# Question 22150: An Omitted Divisor Must Be a Discriminative Missing-Term Witness [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-0100) 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. R0043-22150 · q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0100 ## Original question and givens Use positive integer n=702, exact integer arithmetic function f(k)=a*k+b with a=25, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary). - G001 — Use positive integer n=702, exact integer arithmetic function f(k)=a*k+b with a=25, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-mobius-inversion-finite-divisors](https://ocme.evemisslab.com/en/mko/mko-mobius-inversion-finite-divisors/) ## Step-by-step derivation S001 · impliesBefore source operation=mobinv-incomplete-divisor-boundary;original snapshot and mathematics retained After explicit ordered source output fields=[n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary] Conditions: none Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore Mobius inversion ORIGINAL inverse:actual positive input=702;check complete integer range1..702 After ALL positive divisors INCREASING=[1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702];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 inversion ORIGINAL inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1 After complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search Conditions: none Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search. Step identifiers and dependenciesRule: mobius_complete_UPF_input_role · Uses: G001, S002 S004 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[] After mu=1;empty product case=Yes;repeated prime=No Conditions: none The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities. Step identifiers and dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S003 S005 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 := div(702, 1) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 702 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 · impliesBefore Mobius inversion ORIGINAL inverse F(n/1):actual positive input=702;check complete integer range1..702 After ALL positive divisors INCREASING=[1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702];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, S005 S007 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor := mul(25, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(1) 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, S006 S008 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(25, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = 26 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, S007 S009 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(25, 2) After Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 50 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, S008 S010 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b := sum(50, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 51 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor := mul(25, 3) After Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor = 75 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, S010 S012 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b := sum(75, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b = 76 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 inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor := mul(25, 6) After Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor = 150 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 inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b := sum(150, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b = 151 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 inversion ORIGINAL inverse F(n/1) f(9) ORIGINAL a times divisor := mul(25, 9) After Mobius inversion ORIGINAL inverse F(n/1) f(9) ORIGINAL a times divisor = 225 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 inversion ORIGINAL inverse F(n/1) f(9) plus ORIGINAL b := sum(225, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(9) plus ORIGINAL b = 226 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 inversion ORIGINAL inverse F(n/1) f(13) ORIGINAL a times divisor := mul(25, 13) After Mobius inversion ORIGINAL inverse F(n/1) f(13) ORIGINAL a times divisor = 325 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 inversion ORIGINAL inverse F(n/1) f(13) plus ORIGINAL b := sum(325, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(13) plus ORIGINAL b = 326 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(18) ORIGINAL a times divisor := mul(25, 18) After Mobius inversion ORIGINAL inverse F(n/1) f(18) ORIGINAL a times divisor = 450 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, S018 S020 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(18) plus ORIGINAL b := sum(450, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(18) plus ORIGINAL b = 451 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, S019 S021 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(26) ORIGINAL a times divisor := mul(25, 26) After Mobius inversion ORIGINAL inverse F(n/1) f(26) ORIGINAL a times divisor = 650 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 inversion ORIGINAL inverse F(n/1) f(26) plus ORIGINAL b := sum(650, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(26) plus ORIGINAL b = 651 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 inversion ORIGINAL inverse F(n/1) f(27) ORIGINAL a times divisor := mul(25, 27) After Mobius inversion ORIGINAL inverse F(n/1) f(27) ORIGINAL a times divisor = 675 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 inversion ORIGINAL inverse F(n/1) f(27) plus ORIGINAL b := sum(675, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(27) plus ORIGINAL b = 676 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 inversion ORIGINAL inverse F(n/1) f(39) ORIGINAL a times divisor := mul(25, 39) After Mobius inversion ORIGINAL inverse F(n/1) f(39) ORIGINAL a times divisor = 975 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(39) plus ORIGINAL b := sum(975, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(39) plus ORIGINAL b = 976 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, S025 S027 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(54) ORIGINAL a times divisor := mul(25, 54) After Mobius inversion ORIGINAL inverse F(n/1) f(54) ORIGINAL a times divisor = 1350 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, S026 S028 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(54) plus ORIGINAL b := sum(1350, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(54) plus ORIGINAL b = 1351 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, S027 S029 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(78) ORIGINAL a times divisor := mul(25, 78) After Mobius inversion ORIGINAL inverse F(n/1) f(78) ORIGINAL a times divisor = 1950 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(78) plus ORIGINAL b := sum(1950, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(78) plus ORIGINAL b = 1951 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, S029 S031 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(117) ORIGINAL a times divisor := mul(25, 117) After Mobius inversion ORIGINAL inverse F(n/1) f(117) ORIGINAL a times divisor = 2925 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 inversion ORIGINAL inverse F(n/1) f(117) plus ORIGINAL b := sum(2925, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(117) plus ORIGINAL b = 2926 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 inversion ORIGINAL inverse F(n/1) f(234) ORIGINAL a times divisor := mul(25, 234) After Mobius inversion ORIGINAL inverse F(n/1) f(234) ORIGINAL a times divisor = 5850 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 inversion ORIGINAL inverse F(n/1) f(234) plus ORIGINAL b := sum(5850, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(234) plus ORIGINAL b = 5851 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 inversion ORIGINAL inverse F(n/1) f(351) ORIGINAL a times divisor := mul(25, 351) After Mobius inversion ORIGINAL inverse F(n/1) f(351) ORIGINAL a times divisor = 8775 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 inversion ORIGINAL inverse F(n/1) f(351) plus ORIGINAL b := sum(8775, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(351) plus ORIGINAL b = 8776 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(702) ORIGINAL a times divisor := mul(25, 702) After Mobius inversion ORIGINAL inverse F(n/1) f(702) ORIGINAL a times divisor = 17550 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, S036 S038 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(702) plus ORIGINAL b := sum(17550, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(702) plus ORIGINAL b = 17551 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, S037 S039 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(26, 51, 76, 151, 226, 326, 451, 651, 676, 976, 1351, 1951, 2926, 5851, 8776, 17551) After Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 42016 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, S038 S040 · calculationBefore Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 42016) After Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 42016 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, S039 S041 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor2:actual positive input for mu=2 After complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search Conditions: none Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search. Step identifiers and dependenciesRule: mobius_complete_UPF_input_role · Uses: G001, S040 S042 · impliesBefore ORIGINAL positive integer absolute magnitude=2>1 After initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted Conditions: none Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime. Step identifiers and dependenciesRule: upf_factorization_remaining_product_invariant · Uses: G001, S041 S043 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · impliesBefore exhausted trial divisors below2;terminal remaining=2;d*d>remaining;ORIGINAL n=2 After complete increasing prime multiset=[2];full product=2;a composite terminal remainder would have an already tested smaller divisor Conditions: none If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors. Step identifiers and dependenciesRule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S043 S045 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor2:input=2;COMPLETE prime multiset=[2];distinct primes=[2] After mu=-1;empty product case=No;repeated prime=No Conditions: none The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities. Step identifiers and dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S044 S046 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 := div(702, 2) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 351 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 · impliesBefore Mobius inversion ORIGINAL inverse F(n/2):actual positive input=351;check complete integer range1..351 After ALL positive divisors INCREASING=[1, 3, 9, 13, 27, 39, 117, 351];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, S046 S048 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(25, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(1) 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, S047 S049 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(25, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 26 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 inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor := mul(25, 3) After Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor = 75 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 inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b := sum(75, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b = 76 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) f(9) ORIGINAL a times divisor := mul(25, 9) After Mobius inversion ORIGINAL inverse F(n/2) f(9) ORIGINAL a times divisor = 225 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, S051 S053 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) f(9) plus ORIGINAL b := sum(225, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(9) plus ORIGINAL b = 226 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, S052 S054 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) f(13) ORIGINAL a times divisor := mul(25, 13) After Mobius inversion ORIGINAL inverse F(n/2) f(13) ORIGINAL a times divisor = 325 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 inversion ORIGINAL inverse F(n/2) f(13) plus ORIGINAL b := sum(325, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(13) plus ORIGINAL b = 326 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 inversion ORIGINAL inverse F(n/2) f(27) ORIGINAL a times divisor := mul(25, 27) After Mobius inversion ORIGINAL inverse F(n/2) f(27) ORIGINAL a times divisor = 675 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 inversion ORIGINAL inverse F(n/2) f(27) plus ORIGINAL b := sum(675, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(27) plus ORIGINAL b = 676 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 inversion ORIGINAL inverse F(n/2) f(39) ORIGINAL a times divisor := mul(25, 39) After Mobius inversion ORIGINAL inverse F(n/2) f(39) ORIGINAL a times divisor = 975 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 inversion ORIGINAL inverse F(n/2) f(39) plus ORIGINAL b := sum(975, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(39) plus ORIGINAL b = 976 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 inversion ORIGINAL inverse F(n/2) f(117) ORIGINAL a times divisor := mul(25, 117) After Mobius inversion ORIGINAL inverse F(n/2) f(117) ORIGINAL a times divisor = 2925 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) f(117) plus ORIGINAL b := sum(2925, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(117) plus ORIGINAL b = 2926 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, S060 S062 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) f(351) ORIGINAL a times divisor := mul(25, 351) After Mobius inversion ORIGINAL inverse F(n/2) f(351) ORIGINAL a times divisor = 8775 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, S061 S063 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) f(351) plus ORIGINAL b := sum(8775, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(351) plus ORIGINAL b = 8776 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, S062 S064 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(26, 76, 226, 326, 676, 976, 2926, 8776) After Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 14008 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 · calculationBefore Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 14008) After Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -14008 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, S064 S066 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor3:actual positive input for mu=3 After complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search Conditions: none Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search. Step identifiers and dependenciesRule: mobius_complete_UPF_input_role · Uses: G001, S065 S067 · impliesBefore ORIGINAL positive integer absolute magnitude=3>1 After initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted Conditions: none Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime. Step identifiers and dependenciesRule: upf_factorization_remaining_product_invariant · Uses: G001, S066 S068 · calculationBefore upf canonical complete product factor0 := mul(1, 3) After upf canonical complete product factor0 = 3 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067 S069 · impliesBefore exhausted trial divisors below2;terminal remaining=3;d*d>remaining;ORIGINAL n=3 After complete increasing prime multiset=[3];full product=3;a composite terminal remainder would have an already tested smaller divisor Conditions: none 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, S068 S070 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor3:input=3;COMPLETE prime multiset=[3];distinct primes=[3] After mu=-1;empty product case=No;repeated prime=No Conditions: none The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities. Step identifiers and dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S069 S071 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 := div(702, 3) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 = 234 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 · impliesBefore Mobius inversion ORIGINAL inverse F(n/3):actual positive input=234;check complete integer range1..234 After ALL positive divisors INCREASING=[1, 2, 3, 6, 9, 13, 18, 26, 39, 78, 117, 234];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, S071 S073 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor := mul(25, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(1) 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, S072 S074 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b := sum(25, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b = 26 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, S073 S075 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor := mul(25, 2) After Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor = 50 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, S074 S076 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b := sum(50, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b = 51 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, S075 S077 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(3) ORIGINAL a times divisor := mul(25, 3) After Mobius inversion ORIGINAL inverse F(n/3) f(3) ORIGINAL a times divisor = 75 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, S076 S078 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(3) plus ORIGINAL b := sum(75, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(3) plus ORIGINAL b = 76 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, S077 S079 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(6) ORIGINAL a times divisor := mul(25, 6) After Mobius inversion ORIGINAL inverse F(n/3) f(6) ORIGINAL a times divisor = 150 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, S078 S080 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(6) plus ORIGINAL b := sum(150, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(6) plus ORIGINAL b = 151 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(9) ORIGINAL a times divisor := mul(25, 9) After Mobius inversion ORIGINAL inverse F(n/3) f(9) ORIGINAL a times divisor = 225 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, S080 S082 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(9) plus ORIGINAL b := sum(225, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(9) plus ORIGINAL b = 226 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 inversion ORIGINAL inverse F(n/3) f(13) ORIGINAL a times divisor := mul(25, 13) After Mobius inversion ORIGINAL inverse F(n/3) f(13) ORIGINAL a times divisor = 325 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 inversion ORIGINAL inverse F(n/3) f(13) plus ORIGINAL b := sum(325, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(13) plus ORIGINAL b = 326 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 inversion ORIGINAL inverse F(n/3) f(18) ORIGINAL a times divisor := mul(25, 18) After Mobius inversion ORIGINAL inverse F(n/3) f(18) ORIGINAL a times divisor = 450 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 inversion ORIGINAL inverse F(n/3) f(18) plus ORIGINAL b := sum(450, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(18) plus ORIGINAL b = 451 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 inversion ORIGINAL inverse F(n/3) f(26) ORIGINAL a times divisor := mul(25, 26) After Mobius inversion ORIGINAL inverse F(n/3) f(26) ORIGINAL a times divisor = 650 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(26) plus ORIGINAL b := sum(650, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(26) plus ORIGINAL b = 651 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, S087 S089 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(39) ORIGINAL a times divisor := mul(25, 39) After Mobius inversion ORIGINAL inverse F(n/3) f(39) ORIGINAL a times divisor = 975 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, S088 S090 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(39) plus ORIGINAL b := sum(975, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(39) plus ORIGINAL b = 976 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, S089 S091 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(78) ORIGINAL a times divisor := mul(25, 78) After Mobius inversion ORIGINAL inverse F(n/3) f(78) ORIGINAL a times divisor = 1950 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, S090 S092 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(78) plus ORIGINAL b := sum(1950, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(78) plus ORIGINAL b = 1951 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, S091 S093 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(117) ORIGINAL a times divisor := mul(25, 117) After Mobius inversion ORIGINAL inverse F(n/3) f(117) ORIGINAL a times divisor = 2925 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, S092 S094 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(117) plus ORIGINAL b := sum(2925, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(117) plus ORIGINAL b = 2926 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, S093 S095 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(234) ORIGINAL a times divisor := mul(25, 234) After Mobius inversion ORIGINAL inverse F(n/3) f(234) ORIGINAL a times divisor = 5850 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, S094 S096 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(234) plus ORIGINAL b := sum(5850, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(234) plus ORIGINAL b = 5851 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, S095 S097 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum := sum(26, 51, 76, 151, 226, 326, 451, 651, 976, 1951, 2926, 5851) After Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum = 13662 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, S096 S098 · calculationBefore Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) := mul(-1, 13662) After Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) = -13662 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, S097 S099 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor6:actual positive input for mu=6 After complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search Conditions: none Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search. Step identifiers and dependenciesRule: mobius_complete_UPF_input_role · Uses: G001, S098 S100 · impliesBefore ORIGINAL positive integer absolute magnitude=6>1 After initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted Conditions: none Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime. Step identifiers and dependenciesRule: upf_factorization_remaining_product_invariant · Uses: G001, S099 S101 · calculationBefore upf extract divisor2 occurrence0 := div(6, 2) After upf extract divisor2 occurrence0 = 3 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S100 S102 · impliesBefore next divisor=2;previous remaining=6;all smaller divisors exhausted After extracted multiplicity=1;next remaining=3;remaining modulo2=1;extracted multiset=[2] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S101 S103 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S102 S104 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S103 S105 · impliesBefore exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=6 After complete increasing prime multiset=[2, 3];full product=6;a composite terminal remainder would have an already tested smaller divisor Conditions: none 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, S104 S106 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor6:input=6;COMPLETE prime multiset=[2, 3];distinct primes=[2, 3] After mu=1;empty product case=No;repeated prime=No Conditions: none The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities. Step identifiers and dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S105 S107 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 := div(702, 6) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 = 117 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 · impliesBefore Mobius inversion ORIGINAL inverse F(n/6):actual positive input=117;check complete integer range1..117 After ALL positive divisors INCREASING=[1, 3, 9, 13, 39, 117];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, S107 S109 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor := mul(25, 1) After Mobius inversion ORIGINAL inverse F(n/6) f(1) 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, S108 S110 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b := sum(25, 1) After Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b = 26 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, S109 S111 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(3) ORIGINAL a times divisor := mul(25, 3) After Mobius inversion ORIGINAL inverse F(n/6) f(3) ORIGINAL a times divisor = 75 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, S110 S112 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(3) plus ORIGINAL b := sum(75, 1) After Mobius inversion ORIGINAL inverse F(n/6) f(3) plus ORIGINAL b = 76 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, S111 S113 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(9) ORIGINAL a times divisor := mul(25, 9) After Mobius inversion ORIGINAL inverse F(n/6) f(9) ORIGINAL a times divisor = 225 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, S112 S114 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(9) plus ORIGINAL b := sum(225, 1) After Mobius inversion ORIGINAL inverse F(n/6) f(9) plus ORIGINAL b = 226 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, S113 S115 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(13) ORIGINAL a times divisor := mul(25, 13) After Mobius inversion ORIGINAL inverse F(n/6) f(13) ORIGINAL a times divisor = 325 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, S114 S116 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(13) plus ORIGINAL b := sum(325, 1) After Mobius inversion ORIGINAL inverse F(n/6) f(13) plus ORIGINAL b = 326 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, S115 S117 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(39) ORIGINAL a times divisor := mul(25, 39) After Mobius inversion ORIGINAL inverse F(n/6) f(39) ORIGINAL a times divisor = 975 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, S116 S118 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(39) plus ORIGINAL b := sum(975, 1) After Mobius inversion ORIGINAL inverse F(n/6) f(39) plus ORIGINAL b = 976 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, S117 S119 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(117) ORIGINAL a times divisor := mul(25, 117) After Mobius inversion ORIGINAL inverse F(n/6) f(117) ORIGINAL a times divisor = 2925 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, S118 S120 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(117) plus ORIGINAL b := sum(2925, 1) After Mobius inversion ORIGINAL inverse F(n/6) f(117) plus ORIGINAL b = 2926 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, S119 S121 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum := sum(26, 76, 226, 326, 976, 2926) After Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum = 4556 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, S120 S122 · calculationBefore Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) := mul(1, 4556) After Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) = 4556 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, S121 S123 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor9:actual positive input for mu=9 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, S122 S124 · impliesBefore ORIGINAL positive integer absolute magnitude=9>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, S123 S125 · impliesBefore next divisor=2;previous remaining=9;all smaller divisors exhausted After extracted multiplicity=0;next remaining=9;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, S124 S126 · calculationBefore upf extract divisor3 occurrence0 := div(9, 3) After upf extract divisor3 occurrence0 = 3 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, S125 S127 · calculationBefore upf extract divisor3 occurrence1 := div(3, 3) After upf extract divisor3 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, S126 S128 · impliesBefore next divisor=3;previous remaining=9;all smaller divisors exhausted After extracted multiplicity=2;next remaining=1;remaining modulo3=1;extracted multiset=[3, 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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S127 S129 · calculationBefore upf canonical complete product factor0 := mul(1, 3) After upf canonical complete product factor0 = 3 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S128 S130 · calculationBefore upf canonical complete product factor1 := mul(3, 3) After upf canonical complete product factor1 = 9 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, S129 S131 · impliesBefore exhausted trial divisors below4;terminal remaining=1;d*d>remaining;ORIGINAL n=9 After complete increasing prime multiset=[3, 3];full product=9;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, S130 S132 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor9:input=9;COMPLETE prime multiset=[3, 3];distinct primes=[3] After mu=0;empty product case=No;repeated prime=Yes Conditions: none The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities. Step identifiers and dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S131 S133 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor9 := div(702, 9) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor9 = 78 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, S132 S134 · impliesBefore Mobius inversion ORIGINAL inverse F(n/9):actual positive input=78;check complete integer range1..78 After ALL positive divisors INCREASING=[1, 2, 3, 6, 13, 26, 39, 78];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, S133 S135 · calculationBefore Mobius inversion ORIGINAL inverse F(n/9) f(1) ORIGINAL a times divisor := mul(25, 1) After Mobius inversion ORIGINAL inverse F(n/9) f(1) 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, S134 S136 · calculationBefore Mobius inversion ORIGINAL inverse F(n/9) f(1) plus ORIGINAL b := sum(25, 1) After Mobius inversion ORIGINAL inverse F(n/9) f(1) plus ORIGINAL b = 26 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, S135 S137 · calculationBefore Mobius inversion ORIGINAL inverse F(n/9) f(2) ORIGINAL a times divisor := mul(25, 2) After Mobius inversion ORIGINAL inverse F(n/9) f(2) ORIGINAL a times divisor = 50 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, S136 S138 · calculationBefore Mobius inversion ORIGINAL inverse F(n/9) f(2) plus ORIGINAL b := sum(50, 1) After Mobius inversion ORIGINAL inverse F(n/9) f(2) plus ORIGINAL b = 51 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, S137 S139 · calculationBefore Mobius inversion ORIGINAL inverse F(n/9) f(3) ORIGINAL a times divisor := mul(25, 3) After Mobius inversion ORIGINAL inverse F(n/9) f(3) ORIGINAL a times divisor = 75 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, S138 S140 · calculationBefore Mobius inversion ORIGINAL inverse F(n/9) f(3) plus ORIGINAL b := sum(75, 1) After Mobius inversion ORIGINAL inverse F(n/9) f(3) plus ORIGINAL b = 76 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, S139 S141 · calculationBefore Mobius inversion ORIGINAL inverse F(n/9) f(6) ORIGINAL a times divisor := mul(25, 6) After Mobius inversion ORIGINAL inverse F(n/9) f(6) ORIGINAL a times divisor = 150 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, S140 S142 · calculationBefore Mobius inversion ORIGINAL inverse F(n/9) f(6) plus ORIGINAL b := sum(150, 1) After Mobius inversion ORIGINAL inverse F(n/9) f(6) plus ORIGINAL b = 151 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, S141 S143 · calculationBefore Mobius inversion ORIGINAL inverse F(n/9) f(13) ORIGINAL a times divisor := mul(25, 13) After Mobius inversion ORIGINAL inverse F(n/9) f(13) ORIGINAL a times divisor = 325 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, S142 S144 · calculationBefore Mobius inversion ORIGINAL inverse F(n/9) f(13) plus ORIGINAL b := sum(325, 1) After Mobius inversion ORIGINAL inverse F(n/9) f(13) plus ORIGINAL b = 326 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, S143 S145 · calculationBefore Mobius inversion ORIGINAL inverse F(n/9) f(26) ORIGINAL a times divisor := mul(25, 26) After Mobius inversion ORIGINAL inverse F(n/9) f(26) ORIGINAL a times divisor = 650 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, S144 S146 · calculationBefore Mobius inversion ORIGINAL inverse F(n/9) f(26) plus ORIGINAL b := sum(650, 1) After Mobius inversion ORIGINAL inverse F(n/9) f(26) plus ORIGINAL b = 651 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, S145 S147 · calculationBefore Mobius inversion ORIGINAL inverse F(n/9) f(39) ORIGINAL a times divisor := mul(25, 39) After Mobius inversion ORIGINAL inverse F(n/9) f(39) ORIGINAL a times divisor = 975 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, S146 S148 · calculationBefore Mobius inversion ORIGINAL inverse F(n/9) f(39) plus ORIGINAL b := sum(975, 1) After Mobius inversion ORIGINAL inverse F(n/9) f(39) plus ORIGINAL b = 976 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, S147 S149 · calculationBefore Mobius inversion ORIGINAL inverse F(n/9) f(78) ORIGINAL a times divisor := mul(25, 78) After Mobius inversion ORIGINAL inverse F(n/9) f(78) ORIGINAL a times divisor = 1950 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, S148 S150 · calculationBefore Mobius inversion ORIGINAL inverse F(n/9) f(78) plus ORIGINAL b := sum(1950, 1) After Mobius inversion ORIGINAL inverse F(n/9) f(78) plus ORIGINAL b = 1951 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, S149 S151 · calculationBefore Mobius inversion ORIGINAL inverse F(n/9) full forward divisor sum := sum(26, 51, 76, 151, 326, 651, 976, 1951) After Mobius inversion ORIGINAL inverse F(n/9) full forward divisor sum = 4208 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, S150 S152 · calculationBefore Mobius inversion ORIGINAL inverse mu(9) times SAME F(n/9) := mul(0, 4208) After Mobius inversion ORIGINAL inverse mu(9) times SAME F(n/9) = 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, S151 S153 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor13:actual positive input for mu=13 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, S152 S154 · impliesBefore ORIGINAL positive integer absolute magnitude=13>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, S153 S155 · impliesBefore next divisor=2;previous remaining=13;all smaller divisors exhausted After extracted multiplicity=0;next remaining=13;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, S154 S156 · impliesBefore next divisor=3;previous remaining=13;all smaller divisors exhausted After extracted multiplicity=0;next remaining=13;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, S155 S157 · calculationBefore upf canonical complete product factor0 := mul(1, 13) After upf canonical complete product factor0 = 13 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, S156 S158 · impliesBefore exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=13 After complete increasing prime multiset=[13];full product=13;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, S157 S159 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor13:input=13;COMPLETE prime multiset=[13];distinct primes=[13] 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, S158 S160 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor13 := div(702, 13) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor13 = 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, S159 S161 · impliesBefore Mobius inversion ORIGINAL inverse F(n/13):actual positive input=54;check complete integer range1..54 After ALL positive divisors INCREASING=[1, 2, 3, 6, 9, 18, 27, 54];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, S160 S162 · calculationBefore Mobius inversion ORIGINAL inverse F(n/13) f(1) ORIGINAL a times divisor := mul(25, 1) After Mobius inversion ORIGINAL inverse F(n/13) f(1) 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, S161 S163 · calculationBefore Mobius inversion ORIGINAL inverse F(n/13) f(1) plus ORIGINAL b := sum(25, 1) After Mobius inversion ORIGINAL inverse F(n/13) f(1) plus ORIGINAL b = 26 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, S162 S164 · calculationBefore Mobius inversion ORIGINAL inverse F(n/13) f(2) ORIGINAL a times divisor := mul(25, 2) After Mobius inversion ORIGINAL inverse F(n/13) f(2) ORIGINAL a times divisor = 50 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, S163 S165 · calculationBefore Mobius inversion ORIGINAL inverse F(n/13) f(2) plus ORIGINAL b := sum(50, 1) After Mobius inversion ORIGINAL inverse F(n/13) f(2) plus ORIGINAL b = 51 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, S164 S166 · calculationBefore Mobius inversion ORIGINAL inverse F(n/13) f(3) ORIGINAL a times divisor := mul(25, 3) After Mobius inversion ORIGINAL inverse F(n/13) f(3) ORIGINAL a times divisor = 75 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, S165 S167 · calculationBefore Mobius inversion ORIGINAL inverse F(n/13) f(3) plus ORIGINAL b := sum(75, 1) After Mobius inversion ORIGINAL inverse F(n/13) f(3) plus ORIGINAL b = 76 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, S166 S168 · calculationBefore Mobius inversion ORIGINAL inverse F(n/13) f(6) ORIGINAL a times divisor := mul(25, 6) After Mobius inversion ORIGINAL inverse F(n/13) f(6) ORIGINAL a times divisor = 150 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, S167 S169 · calculationBefore Mobius inversion ORIGINAL inverse F(n/13) f(6) plus ORIGINAL b := sum(150, 1) After Mobius inversion ORIGINAL inverse F(n/13) f(6) plus ORIGINAL b = 151 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, S168 S170 · calculationBefore Mobius inversion ORIGINAL inverse F(n/13) f(9) ORIGINAL a times divisor := mul(25, 9) After Mobius inversion ORIGINAL inverse F(n/13) f(9) ORIGINAL a times divisor = 225 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, S169 S171 · calculationBefore Mobius inversion ORIGINAL inverse F(n/13) f(9) plus ORIGINAL b := sum(225, 1) After Mobius inversion ORIGINAL inverse F(n/13) f(9) plus ORIGINAL b = 226 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, S170 S172 · calculationBefore Mobius inversion ORIGINAL inverse F(n/13) f(18) ORIGINAL a times divisor := mul(25, 18) After Mobius inversion ORIGINAL inverse F(n/13) f(18) ORIGINAL a times divisor = 450 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, S171 S173 · calculationBefore Mobius inversion ORIGINAL inverse F(n/13) f(18) plus ORIGINAL b := sum(450, 1) After Mobius inversion ORIGINAL inverse F(n/13) f(18) plus ORIGINAL b = 451 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, S172 S174 · calculationBefore Mobius inversion ORIGINAL inverse F(n/13) f(27) ORIGINAL a times divisor := mul(25, 27) After Mobius inversion ORIGINAL inverse F(n/13) f(27) ORIGINAL a times divisor = 675 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, S173 S175 · calculationBefore Mobius inversion ORIGINAL inverse F(n/13) f(27) plus ORIGINAL b := sum(675, 1) After Mobius inversion ORIGINAL inverse F(n/13) f(27) plus ORIGINAL b = 676 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, S174 S176 · calculationBefore Mobius inversion ORIGINAL inverse F(n/13) f(54) ORIGINAL a times divisor := mul(25, 54) After Mobius inversion ORIGINAL inverse F(n/13) f(54) ORIGINAL a times divisor = 1350 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, S175 S177 · calculationBefore Mobius inversion ORIGINAL inverse F(n/13) f(54) plus ORIGINAL b := sum(1350, 1) After Mobius inversion ORIGINAL inverse F(n/13) f(54) plus ORIGINAL b = 1351 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, S176 S178 · calculationBefore Mobius inversion ORIGINAL inverse F(n/13) full forward divisor sum := sum(26, 51, 76, 151, 226, 451, 676, 1351) After Mobius inversion ORIGINAL inverse F(n/13) full forward divisor sum = 3008 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, S177 S179 · calculationBefore Mobius inversion ORIGINAL inverse mu(13) times SAME F(n/13) := mul(-1, 3008) After Mobius inversion ORIGINAL inverse mu(13) times SAME F(n/13) = -3008 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, S178 S180 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor18:actual positive input for mu=18 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, S179 S181 · impliesBefore ORIGINAL positive integer absolute magnitude=18>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, S180 S182 · calculationBefore upf extract divisor2 occurrence0 := div(18, 2) After upf extract divisor2 occurrence0 = 9 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, S181 S183 · impliesBefore next divisor=2;previous remaining=18;all smaller divisors exhausted After extracted multiplicity=1;next remaining=9;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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S182 S184 · calculationBefore upf extract divisor3 occurrence1 := div(9, 3) After upf extract divisor3 occurrence1 = 3 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, S183 S185 · calculationBefore upf extract divisor3 occurrence2 := div(3, 3) After upf extract divisor3 occurrence2 = 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, S184 S186 · impliesBefore next divisor=3;previous remaining=9;all smaller divisors exhausted After extracted multiplicity=2;next remaining=1;remaining modulo3=1;extracted multiset=[2, 3, 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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S185 S187 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S186 S188 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S187 S189 · calculationBefore upf canonical complete product factor2 := mul(6, 3) After upf canonical complete product factor2 = 18 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, S188 S190 · impliesBefore exhausted trial divisors below4;terminal remaining=1;d*d>remaining;ORIGINAL n=18 After complete increasing prime multiset=[2, 3, 3];full product=18;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, S189 S191 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor18:input=18;COMPLETE prime multiset=[2, 3, 3];distinct primes=[2, 3] After mu=0;empty product case=No;repeated prime=Yes Conditions: none The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities. Step identifiers and dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S190 S192 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor18 := div(702, 18) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor18 = 39 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, S191 S193 · impliesBefore Mobius inversion ORIGINAL inverse F(n/18):actual positive input=39;check complete integer range1..39 After ALL positive divisors INCREASING=[1, 3, 13, 39];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, S192 S194 · calculationBefore Mobius inversion ORIGINAL inverse F(n/18) f(1) ORIGINAL a times divisor := mul(25, 1) After Mobius inversion ORIGINAL inverse F(n/18) f(1) 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, S193 S195 · calculationBefore Mobius inversion ORIGINAL inverse F(n/18) f(1) plus ORIGINAL b := sum(25, 1) After Mobius inversion ORIGINAL inverse F(n/18) f(1) plus ORIGINAL b = 26 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, S194 S196 · calculationBefore Mobius inversion ORIGINAL inverse F(n/18) f(3) ORIGINAL a times divisor := mul(25, 3) After Mobius inversion ORIGINAL inverse F(n/18) f(3) ORIGINAL a times divisor = 75 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, S195 S197 · calculationBefore Mobius inversion ORIGINAL inverse F(n/18) f(3) plus ORIGINAL b := sum(75, 1) After Mobius inversion ORIGINAL inverse F(n/18) f(3) plus ORIGINAL b = 76 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, S196 S198 · calculationBefore Mobius inversion ORIGINAL inverse F(n/18) f(13) ORIGINAL a times divisor := mul(25, 13) After Mobius inversion ORIGINAL inverse F(n/18) f(13) ORIGINAL a times divisor = 325 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, S197 S199 · calculationBefore Mobius inversion ORIGINAL inverse F(n/18) f(13) plus ORIGINAL b := sum(325, 1) After Mobius inversion ORIGINAL inverse F(n/18) f(13) plus ORIGINAL b = 326 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, S198 S200 · calculationBefore Mobius inversion ORIGINAL inverse F(n/18) f(39) ORIGINAL a times divisor := mul(25, 39) After Mobius inversion ORIGINAL inverse F(n/18) f(39) ORIGINAL a times divisor = 975 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, S199 S201 · calculationBefore Mobius inversion ORIGINAL inverse F(n/18) f(39) plus ORIGINAL b := sum(975, 1) After Mobius inversion ORIGINAL inverse F(n/18) f(39) plus ORIGINAL b = 976 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, S200 S202 · calculationBefore Mobius inversion ORIGINAL inverse F(n/18) full forward divisor sum := sum(26, 76, 326, 976) After Mobius inversion ORIGINAL inverse F(n/18) full forward divisor sum = 1404 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, S201 S203 · calculationBefore Mobius inversion ORIGINAL inverse mu(18) times SAME F(n/18) := mul(0, 1404) After Mobius inversion ORIGINAL inverse mu(18) times SAME F(n/18) = 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, S202 S204 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor26:actual positive input for mu=26 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, S203 S205 · impliesBefore ORIGINAL positive integer absolute magnitude=26>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, S204 S206 · calculationBefore upf extract divisor2 occurrence0 := div(26, 2) After upf extract divisor2 occurrence0 = 13 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, S205 S207 · impliesBefore next divisor=2;previous remaining=26;all smaller divisors exhausted After extracted multiplicity=1;next remaining=13;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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S206 S208 · impliesBefore next divisor=3;previous remaining=13;all smaller divisors exhausted After extracted multiplicity=0;next remaining=13;remaining modulo3=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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S207 S209 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S208 S210 · calculationBefore upf canonical complete product factor1 := mul(2, 13) After upf canonical complete product factor1 = 26 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, S209 S211 · impliesBefore exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=26 After complete increasing prime multiset=[2, 13];full product=26;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, S210 S212 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor26:input=26;COMPLETE prime multiset=[2, 13];distinct primes=[2, 13] 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, S211 S213 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor26 := div(702, 26) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor26 = 27 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, S212 S214 · impliesBefore Mobius inversion ORIGINAL inverse F(n/26):actual positive input=27;check complete integer range1..27 After ALL positive divisors INCREASING=[1, 3, 9, 27];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, S213 S215 · calculationBefore Mobius inversion ORIGINAL inverse F(n/26) f(1) ORIGINAL a times divisor := mul(25, 1) After Mobius inversion ORIGINAL inverse F(n/26) f(1) 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, S214 S216 · calculationBefore Mobius inversion ORIGINAL inverse F(n/26) f(1) plus ORIGINAL b := sum(25, 1) After Mobius inversion ORIGINAL inverse F(n/26) f(1) plus ORIGINAL b = 26 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, S215 S217 · calculationBefore Mobius inversion ORIGINAL inverse F(n/26) f(3) ORIGINAL a times divisor := mul(25, 3) After Mobius inversion ORIGINAL inverse F(n/26) f(3) ORIGINAL a times divisor = 75 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, S216 S218 · calculationBefore Mobius inversion ORIGINAL inverse F(n/26) f(3) plus ORIGINAL b := sum(75, 1) After Mobius inversion ORIGINAL inverse F(n/26) f(3) plus ORIGINAL b = 76 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, S217 S219 · calculationBefore Mobius inversion ORIGINAL inverse F(n/26) f(9) ORIGINAL a times divisor := mul(25, 9) After Mobius inversion ORIGINAL inverse F(n/26) f(9) ORIGINAL a times divisor = 225 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, S218 S220 · calculationBefore Mobius inversion ORIGINAL inverse F(n/26) f(9) plus ORIGINAL b := sum(225, 1) After Mobius inversion ORIGINAL inverse F(n/26) f(9) plus ORIGINAL b = 226 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, S219 S221 · calculationBefore Mobius inversion ORIGINAL inverse F(n/26) f(27) ORIGINAL a times divisor := mul(25, 27) After Mobius inversion ORIGINAL inverse F(n/26) f(27) ORIGINAL a times divisor = 675 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, S220 S222 · calculationBefore Mobius inversion ORIGINAL inverse F(n/26) f(27) plus ORIGINAL b := sum(675, 1) After Mobius inversion ORIGINAL inverse F(n/26) f(27) plus ORIGINAL b = 676 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, S221 S223 · calculationBefore Mobius inversion ORIGINAL inverse F(n/26) full forward divisor sum := sum(26, 76, 226, 676) After Mobius inversion ORIGINAL inverse F(n/26) full forward divisor sum = 1004 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, S222 S224 · calculationBefore Mobius inversion ORIGINAL inverse mu(26) times SAME F(n/26) := mul(1, 1004) After Mobius inversion ORIGINAL inverse mu(26) times SAME F(n/26) = 1004 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, S223 S225 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor27:actual positive input for mu=27 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, S224 S226 · impliesBefore ORIGINAL positive integer absolute magnitude=27>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, S225 S227 · impliesBefore next divisor=2;previous remaining=27;all smaller divisors exhausted After extracted multiplicity=0;next remaining=27;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, S226 S228 · calculationBefore upf extract divisor3 occurrence0 := div(27, 3) After upf extract divisor3 occurrence0 = 9 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, S227 S229 · calculationBefore upf extract divisor3 occurrence1 := div(9, 3) After upf extract divisor3 occurrence1 = 3 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, S228 S230 · calculationBefore upf extract divisor3 occurrence2 := div(3, 3) After upf extract divisor3 occurrence2 = 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, S229 S231 · impliesBefore next divisor=3;previous remaining=27;all smaller divisors exhausted After extracted multiplicity=3;next remaining=1;remaining modulo3=1;extracted multiset=[3, 3, 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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S230 S232 · calculationBefore upf canonical complete product factor0 := mul(1, 3) After upf canonical complete product factor0 = 3 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S231 S233 · calculationBefore upf canonical complete product factor1 := mul(3, 3) After upf canonical complete product factor1 = 9 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, S232 S234 · calculationBefore upf canonical complete product factor2 := mul(9, 3) After upf canonical complete product factor2 = 27 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, S233 S235 · impliesBefore exhausted trial divisors below4;terminal remaining=1;d*d>remaining;ORIGINAL n=27 After complete increasing prime multiset=[3, 3, 3];full product=27;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, S234 S236 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor27:input=27;COMPLETE prime multiset=[3, 3, 3];distinct primes=[3] After mu=0;empty product case=No;repeated prime=Yes Conditions: none The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities. Step identifiers and dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S235 S237 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor27 := div(702, 27) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor27 = 26 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, S236 S238 · impliesBefore Mobius inversion ORIGINAL inverse F(n/27):actual positive input=26;check complete integer range1..26 After ALL positive divisors INCREASING=[1, 2, 13, 26];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, S237 S239 · calculationBefore Mobius inversion ORIGINAL inverse F(n/27) f(1) ORIGINAL a times divisor := mul(25, 1) After Mobius inversion ORIGINAL inverse F(n/27) f(1) 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, S238 S240 · calculationBefore Mobius inversion ORIGINAL inverse F(n/27) f(1) plus ORIGINAL b := sum(25, 1) After Mobius inversion ORIGINAL inverse F(n/27) f(1) plus ORIGINAL b = 26 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, S239 S241 · calculationBefore Mobius inversion ORIGINAL inverse F(n/27) f(2) ORIGINAL a times divisor := mul(25, 2) After Mobius inversion ORIGINAL inverse F(n/27) f(2) ORIGINAL a times divisor = 50 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, S240 S242 · calculationBefore Mobius inversion ORIGINAL inverse F(n/27) f(2) plus ORIGINAL b := sum(50, 1) After Mobius inversion ORIGINAL inverse F(n/27) f(2) plus ORIGINAL b = 51 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, S241 S243 · calculationBefore Mobius inversion ORIGINAL inverse F(n/27) f(13) ORIGINAL a times divisor := mul(25, 13) After Mobius inversion ORIGINAL inverse F(n/27) f(13) ORIGINAL a times divisor = 325 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, S242 S244 · calculationBefore Mobius inversion ORIGINAL inverse F(n/27) f(13) plus ORIGINAL b := sum(325, 1) After Mobius inversion ORIGINAL inverse F(n/27) f(13) plus ORIGINAL b = 326 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, S243 S245 · calculationBefore Mobius inversion ORIGINAL inverse F(n/27) f(26) ORIGINAL a times divisor := mul(25, 26) After Mobius inversion ORIGINAL inverse F(n/27) f(26) ORIGINAL a times divisor = 650 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, S244 S246 · calculationBefore Mobius inversion ORIGINAL inverse F(n/27) f(26) plus ORIGINAL b := sum(650, 1) After Mobius inversion ORIGINAL inverse F(n/27) f(26) plus ORIGINAL b = 651 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, S245 S247 · calculationBefore Mobius inversion ORIGINAL inverse F(n/27) full forward divisor sum := sum(26, 51, 326, 651) After Mobius inversion ORIGINAL inverse F(n/27) full forward divisor sum = 1054 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, S246 S248 · calculationBefore Mobius inversion ORIGINAL inverse mu(27) times SAME F(n/27) := mul(0, 1054) After Mobius inversion ORIGINAL inverse mu(27) times SAME F(n/27) = 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, S247 S249 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor39:actual positive input for mu=39 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, S248 S250 · impliesBefore ORIGINAL positive integer absolute magnitude=39>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, S249 S251 · impliesBefore next divisor=2;previous remaining=39;all smaller divisors exhausted After extracted multiplicity=0;next remaining=39;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, S250 S252 · calculationBefore upf extract divisor3 occurrence0 := div(39, 3) After upf extract divisor3 occurrence0 = 13 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, S251 S253 · impliesBefore next divisor=3;previous remaining=39;all smaller divisors exhausted After extracted multiplicity=1;next remaining=13;remaining modulo3=1;extracted multiset=[3] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S252 S254 · calculationBefore upf canonical complete product factor0 := mul(1, 3) After upf canonical complete product factor0 = 3 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S253 S255 · calculationBefore upf canonical complete product factor1 := mul(3, 13) After upf canonical complete product factor1 = 39 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, S254 S256 · impliesBefore exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=39 After complete increasing prime multiset=[3, 13];full product=39;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, S255 S257 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor39:input=39;COMPLETE prime multiset=[3, 13];distinct primes=[3, 13] 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, S256 S258 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor39 := div(702, 39) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor39 = 18 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, S257 S259 · impliesBefore Mobius inversion ORIGINAL inverse F(n/39):actual positive input=18;check complete integer range1..18 After ALL positive divisors INCREASING=[1, 2, 3, 6, 9, 18];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, S258 S260 · calculationBefore Mobius inversion ORIGINAL inverse F(n/39) f(1) ORIGINAL a times divisor := mul(25, 1) After Mobius inversion ORIGINAL inverse F(n/39) f(1) 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, S259 S261 · calculationBefore Mobius inversion ORIGINAL inverse F(n/39) f(1) plus ORIGINAL b := sum(25, 1) After Mobius inversion ORIGINAL inverse F(n/39) f(1) plus ORIGINAL b = 26 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, S260 S262 · calculationBefore Mobius inversion ORIGINAL inverse F(n/39) f(2) ORIGINAL a times divisor := mul(25, 2) After Mobius inversion ORIGINAL inverse F(n/39) f(2) ORIGINAL a times divisor = 50 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, S261 S263 · calculationBefore Mobius inversion ORIGINAL inverse F(n/39) f(2) plus ORIGINAL b := sum(50, 1) After Mobius inversion ORIGINAL inverse F(n/39) f(2) plus ORIGINAL b = 51 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, S262 S264 · calculationBefore Mobius inversion ORIGINAL inverse F(n/39) f(3) ORIGINAL a times divisor := mul(25, 3) After Mobius inversion ORIGINAL inverse F(n/39) f(3) ORIGINAL a times divisor = 75 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, S263 S265 · calculationBefore Mobius inversion ORIGINAL inverse F(n/39) f(3) plus ORIGINAL b := sum(75, 1) After Mobius inversion ORIGINAL inverse F(n/39) f(3) plus ORIGINAL b = 76 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, S264 S266 · calculationBefore Mobius inversion ORIGINAL inverse F(n/39) f(6) ORIGINAL a times divisor := mul(25, 6) After Mobius inversion ORIGINAL inverse F(n/39) f(6) ORIGINAL a times divisor = 150 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, S265 S267 · calculationBefore Mobius inversion ORIGINAL inverse F(n/39) f(6) plus ORIGINAL b := sum(150, 1) After Mobius inversion ORIGINAL inverse F(n/39) f(6) plus ORIGINAL b = 151 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, S266 S268 · calculationBefore Mobius inversion ORIGINAL inverse F(n/39) f(9) ORIGINAL a times divisor := mul(25, 9) After Mobius inversion ORIGINAL inverse F(n/39) f(9) ORIGINAL a times divisor = 225 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, S267 S269 · calculationBefore Mobius inversion ORIGINAL inverse F(n/39) f(9) plus ORIGINAL b := sum(225, 1) After Mobius inversion ORIGINAL inverse F(n/39) f(9) plus ORIGINAL b = 226 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, S268 S270 · calculationBefore Mobius inversion ORIGINAL inverse F(n/39) f(18) ORIGINAL a times divisor := mul(25, 18) After Mobius inversion ORIGINAL inverse F(n/39) f(18) ORIGINAL a times divisor = 450 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, S269 S271 · calculationBefore Mobius inversion ORIGINAL inverse F(n/39) f(18) plus ORIGINAL b := sum(450, 1) After Mobius inversion ORIGINAL inverse F(n/39) f(18) plus ORIGINAL b = 451 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, S270 S272 · calculationBefore Mobius inversion ORIGINAL inverse F(n/39) full forward divisor sum := sum(26, 51, 76, 151, 226, 451) After Mobius inversion ORIGINAL inverse F(n/39) full forward divisor sum = 981 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, S271 S273 · calculationBefore Mobius inversion ORIGINAL inverse mu(39) times SAME F(n/39) := mul(1, 981) After Mobius inversion ORIGINAL inverse mu(39) times SAME F(n/39) = 981 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, S272 S274 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor54:actual positive input for mu=54 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, S273 S275 · impliesBefore ORIGINAL positive integer absolute magnitude=54>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, S274 S276 · calculationBefore upf extract divisor2 occurrence0 := div(54, 2) After upf extract divisor2 occurrence0 = 27 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, S275 S277 · impliesBefore next divisor=2;previous remaining=54;all smaller divisors exhausted After extracted multiplicity=1;next remaining=27;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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S276 S278 · calculationBefore upf extract divisor3 occurrence1 := div(27, 3) After upf extract divisor3 occurrence1 = 9 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, S277 S279 · calculationBefore upf extract divisor3 occurrence2 := div(9, 3) After upf extract divisor3 occurrence2 = 3 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, S278 S280 · calculationBefore upf extract divisor3 occurrence3 := div(3, 3) After upf extract divisor3 occurrence3 = 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, S279 S281 · impliesBefore next divisor=3;previous remaining=27;all smaller divisors exhausted After extracted multiplicity=3;next remaining=1;remaining modulo3=1;extracted multiset=[2, 3, 3, 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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S280 S282 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S281 S283 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S282 S284 · calculationBefore upf canonical complete product factor2 := mul(6, 3) After upf canonical complete product factor2 = 18 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, S283 S285 · calculationBefore upf canonical complete product factor3 := mul(18, 3) After upf canonical complete product factor3 = 54 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, S284 S286 · impliesBefore exhausted trial divisors below4;terminal remaining=1;d*d>remaining;ORIGINAL n=54 After complete increasing prime multiset=[2, 3, 3, 3];full product=54;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, S285 S287 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor54:input=54;COMPLETE prime multiset=[2, 3, 3, 3];distinct primes=[2, 3] After mu=0;empty product case=No;repeated prime=Yes Conditions: none The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities. Step identifiers and dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S286 S288 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor54 := div(702, 54) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor54 = 13 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, S287 S289 · impliesBefore Mobius inversion ORIGINAL inverse F(n/54):actual positive input=13;check complete integer range1..13 After ALL positive divisors INCREASING=[1, 13];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, S288 S290 · calculationBefore Mobius inversion ORIGINAL inverse F(n/54) f(1) ORIGINAL a times divisor := mul(25, 1) After Mobius inversion ORIGINAL inverse F(n/54) f(1) 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, S289 S291 · calculationBefore Mobius inversion ORIGINAL inverse F(n/54) f(1) plus ORIGINAL b := sum(25, 1) After Mobius inversion ORIGINAL inverse F(n/54) f(1) plus ORIGINAL b = 26 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, S290 S292 · calculationBefore Mobius inversion ORIGINAL inverse F(n/54) f(13) ORIGINAL a times divisor := mul(25, 13) After Mobius inversion ORIGINAL inverse F(n/54) f(13) ORIGINAL a times divisor = 325 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, S291 S293 · calculationBefore Mobius inversion ORIGINAL inverse F(n/54) f(13) plus ORIGINAL b := sum(325, 1) After Mobius inversion ORIGINAL inverse F(n/54) f(13) plus ORIGINAL b = 326 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, S292 S294 · calculationBefore Mobius inversion ORIGINAL inverse F(n/54) full forward divisor sum := sum(26, 326) After Mobius inversion ORIGINAL inverse F(n/54) full forward divisor sum = 352 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, S293 S295 · calculationBefore Mobius inversion ORIGINAL inverse mu(54) times SAME F(n/54) := mul(0, 352) After Mobius inversion ORIGINAL inverse mu(54) times SAME F(n/54) = 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, S294 S296 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor78:actual positive input for mu=78 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, S295 S297 · impliesBefore ORIGINAL positive integer absolute magnitude=78>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, S296 S298 · calculationBefore upf extract divisor2 occurrence0 := div(78, 2) After upf extract divisor2 occurrence0 = 39 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, S297 S299 · impliesBefore next divisor=2;previous remaining=78;all smaller divisors exhausted After extracted multiplicity=1;next remaining=39;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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S298 S300 · calculationBefore upf extract divisor3 occurrence1 := div(39, 3) After upf extract divisor3 occurrence1 = 13 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, S299 S301 · impliesBefore next divisor=3;previous remaining=39;all smaller divisors exhausted After extracted multiplicity=1;next remaining=13;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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S300 S302 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S301 S303 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S302 S304 · calculationBefore upf canonical complete product factor2 := mul(6, 13) After upf canonical complete product factor2 = 78 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, S303 S305 · impliesBefore exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=78 After complete increasing prime multiset=[2, 3, 13];full product=78;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, S304 S306 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor78:input=78;COMPLETE prime multiset=[2, 3, 13];distinct primes=[2, 3, 13] 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, S305 S307 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor78 := div(702, 78) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor78 = 9 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, S306 S308 · impliesBefore Mobius inversion ORIGINAL inverse F(n/78):actual positive input=9;check complete integer range1..9 After ALL positive divisors INCREASING=[1, 3, 9];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, S307 S309 · calculationBefore Mobius inversion ORIGINAL inverse F(n/78) f(1) ORIGINAL a times divisor := mul(25, 1) After Mobius inversion ORIGINAL inverse F(n/78) f(1) 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, S308 S310 · calculationBefore Mobius inversion ORIGINAL inverse F(n/78) f(1) plus ORIGINAL b := sum(25, 1) After Mobius inversion ORIGINAL inverse F(n/78) f(1) plus ORIGINAL b = 26 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, S309 S311 · calculationBefore Mobius inversion ORIGINAL inverse F(n/78) f(3) ORIGINAL a times divisor := mul(25, 3) After Mobius inversion ORIGINAL inverse F(n/78) f(3) ORIGINAL a times divisor = 75 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, S310 S312 · calculationBefore Mobius inversion ORIGINAL inverse F(n/78) f(3) plus ORIGINAL b := sum(75, 1) After Mobius inversion ORIGINAL inverse F(n/78) f(3) plus ORIGINAL b = 76 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, S311 S313 · calculationBefore Mobius inversion ORIGINAL inverse F(n/78) f(9) ORIGINAL a times divisor := mul(25, 9) After Mobius inversion ORIGINAL inverse F(n/78) f(9) ORIGINAL a times divisor = 225 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, S312 S314 · calculationBefore Mobius inversion ORIGINAL inverse F(n/78) f(9) plus ORIGINAL b := sum(225, 1) After Mobius inversion ORIGINAL inverse F(n/78) f(9) plus ORIGINAL b = 226 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, S313 S315 · calculationBefore Mobius inversion ORIGINAL inverse F(n/78) full forward divisor sum := sum(26, 76, 226) After Mobius inversion ORIGINAL inverse F(n/78) full forward divisor sum = 328 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, S314 S316 · calculationBefore Mobius inversion ORIGINAL inverse mu(78) times SAME F(n/78) := mul(-1, 328) After Mobius inversion ORIGINAL inverse mu(78) times SAME F(n/78) = -328 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, S315 S317 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor117:actual positive input for mu=117 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, S316 S318 · impliesBefore ORIGINAL positive integer absolute magnitude=117>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, S317 S319 · impliesBefore next divisor=2;previous remaining=117;all smaller divisors exhausted After extracted multiplicity=0;next remaining=117;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, S318 S320 · calculationBefore upf extract divisor3 occurrence0 := div(117, 3) After upf extract divisor3 occurrence0 = 39 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, S319 S321 · calculationBefore upf extract divisor3 occurrence1 := div(39, 3) After upf extract divisor3 occurrence1 = 13 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, S320 S322 · impliesBefore next divisor=3;previous remaining=117;all smaller divisors exhausted After extracted multiplicity=2;next remaining=13;remaining modulo3=1;extracted multiset=[3, 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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S321 S323 · calculationBefore upf canonical complete product factor0 := mul(1, 3) After upf canonical complete product factor0 = 3 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S322 S324 · calculationBefore upf canonical complete product factor1 := mul(3, 3) After upf canonical complete product factor1 = 9 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, S323 S325 · calculationBefore upf canonical complete product factor2 := mul(9, 13) After upf canonical complete product factor2 = 117 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, S324 S326 · impliesBefore exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=117 After complete increasing prime multiset=[3, 3, 13];full product=117;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, S325 S327 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor117:input=117;COMPLETE prime multiset=[3, 3, 13];distinct primes=[3, 13] 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, S326 S328 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor117 := div(702, 117) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor117 = 6 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, S327 S329 · impliesBefore Mobius inversion ORIGINAL inverse F(n/117):actual positive input=6;check complete integer range1..6 After ALL positive divisors INCREASING=[1, 2, 3, 6];not only primes/proper divisors or all integers<=n Conditions: none Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction. Step identifiers and dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S328 S330 · calculationBefore Mobius inversion ORIGINAL inverse F(n/117) f(1) ORIGINAL a times divisor := mul(25, 1) After Mobius inversion ORIGINAL inverse F(n/117) f(1) 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, S329 S331 · calculationBefore Mobius inversion ORIGINAL inverse F(n/117) f(1) plus ORIGINAL b := sum(25, 1) After Mobius inversion ORIGINAL inverse F(n/117) f(1) plus ORIGINAL b = 26 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, S330 S332 · calculationBefore Mobius inversion ORIGINAL inverse F(n/117) f(2) ORIGINAL a times divisor := mul(25, 2) After Mobius inversion ORIGINAL inverse F(n/117) f(2) ORIGINAL a times divisor = 50 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, S331 S333 · calculationBefore Mobius inversion ORIGINAL inverse F(n/117) f(2) plus ORIGINAL b := sum(50, 1) After Mobius inversion ORIGINAL inverse F(n/117) f(2) plus ORIGINAL b = 51 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, S332 S334 · calculationBefore Mobius inversion ORIGINAL inverse F(n/117) f(3) ORIGINAL a times divisor := mul(25, 3) After Mobius inversion ORIGINAL inverse F(n/117) f(3) ORIGINAL a times divisor = 75 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, S333 S335 · calculationBefore Mobius inversion ORIGINAL inverse F(n/117) f(3) plus ORIGINAL b := sum(75, 1) After Mobius inversion ORIGINAL inverse F(n/117) f(3) plus ORIGINAL b = 76 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, S334 S336 · calculationBefore Mobius inversion ORIGINAL inverse F(n/117) f(6) ORIGINAL a times divisor := mul(25, 6) After Mobius inversion ORIGINAL inverse F(n/117) f(6) ORIGINAL a times divisor = 150 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, S335 S337 · calculationBefore Mobius inversion ORIGINAL inverse F(n/117) f(6) plus ORIGINAL b := sum(150, 1) After Mobius inversion ORIGINAL inverse F(n/117) f(6) plus ORIGINAL b = 151 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, S336 S338 · calculationBefore Mobius inversion ORIGINAL inverse F(n/117) full forward divisor sum := sum(26, 51, 76, 151) After Mobius inversion ORIGINAL inverse F(n/117) full forward divisor sum = 304 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, S337 S339 · calculationBefore Mobius inversion ORIGINAL inverse mu(117) times SAME F(n/117) := mul(0, 304) After Mobius inversion ORIGINAL inverse mu(117) times SAME F(n/117) = 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, S338 S340 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor234:actual positive input for mu=234 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, S339 S341 · impliesBefore ORIGINAL positive integer absolute magnitude=234>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, S340 S342 · calculationBefore upf extract divisor2 occurrence0 := div(234, 2) After upf extract divisor2 occurrence0 = 117 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, S341 S343 · impliesBefore next divisor=2;previous remaining=234;all smaller divisors exhausted After extracted multiplicity=1;next remaining=117;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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S342 S344 · calculationBefore upf extract divisor3 occurrence1 := div(117, 3) After upf extract divisor3 occurrence1 = 39 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, S343 S345 · calculationBefore upf extract divisor3 occurrence2 := div(39, 3) After upf extract divisor3 occurrence2 = 13 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, S344 S346 · impliesBefore next divisor=3;previous remaining=117;all smaller divisors exhausted After extracted multiplicity=2;next remaining=13;remaining modulo3=1;extracted multiset=[2, 3, 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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S345 S347 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S346 S348 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S347 S349 · calculationBefore upf canonical complete product factor2 := mul(6, 3) After upf canonical complete product factor2 = 18 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, S348 S350 · calculationBefore upf canonical complete product factor3 := mul(18, 13) After upf canonical complete product factor3 = 234 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, S349 S351 · impliesBefore exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=234 After complete increasing prime multiset=[2, 3, 3, 13];full product=234;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, S350 S352 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor234:input=234;COMPLETE prime multiset=[2, 3, 3, 13];distinct primes=[2, 3, 13] 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, S351 S353 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor234 := div(702, 234) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor234 = 3 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, S352 S354 · impliesBefore Mobius inversion ORIGINAL inverse F(n/234):actual positive input=3;check complete integer range1..3 After ALL positive divisors INCREASING=[1, 3];not only primes/proper divisors or all integers<=n Conditions: none Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction. Step identifiers and dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S353 S355 · calculationBefore Mobius inversion ORIGINAL inverse F(n/234) f(1) ORIGINAL a times divisor := mul(25, 1) After Mobius inversion ORIGINAL inverse F(n/234) f(1) 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, S354 S356 · calculationBefore Mobius inversion ORIGINAL inverse F(n/234) f(1) plus ORIGINAL b := sum(25, 1) After Mobius inversion ORIGINAL inverse F(n/234) f(1) plus ORIGINAL b = 26 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, S355 S357 · calculationBefore Mobius inversion ORIGINAL inverse F(n/234) f(3) ORIGINAL a times divisor := mul(25, 3) After Mobius inversion ORIGINAL inverse F(n/234) f(3) ORIGINAL a times divisor = 75 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, S356 S358 · calculationBefore Mobius inversion ORIGINAL inverse F(n/234) f(3) plus ORIGINAL b := sum(75, 1) After Mobius inversion ORIGINAL inverse F(n/234) f(3) plus ORIGINAL b = 76 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, S357 S359 · calculationBefore Mobius inversion ORIGINAL inverse F(n/234) full forward divisor sum := sum(26, 76) After Mobius inversion ORIGINAL inverse F(n/234) full forward divisor sum = 102 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, S358 S360 · calculationBefore Mobius inversion ORIGINAL inverse mu(234) times SAME F(n/234) := mul(0, 102) After Mobius inversion ORIGINAL inverse mu(234) times SAME F(n/234) = 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, S359 S361 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor351:actual positive input for mu=351 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, S360 S362 · impliesBefore ORIGINAL positive integer absolute magnitude=351>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, S361 S363 · impliesBefore next divisor=2;previous remaining=351;all smaller divisors exhausted After extracted multiplicity=0;next remaining=351;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, S362 S364 · calculationBefore upf extract divisor3 occurrence0 := div(351, 3) After upf extract divisor3 occurrence0 = 117 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, S363 S365 · calculationBefore upf extract divisor3 occurrence1 := div(117, 3) After upf extract divisor3 occurrence1 = 39 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, S364 S366 · calculationBefore upf extract divisor3 occurrence2 := div(39, 3) After upf extract divisor3 occurrence2 = 13 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, S365 S367 · impliesBefore next divisor=3;previous remaining=351;all smaller divisors exhausted After extracted multiplicity=3;next remaining=13;remaining modulo3=1;extracted multiset=[3, 3, 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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S366 S368 · calculationBefore upf canonical complete product factor0 := mul(1, 3) After upf canonical complete product factor0 = 3 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S367 S369 · calculationBefore upf canonical complete product factor1 := mul(3, 3) After upf canonical complete product factor1 = 9 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, S368 S370 · calculationBefore upf canonical complete product factor2 := mul(9, 3) After upf canonical complete product factor2 = 27 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, S369 S371 · calculationBefore upf canonical complete product factor3 := mul(27, 13) After upf canonical complete product factor3 = 351 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, S370 S372 · impliesBefore exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=351 After complete increasing prime multiset=[3, 3, 3, 13];full product=351;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, S371 S373 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor351:input=351;COMPLETE prime multiset=[3, 3, 3, 13];distinct primes=[3, 13] 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, S372 S374 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor351 := div(702, 351) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor351 = 2 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, S373 S375 · impliesBefore Mobius inversion ORIGINAL inverse F(n/351):actual positive input=2;check complete integer range1..2 After ALL positive divisors INCREASING=[1, 2];not only primes/proper divisors or all integers<=n Conditions: none Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction. Step identifiers and dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S374 S376 · calculationBefore Mobius inversion ORIGINAL inverse F(n/351) f(1) ORIGINAL a times divisor := mul(25, 1) After Mobius inversion ORIGINAL inverse F(n/351) f(1) 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, S375 S377 · calculationBefore Mobius inversion ORIGINAL inverse F(n/351) f(1) plus ORIGINAL b := sum(25, 1) After Mobius inversion ORIGINAL inverse F(n/351) f(1) plus ORIGINAL b = 26 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, S376 S378 · calculationBefore Mobius inversion ORIGINAL inverse F(n/351) f(2) ORIGINAL a times divisor := mul(25, 2) After Mobius inversion ORIGINAL inverse F(n/351) f(2) ORIGINAL a times divisor = 50 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, S377 S379 · calculationBefore Mobius inversion ORIGINAL inverse F(n/351) f(2) plus ORIGINAL b := sum(50, 1) After Mobius inversion ORIGINAL inverse F(n/351) f(2) plus ORIGINAL b = 51 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, S378 S380 · calculationBefore Mobius inversion ORIGINAL inverse F(n/351) full forward divisor sum := sum(26, 51) After Mobius inversion ORIGINAL inverse F(n/351) full forward divisor sum = 77 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, S379 S381 · calculationBefore Mobius inversion ORIGINAL inverse mu(351) times SAME F(n/351) := mul(0, 77) After Mobius inversion ORIGINAL inverse mu(351) times SAME F(n/351) = 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, S380 S382 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor702:actual positive input for mu=702 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, S381 S383 · impliesBefore ORIGINAL positive integer absolute magnitude=702>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, S382 S384 · calculationBefore upf extract divisor2 occurrence0 := div(702, 2) After upf extract divisor2 occurrence0 = 351 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, S383 S385 · impliesBefore next divisor=2;previous remaining=702;all smaller divisors exhausted After extracted multiplicity=1;next remaining=351;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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S384 S386 · calculationBefore upf extract divisor3 occurrence1 := div(351, 3) After upf extract divisor3 occurrence1 = 117 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, S385 S387 · calculationBefore upf extract divisor3 occurrence2 := div(117, 3) After upf extract divisor3 occurrence2 = 39 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, S386 S388 · calculationBefore upf extract divisor3 occurrence3 := div(39, 3) After upf extract divisor3 occurrence3 = 13 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, S387 S389 · impliesBefore next divisor=3;previous remaining=351;all smaller divisors exhausted After extracted multiplicity=3;next remaining=13;remaining modulo3=1;extracted multiset=[2, 3, 3, 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 dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S388 S390 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S389 S391 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S390 S392 · calculationBefore upf canonical complete product factor2 := mul(6, 3) After upf canonical complete product factor2 = 18 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, S391 S393 · calculationBefore upf canonical complete product factor3 := mul(18, 3) After upf canonical complete product factor3 = 54 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, S392 S394 · calculationBefore upf canonical complete product factor4 := mul(54, 13) After upf canonical complete product factor4 = 702 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, S393 S395 · impliesBefore exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=702 After complete increasing prime multiset=[2, 3, 3, 3, 13];full product=702;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, S394 S396 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor702:input=702;COMPLETE prime multiset=[2, 3, 3, 3, 13];distinct primes=[2, 3, 13] 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, S395 S397 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor702 := div(702, 702) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor702 = 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, S396 S398 · impliesBefore Mobius inversion ORIGINAL inverse F(n/702):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, S397 S399 · calculationBefore Mobius inversion ORIGINAL inverse F(n/702) f(1) ORIGINAL a times divisor := mul(25, 1) After Mobius inversion ORIGINAL inverse F(n/702) f(1) 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, S398 S400 · calculationBefore Mobius inversion ORIGINAL inverse F(n/702) f(1) plus ORIGINAL b := sum(25, 1) After Mobius inversion ORIGINAL inverse F(n/702) f(1) plus ORIGINAL b = 26 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, S399 S401 · calculationBefore Mobius inversion ORIGINAL inverse F(n/702) full forward divisor sum := sum(26) After Mobius inversion ORIGINAL inverse F(n/702) full forward divisor sum = 26 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, S400 S402 · calculationBefore Mobius inversion ORIGINAL inverse mu(702) times SAME F(n/702) := mul(0, 26) After Mobius inversion ORIGINAL inverse mu(702) times SAME F(n/702) = 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, S401 S403 · calculationBefore Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(42016, -14008, -13662, 4556, 0, -3008, 0, 1004, 0, 981, 0, -328, 0, 0, 0, 0) After Mobius inversion ORIGINAL inverse complete inverse-term sum = 17551 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, S402 S404 · calculationBefore Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(25, 702) After Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 17550 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, S403 S405 · calculationBefore Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(17550, 1) After Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 17551 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, S404 S406 · impliesBefore Mobius inversion ORIGINAL inverse:actual n=702;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 702, 42016, 42016], [2, -1, 351, 14008, -14008], [3, -1, 234, 13662, -13662], [6, 1, 117, 4556, 4556], [9, 0, 78, 4208, 0], [13, -1, 54, 3008, -3008], [18, 0, 39, 1404, 0], [26, 1, 27, 1004, 1004], [27, 0, 26, 1054, 0], [39, 1, 18, 981, 981], [54, 0, 13, 352, 0], [78, -1, 9, 328, -328], [117, 0, 6, 304, 0], [234, 0, 3, 102, 0], [351, 0, 2, 77, 0], [702, 0, 1, 26, 0]] After complete reconstructed=17551;direct f(n)=17551;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, S405 S407 · calculationBefore Mobius inversion ORIGINAL omitted-divisor partial sum := sum(-14008, -13662, 4556, 0, -3008, 0, 1004, 0, 981, 0, -328, 0, 0, 0, 0) After Mobius inversion ORIGINAL omitted-divisor partial sum = -24465 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, S406 S408 · calculationBefore Mobius inversion complete minus partial := sub(17551, -24465) After Mobius inversion complete minus partial = 42016 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, S407 S409 · impliesBefore ORIGINAL full divisors=[1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702] After complete=17551=f(n)=17551;partial=-24465;lost term=42016!=0;incomplete_certificate fixed No Conditions: none The omitted value must be an actual divisor with nonzero contribution,so the partial result differs from the target.Omitting a zero term does not qualify this discriminative boundary.The No rejects the incomplete certificate,not the complete inversion. Step identifiers and dependenciesRule: mobinv_omission_actual_nonzero_discriminative_term · Uses: G001, S408 ## Reverse checks - S409 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S408 · premise_and_result_audit — Recompute Mobius inversion complete minus partial with exact rational arithmetic, not rounded decimals. - S407 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL omitted-divisor partial sum with exact rational arithmetic, not rounded decimals. - S406 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S405 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S404 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals. - S403 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals. - S402 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(702) times SAME F(n/702) with exact rational arithmetic, not rounded decimals. - S401 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/702) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S400 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/702) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S399 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/702) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S398 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S397 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor702 with exact rational arithmetic, not rounded decimals. - S396 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S395 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S394 · premise_and_result_audit — Recompute upf canonical complete product factor4 with exact rational arithmetic, not rounded decimals. - S393 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals. - S392 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S391 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S390 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S389 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S388 · premise_and_result_audit — Recompute upf extract divisor3 occurrence3 with exact rational arithmetic, not rounded decimals. - S387 · premise_and_result_audit — Recompute upf extract divisor3 occurrence2 with exact rational arithmetic, not rounded decimals. - S386 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals. - S385 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S384 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S383 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S382 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S381 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(351) times SAME F(n/351) with exact rational arithmetic, not rounded decimals. - S380 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/351) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S379 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/351) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S378 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/351) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S377 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/351) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S376 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/351) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S375 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S374 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor351 with exact rational arithmetic, not rounded decimals. - S373 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S372 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S371 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals. - S370 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S369 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S368 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S367 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S366 · premise_and_result_audit — Recompute upf extract divisor3 occurrence2 with exact rational arithmetic, not rounded decimals. - S365 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals. - S364 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals. - S363 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S362 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S361 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S360 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(234) times SAME F(n/234) with exact rational arithmetic, not rounded decimals. - S359 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/234) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S358 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/234) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S357 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/234) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S356 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/234) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S355 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/234) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S354 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S353 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor234 with exact rational arithmetic, not rounded decimals. - S352 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S351 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S350 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals. - S349 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S348 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S347 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S346 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S345 · premise_and_result_audit — Recompute upf extract divisor3 occurrence2 with exact rational arithmetic, not rounded decimals. - S344 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals. - S343 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S342 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S341 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S340 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S339 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(117) times SAME F(n/117) with exact rational arithmetic, not rounded decimals. - S338 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/117) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S337 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/117) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S336 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/117) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S335 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/117) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S334 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/117) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S333 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/117) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S332 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/117) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S331 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/117) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S330 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/117) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S329 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S328 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor117 with exact rational arithmetic, not rounded decimals. - S327 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S326 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S325 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S324 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S323 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S322 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S321 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals. - S320 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals. - S319 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S318 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S317 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S316 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(78) times SAME F(n/78) with exact rational arithmetic, not rounded decimals. - S315 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/78) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S314 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/78) f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S313 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/78) f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S312 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/78) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S311 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/78) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S310 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/78) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S309 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/78) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S308 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S307 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor78 with exact rational arithmetic, not rounded decimals. - S306 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S305 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S304 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S303 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S302 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S301 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S300 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals. - S299 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S298 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S297 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S296 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S295 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(54) times SAME F(n/54) with exact rational arithmetic, not rounded decimals. - S294 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/54) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S293 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/54) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S292 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/54) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S291 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/54) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S290 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/54) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S289 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S288 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor54 with exact rational arithmetic, not rounded decimals. - S287 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S286 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S285 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals. - S284 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S283 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S282 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S281 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S280 · premise_and_result_audit — Recompute upf extract divisor3 occurrence3 with exact rational arithmetic, not rounded decimals. - S279 · premise_and_result_audit — Recompute upf extract divisor3 occurrence2 with exact rational arithmetic, not rounded decimals. - S278 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals. - S277 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S276 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S275 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S274 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S273 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(39) times SAME F(n/39) with exact rational arithmetic, not rounded decimals. - S272 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S271 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) f(18) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S270 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) f(18) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S269 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S268 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S267 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S266 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S265 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S264 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S263 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S262 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S261 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S260 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S259 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S258 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor39 with exact rational arithmetic, not rounded decimals. - S257 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S256 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S255 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S254 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S253 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S252 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals. - S251 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S250 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S249 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S248 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(27) times SAME F(n/27) with exact rational arithmetic, not rounded decimals. - S247 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/27) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S246 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/27) f(26) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S245 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/27) f(26) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S244 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/27) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S243 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/27) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S242 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/27) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S241 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/27) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S240 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/27) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S239 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/27) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S238 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S237 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor27 with exact rational arithmetic, not rounded decimals. - S236 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S235 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S234 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S233 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S232 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S231 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S230 · premise_and_result_audit — Recompute upf extract divisor3 occurrence2 with exact rational arithmetic, not rounded decimals. - S229 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals. - S228 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals. - S227 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S226 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S225 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S224 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(26) times SAME F(n/26) with exact rational arithmetic, not rounded decimals. - S223 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S222 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(27) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S221 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(27) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S220 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S219 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S218 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S217 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S216 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S215 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S214 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S213 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor26 with exact rational arithmetic, not rounded decimals. - S212 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S211 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S210 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S209 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S208 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S207 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S206 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S205 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S204 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S203 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(18) times SAME F(n/18) with exact rational arithmetic, not rounded decimals. - S202 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S201 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(39) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S200 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(39) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S199 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S198 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S197 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S196 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S195 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S194 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S193 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S192 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor18 with exact rational arithmetic, not rounded decimals. - S191 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S190 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S189 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S188 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S187 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S186 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S185 · premise_and_result_audit — Recompute upf extract divisor3 occurrence2 with exact rational arithmetic, not rounded decimals. - S184 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals. - S183 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S182 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S181 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S180 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S179 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(13) times SAME F(n/13) with exact rational arithmetic, not rounded decimals. - S178 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S177 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(54) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S176 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(54) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S175 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(27) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S174 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(27) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S173 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(18) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S172 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(18) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S171 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S170 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S169 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S168 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S167 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S166 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S165 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S164 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S163 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S162 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S161 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S160 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor13 with exact rational arithmetic, not rounded decimals. - S159 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S158 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S157 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S156 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S155 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S154 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S153 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S152 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(9) times SAME F(n/9) with exact rational arithmetic, not rounded decimals. - S151 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S150 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(78) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S149 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(78) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S148 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(39) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S147 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(39) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S146 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(26) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S145 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(26) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S144 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S143 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S142 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S141 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S140 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S139 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S138 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S137 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S136 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S135 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S133 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor9 with exact rational arithmetic, not rounded decimals. - S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S131 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S130 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S129 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S127 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals. - S126 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals. - S125 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S122 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) with exact rational arithmetic, not rounded decimals. - S121 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S120 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(117) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S119 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(117) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S118 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(39) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S117 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(39) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S116 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S115 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S114 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S113 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S112 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S111 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S110 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S109 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S107 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 with exact rational arithmetic, not rounded decimals. - S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S104 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S103 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S101 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Recompute Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) with exact rational arithmetic, not rounded decimals. - S097 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S096 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(234) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S095 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(234) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S094 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(117) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S093 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(117) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S092 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(78) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S091 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(78) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S090 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(39) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S089 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(39) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S088 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(26) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S087 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(26) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S086 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(18) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S085 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(18) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S084 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S083 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S082 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S081 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S080 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S079 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S078 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S077 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S075 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S074 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S073 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S071 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 with exact rational arithmetic, not rounded decimals. - S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S068 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S065 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals. - S064 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S063 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(351) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(351) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(117) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(117) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(39) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(39) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(27) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(27) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S046 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 upf canonical complete product factor0 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 Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(702) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(702) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(351) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(351) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(234) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(234) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(117) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(117) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(78) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(78) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(54) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(54) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(39) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(39) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(27) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(27) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(26) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(26) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(18) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(18) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (702, [1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702], 1, [2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702], 17551, -24465, 17551, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 3c8c6f3057c0dcaf3f07e4b509df351720ab4c7ce8a5f6d017b8b9deae512fd5 - candidate record SHA-256: 4d9581fdeaba4edf463f815d6119fe6b97dfcbad3b3a6eb0949f08ca65905f59 - 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-0100.json)