:actual n=1;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 1, 17, 17]] After complete reconstructed=17;direct f(n)=17;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 dependenciesRule: mobinv_general_finite_double_sum_and_Mobius_cancellation · Uses: G001, S018 S020 · impliesBefore Mobius roundtrip m=5 forward: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 dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S019 S021 · calculationBefore Mobius roundtrip m=5 forward f(1) ORIGINAL a times divisor := mul(5, 1) After Mobius roundtrip m=5 forward f(1) ORIGINAL a times divisor = 5 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Mobius roundtrip m=5 forward f(1) plus ORIGINAL b := sum(5, 12) After Mobius roundtrip m=5 forward f(1) plus ORIGINAL b = 17 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Mobius roundtrip m=5 forward f(5) ORIGINAL a times divisor := mul(5, 5) After Mobius roundtrip m=5 forward f(5) ORIGINAL a times divisor = 25 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Mobius roundtrip m=5 forward f(5) plus ORIGINAL b := sum(25, 12) After Mobius roundtrip m=5 forward f(5) plus ORIGINAL b = 37 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Mobius roundtrip m=5 forward full forward divisor sum := sum(17, 37) After Mobius roundtrip m=5 forward full forward divisor sum = 54 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · impliesBefore Mobius roundtrip m=5 inverse: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 dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S025 S027 · impliesBefore Mobius roundtrip m=5 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 dependenciesRule: mobius_complete_UPF_input_role · Uses: G001, S026 S028 · impliesBefore Mobius roundtrip m=5 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 dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S027 S029 · calculationBefore Mobius roundtrip m=5 inverse n divided by ORIGINAL divisor1 := div(5, 1) After Mobius roundtrip m=5 inverse n divided by ORIGINAL divisor1 = 5 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · impliesBefore Mobius roundtrip m=5 inverse F(n/1):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 dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S029 S031 · calculationBefore Mobius roundtrip m=5 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(5, 1) After Mobius roundtrip m=5 inverse F(n/1) f(1) ORIGINAL a times divisor = 5 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore Mobius roundtrip m=5 inverse F(n/1) f(1) plus ORIGINAL b := sum(5, 12) After Mobius roundtrip m=5 inverse F(n/1) f(1) plus ORIGINAL b = 17 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore Mobius roundtrip m=5 inverse F(n/1) f(5) ORIGINAL a times divisor := mul(5, 5) After Mobius roundtrip m=5 inverse F(n/1) f(5) ORIGINAL a times divisor = 25 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore Mobius roundtrip m=5 inverse F(n/1) f(5) plus ORIGINAL b := sum(25, 12) After Mobius roundtrip m=5 inverse F(n/1) f(5) plus ORIGINAL b = 37 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore Mobius roundtrip m=5 inverse F(n/1) full forward divisor sum := sum(17, 37) After Mobius roundtrip m=5 inverse F(n/1) full forward divisor sum = 54 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore Mobius roundtrip m=5 inverse mu(1) times SAME F(n/1) := mul(1, 54) After Mobius roundtrip m=5 inverse mu(1) times SAME F(n/1) = 54 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · impliesBefore Mobius roundtrip m=5 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 dependenciesRule: mobius_complete_UPF_input_role · Uses: G001, S036 S038 · impliesBefore 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 dependenciesRule: upf_factorization_remaining_product_invariant · Uses: G001, S037 S039 · impliesBefore 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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S038 S040 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · impliesBefore 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 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 dependenciesRule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S040 S042 · impliesBefore Mobius roundtrip m=5 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 dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S041 S043 · calculationBefore Mobius roundtrip m=5 inverse n divided by ORIGINAL divisor5 := div(5, 5) After Mobius roundtrip m=5 inverse n divided by ORIGINAL divisor5 = 1 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · impliesBefore Mobius roundtrip m=5 inverse F(n/5):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 dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S043 S045 · calculationBefore Mobius roundtrip m=5 inverse F(n/5) f(1) ORIGINAL a times divisor := mul(5, 1) After Mobius roundtrip m=5 inverse F(n/5) f(1) ORIGINAL a times divisor = 5 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore Mobius roundtrip m=5 inverse F(n/5) f(1) plus ORIGINAL b := sum(5, 12) After Mobius roundtrip m=5 inverse F(n/5) f(1) plus ORIGINAL b = 17 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore Mobius roundtrip m=5 inverse F(n/5) full forward divisor sum := sum(17) After Mobius roundtrip m=5 inverse F(n/5) full forward divisor sum = 17 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore Mobius roundtrip m=5 inverse mu(5) times SAME F(n/5) := mul(-1, 17) After Mobius roundtrip m=5 inverse mu(5) times SAME F(n/5) = -17 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · calculationBefore Mobius roundtrip m=5 inverse complete inverse-term sum := sum(54, -17) After Mobius roundtrip m=5 inverse complete inverse-term sum = 37 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · calculationBefore Mobius roundtrip m=5 inverse direct ORIGINAL a times n := mul(5, 5) After Mobius roundtrip m=5 inverse direct ORIGINAL a times n = 25 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · calculationBefore Mobius roundtrip m=5 inverse direct plus ORIGINAL b := sum(25, 12) After Mobius roundtrip m=5 inverse direct plus ORIGINAL b = 37 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · impliesBefore Mobius roundtrip m=5 inverse:actual n=5;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 5, 54, 54], [5, -1, 1, 17, -17]] After complete reconstructed=37;direct f(n)=37;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 dependenciesRule: mobinv_general_finite_double_sum_and_Mobius_cancellation · Uses: G001, S051 S053 · impliesBefore Mobius roundtrip m=25 forward:actual positive input=25;check complete integer range1..25 After ALL positive divisors INCREASING=[1, 5, 25];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 dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S052 S054 · calculationBefore Mobius roundtrip m=25 forward f(1) ORIGINAL a times divisor := mul(5, 1) After Mobius roundtrip m=25 forward f(1) ORIGINAL a times divisor = 5 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S053 S055 · calculationBefore Mobius roundtrip m=25 forward f(1) plus ORIGINAL b := sum(5, 12) After Mobius roundtrip m=25 forward f(1) plus ORIGINAL b = 17 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054 S056 · calculationBefore Mobius roundtrip m=25 forward f(5) ORIGINAL a times divisor := mul(5, 5) After Mobius roundtrip m=25 forward f(5) ORIGINAL a times divisor = 25 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S055 S057 · calculationBefore Mobius roundtrip m=25 forward f(5) plus ORIGINAL b := sum(25, 12) After Mobius roundtrip m=25 forward f(5) plus ORIGINAL b = 37 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056 S058 · calculationBefore Mobius roundtrip m=25 forward f(25) ORIGINAL a times divisor := mul(5, 25) After Mobius roundtrip m=25 forward f(25) ORIGINAL a times divisor = 125 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S057 S059 · calculationBefore Mobius roundtrip m=25 forward f(25) plus ORIGINAL b := sum(125, 12) After Mobius roundtrip m=25 forward f(25) plus ORIGINAL b = 137 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058 S060 · calculationBefore Mobius roundtrip m=25 forward full forward divisor sum := sum(17, 37, 137) After Mobius roundtrip m=25 forward full forward divisor sum = 191 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S059 S061 · impliesBefore Mobius roundtrip m=25 inverse:actual positive input=25;check complete integer range1..25 After ALL positive divisors INCREASING=[1, 5, 25];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 dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S060 S062 · impliesBefore Mobius roundtrip m=25 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 dependenciesRule: mobius_complete_UPF_input_role · Uses: G001, S061 S063 · impliesBefore Mobius roundtrip m=25 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 dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S062 S064 · calculationBefore Mobius roundtrip m=25 inverse n divided by ORIGINAL divisor1 := div(25, 1) After Mobius roundtrip m=25 inverse n divided by ORIGINAL divisor1 = 25 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S063 S065 · impliesBefore Mobius roundtrip m=25 inverse F(n/1):actual positive input=25;check complete integer range1..25 After ALL positive divisors INCREASING=[1, 5, 25];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 dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S064 S066 · calculationBefore Mobius roundtrip m=25 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(5, 1) After Mobius roundtrip m=25 inverse F(n/1) f(1) ORIGINAL a times divisor = 5 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S065 S067 · calculationBefore Mobius roundtrip m=25 inverse F(n/1) f(1) plus ORIGINAL b := sum(5, 12) After Mobius roundtrip m=25 inverse F(n/1) f(1) plus ORIGINAL b = 17 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S066 S068 · calculationBefore Mobius roundtrip m=25 inverse F(n/1) f(5) ORIGINAL a times divisor := mul(5, 5) After Mobius roundtrip m=25 inverse F(n/1) f(5) ORIGINAL a times divisor = 25 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067 S069 · calculationBefore Mobius roundtrip m=25 inverse F(n/1) f(5) plus ORIGINAL b := sum(25, 12) After Mobius roundtrip m=25 inverse F(n/1) f(5) plus ORIGINAL b = 37 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S068 S070 · calculationBefore Mobius roundtrip m=25 inverse F(n/1) f(25) ORIGINAL a times divisor := mul(5, 25) After Mobius roundtrip m=25 inverse F(n/1) f(25) ORIGINAL a times divisor = 125 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069 S071 · calculationBefore Mobius roundtrip m=25 inverse F(n/1) f(25) plus ORIGINAL b := sum(125, 12) After Mobius roundtrip m=25 inverse F(n/1) f(25) plus ORIGINAL b = 137 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S070 S072 · calculationBefore Mobius roundtrip m=25 inverse F(n/1) full forward divisor sum := sum(17, 37, 137) After Mobius roundtrip m=25 inverse F(n/1) full forward divisor sum = 191 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071 S073 · calculationBefore Mobius roundtrip m=25 inverse mu(1) times SAME F(n/1) := mul(1, 191) After Mobius roundtrip m=25 inverse mu(1) times SAME F(n/1) = 191 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S072 S074 · impliesBefore Mobius roundtrip m=25 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 dependenciesRule: mobius_complete_UPF_input_role · Uses: G001, S073 S075 · impliesBefore 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 dependenciesRule: upf_factorization_remaining_product_invariant · Uses: G001, S074 S076 · impliesBefore 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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S075 S077 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S076 S078 · impliesBefore 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 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 dependenciesRule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S077 S079 · impliesBefore Mobius roundtrip m=25 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 dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S078 S080 · calculationBefore Mobius roundtrip m=25 inverse n divided by ORIGINAL divisor5 := div(25, 5) After Mobius roundtrip m=25 inverse n divided by ORIGINAL divisor5 = 5 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S079 S081 · impliesBefore Mobius roundtrip m=25 inverse F(n/5):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 dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S080 S082 · calculationBefore Mobius roundtrip m=25 inverse F(n/5) f(1) ORIGINAL a times divisor := mul(5, 1) After Mobius roundtrip m=25 inverse F(n/5) f(1) ORIGINAL a times divisor = 5 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S081 S083 · calculationBefore Mobius roundtrip m=25 inverse F(n/5) f(1) plus ORIGINAL b := sum(5, 12) After Mobius roundtrip m=25 inverse F(n/5) f(1) plus ORIGINAL b = 17 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S082 S084 · calculationBefore Mobius roundtrip m=25 inverse F(n/5) f(5) ORIGINAL a times divisor := mul(5, 5) After Mobius roundtrip m=25 inverse F(n/5) f(5) ORIGINAL a times divisor = 25 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S083 S085 · calculationBefore Mobius roundtrip m=25 inverse F(n/5) f(5) plus ORIGINAL b := sum(25, 12) After Mobius roundtrip m=25 inverse F(n/5) f(5) plus ORIGINAL b = 37 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S084 S086 · calculationBefore Mobius roundtrip m=25 inverse F(n/5) full forward divisor sum := sum(17, 37) After Mobius roundtrip m=25 inverse F(n/5) full forward divisor sum = 54 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S085 S087 · calculationBefore Mobius roundtrip m=25 inverse mu(5) times SAME F(n/5) := mul(-1, 54) After Mobius roundtrip m=25 inverse mu(5) times SAME F(n/5) = -54 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S086 S088 · impliesBefore Mobius roundtrip m=25 inverse Mobius at ORIGINAL divisor25:actual positive input for mu=25 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 dependenciesRule: mobius_complete_UPF_input_role · Uses: G001, S087 S089 · impliesBefore ORIGINAL positive integer absolute magnitude=25>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 dependenciesRule: upf_factorization_remaining_product_invariant · Uses: G001, S088 S090 · impliesBefore next divisor=2;previous remaining=25;all smaller divisors exhausted After extracted multiplicity=0;next remaining=25;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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S089 S091 · impliesBefore next divisor=3;previous remaining=25;all smaller divisors exhausted After extracted multiplicity=0;next remaining=25;remaining modulo3=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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S090 S092 · impliesBefore next divisor=4;previous remaining=25;all smaller divisors exhausted After extracted multiplicity=0;next remaining=25;remaining modulo4=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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S091 S093 · calculationBefore upf extract divisor5 occurrence0 := div(25, 5) After upf extract divisor5 occurrence0 = 5 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S092 S094 · calculationBefore upf extract divisor5 occurrence1 := div(5, 5) After upf extract divisor5 occurrence1 = 1 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S093 S095 · impliesBefore next divisor=5;previous remaining=25;all smaller divisors exhausted After extracted multiplicity=2;next remaining=1;remaining modulo5=1;extracted multiset=[5, 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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S094 S096 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S095 S097 · calculationBefore upf canonical complete product factor1 := mul(5, 5) After upf canonical complete product factor1 = 25 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S096 S098 · impliesBefore exhausted trial divisors below6;terminal remaining=1;d*d>remaining;ORIGINAL n=25 After complete increasing prime multiset=[5, 5];full product=25;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 dependenciesRule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S097 S099 · impliesBefore Mobius roundtrip m=25 inverse Mobius at ORIGINAL divisor25:input=25;COMPLETE prime multiset=[5, 5];distinct primes=[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 dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S098 S100 · calculationBefore Mobius roundtrip m=25 inverse n divided by ORIGINAL divisor25 := div(25, 25) After Mobius roundtrip m=25 inverse n divided by ORIGINAL divisor25 = 1 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S099 S101 · impliesBefore Mobius roundtrip m=25 inverse F(n/25):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 dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S100 S102 · calculationBefore Mobius roundtrip m=25 inverse F(n/25) f(1) ORIGINAL a times divisor := mul(5, 1) After Mobius roundtrip m=25 inverse F(n/25) f(1) ORIGINAL a times divisor = 5 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S101 S103 · calculationBefore Mobius roundtrip m=25 inverse F(n/25) f(1) plus ORIGINAL b := sum(5, 12) After Mobius roundtrip m=25 inverse F(n/25) f(1) plus ORIGINAL b = 17 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S102 S104 · calculationBefore Mobius roundtrip m=25 inverse F(n/25) full forward divisor sum := sum(17) After Mobius roundtrip m=25 inverse F(n/25) full forward divisor sum = 17 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S103 S105 · calculationBefore Mobius roundtrip m=25 inverse mu(25) times SAME F(n/25) := mul(0, 17) After Mobius roundtrip m=25 inverse mu(25) times SAME F(n/25) = 0 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S104 S106 · calculationBefore Mobius roundtrip m=25 inverse complete inverse-term sum := sum(191, -54, 0) After Mobius roundtrip m=25 inverse complete inverse-term sum = 137 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S105 S107 · calculationBefore Mobius roundtrip m=25 inverse direct ORIGINAL a times n := mul(5, 25) After Mobius roundtrip m=25 inverse direct ORIGINAL a times n = 125 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S106 S108 · calculationBefore Mobius roundtrip m=25 inverse direct plus ORIGINAL b := sum(125, 12) After Mobius roundtrip m=25 inverse direct plus ORIGINAL b = 137 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S107 S109 · impliesBefore Mobius roundtrip m=25 inverse:actual n=25;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 25, 191, 191], [5, -1, 5, 54, -54], [25, 0, 1, 17, 0]] After complete reconstructed=137;direct f(n)=137;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 dependenciesRule: mobinv_general_finite_double_sum_and_Mobius_cancellation · Uses: G001, S108 ## Reverse checks - S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S108 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S107 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals. - S106 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals. - S105 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse mu(25) times SAME F(n/25) with exact rational arithmetic, not rounded decimals. - S104 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse F(n/25) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S103 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse F(n/25) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S102 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse F(n/25) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - 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 Mobius roundtrip m=25 inverse n divided by ORIGINAL divisor25 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 upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S096 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S094 · premise_and_result_audit — Recompute upf extract divisor5 occurrence1 with exact rational arithmetic, not rounded decimals. - S093 · premise_and_result_audit — Recompute upf extract divisor5 occurrence0 with exact rational arithmetic, not rounded decimals. - S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S087 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse mu(5) times SAME F(n/5) with exact rational arithmetic, not rounded decimals. - S086 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse F(n/5) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S085 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse F(n/5) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S084 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse F(n/5) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S083 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse F(n/5) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S082 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse F(n/5) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S080 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse n divided by ORIGINAL divisor5 with exact rational arithmetic, not rounded decimals. - S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S077 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S073 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse F(n/1) f(25) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S070 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse F(n/1) f(25) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse F(n/1) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S068 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse F(n/1) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse F(n/1) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S064 · premise_and_result_audit — Recompute Mobius roundtrip m=25 inverse n divided by ORIGINAL divisor1 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S060 · premise_and_result_audit — Recompute Mobius roundtrip m=25 forward full forward divisor sum with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Recompute Mobius roundtrip m=25 forward f(25) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Recompute Mobius roundtrip m=25 forward f(25) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute Mobius roundtrip m=25 forward f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute Mobius roundtrip m=25 forward f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute Mobius roundtrip m=25 forward f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Recompute Mobius roundtrip m=25 forward f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S051 · premise_and_result_audit — Recompute Mobius roundtrip m=5 inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute Mobius roundtrip m=5 inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute Mobius roundtrip m=5 inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute Mobius roundtrip m=5 inverse mu(5) times SAME F(n/5) with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute Mobius roundtrip m=5 inverse F(n/5) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute Mobius roundtrip m=5 inverse F(n/5) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute Mobius roundtrip m=5 inverse F(n/5) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - 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 roundtrip m=5 inverse n divided by ORIGINAL divisor5 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 factor0 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Recompute Mobius roundtrip m=5 inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Mobius roundtrip m=5 inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute Mobius roundtrip m=5 inverse F(n/1) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Mobius roundtrip m=5 inverse F(n/1) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Mobius roundtrip m=5 inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Mobius roundtrip m=5 inverse F(n/1) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - 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 — Recompute Mobius roundtrip m=5 inverse n divided by ORIGINAL divisor1 with exact rational arithmetic, not rounded decimals. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Recompute Mobius roundtrip m=5 forward full forward divisor sum with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Mobius roundtrip m=5 forward f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Mobius roundtrip m=5 forward f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Mobius roundtrip m=5 forward f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Mobius roundtrip m=5 forward f(1) ORIGINAL a times divisor 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 — Recompute Mobius roundtrip m=1 inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse F(n/1) f(1) ORIGINAL a times divisor 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 — Recompute Mobius roundtrip m=1 inverse n divided by ORIGINAL divisor1 with exact rational arithmetic, not rounded decimals. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute Mobius roundtrip m=1 forward full forward divisor sum with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Mobius roundtrip m=1 forward f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Mobius roundtrip m=1 forward f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - 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 (25, [[1, 17, 17, 17], [5, 54, 37, 37], [25, 191, 137, 137]], Yes, finite_divisor_poset_roundtrip) 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: bedf155c587bab20ceb58ee7f815ee737cc3039a290b430b415169035ba2f4f7 - candidate record SHA-256: 311287f20e897431fee0ce897beac3f2aab78da36b9bec26f191c9224865595d - semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5 - question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0075.json)