# Question 22144: 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-0094) 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. R0043-22144 · q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0094 ## Original question and givens Use positive integer n=420, exact integer arithmetic function f(k)=a*k+b with a=19, 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=420, exact integer arithmetic function f(k)=a*k+b with a=19, 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=420;check complete integer range1..420 After ALL positive divisors INCREASING=[1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420];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(420, 1) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 420 Conditions: none Substitute original 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=420;check complete integer range1..420 After ALL positive divisors INCREASING=[1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420];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(19, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original 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(19, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original 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(19, 2) After Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 38 Conditions: none Substitute original 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(38, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 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, S009 S011 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor := mul(19, 3) After Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor = 57 Conditions: none Substitute original 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(57, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b = 58 Conditions: none Substitute original 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(4) ORIGINAL a times divisor := mul(19, 4) After Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor = 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, S012 S014 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b := sum(76, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b = 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, S013 S015 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor := mul(19, 5) After Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor = 95 Conditions: none Substitute original 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(5) plus ORIGINAL b := sum(95, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b = 96 Conditions: none Substitute original 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(6) ORIGINAL a times divisor := mul(19, 6) After Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor = 114 Conditions: none Substitute original 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(6) plus ORIGINAL b := sum(114, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b = 115 Conditions: none Substitute original 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(7) ORIGINAL a times divisor := mul(19, 7) After Mobius inversion ORIGINAL inverse F(n/1) f(7) ORIGINAL a times divisor = 133 Conditions: none Substitute original 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(7) plus ORIGINAL b := sum(133, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b = 134 Conditions: none Substitute original 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(10) ORIGINAL a times divisor := mul(19, 10) After Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor = 190 Conditions: none Substitute original 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(10) plus ORIGINAL b := sum(190, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b = 191 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(12) ORIGINAL a times divisor := mul(19, 12) After Mobius inversion ORIGINAL inverse F(n/1) f(12) ORIGINAL a times divisor = 228 Conditions: none Substitute original 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(12) plus ORIGINAL b := sum(228, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(12) plus ORIGINAL b = 229 Conditions: none Substitute original 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(14) ORIGINAL a times divisor := mul(19, 14) After Mobius inversion ORIGINAL inverse F(n/1) f(14) ORIGINAL a times divisor = 266 Conditions: none Substitute original 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(14) plus ORIGINAL b := sum(266, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(14) plus ORIGINAL b = 267 Conditions: none Substitute original 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(15) ORIGINAL a times divisor := mul(19, 15) After Mobius inversion ORIGINAL inverse F(n/1) f(15) ORIGINAL a times divisor = 285 Conditions: none Substitute original 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(15) plus ORIGINAL b := sum(285, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(15) plus ORIGINAL b = 286 Conditions: none Substitute original 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(20) ORIGINAL a times divisor := mul(19, 20) After Mobius inversion ORIGINAL inverse F(n/1) f(20) ORIGINAL a times divisor = 380 Conditions: none Substitute original 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(20) plus ORIGINAL b := sum(380, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(20) plus ORIGINAL b = 381 Conditions: none Substitute original 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(21) ORIGINAL a times divisor := mul(19, 21) After Mobius inversion ORIGINAL inverse F(n/1) f(21) ORIGINAL a times divisor = 399 Conditions: none Substitute original 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(21) plus ORIGINAL b := sum(399, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(21) plus ORIGINAL b = 400 Conditions: none Substitute original 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(28) ORIGINAL a times divisor := mul(19, 28) After Mobius inversion ORIGINAL inverse F(n/1) f(28) ORIGINAL a times divisor = 532 Conditions: none Substitute original 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(28) plus ORIGINAL b := sum(532, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(28) plus ORIGINAL b = 533 Conditions: none Substitute original 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(30) ORIGINAL a times divisor := mul(19, 30) After Mobius inversion ORIGINAL inverse F(n/1) f(30) ORIGINAL a times divisor = 570 Conditions: none Substitute original 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(30) plus ORIGINAL b := sum(570, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(30) plus ORIGINAL b = 571 Conditions: none Substitute original 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(35) ORIGINAL a times divisor := mul(19, 35) After Mobius inversion ORIGINAL inverse F(n/1) f(35) ORIGINAL a times divisor = 665 Conditions: none Substitute original 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(35) plus ORIGINAL b := sum(665, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(35) plus ORIGINAL b = 666 Conditions: none Substitute original 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) f(42) ORIGINAL a times divisor := mul(19, 42) After Mobius inversion ORIGINAL inverse F(n/1) f(42) ORIGINAL a times divisor = 798 Conditions: none Substitute original 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 F(n/1) f(42) plus ORIGINAL b := sum(798, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(42) plus ORIGINAL b = 799 Conditions: none Substitute original 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(60) ORIGINAL a times divisor := mul(19, 60) After Mobius inversion ORIGINAL inverse F(n/1) f(60) ORIGINAL a times divisor = 1140 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(60) plus ORIGINAL b := sum(1140, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(60) plus ORIGINAL b = 1141 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(70) ORIGINAL a times divisor := mul(19, 70) After Mobius inversion ORIGINAL inverse F(n/1) f(70) ORIGINAL a times divisor = 1330 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(70) plus ORIGINAL b := sum(1330, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(70) plus ORIGINAL b = 1331 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(84) ORIGINAL a times divisor := mul(19, 84) After Mobius inversion ORIGINAL inverse F(n/1) f(84) ORIGINAL a times divisor = 1596 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(84) plus ORIGINAL b := sum(1596, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(84) plus ORIGINAL b = 1597 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(105) ORIGINAL a times divisor := mul(19, 105) After Mobius inversion ORIGINAL inverse F(n/1) f(105) ORIGINAL a times divisor = 1995 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore Mobius inversion ORIGINAL inverse F(n/1) f(105) plus ORIGINAL b := sum(1995, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(105) plus ORIGINAL b = 1996 Conditions: none Substitute original 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/1) f(140) ORIGINAL a times divisor := mul(19, 140) After Mobius inversion ORIGINAL inverse F(n/1) f(140) ORIGINAL a times divisor = 2660 Conditions: none Substitute original 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/1) f(140) plus ORIGINAL b := sum(2660, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(140) plus ORIGINAL b = 2661 Conditions: none Substitute original 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/1) f(210) ORIGINAL a times divisor := mul(19, 210) After Mobius inversion ORIGINAL inverse F(n/1) f(210) ORIGINAL a times divisor = 3990 Conditions: none Substitute original 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/1) f(210) plus ORIGINAL b := sum(3990, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(210) plus ORIGINAL b = 3991 Conditions: none Substitute original 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/1) f(420) ORIGINAL a times divisor := mul(19, 420) After Mobius inversion ORIGINAL inverse F(n/1) f(420) ORIGINAL a times divisor = 7980 Conditions: none Substitute original 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/1) f(420) plus ORIGINAL b := sum(7980, 1) After Mobius inversion ORIGINAL inverse F(n/1) f(420) plus ORIGINAL b = 7981 Conditions: none Substitute original 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/1) full forward divisor sum := sum(20, 39, 58, 77, 96, 115, 134, 191, 229, 267, 286, 381, 400, 533, 571, 666, 799, 1141, 1331, 1597, 1996, 2661, 3991, 7981) After Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 25560 Conditions: none Substitute original 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 mu(1) times SAME F(n/1) := mul(1, 25560) After Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 25560 Conditions: none Substitute original 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 · 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, S056 S058 · 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, S057 S059 · 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, S058 S060 · 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, S059 S061 · 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, S060 S062 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 := div(420, 2) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 210 Conditions: none Substitute original 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 · impliesBefore Mobius inversion ORIGINAL inverse F(n/2):actual positive input=210;check complete integer range1..210 After ALL positive divisors INCREASING=[1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210];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, S062 S064 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original 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 F(n/2) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor := mul(19, 2) After Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor = 38 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S065 S067 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b := sum(38, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b = 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, S066 S068 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor := mul(19, 3) After Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor = 57 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067 S069 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b := sum(57, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b = 58 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S068 S070 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor := mul(19, 5) After Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor = 95 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069 S071 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b := sum(95, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b = 96 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S070 S072 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) f(6) ORIGINAL a times divisor := mul(19, 6) After Mobius inversion ORIGINAL inverse F(n/2) f(6) ORIGINAL a times divisor = 114 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071 S073 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) f(6) plus ORIGINAL b := sum(114, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(6) plus ORIGINAL b = 115 Conditions: none Substitute original 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/2) f(7) ORIGINAL a times divisor := mul(19, 7) After Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor = 133 Conditions: none Substitute original 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/2) f(7) plus ORIGINAL b := sum(133, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b = 134 Conditions: none Substitute original 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/2) f(10) ORIGINAL a times divisor := mul(19, 10) After Mobius inversion ORIGINAL inverse F(n/2) f(10) ORIGINAL a times divisor = 190 Conditions: none Substitute original 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/2) f(10) plus ORIGINAL b := sum(190, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(10) plus ORIGINAL b = 191 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S076 S078 · calculationBefore Mobius inversion ORIGINAL inverse F(n/2) f(14) ORIGINAL a times divisor := mul(19, 14) After Mobius inversion ORIGINAL inverse F(n/2) f(14) ORIGINAL a times divisor = 266 Conditions: none Substitute original 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/2) f(14) plus ORIGINAL b := sum(266, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(14) plus ORIGINAL b = 267 Conditions: none Substitute original 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/2) f(15) ORIGINAL a times divisor := mul(19, 15) After Mobius inversion ORIGINAL inverse F(n/2) f(15) ORIGINAL a times divisor = 285 Conditions: none Substitute original 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/2) f(15) plus ORIGINAL b := sum(285, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(15) plus ORIGINAL b = 286 Conditions: none Substitute original 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/2) f(21) ORIGINAL a times divisor := mul(19, 21) After Mobius inversion ORIGINAL inverse F(n/2) f(21) ORIGINAL a times divisor = 399 Conditions: none Substitute original 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/2) f(21) plus ORIGINAL b := sum(399, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(21) plus ORIGINAL b = 400 Conditions: none Substitute original 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/2) f(30) ORIGINAL a times divisor := mul(19, 30) After Mobius inversion ORIGINAL inverse F(n/2) f(30) ORIGINAL a times divisor = 570 Conditions: none Substitute original 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/2) f(30) plus ORIGINAL b := sum(570, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(30) plus ORIGINAL b = 571 Conditions: none Substitute original 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/2) f(35) ORIGINAL a times divisor := mul(19, 35) After Mobius inversion ORIGINAL inverse F(n/2) f(35) ORIGINAL a times divisor = 665 Conditions: none Substitute original 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/2) f(35) plus ORIGINAL b := sum(665, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(35) plus ORIGINAL b = 666 Conditions: none Substitute original 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/2) f(42) ORIGINAL a times divisor := mul(19, 42) After Mobius inversion ORIGINAL inverse F(n/2) f(42) ORIGINAL a times divisor = 798 Conditions: none Substitute original 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/2) f(42) plus ORIGINAL b := sum(798, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(42) plus ORIGINAL b = 799 Conditions: none Substitute original 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/2) f(70) ORIGINAL a times divisor := mul(19, 70) After Mobius inversion ORIGINAL inverse F(n/2) f(70) ORIGINAL a times divisor = 1330 Conditions: none Substitute original 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/2) f(70) plus ORIGINAL b := sum(1330, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(70) plus ORIGINAL b = 1331 Conditions: none Substitute original 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/2) f(105) ORIGINAL a times divisor := mul(19, 105) After Mobius inversion ORIGINAL inverse F(n/2) f(105) ORIGINAL a times divisor = 1995 Conditions: none Substitute original 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/2) f(105) plus ORIGINAL b := sum(1995, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(105) plus ORIGINAL b = 1996 Conditions: none Substitute original 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/2) f(210) ORIGINAL a times divisor := mul(19, 210) After Mobius inversion ORIGINAL inverse F(n/2) f(210) ORIGINAL a times divisor = 3990 Conditions: none Substitute original 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/2) f(210) plus ORIGINAL b := sum(3990, 1) After Mobius inversion ORIGINAL inverse F(n/2) f(210) plus ORIGINAL b = 3991 Conditions: none Substitute original 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/2) full forward divisor sum := sum(20, 39, 58, 96, 115, 134, 191, 267, 286, 400, 571, 666, 799, 1331, 1996, 3991) After Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 10960 Conditions: none Substitute original 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 mu(2) times SAME F(n/2) := mul(-1, 10960) After Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -10960 Conditions: none Substitute original 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 · 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, S097 S099 · 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, S098 S100 · 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, S099 S101 · 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, S100 S102 · 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, S101 S103 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 := div(420, 3) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 = 140 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S102 S104 · impliesBefore Mobius inversion ORIGINAL inverse F(n/3):actual positive input=140;check complete integer range1..140 After ALL positive divisors INCREASING=[1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140];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, S103 S105 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S104 S106 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S105 S107 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor := mul(19, 2) After Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor = 38 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S106 S108 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b := sum(38, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b = 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, S107 S109 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(4) ORIGINAL a times divisor := mul(19, 4) After Mobius inversion ORIGINAL inverse F(n/3) f(4) ORIGINAL a times divisor = 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, S108 S110 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b := sum(76, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b = 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, S109 S111 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(5) ORIGINAL a times divisor := mul(19, 5) After Mobius inversion ORIGINAL inverse F(n/3) f(5) ORIGINAL a times divisor = 95 Conditions: none Substitute original 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/3) f(5) plus ORIGINAL b := sum(95, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(5) plus ORIGINAL b = 96 Conditions: none Substitute original 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/3) f(7) ORIGINAL a times divisor := mul(19, 7) After Mobius inversion ORIGINAL inverse F(n/3) f(7) ORIGINAL a times divisor = 133 Conditions: none Substitute original 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/3) f(7) plus ORIGINAL b := sum(133, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(7) plus ORIGINAL b = 134 Conditions: none Substitute original 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/3) f(10) ORIGINAL a times divisor := mul(19, 10) After Mobius inversion ORIGINAL inverse F(n/3) f(10) ORIGINAL a times divisor = 190 Conditions: none Substitute original 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/3) f(10) plus ORIGINAL b := sum(190, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(10) plus ORIGINAL b = 191 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S115 S117 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(14) ORIGINAL a times divisor := mul(19, 14) After Mobius inversion ORIGINAL inverse F(n/3) f(14) ORIGINAL a times divisor = 266 Conditions: none Substitute original 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/3) f(14) plus ORIGINAL b := sum(266, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(14) plus ORIGINAL b = 267 Conditions: none Substitute original 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/3) f(20) ORIGINAL a times divisor := mul(19, 20) After Mobius inversion ORIGINAL inverse F(n/3) f(20) ORIGINAL a times divisor = 380 Conditions: none Substitute original 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/3) f(20) plus ORIGINAL b := sum(380, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(20) plus ORIGINAL b = 381 Conditions: none Substitute original 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/3) f(28) ORIGINAL a times divisor := mul(19, 28) After Mobius inversion ORIGINAL inverse F(n/3) f(28) ORIGINAL a times divisor = 532 Conditions: none Substitute original 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 F(n/3) f(28) plus ORIGINAL b := sum(532, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(28) plus ORIGINAL b = 533 Conditions: none Substitute original 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(35) ORIGINAL a times divisor := mul(19, 35) After Mobius inversion ORIGINAL inverse F(n/3) f(35) ORIGINAL a times divisor = 665 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S122 S124 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(35) plus ORIGINAL b := sum(665, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(35) plus ORIGINAL b = 666 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S123 S125 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(70) ORIGINAL a times divisor := mul(19, 70) After Mobius inversion ORIGINAL inverse F(n/3) f(70) ORIGINAL a times divisor = 1330 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S124 S126 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(70) plus ORIGINAL b := sum(1330, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(70) plus ORIGINAL b = 1331 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S125 S127 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(140) ORIGINAL a times divisor := mul(19, 140) After Mobius inversion ORIGINAL inverse F(n/3) f(140) ORIGINAL a times divisor = 2660 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S126 S128 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) f(140) plus ORIGINAL b := sum(2660, 1) After Mobius inversion ORIGINAL inverse F(n/3) f(140) plus ORIGINAL b = 2661 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S127 S129 · calculationBefore Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum := sum(20, 39, 77, 96, 134, 191, 267, 381, 533, 666, 1331, 2661) After Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum = 6396 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S128 S130 · calculationBefore Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) := mul(-1, 6396) After Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) = -6396 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S129 S131 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor4:actual positive input for mu=4 After complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search Conditions: none Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search. Step identifiers and dependenciesRule: mobius_complete_UPF_input_role · Uses: G001, S130 S132 · impliesBefore ORIGINAL positive integer absolute magnitude=4>1 After initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted Conditions: none Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime. Step identifiers and dependenciesRule: upf_factorization_remaining_product_invariant · Uses: G001, S131 S133 · calculationBefore upf extract divisor2 occurrence0 := div(4, 2) After upf extract divisor2 occurrence0 = 2 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S132 S134 · calculationBefore upf extract divisor2 occurrence1 := div(2, 2) After upf extract divisor2 occurrence1 = 1 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S133 S135 · impliesBefore next divisor=2;previous remaining=4;all smaller divisors exhausted After extracted multiplicity=2;next remaining=1;remaining modulo2=1;extracted multiset=[2, 2] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S134 S136 · 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, S135 S137 · calculationBefore upf canonical complete product factor1 := mul(2, 2) After upf canonical complete product factor1 = 4 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S136 S138 · impliesBefore exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=4 After complete increasing prime multiset=[2, 2];full product=4;a composite terminal remainder would have an already tested smaller divisor Conditions: none If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors. Step identifiers and dependenciesRule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S137 S139 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor4:input=4;COMPLETE prime multiset=[2, 2];distinct primes=[2] After mu=0;empty product case=No;repeated prime=Yes Conditions: none The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities. Step identifiers and dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S138 S140 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 := div(420, 4) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 = 105 Conditions: none Substitute original 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 · impliesBefore Mobius inversion ORIGINAL inverse F(n/4):actual positive input=105;check complete integer range1..105 After ALL positive divisors INCREASING=[1, 3, 5, 7, 15, 21, 35, 105];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, S140 S142 · calculationBefore Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original 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/4) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original 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/4) f(3) ORIGINAL a times divisor := mul(19, 3) After Mobius inversion ORIGINAL inverse F(n/4) f(3) ORIGINAL a times divisor = 57 Conditions: none Substitute original 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/4) f(3) plus ORIGINAL b := sum(57, 1) After Mobius inversion ORIGINAL inverse F(n/4) f(3) plus ORIGINAL b = 58 Conditions: none Substitute original 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/4) f(5) ORIGINAL a times divisor := mul(19, 5) After Mobius inversion ORIGINAL inverse F(n/4) f(5) ORIGINAL a times divisor = 95 Conditions: none Substitute original 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/4) f(5) plus ORIGINAL b := sum(95, 1) After Mobius inversion ORIGINAL inverse F(n/4) f(5) plus ORIGINAL b = 96 Conditions: none Substitute original 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/4) f(7) ORIGINAL a times divisor := mul(19, 7) After Mobius inversion ORIGINAL inverse F(n/4) f(7) ORIGINAL a times divisor = 133 Conditions: none Substitute original 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/4) f(7) plus ORIGINAL b := sum(133, 1) After Mobius inversion ORIGINAL inverse F(n/4) f(7) plus ORIGINAL b = 134 Conditions: none Substitute original 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/4) f(15) ORIGINAL a times divisor := mul(19, 15) After Mobius inversion ORIGINAL inverse F(n/4) f(15) ORIGINAL a times divisor = 285 Conditions: none Substitute original 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/4) f(15) plus ORIGINAL b := sum(285, 1) After Mobius inversion ORIGINAL inverse F(n/4) f(15) plus ORIGINAL b = 286 Conditions: none Substitute original 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 F(n/4) f(21) ORIGINAL a times divisor := mul(19, 21) After Mobius inversion ORIGINAL inverse F(n/4) f(21) ORIGINAL a times divisor = 399 Conditions: none Substitute original 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/4) f(21) plus ORIGINAL b := sum(399, 1) After Mobius inversion ORIGINAL inverse F(n/4) f(21) plus ORIGINAL b = 400 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S152 S154 · calculationBefore Mobius inversion ORIGINAL inverse F(n/4) f(35) ORIGINAL a times divisor := mul(19, 35) After Mobius inversion ORIGINAL inverse F(n/4) f(35) ORIGINAL a times divisor = 665 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S153 S155 · calculationBefore Mobius inversion ORIGINAL inverse F(n/4) f(35) plus ORIGINAL b := sum(665, 1) After Mobius inversion ORIGINAL inverse F(n/4) f(35) plus ORIGINAL b = 666 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S154 S156 · calculationBefore Mobius inversion ORIGINAL inverse F(n/4) f(105) ORIGINAL a times divisor := mul(19, 105) After Mobius inversion ORIGINAL inverse F(n/4) f(105) ORIGINAL a times divisor = 1995 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S155 S157 · calculationBefore Mobius inversion ORIGINAL inverse F(n/4) f(105) plus ORIGINAL b := sum(1995, 1) After Mobius inversion ORIGINAL inverse F(n/4) f(105) plus ORIGINAL b = 1996 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S156 S158 · calculationBefore Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum := sum(20, 58, 96, 134, 286, 400, 666, 1996) After Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum = 3656 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S157 S159 · calculationBefore Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) := mul(0, 3656) After Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) = 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, S158 S160 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor5:actual positive input for mu=5 After complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search Conditions: none Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search. Step identifiers and dependenciesRule: mobius_complete_UPF_input_role · Uses: G001, S159 S161 · impliesBefore ORIGINAL positive integer absolute magnitude=5>1 After initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted Conditions: none Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime. Step identifiers and dependenciesRule: upf_factorization_remaining_product_invariant · Uses: G001, S160 S162 · impliesBefore next divisor=2;previous remaining=5;all smaller divisors exhausted After extracted multiplicity=0;next remaining=5;remaining modulo2=1;extracted multiset=[] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S161 S163 · calculationBefore upf canonical complete product factor0 := mul(1, 5) After upf canonical complete product factor0 = 5 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S162 S164 · impliesBefore exhausted trial divisors below3;terminal remaining=5;d*d>remaining;ORIGINAL n=5 After complete increasing prime multiset=[5];full product=5;a composite terminal remainder would have an already tested smaller divisor Conditions: none If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors. Step identifiers and dependenciesRule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S163 S165 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor5:input=5;COMPLETE prime multiset=[5];distinct primes=[5] After mu=-1;empty product case=No;repeated prime=No Conditions: none The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities. Step identifiers and dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S164 S166 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 := div(420, 5) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 = 84 Conditions: none Substitute original 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 · impliesBefore Mobius inversion ORIGINAL inverse F(n/5):actual positive input=84;check complete integer range1..84 After ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84];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, S166 S168 · calculationBefore Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original 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/5) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original 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/5) f(2) ORIGINAL a times divisor := mul(19, 2) After Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor = 38 Conditions: none Substitute original 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/5) f(2) plus ORIGINAL b := sum(38, 1) After Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b = 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, S170 S172 · calculationBefore Mobius inversion ORIGINAL inverse F(n/5) f(3) ORIGINAL a times divisor := mul(19, 3) After Mobius inversion ORIGINAL inverse F(n/5) f(3) ORIGINAL a times divisor = 57 Conditions: none Substitute original 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/5) f(3) plus ORIGINAL b := sum(57, 1) After Mobius inversion ORIGINAL inverse F(n/5) f(3) plus ORIGINAL b = 58 Conditions: none Substitute original 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/5) f(4) ORIGINAL a times divisor := mul(19, 4) After Mobius inversion ORIGINAL inverse F(n/5) f(4) ORIGINAL a times divisor = 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, S173 S175 · calculationBefore Mobius inversion ORIGINAL inverse F(n/5) f(4) plus ORIGINAL b := sum(76, 1) After Mobius inversion ORIGINAL inverse F(n/5) f(4) plus ORIGINAL b = 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, S174 S176 · calculationBefore Mobius inversion ORIGINAL inverse F(n/5) f(6) ORIGINAL a times divisor := mul(19, 6) After Mobius inversion ORIGINAL inverse F(n/5) f(6) ORIGINAL a times divisor = 114 Conditions: none Substitute original 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/5) f(6) plus ORIGINAL b := sum(114, 1) After Mobius inversion ORIGINAL inverse F(n/5) f(6) plus ORIGINAL b = 115 Conditions: none Substitute original 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/5) f(7) ORIGINAL a times divisor := mul(19, 7) After Mobius inversion ORIGINAL inverse F(n/5) f(7) ORIGINAL a times divisor = 133 Conditions: none Substitute original 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 F(n/5) f(7) plus ORIGINAL b := sum(133, 1) After Mobius inversion ORIGINAL inverse F(n/5) f(7) plus ORIGINAL b = 134 Conditions: none Substitute original 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/5) f(12) ORIGINAL a times divisor := mul(19, 12) After Mobius inversion ORIGINAL inverse F(n/5) f(12) ORIGINAL a times divisor = 228 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S179 S181 · calculationBefore Mobius inversion ORIGINAL inverse F(n/5) f(12) plus ORIGINAL b := sum(228, 1) After Mobius inversion ORIGINAL inverse F(n/5) f(12) plus ORIGINAL b = 229 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S180 S182 · calculationBefore Mobius inversion ORIGINAL inverse F(n/5) f(14) ORIGINAL a times divisor := mul(19, 14) After Mobius inversion ORIGINAL inverse F(n/5) f(14) ORIGINAL a times divisor = 266 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S181 S183 · calculationBefore Mobius inversion ORIGINAL inverse F(n/5) f(14) plus ORIGINAL b := sum(266, 1) After Mobius inversion ORIGINAL inverse F(n/5) f(14) plus ORIGINAL b = 267 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S182 S184 · calculationBefore Mobius inversion ORIGINAL inverse F(n/5) f(21) ORIGINAL a times divisor := mul(19, 21) After Mobius inversion ORIGINAL inverse F(n/5) f(21) ORIGINAL a times divisor = 399 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S183 S185 · calculationBefore Mobius inversion ORIGINAL inverse F(n/5) f(21) plus ORIGINAL b := sum(399, 1) After Mobius inversion ORIGINAL inverse F(n/5) f(21) plus ORIGINAL b = 400 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S184 S186 · calculationBefore Mobius inversion ORIGINAL inverse F(n/5) f(28) ORIGINAL a times divisor := mul(19, 28) After Mobius inversion ORIGINAL inverse F(n/5) f(28) ORIGINAL a times divisor = 532 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S185 S187 · calculationBefore Mobius inversion ORIGINAL inverse F(n/5) f(28) plus ORIGINAL b := sum(532, 1) After Mobius inversion ORIGINAL inverse F(n/5) f(28) plus ORIGINAL b = 533 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S186 S188 · calculationBefore Mobius inversion ORIGINAL inverse F(n/5) f(42) ORIGINAL a times divisor := mul(19, 42) After Mobius inversion ORIGINAL inverse F(n/5) f(42) ORIGINAL a times divisor = 798 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S187 S189 · calculationBefore Mobius inversion ORIGINAL inverse F(n/5) f(42) plus ORIGINAL b := sum(798, 1) After Mobius inversion ORIGINAL inverse F(n/5) f(42) plus ORIGINAL b = 799 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S188 S190 · calculationBefore Mobius inversion ORIGINAL inverse F(n/5) f(84) ORIGINAL a times divisor := mul(19, 84) After Mobius inversion ORIGINAL inverse F(n/5) f(84) ORIGINAL a times divisor = 1596 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S189 S191 · calculationBefore Mobius inversion ORIGINAL inverse F(n/5) f(84) plus ORIGINAL b := sum(1596, 1) After Mobius inversion ORIGINAL inverse F(n/5) f(84) plus ORIGINAL b = 1597 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S190 S192 · calculationBefore Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum := sum(20, 39, 58, 77, 115, 134, 229, 267, 400, 533, 799, 1597) After Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum = 4268 Conditions: none Substitute original 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 · calculationBefore Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) := mul(-1, 4268) After Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) = -4268 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S192 S194 · 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, S193 S195 · 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, S194 S196 · 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, S195 S197 · 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, S196 S198 · 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, S197 S199 · 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, S198 S200 · 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, S199 S201 · 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, S200 S202 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 := div(420, 6) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 = 70 Conditions: none Substitute original 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 · impliesBefore Mobius inversion ORIGINAL inverse F(n/6):actual positive input=70;check complete integer range1..70 After ALL positive divisors INCREASING=[1, 2, 5, 7, 10, 14, 35, 70];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, S202 S204 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S203 S205 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S204 S206 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor := mul(19, 2) After Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor = 38 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S205 S207 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b := sum(38, 1) After Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b = 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, S206 S208 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(5) ORIGINAL a times divisor := mul(19, 5) After Mobius inversion ORIGINAL inverse F(n/6) f(5) ORIGINAL a times divisor = 95 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S207 S209 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(5) plus ORIGINAL b := sum(95, 1) After Mobius inversion ORIGINAL inverse F(n/6) f(5) plus ORIGINAL b = 96 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S208 S210 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor := mul(19, 7) After Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor = 133 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S209 S211 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b := sum(133, 1) After Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b = 134 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S210 S212 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(10) ORIGINAL a times divisor := mul(19, 10) After Mobius inversion ORIGINAL inverse F(n/6) f(10) ORIGINAL a times divisor = 190 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S211 S213 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(10) plus ORIGINAL b := sum(190, 1) After Mobius inversion ORIGINAL inverse F(n/6) f(10) plus ORIGINAL b = 191 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S212 S214 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(14) ORIGINAL a times divisor := mul(19, 14) After Mobius inversion ORIGINAL inverse F(n/6) f(14) ORIGINAL a times divisor = 266 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S213 S215 · calculationBefore Mobius inversion ORIGINAL inverse F(n/6) f(14) plus ORIGINAL b := sum(266, 1) After Mobius inversion ORIGINAL inverse F(n/6) f(14) plus ORIGINAL b = 267 Conditions: none Substitute original 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/6) f(35) ORIGINAL a times divisor := mul(19, 35) After Mobius inversion ORIGINAL inverse F(n/6) f(35) ORIGINAL a times divisor = 665 Conditions: none Substitute original 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/6) f(35) plus ORIGINAL b := sum(665, 1) After Mobius inversion ORIGINAL inverse F(n/6) f(35) plus ORIGINAL b = 666 Conditions: none Substitute original 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/6) f(70) ORIGINAL a times divisor := mul(19, 70) After Mobius inversion ORIGINAL inverse F(n/6) f(70) ORIGINAL a times divisor = 1330 Conditions: none Substitute original 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/6) f(70) plus ORIGINAL b := sum(1330, 1) After Mobius inversion ORIGINAL inverse F(n/6) f(70) plus ORIGINAL b = 1331 Conditions: none Substitute original 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/6) full forward divisor sum := sum(20, 39, 96, 134, 191, 267, 666, 1331) After Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum = 2744 Conditions: none Substitute original 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 mu(6) times SAME F(n/6) := mul(1, 2744) After Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) = 2744 Conditions: none Substitute original 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 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor7:actual positive input for mu=7 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, S221 S223 · impliesBefore ORIGINAL positive integer absolute magnitude=7>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, S222 S224 · impliesBefore next divisor=2;previous remaining=7;all smaller divisors exhausted After extracted multiplicity=0;next remaining=7;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, S223 S225 · calculationBefore upf canonical complete product factor0 := mul(1, 7) After upf canonical complete product factor0 = 7 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, S224 S226 · impliesBefore exhausted trial divisors below3;terminal remaining=7;d*d>remaining;ORIGINAL n=7 After complete increasing prime multiset=[7];full product=7;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, S225 S227 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor7:input=7;COMPLETE prime multiset=[7];distinct primes=[7] 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, S226 S228 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 := div(420, 7) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 = 60 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S227 S229 · impliesBefore Mobius inversion ORIGINAL inverse F(n/7):actual positive input=60;check complete integer range1..60 After ALL positive divisors INCREASING=[1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60];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, S228 S230 · calculationBefore Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S229 S231 · calculationBefore Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S230 S232 · calculationBefore Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor := mul(19, 2) After Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor = 38 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S231 S233 · calculationBefore Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b := sum(38, 1) After Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b = 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, S232 S234 · calculationBefore Mobius inversion ORIGINAL inverse F(n/7) f(3) ORIGINAL a times divisor := mul(19, 3) After Mobius inversion ORIGINAL inverse F(n/7) f(3) ORIGINAL a times divisor = 57 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S233 S235 · calculationBefore Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b := sum(57, 1) After Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b = 58 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S234 S236 · calculationBefore Mobius inversion ORIGINAL inverse F(n/7) f(4) ORIGINAL a times divisor := mul(19, 4) After Mobius inversion ORIGINAL inverse F(n/7) f(4) ORIGINAL a times divisor = 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, S235 S237 · calculationBefore Mobius inversion ORIGINAL inverse F(n/7) f(4) plus ORIGINAL b := sum(76, 1) After Mobius inversion ORIGINAL inverse F(n/7) f(4) plus ORIGINAL b = 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, S236 S238 · calculationBefore Mobius inversion ORIGINAL inverse F(n/7) f(5) ORIGINAL a times divisor := mul(19, 5) After Mobius inversion ORIGINAL inverse F(n/7) f(5) ORIGINAL a times divisor = 95 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S237 S239 · calculationBefore Mobius inversion ORIGINAL inverse F(n/7) f(5) plus ORIGINAL b := sum(95, 1) After Mobius inversion ORIGINAL inverse F(n/7) f(5) plus ORIGINAL b = 96 Conditions: none Substitute original 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/7) f(6) ORIGINAL a times divisor := mul(19, 6) After Mobius inversion ORIGINAL inverse F(n/7) f(6) ORIGINAL a times divisor = 114 Conditions: none Substitute original 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/7) f(6) plus ORIGINAL b := sum(114, 1) After Mobius inversion ORIGINAL inverse F(n/7) f(6) plus ORIGINAL b = 115 Conditions: none Substitute original 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/7) f(10) ORIGINAL a times divisor := mul(19, 10) After Mobius inversion ORIGINAL inverse F(n/7) f(10) ORIGINAL a times divisor = 190 Conditions: none Substitute original 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/7) f(10) plus ORIGINAL b := sum(190, 1) After Mobius inversion ORIGINAL inverse F(n/7) f(10) plus ORIGINAL b = 191 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S242 S244 · calculationBefore Mobius inversion ORIGINAL inverse F(n/7) f(12) ORIGINAL a times divisor := mul(19, 12) After Mobius inversion ORIGINAL inverse F(n/7) f(12) ORIGINAL a times divisor = 228 Conditions: none Substitute original 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/7) f(12) plus ORIGINAL b := sum(228, 1) After Mobius inversion ORIGINAL inverse F(n/7) f(12) plus ORIGINAL b = 229 Conditions: none Substitute original 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/7) f(15) ORIGINAL a times divisor := mul(19, 15) After Mobius inversion ORIGINAL inverse F(n/7) f(15) ORIGINAL a times divisor = 285 Conditions: none Substitute original 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/7) f(15) plus ORIGINAL b := sum(285, 1) After Mobius inversion ORIGINAL inverse F(n/7) f(15) plus ORIGINAL b = 286 Conditions: none Substitute original 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 F(n/7) f(20) ORIGINAL a times divisor := mul(19, 20) After Mobius inversion ORIGINAL inverse F(n/7) f(20) ORIGINAL a times divisor = 380 Conditions: none Substitute original 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/7) f(20) plus ORIGINAL b := sum(380, 1) After Mobius inversion ORIGINAL inverse F(n/7) f(20) plus ORIGINAL b = 381 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S248 S250 · calculationBefore Mobius inversion ORIGINAL inverse F(n/7) f(30) ORIGINAL a times divisor := mul(19, 30) After Mobius inversion ORIGINAL inverse F(n/7) f(30) ORIGINAL a times divisor = 570 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S249 S251 · calculationBefore Mobius inversion ORIGINAL inverse F(n/7) f(30) plus ORIGINAL b := sum(570, 1) After Mobius inversion ORIGINAL inverse F(n/7) f(30) plus ORIGINAL b = 571 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S250 S252 · calculationBefore Mobius inversion ORIGINAL inverse F(n/7) f(60) ORIGINAL a times divisor := mul(19, 60) After Mobius inversion ORIGINAL inverse F(n/7) f(60) ORIGINAL a times divisor = 1140 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S251 S253 · calculationBefore Mobius inversion ORIGINAL inverse F(n/7) f(60) plus ORIGINAL b := sum(1140, 1) After Mobius inversion ORIGINAL inverse F(n/7) f(60) plus ORIGINAL b = 1141 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S252 S254 · calculationBefore Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum := sum(20, 39, 58, 77, 96, 115, 191, 229, 286, 381, 571, 1141) After Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum = 3204 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S253 S255 · calculationBefore Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) := mul(-1, 3204) After Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) = -3204 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S254 S256 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor10:actual positive input for mu=10 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, S255 S257 · impliesBefore ORIGINAL positive integer absolute magnitude=10>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, S256 S258 · calculationBefore upf extract divisor2 occurrence0 := div(10, 2) After upf extract divisor2 occurrence0 = 5 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S257 S259 · impliesBefore next divisor=2;previous remaining=10;all smaller divisors exhausted After extracted multiplicity=1;next remaining=5;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, S258 S260 · 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, S259 S261 · calculationBefore upf canonical complete product factor1 := mul(2, 5) After upf canonical complete product factor1 = 10 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, S260 S262 · impliesBefore exhausted trial divisors below3;terminal remaining=5;d*d>remaining;ORIGINAL n=10 After complete increasing prime multiset=[2, 5];full product=10;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, S261 S263 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor10:input=10;COMPLETE prime multiset=[2, 5];distinct primes=[2, 5] After mu=1;empty product case=No;repeated prime=No Conditions: none The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities. Step identifiers and dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S262 S264 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 := div(420, 10) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 = 42 Conditions: none Substitute original 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 · impliesBefore Mobius inversion ORIGINAL inverse F(n/10):actual positive input=42;check complete integer range1..42 After ALL positive divisors INCREASING=[1, 2, 3, 6, 7, 14, 21, 42];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, S264 S266 · calculationBefore Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original 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/10) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original 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/10) f(2) ORIGINAL a times divisor := mul(19, 2) After Mobius inversion ORIGINAL inverse F(n/10) f(2) ORIGINAL a times divisor = 38 Conditions: none Substitute original 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/10) f(2) plus ORIGINAL b := sum(38, 1) After Mobius inversion ORIGINAL inverse F(n/10) f(2) plus ORIGINAL b = 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, S268 S270 · calculationBefore Mobius inversion ORIGINAL inverse F(n/10) f(3) ORIGINAL a times divisor := mul(19, 3) After Mobius inversion ORIGINAL inverse F(n/10) f(3) ORIGINAL a times divisor = 57 Conditions: none Substitute original 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/10) f(3) plus ORIGINAL b := sum(57, 1) After Mobius inversion ORIGINAL inverse F(n/10) f(3) plus ORIGINAL b = 58 Conditions: none Substitute original 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/10) f(6) ORIGINAL a times divisor := mul(19, 6) After Mobius inversion ORIGINAL inverse F(n/10) f(6) ORIGINAL a times divisor = 114 Conditions: none Substitute original 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 F(n/10) f(6) plus ORIGINAL b := sum(114, 1) After Mobius inversion ORIGINAL inverse F(n/10) f(6) plus ORIGINAL b = 115 Conditions: none Substitute original 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/10) f(7) ORIGINAL a times divisor := mul(19, 7) After Mobius inversion ORIGINAL inverse F(n/10) f(7) ORIGINAL a times divisor = 133 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S273 S275 · calculationBefore Mobius inversion ORIGINAL inverse F(n/10) f(7) plus ORIGINAL b := sum(133, 1) After Mobius inversion ORIGINAL inverse F(n/10) f(7) plus ORIGINAL b = 134 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S274 S276 · calculationBefore Mobius inversion ORIGINAL inverse F(n/10) f(14) ORIGINAL a times divisor := mul(19, 14) After Mobius inversion ORIGINAL inverse F(n/10) f(14) ORIGINAL a times divisor = 266 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S275 S277 · calculationBefore Mobius inversion ORIGINAL inverse F(n/10) f(14) plus ORIGINAL b := sum(266, 1) After Mobius inversion ORIGINAL inverse F(n/10) f(14) plus ORIGINAL b = 267 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S276 S278 · calculationBefore Mobius inversion ORIGINAL inverse F(n/10) f(21) ORIGINAL a times divisor := mul(19, 21) After Mobius inversion ORIGINAL inverse F(n/10) f(21) ORIGINAL a times divisor = 399 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S277 S279 · calculationBefore Mobius inversion ORIGINAL inverse F(n/10) f(21) plus ORIGINAL b := sum(399, 1) After Mobius inversion ORIGINAL inverse F(n/10) f(21) plus ORIGINAL b = 400 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S278 S280 · calculationBefore Mobius inversion ORIGINAL inverse F(n/10) f(42) ORIGINAL a times divisor := mul(19, 42) After Mobius inversion ORIGINAL inverse F(n/10) f(42) ORIGINAL a times divisor = 798 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S279 S281 · calculationBefore Mobius inversion ORIGINAL inverse F(n/10) f(42) plus ORIGINAL b := sum(798, 1) After Mobius inversion ORIGINAL inverse F(n/10) f(42) plus ORIGINAL b = 799 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S280 S282 · calculationBefore Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum := sum(20, 39, 58, 115, 134, 267, 400, 799) After Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum = 1832 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S281 S283 · calculationBefore Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) := mul(1, 1832) After Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) = 1832 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S282 S284 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor12:actual positive input for mu=12 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, S283 S285 · impliesBefore ORIGINAL positive integer absolute magnitude=12>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, S284 S286 · calculationBefore upf extract divisor2 occurrence0 := div(12, 2) After upf extract divisor2 occurrence0 = 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, S285 S287 · calculationBefore upf extract divisor2 occurrence1 := div(6, 2) After upf extract divisor2 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, S286 S288 · impliesBefore next divisor=2;previous remaining=12;all smaller divisors exhausted After extracted multiplicity=2;next remaining=3;remaining modulo2=1;extracted multiset=[2, 2] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S287 S289 · 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, S288 S290 · calculationBefore upf canonical complete product factor1 := mul(2, 2) After upf canonical complete product factor1 = 4 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S289 S291 · calculationBefore upf canonical complete product factor2 := mul(4, 3) After upf canonical complete product factor2 = 12 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, S290 S292 · impliesBefore exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=12 After complete increasing prime multiset=[2, 2, 3];full product=12;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, S291 S293 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor12:input=12;COMPLETE prime multiset=[2, 2, 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, S292 S294 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12 := div(420, 12) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12 = 35 Conditions: none Substitute original 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 · impliesBefore Mobius inversion ORIGINAL inverse F(n/12):actual positive input=35;check complete integer range1..35 After ALL positive divisors INCREASING=[1, 5, 7, 35];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, S294 S296 · calculationBefore Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S295 S297 · calculationBefore Mobius inversion ORIGINAL inverse F(n/12) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/12) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S296 S298 · calculationBefore Mobius inversion ORIGINAL inverse F(n/12) f(5) ORIGINAL a times divisor := mul(19, 5) After Mobius inversion ORIGINAL inverse F(n/12) f(5) ORIGINAL a times divisor = 95 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S297 S299 · calculationBefore Mobius inversion ORIGINAL inverse F(n/12) f(5) plus ORIGINAL b := sum(95, 1) After Mobius inversion ORIGINAL inverse F(n/12) f(5) plus ORIGINAL b = 96 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S298 S300 · calculationBefore Mobius inversion ORIGINAL inverse F(n/12) f(7) ORIGINAL a times divisor := mul(19, 7) After Mobius inversion ORIGINAL inverse F(n/12) f(7) ORIGINAL a times divisor = 133 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S299 S301 · calculationBefore Mobius inversion ORIGINAL inverse F(n/12) f(7) plus ORIGINAL b := sum(133, 1) After Mobius inversion ORIGINAL inverse F(n/12) f(7) plus ORIGINAL b = 134 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S300 S302 · calculationBefore Mobius inversion ORIGINAL inverse F(n/12) f(35) ORIGINAL a times divisor := mul(19, 35) After Mobius inversion ORIGINAL inverse F(n/12) f(35) ORIGINAL a times divisor = 665 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S301 S303 · calculationBefore Mobius inversion ORIGINAL inverse F(n/12) f(35) plus ORIGINAL b := sum(665, 1) After Mobius inversion ORIGINAL inverse F(n/12) f(35) plus ORIGINAL b = 666 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S302 S304 · calculationBefore Mobius inversion ORIGINAL inverse F(n/12) full forward divisor sum := sum(20, 96, 134, 666) After Mobius inversion ORIGINAL inverse F(n/12) full forward divisor sum = 916 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S303 S305 · calculationBefore Mobius inversion ORIGINAL inverse mu(12) times SAME F(n/12) := mul(0, 916) After Mobius inversion ORIGINAL inverse mu(12) times SAME F(n/12) = 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, S304 S306 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor14:actual positive input for mu=14 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, S305 S307 · impliesBefore ORIGINAL positive integer absolute magnitude=14>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, S306 S308 · calculationBefore upf extract divisor2 occurrence0 := div(14, 2) After upf extract divisor2 occurrence0 = 7 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, S307 S309 · impliesBefore next divisor=2;previous remaining=14;all smaller divisors exhausted After extracted multiplicity=1;next remaining=7;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, S308 S310 · 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, S309 S311 · calculationBefore upf canonical complete product factor1 := mul(2, 7) After upf canonical complete product factor1 = 14 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, S310 S312 · impliesBefore exhausted trial divisors below3;terminal remaining=7;d*d>remaining;ORIGINAL n=14 After complete increasing prime multiset=[2, 7];full product=14;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, S311 S313 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor14:input=14;COMPLETE prime multiset=[2, 7];distinct primes=[2, 7] 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, S312 S314 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 := div(420, 14) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 = 30 Conditions: none Substitute original 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 · impliesBefore Mobius inversion ORIGINAL inverse F(n/14):actual positive input=30;check complete integer range1..30 After ALL positive divisors INCREASING=[1, 2, 3, 5, 6, 10, 15, 30];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, S314 S316 · calculationBefore Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S316 S318 · calculationBefore Mobius inversion ORIGINAL inverse F(n/14) f(2) ORIGINAL a times divisor := mul(19, 2) After Mobius inversion ORIGINAL inverse F(n/14) f(2) ORIGINAL a times divisor = 38 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S317 S319 · calculationBefore Mobius inversion ORIGINAL inverse F(n/14) f(2) plus ORIGINAL b := sum(38, 1) After Mobius inversion ORIGINAL inverse F(n/14) f(2) plus ORIGINAL b = 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, S318 S320 · calculationBefore Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor := mul(19, 3) After Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor = 57 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S319 S321 · calculationBefore Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b := sum(57, 1) After Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b = 58 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S320 S322 · calculationBefore Mobius inversion ORIGINAL inverse F(n/14) f(5) ORIGINAL a times divisor := mul(19, 5) After Mobius inversion ORIGINAL inverse F(n/14) f(5) ORIGINAL a times divisor = 95 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S321 S323 · calculationBefore Mobius inversion ORIGINAL inverse F(n/14) f(5) plus ORIGINAL b := sum(95, 1) After Mobius inversion ORIGINAL inverse F(n/14) f(5) plus ORIGINAL b = 96 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S322 S324 · calculationBefore Mobius inversion ORIGINAL inverse F(n/14) f(6) ORIGINAL a times divisor := mul(19, 6) After Mobius inversion ORIGINAL inverse F(n/14) f(6) ORIGINAL a times divisor = 114 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S323 S325 · calculationBefore Mobius inversion ORIGINAL inverse F(n/14) f(6) plus ORIGINAL b := sum(114, 1) After Mobius inversion ORIGINAL inverse F(n/14) f(6) plus ORIGINAL b = 115 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S324 S326 · calculationBefore Mobius inversion ORIGINAL inverse F(n/14) f(10) ORIGINAL a times divisor := mul(19, 10) After Mobius inversion ORIGINAL inverse F(n/14) f(10) ORIGINAL a times divisor = 190 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S325 S327 · calculationBefore Mobius inversion ORIGINAL inverse F(n/14) f(10) plus ORIGINAL b := sum(190, 1) After Mobius inversion ORIGINAL inverse F(n/14) f(10) plus ORIGINAL b = 191 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S326 S328 · calculationBefore Mobius inversion ORIGINAL inverse F(n/14) f(15) ORIGINAL a times divisor := mul(19, 15) After Mobius inversion ORIGINAL inverse F(n/14) f(15) ORIGINAL a times divisor = 285 Conditions: none Substitute original 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/14) f(15) plus ORIGINAL b := sum(285, 1) After Mobius inversion ORIGINAL inverse F(n/14) f(15) plus ORIGINAL b = 286 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S328 S330 · calculationBefore Mobius inversion ORIGINAL inverse F(n/14) f(30) ORIGINAL a times divisor := mul(19, 30) After Mobius inversion ORIGINAL inverse F(n/14) f(30) ORIGINAL a times divisor = 570 Conditions: none Substitute original 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/14) f(30) plus ORIGINAL b := sum(570, 1) After Mobius inversion ORIGINAL inverse F(n/14) f(30) plus ORIGINAL b = 571 Conditions: none Substitute original 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/14) full forward divisor sum := sum(20, 39, 58, 96, 115, 191, 286, 571) After Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum = 1376 Conditions: none Substitute original 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 mu(14) times SAME F(n/14) := mul(1, 1376) After Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) = 1376 Conditions: none Substitute original 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 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor15:actual positive input for mu=15 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, S333 S335 · impliesBefore ORIGINAL positive integer absolute magnitude=15>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, S334 S336 · impliesBefore next divisor=2;previous remaining=15;all smaller divisors exhausted After extracted multiplicity=0;next remaining=15;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, S335 S337 · calculationBefore upf extract divisor3 occurrence0 := div(15, 3) After upf extract divisor3 occurrence0 = 5 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S336 S338 · impliesBefore next divisor=3;previous remaining=15;all smaller divisors exhausted After extracted multiplicity=1;next remaining=5;remaining modulo3=2;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, S337 S339 · 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, S338 S340 · calculationBefore upf canonical complete product factor1 := mul(3, 5) After upf canonical complete product factor1 = 15 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, S339 S341 · impliesBefore exhausted trial divisors below4;terminal remaining=5;d*d>remaining;ORIGINAL n=15 After complete increasing prime multiset=[3, 5];full product=15;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, S340 S342 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor15:input=15;COMPLETE prime multiset=[3, 5];distinct primes=[3, 5] After mu=1;empty product case=No;repeated prime=No Conditions: none The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities. Step identifiers and dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S341 S343 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor15 := div(420, 15) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor15 = 28 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S342 S344 · impliesBefore Mobius inversion ORIGINAL inverse F(n/15):actual positive input=28;check complete integer range1..28 After ALL positive divisors INCREASING=[1, 2, 4, 7, 14, 28];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, S343 S345 · calculationBefore Mobius inversion ORIGINAL inverse F(n/15) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/15) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S344 S346 · calculationBefore Mobius inversion ORIGINAL inverse F(n/15) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/15) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S345 S347 · calculationBefore Mobius inversion ORIGINAL inverse F(n/15) f(2) ORIGINAL a times divisor := mul(19, 2) After Mobius inversion ORIGINAL inverse F(n/15) f(2) ORIGINAL a times divisor = 38 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S346 S348 · calculationBefore Mobius inversion ORIGINAL inverse F(n/15) f(2) plus ORIGINAL b := sum(38, 1) After Mobius inversion ORIGINAL inverse F(n/15) f(2) plus ORIGINAL b = 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, S347 S349 · calculationBefore Mobius inversion ORIGINAL inverse F(n/15) f(4) ORIGINAL a times divisor := mul(19, 4) After Mobius inversion ORIGINAL inverse F(n/15) f(4) ORIGINAL a times divisor = 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, S348 S350 · calculationBefore Mobius inversion ORIGINAL inverse F(n/15) f(4) plus ORIGINAL b := sum(76, 1) After Mobius inversion ORIGINAL inverse F(n/15) f(4) plus ORIGINAL b = 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, S349 S351 · calculationBefore Mobius inversion ORIGINAL inverse F(n/15) f(7) ORIGINAL a times divisor := mul(19, 7) After Mobius inversion ORIGINAL inverse F(n/15) f(7) ORIGINAL a times divisor = 133 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S350 S352 · calculationBefore Mobius inversion ORIGINAL inverse F(n/15) f(7) plus ORIGINAL b := sum(133, 1) After Mobius inversion ORIGINAL inverse F(n/15) f(7) plus ORIGINAL b = 134 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S351 S353 · calculationBefore Mobius inversion ORIGINAL inverse F(n/15) f(14) ORIGINAL a times divisor := mul(19, 14) After Mobius inversion ORIGINAL inverse F(n/15) f(14) ORIGINAL a times divisor = 266 Conditions: none Substitute original 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/15) f(14) plus ORIGINAL b := sum(266, 1) After Mobius inversion ORIGINAL inverse F(n/15) f(14) plus ORIGINAL b = 267 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S353 S355 · calculationBefore Mobius inversion ORIGINAL inverse F(n/15) f(28) ORIGINAL a times divisor := mul(19, 28) After Mobius inversion ORIGINAL inverse F(n/15) f(28) ORIGINAL a times divisor = 532 Conditions: none Substitute original 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/15) f(28) plus ORIGINAL b := sum(532, 1) After Mobius inversion ORIGINAL inverse F(n/15) f(28) plus ORIGINAL b = 533 Conditions: none Substitute original 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/15) full forward divisor sum := sum(20, 39, 77, 134, 267, 533) After Mobius inversion ORIGINAL inverse F(n/15) full forward divisor sum = 1070 Conditions: none Substitute original 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 mu(15) times SAME F(n/15) := mul(1, 1070) After Mobius inversion ORIGINAL inverse mu(15) times SAME F(n/15) = 1070 Conditions: none Substitute original 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 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor20:actual positive input for mu=20 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, S358 S360 · impliesBefore ORIGINAL positive integer absolute magnitude=20>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, S359 S361 · calculationBefore upf extract divisor2 occurrence0 := div(20, 2) After upf extract divisor2 occurrence0 = 10 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, S360 S362 · calculationBefore upf extract divisor2 occurrence1 := div(10, 2) After upf extract divisor2 occurrence1 = 5 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S361 S363 · impliesBefore next divisor=2;previous remaining=20;all smaller divisors exhausted After extracted multiplicity=2;next remaining=5;remaining modulo2=1;extracted multiset=[2, 2] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S362 S364 · 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, S363 S365 · calculationBefore upf canonical complete product factor1 := mul(2, 2) After upf canonical complete product factor1 = 4 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S364 S366 · calculationBefore upf canonical complete product factor2 := mul(4, 5) After upf canonical complete product factor2 = 20 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 exhausted trial divisors below3;terminal remaining=5;d*d>remaining;ORIGINAL n=20 After complete increasing prime multiset=[2, 2, 5];full product=20;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, S366 S368 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor20:input=20;COMPLETE prime multiset=[2, 2, 5];distinct primes=[2, 5] After mu=0;empty product case=No;repeated prime=Yes Conditions: none The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities. Step identifiers and dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S367 S369 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor20 := div(420, 20) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor20 = 21 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S368 S370 · impliesBefore Mobius inversion ORIGINAL inverse F(n/20):actual positive input=21;check complete integer range1..21 After ALL positive divisors INCREASING=[1, 3, 7, 21];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, S369 S371 · calculationBefore Mobius inversion ORIGINAL inverse F(n/20) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/20) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S370 S372 · calculationBefore Mobius inversion ORIGINAL inverse F(n/20) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/20) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S371 S373 · calculationBefore Mobius inversion ORIGINAL inverse F(n/20) f(3) ORIGINAL a times divisor := mul(19, 3) After Mobius inversion ORIGINAL inverse F(n/20) f(3) ORIGINAL a times divisor = 57 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S372 S374 · calculationBefore Mobius inversion ORIGINAL inverse F(n/20) f(3) plus ORIGINAL b := sum(57, 1) After Mobius inversion ORIGINAL inverse F(n/20) f(3) plus ORIGINAL b = 58 Conditions: none Substitute original 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 · calculationBefore Mobius inversion ORIGINAL inverse F(n/20) f(7) ORIGINAL a times divisor := mul(19, 7) After Mobius inversion ORIGINAL inverse F(n/20) f(7) ORIGINAL a times divisor = 133 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S374 S376 · calculationBefore Mobius inversion ORIGINAL inverse F(n/20) f(7) plus ORIGINAL b := sum(133, 1) After Mobius inversion ORIGINAL inverse F(n/20) f(7) plus ORIGINAL b = 134 Conditions: none Substitute original 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/20) f(21) ORIGINAL a times divisor := mul(19, 21) After Mobius inversion ORIGINAL inverse F(n/20) f(21) ORIGINAL a times divisor = 399 Conditions: none Substitute original 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/20) f(21) plus ORIGINAL b := sum(399, 1) After Mobius inversion ORIGINAL inverse F(n/20) f(21) plus ORIGINAL b = 400 Conditions: none Substitute original 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/20) full forward divisor sum := sum(20, 58, 134, 400) After Mobius inversion ORIGINAL inverse F(n/20) full forward divisor sum = 612 Conditions: none Substitute original 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 mu(20) times SAME F(n/20) := mul(0, 612) After Mobius inversion ORIGINAL inverse mu(20) times SAME F(n/20) = 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, S379 S381 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor21:actual positive input for mu=21 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, S380 S382 · impliesBefore ORIGINAL positive integer absolute magnitude=21>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, S381 S383 · impliesBefore next divisor=2;previous remaining=21;all smaller divisors exhausted After extracted multiplicity=0;next remaining=21;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, S382 S384 · calculationBefore upf extract divisor3 occurrence0 := div(21, 3) After upf extract divisor3 occurrence0 = 7 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=3;previous remaining=21;all smaller divisors exhausted After extracted multiplicity=1;next remaining=7;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, S384 S386 · 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, S385 S387 · calculationBefore upf canonical complete product factor1 := mul(3, 7) After upf canonical complete product factor1 = 21 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 · impliesBefore exhausted trial divisors below4;terminal remaining=7;d*d>remaining;ORIGINAL n=21 After complete increasing prime multiset=[3, 7];full product=21;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, S387 S389 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor21:input=21;COMPLETE prime multiset=[3, 7];distinct primes=[3, 7] 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, S388 S390 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21 := div(420, 21) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21 = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S389 S391 · impliesBefore Mobius inversion ORIGINAL inverse F(n/21):actual positive input=20;check complete integer range1..20 After ALL positive divisors INCREASING=[1, 2, 4, 5, 10, 20];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, S390 S392 · calculationBefore Mobius inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S391 S393 · calculationBefore Mobius inversion ORIGINAL inverse F(n/21) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/21) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S392 S394 · calculationBefore Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor := mul(19, 2) After Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor = 38 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S393 S395 · calculationBefore Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b := sum(38, 1) After Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b = 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, S394 S396 · calculationBefore Mobius inversion ORIGINAL inverse F(n/21) f(4) ORIGINAL a times divisor := mul(19, 4) After Mobius inversion ORIGINAL inverse F(n/21) f(4) ORIGINAL a times divisor = 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, S395 S397 · calculationBefore Mobius inversion ORIGINAL inverse F(n/21) f(4) plus ORIGINAL b := sum(76, 1) After Mobius inversion ORIGINAL inverse F(n/21) f(4) plus ORIGINAL b = 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, S396 S398 · calculationBefore Mobius inversion ORIGINAL inverse F(n/21) f(5) ORIGINAL a times divisor := mul(19, 5) After Mobius inversion ORIGINAL inverse F(n/21) f(5) ORIGINAL a times divisor = 95 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S397 S399 · calculationBefore Mobius inversion ORIGINAL inverse F(n/21) f(5) plus ORIGINAL b := sum(95, 1) After Mobius inversion ORIGINAL inverse F(n/21) f(5) plus ORIGINAL b = 96 Conditions: none Substitute original 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/21) f(10) ORIGINAL a times divisor := mul(19, 10) After Mobius inversion ORIGINAL inverse F(n/21) f(10) ORIGINAL a times divisor = 190 Conditions: none Substitute original 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/21) f(10) plus ORIGINAL b := sum(190, 1) After Mobius inversion ORIGINAL inverse F(n/21) f(10) plus ORIGINAL b = 191 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S400 S402 · calculationBefore Mobius inversion ORIGINAL inverse F(n/21) f(20) ORIGINAL a times divisor := mul(19, 20) After Mobius inversion ORIGINAL inverse F(n/21) f(20) ORIGINAL a times divisor = 380 Conditions: none Substitute original 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 F(n/21) f(20) plus ORIGINAL b := sum(380, 1) After Mobius inversion ORIGINAL inverse F(n/21) f(20) plus ORIGINAL b = 381 Conditions: none Substitute original 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 F(n/21) full forward divisor sum := sum(20, 39, 77, 96, 191, 381) After Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum = 804 Conditions: none Substitute original 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 mu(21) times SAME F(n/21) := mul(1, 804) After Mobius inversion ORIGINAL inverse mu(21) times SAME F(n/21) = 804 Conditions: none Substitute original 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 Mobius at ORIGINAL divisor28:actual positive input for mu=28 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, S405 S407 · impliesBefore ORIGINAL positive integer absolute magnitude=28>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, S406 S408 · calculationBefore upf extract divisor2 occurrence0 := div(28, 2) After upf extract divisor2 occurrence0 = 14 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, S407 S409 · calculationBefore upf extract divisor2 occurrence1 := div(14, 2) After upf extract divisor2 occurrence1 = 7 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, S408 S410 · impliesBefore next divisor=2;previous remaining=28;all smaller divisors exhausted After extracted multiplicity=2;next remaining=7;remaining modulo2=1;extracted multiset=[2, 2] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S409 S411 · 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, S410 S412 · calculationBefore upf canonical complete product factor1 := mul(2, 2) After upf canonical complete product factor1 = 4 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S411 S413 · calculationBefore upf canonical complete product factor2 := mul(4, 7) After upf canonical complete product factor2 = 28 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, S412 S414 · impliesBefore exhausted trial divisors below3;terminal remaining=7;d*d>remaining;ORIGINAL n=28 After complete increasing prime multiset=[2, 2, 7];full product=28;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, S413 S415 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor28:input=28;COMPLETE prime multiset=[2, 2, 7];distinct primes=[2, 7] 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, S414 S416 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor28 := div(420, 28) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor28 = 15 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S415 S417 · impliesBefore Mobius inversion ORIGINAL inverse F(n/28):actual positive input=15;check complete integer range1..15 After ALL positive divisors INCREASING=[1, 3, 5, 15];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, S416 S418 · calculationBefore Mobius inversion ORIGINAL inverse F(n/28) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/28) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S417 S419 · calculationBefore Mobius inversion ORIGINAL inverse F(n/28) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/28) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S418 S420 · calculationBefore Mobius inversion ORIGINAL inverse F(n/28) f(3) ORIGINAL a times divisor := mul(19, 3) After Mobius inversion ORIGINAL inverse F(n/28) f(3) ORIGINAL a times divisor = 57 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S419 S421 · calculationBefore Mobius inversion ORIGINAL inverse F(n/28) f(3) plus ORIGINAL b := sum(57, 1) After Mobius inversion ORIGINAL inverse F(n/28) f(3) plus ORIGINAL b = 58 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S420 S422 · calculationBefore Mobius inversion ORIGINAL inverse F(n/28) f(5) ORIGINAL a times divisor := mul(19, 5) After Mobius inversion ORIGINAL inverse F(n/28) f(5) ORIGINAL a times divisor = 95 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S421 S423 · calculationBefore Mobius inversion ORIGINAL inverse F(n/28) f(5) plus ORIGINAL b := sum(95, 1) After Mobius inversion ORIGINAL inverse F(n/28) f(5) plus ORIGINAL b = 96 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S422 S424 · calculationBefore Mobius inversion ORIGINAL inverse F(n/28) f(15) ORIGINAL a times divisor := mul(19, 15) After Mobius inversion ORIGINAL inverse F(n/28) f(15) ORIGINAL a times divisor = 285 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S423 S425 · calculationBefore Mobius inversion ORIGINAL inverse F(n/28) f(15) plus ORIGINAL b := sum(285, 1) After Mobius inversion ORIGINAL inverse F(n/28) f(15) plus ORIGINAL b = 286 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S424 S426 · calculationBefore Mobius inversion ORIGINAL inverse F(n/28) full forward divisor sum := sum(20, 58, 96, 286) After Mobius inversion ORIGINAL inverse F(n/28) full forward divisor sum = 460 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S425 S427 · calculationBefore Mobius inversion ORIGINAL inverse mu(28) times SAME F(n/28) := mul(0, 460) After Mobius inversion ORIGINAL inverse mu(28) times SAME F(n/28) = 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, S426 S428 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor30:actual positive input for mu=30 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, S427 S429 · impliesBefore ORIGINAL positive integer absolute magnitude=30>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, S428 S430 · calculationBefore upf extract divisor2 occurrence0 := div(30, 2) After upf extract divisor2 occurrence0 = 15 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, S429 S431 · impliesBefore next divisor=2;previous remaining=30;all smaller divisors exhausted After extracted multiplicity=1;next remaining=15;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, S430 S432 · calculationBefore upf extract divisor3 occurrence1 := div(15, 3) After upf extract divisor3 occurrence1 = 5 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S431 S433 · impliesBefore next divisor=3;previous remaining=15;all smaller divisors exhausted After extracted multiplicity=1;next remaining=5;remaining modulo3=2;extracted multiset=[2, 3] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S432 S434 · 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, S433 S435 · 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, S434 S436 · calculationBefore upf canonical complete product factor2 := mul(6, 5) After upf canonical complete product factor2 = 30 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, S435 S437 · impliesBefore exhausted trial divisors below4;terminal remaining=5;d*d>remaining;ORIGINAL n=30 After complete increasing prime multiset=[2, 3, 5];full product=30;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, S436 S438 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor30:input=30;COMPLETE prime multiset=[2, 3, 5];distinct primes=[2, 3, 5] After mu=-1;empty product case=No;repeated prime=No Conditions: none The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities. Step identifiers and dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S437 S439 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor30 := div(420, 30) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor30 = 14 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S438 S440 · impliesBefore Mobius inversion ORIGINAL inverse F(n/30):actual positive input=14;check complete integer range1..14 After ALL positive divisors INCREASING=[1, 2, 7, 14];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, S439 S441 · calculationBefore Mobius inversion ORIGINAL inverse F(n/30) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/30) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S440 S442 · calculationBefore Mobius inversion ORIGINAL inverse F(n/30) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/30) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S441 S443 · calculationBefore Mobius inversion ORIGINAL inverse F(n/30) f(2) ORIGINAL a times divisor := mul(19, 2) After Mobius inversion ORIGINAL inverse F(n/30) f(2) ORIGINAL a times divisor = 38 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S442 S444 · calculationBefore Mobius inversion ORIGINAL inverse F(n/30) f(2) plus ORIGINAL b := sum(38, 1) After Mobius inversion ORIGINAL inverse F(n/30) f(2) plus ORIGINAL b = 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, S443 S445 · calculationBefore Mobius inversion ORIGINAL inverse F(n/30) f(7) ORIGINAL a times divisor := mul(19, 7) After Mobius inversion ORIGINAL inverse F(n/30) f(7) ORIGINAL a times divisor = 133 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S444 S446 · calculationBefore Mobius inversion ORIGINAL inverse F(n/30) f(7) plus ORIGINAL b := sum(133, 1) After Mobius inversion ORIGINAL inverse F(n/30) f(7) plus ORIGINAL b = 134 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S445 S447 · calculationBefore Mobius inversion ORIGINAL inverse F(n/30) f(14) ORIGINAL a times divisor := mul(19, 14) After Mobius inversion ORIGINAL inverse F(n/30) f(14) ORIGINAL a times divisor = 266 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S446 S448 · calculationBefore Mobius inversion ORIGINAL inverse F(n/30) f(14) plus ORIGINAL b := sum(266, 1) After Mobius inversion ORIGINAL inverse F(n/30) f(14) plus ORIGINAL b = 267 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S447 S449 · calculationBefore Mobius inversion ORIGINAL inverse F(n/30) full forward divisor sum := sum(20, 39, 134, 267) After Mobius inversion ORIGINAL inverse F(n/30) full forward divisor sum = 460 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S448 S450 · calculationBefore Mobius inversion ORIGINAL inverse mu(30) times SAME F(n/30) := mul(-1, 460) After Mobius inversion ORIGINAL inverse mu(30) times SAME F(n/30) = -460 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S449 S451 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor35:actual positive input for mu=35 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, S450 S452 · impliesBefore ORIGINAL positive integer absolute magnitude=35>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, S451 S453 · impliesBefore next divisor=2;previous remaining=35;all smaller divisors exhausted After extracted multiplicity=0;next remaining=35;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, S452 S454 · impliesBefore next divisor=3;previous remaining=35;all smaller divisors exhausted After extracted multiplicity=0;next remaining=35;remaining modulo3=2;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, S453 S455 · impliesBefore next divisor=4;previous remaining=35;all smaller divisors exhausted After extracted multiplicity=0;next remaining=35;remaining modulo4=3;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, S454 S456 · calculationBefore upf extract divisor5 occurrence0 := div(35, 5) After upf extract divisor5 occurrence0 = 7 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, S455 S457 · impliesBefore next divisor=5;previous remaining=35;all smaller divisors exhausted After extracted multiplicity=1;next remaining=7;remaining modulo5=2;extracted multiset=[5] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S456 S458 · calculationBefore upf canonical complete product factor0 := mul(1, 5) After upf canonical complete product factor0 = 5 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S457 S459 · calculationBefore upf canonical complete product factor1 := mul(5, 7) After upf canonical complete product factor1 = 35 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, S458 S460 · impliesBefore exhausted trial divisors below6;terminal remaining=7;d*d>remaining;ORIGINAL n=35 After complete increasing prime multiset=[5, 7];full product=35;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, S459 S461 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor35:input=35;COMPLETE prime multiset=[5, 7];distinct primes=[5, 7] 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, S460 S462 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor35 := div(420, 35) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor35 = 12 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S461 S463 · impliesBefore Mobius inversion ORIGINAL inverse F(n/35):actual positive input=12;check complete integer range1..12 After ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 12];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, S462 S464 · calculationBefore Mobius inversion ORIGINAL inverse F(n/35) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/35) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S463 S465 · calculationBefore Mobius inversion ORIGINAL inverse F(n/35) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/35) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S464 S466 · calculationBefore Mobius inversion ORIGINAL inverse F(n/35) f(2) ORIGINAL a times divisor := mul(19, 2) After Mobius inversion ORIGINAL inverse F(n/35) f(2) ORIGINAL a times divisor = 38 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S465 S467 · calculationBefore Mobius inversion ORIGINAL inverse F(n/35) f(2) plus ORIGINAL b := sum(38, 1) After Mobius inversion ORIGINAL inverse F(n/35) f(2) plus ORIGINAL b = 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, S466 S468 · calculationBefore Mobius inversion ORIGINAL inverse F(n/35) f(3) ORIGINAL a times divisor := mul(19, 3) After Mobius inversion ORIGINAL inverse F(n/35) f(3) ORIGINAL a times divisor = 57 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S467 S469 · calculationBefore Mobius inversion ORIGINAL inverse F(n/35) f(3) plus ORIGINAL b := sum(57, 1) After Mobius inversion ORIGINAL inverse F(n/35) f(3) plus ORIGINAL b = 58 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S468 S470 · calculationBefore Mobius inversion ORIGINAL inverse F(n/35) f(4) ORIGINAL a times divisor := mul(19, 4) After Mobius inversion ORIGINAL inverse F(n/35) f(4) ORIGINAL a times divisor = 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, S469 S471 · calculationBefore Mobius inversion ORIGINAL inverse F(n/35) f(4) plus ORIGINAL b := sum(76, 1) After Mobius inversion ORIGINAL inverse F(n/35) f(4) plus ORIGINAL b = 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, S470 S472 · calculationBefore Mobius inversion ORIGINAL inverse F(n/35) f(6) ORIGINAL a times divisor := mul(19, 6) After Mobius inversion ORIGINAL inverse F(n/35) f(6) ORIGINAL a times divisor = 114 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S471 S473 · calculationBefore Mobius inversion ORIGINAL inverse F(n/35) f(6) plus ORIGINAL b := sum(114, 1) After Mobius inversion ORIGINAL inverse F(n/35) f(6) plus ORIGINAL b = 115 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S472 S474 · calculationBefore Mobius inversion ORIGINAL inverse F(n/35) f(12) ORIGINAL a times divisor := mul(19, 12) After Mobius inversion ORIGINAL inverse F(n/35) f(12) ORIGINAL a times divisor = 228 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S473 S475 · calculationBefore Mobius inversion ORIGINAL inverse F(n/35) f(12) plus ORIGINAL b := sum(228, 1) After Mobius inversion ORIGINAL inverse F(n/35) f(12) plus ORIGINAL b = 229 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S474 S476 · calculationBefore Mobius inversion ORIGINAL inverse F(n/35) full forward divisor sum := sum(20, 39, 58, 77, 115, 229) After Mobius inversion ORIGINAL inverse F(n/35) full forward divisor sum = 538 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S475 S477 · calculationBefore Mobius inversion ORIGINAL inverse mu(35) times SAME F(n/35) := mul(1, 538) After Mobius inversion ORIGINAL inverse mu(35) times SAME F(n/35) = 538 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S476 S478 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor42:actual positive input for mu=42 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, S477 S479 · impliesBefore ORIGINAL positive integer absolute magnitude=42>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, S478 S480 · calculationBefore upf extract divisor2 occurrence0 := div(42, 2) After upf extract divisor2 occurrence0 = 21 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, S479 S481 · impliesBefore next divisor=2;previous remaining=42;all smaller divisors exhausted After extracted multiplicity=1;next remaining=21;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, S480 S482 · calculationBefore upf extract divisor3 occurrence1 := div(21, 3) After upf extract divisor3 occurrence1 = 7 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, S481 S483 · impliesBefore next divisor=3;previous remaining=21;all smaller divisors exhausted After extracted multiplicity=1;next remaining=7;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, S482 S484 · 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, S483 S485 · 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, S484 S486 · calculationBefore upf canonical complete product factor2 := mul(6, 7) After upf canonical complete product factor2 = 42 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, S485 S487 · impliesBefore exhausted trial divisors below4;terminal remaining=7;d*d>remaining;ORIGINAL n=42 After complete increasing prime multiset=[2, 3, 7];full product=42;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, S486 S488 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor42:input=42;COMPLETE prime multiset=[2, 3, 7];distinct primes=[2, 3, 7] 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, S487 S489 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor42 := div(420, 42) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor42 = 10 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S488 S490 · impliesBefore Mobius inversion ORIGINAL inverse F(n/42):actual positive input=10;check complete integer range1..10 After ALL positive divisors INCREASING=[1, 2, 5, 10];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, S489 S491 · calculationBefore Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S490 S492 · calculationBefore Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S491 S493 · calculationBefore Mobius inversion ORIGINAL inverse F(n/42) f(2) ORIGINAL a times divisor := mul(19, 2) After Mobius inversion ORIGINAL inverse F(n/42) f(2) ORIGINAL a times divisor = 38 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S492 S494 · calculationBefore Mobius inversion ORIGINAL inverse F(n/42) f(2) plus ORIGINAL b := sum(38, 1) After Mobius inversion ORIGINAL inverse F(n/42) f(2) plus ORIGINAL b = 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, S493 S495 · calculationBefore Mobius inversion ORIGINAL inverse F(n/42) f(5) ORIGINAL a times divisor := mul(19, 5) After Mobius inversion ORIGINAL inverse F(n/42) f(5) ORIGINAL a times divisor = 95 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S494 S496 · calculationBefore Mobius inversion ORIGINAL inverse F(n/42) f(5) plus ORIGINAL b := sum(95, 1) After Mobius inversion ORIGINAL inverse F(n/42) f(5) plus ORIGINAL b = 96 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S495 S497 · calculationBefore Mobius inversion ORIGINAL inverse F(n/42) f(10) ORIGINAL a times divisor := mul(19, 10) After Mobius inversion ORIGINAL inverse F(n/42) f(10) ORIGINAL a times divisor = 190 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S496 S498 · calculationBefore Mobius inversion ORIGINAL inverse F(n/42) f(10) plus ORIGINAL b := sum(190, 1) After Mobius inversion ORIGINAL inverse F(n/42) f(10) plus ORIGINAL b = 191 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S497 S499 · calculationBefore Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum := sum(20, 39, 96, 191) After Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum = 346 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S498 S500 · calculationBefore Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42) := mul(-1, 346) After Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42) = -346 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S499 S501 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor60:actual positive input for mu=60 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, S500 S502 · impliesBefore ORIGINAL positive integer absolute magnitude=60>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, S501 S503 · calculationBefore upf extract divisor2 occurrence0 := div(60, 2) After upf extract divisor2 occurrence0 = 30 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, S502 S504 · calculationBefore upf extract divisor2 occurrence1 := div(30, 2) After upf extract divisor2 occurrence1 = 15 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, S503 S505 · impliesBefore next divisor=2;previous remaining=60;all smaller divisors exhausted After extracted multiplicity=2;next remaining=15;remaining modulo2=1;extracted multiset=[2, 2] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S504 S506 · calculationBefore upf extract divisor3 occurrence2 := div(15, 3) After upf extract divisor3 occurrence2 = 5 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S505 S507 · impliesBefore next divisor=3;previous remaining=15;all smaller divisors exhausted After extracted multiplicity=1;next remaining=5;remaining modulo3=2;extracted multiset=[2, 2, 3] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S506 S508 · 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, S507 S509 · calculationBefore upf canonical complete product factor1 := mul(2, 2) After upf canonical complete product factor1 = 4 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S508 S510 · calculationBefore upf canonical complete product factor2 := mul(4, 3) After upf canonical complete product factor2 = 12 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, S509 S511 · calculationBefore upf canonical complete product factor3 := mul(12, 5) After upf canonical complete product factor3 = 60 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, S510 S512 · impliesBefore exhausted trial divisors below4;terminal remaining=5;d*d>remaining;ORIGINAL n=60 After complete increasing prime multiset=[2, 2, 3, 5];full product=60;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, S511 S513 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor60:input=60;COMPLETE prime multiset=[2, 2, 3, 5];distinct primes=[2, 3, 5] After mu=0;empty product case=No;repeated prime=Yes Conditions: none The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities. Step identifiers and dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S512 S514 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor60 := div(420, 60) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor60 = 7 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S513 S515 · impliesBefore Mobius inversion ORIGINAL inverse F(n/60):actual positive input=7;check complete integer range1..7 After ALL positive divisors INCREASING=[1, 7];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, S514 S516 · calculationBefore Mobius inversion ORIGINAL inverse F(n/60) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/60) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S515 S517 · calculationBefore Mobius inversion ORIGINAL inverse F(n/60) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/60) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S516 S518 · calculationBefore Mobius inversion ORIGINAL inverse F(n/60) f(7) ORIGINAL a times divisor := mul(19, 7) After Mobius inversion ORIGINAL inverse F(n/60) f(7) ORIGINAL a times divisor = 133 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S517 S519 · calculationBefore Mobius inversion ORIGINAL inverse F(n/60) f(7) plus ORIGINAL b := sum(133, 1) After Mobius inversion ORIGINAL inverse F(n/60) f(7) plus ORIGINAL b = 134 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S518 S520 · calculationBefore Mobius inversion ORIGINAL inverse F(n/60) full forward divisor sum := sum(20, 134) After Mobius inversion ORIGINAL inverse F(n/60) full forward divisor sum = 154 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S519 S521 · calculationBefore Mobius inversion ORIGINAL inverse mu(60) times SAME F(n/60) := mul(0, 154) After Mobius inversion ORIGINAL inverse mu(60) times SAME F(n/60) = 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, S520 S522 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor70:actual positive input for mu=70 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, S521 S523 · impliesBefore ORIGINAL positive integer absolute magnitude=70>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, S522 S524 · calculationBefore upf extract divisor2 occurrence0 := div(70, 2) After upf extract divisor2 occurrence0 = 35 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, S523 S525 · impliesBefore next divisor=2;previous remaining=70;all smaller divisors exhausted After extracted multiplicity=1;next remaining=35;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, S524 S526 · impliesBefore next divisor=3;previous remaining=35;all smaller divisors exhausted After extracted multiplicity=0;next remaining=35;remaining modulo3=2;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, S525 S527 · impliesBefore next divisor=4;previous remaining=35;all smaller divisors exhausted After extracted multiplicity=0;next remaining=35;remaining modulo4=3;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, S526 S528 · calculationBefore upf extract divisor5 occurrence1 := div(35, 5) After upf extract divisor5 occurrence1 = 7 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, S527 S529 · impliesBefore next divisor=5;previous remaining=35;all smaller divisors exhausted After extracted multiplicity=1;next remaining=7;remaining modulo5=2;extracted multiset=[2, 5] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S528 S530 · 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, S529 S531 · calculationBefore upf canonical complete product factor1 := mul(2, 5) After upf canonical complete product factor1 = 10 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, S530 S532 · calculationBefore upf canonical complete product factor2 := mul(10, 7) After upf canonical complete product factor2 = 70 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, S531 S533 · impliesBefore exhausted trial divisors below6;terminal remaining=7;d*d>remaining;ORIGINAL n=70 After complete increasing prime multiset=[2, 5, 7];full product=70;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, S532 S534 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor70:input=70;COMPLETE prime multiset=[2, 5, 7];distinct primes=[2, 5, 7] 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, S533 S535 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor70 := div(420, 70) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor70 = 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, S534 S536 · impliesBefore Mobius inversion ORIGINAL inverse F(n/70):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, S535 S537 · calculationBefore Mobius inversion ORIGINAL inverse F(n/70) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/70) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S536 S538 · calculationBefore Mobius inversion ORIGINAL inverse F(n/70) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/70) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S537 S539 · calculationBefore Mobius inversion ORIGINAL inverse F(n/70) f(2) ORIGINAL a times divisor := mul(19, 2) After Mobius inversion ORIGINAL inverse F(n/70) f(2) ORIGINAL a times divisor = 38 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S538 S540 · calculationBefore Mobius inversion ORIGINAL inverse F(n/70) f(2) plus ORIGINAL b := sum(38, 1) After Mobius inversion ORIGINAL inverse F(n/70) f(2) plus ORIGINAL b = 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, S539 S541 · calculationBefore Mobius inversion ORIGINAL inverse F(n/70) f(3) ORIGINAL a times divisor := mul(19, 3) After Mobius inversion ORIGINAL inverse F(n/70) f(3) ORIGINAL a times divisor = 57 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S540 S542 · calculationBefore Mobius inversion ORIGINAL inverse F(n/70) f(3) plus ORIGINAL b := sum(57, 1) After Mobius inversion ORIGINAL inverse F(n/70) f(3) plus ORIGINAL b = 58 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S541 S543 · calculationBefore Mobius inversion ORIGINAL inverse F(n/70) f(6) ORIGINAL a times divisor := mul(19, 6) After Mobius inversion ORIGINAL inverse F(n/70) f(6) ORIGINAL a times divisor = 114 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S542 S544 · calculationBefore Mobius inversion ORIGINAL inverse F(n/70) f(6) plus ORIGINAL b := sum(114, 1) After Mobius inversion ORIGINAL inverse F(n/70) f(6) plus ORIGINAL b = 115 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S543 S545 · calculationBefore Mobius inversion ORIGINAL inverse F(n/70) full forward divisor sum := sum(20, 39, 58, 115) After Mobius inversion ORIGINAL inverse F(n/70) full forward divisor sum = 232 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S544 S546 · calculationBefore Mobius inversion ORIGINAL inverse mu(70) times SAME F(n/70) := mul(-1, 232) After Mobius inversion ORIGINAL inverse mu(70) times SAME F(n/70) = -232 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S545 S547 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor84:actual positive input for mu=84 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, S546 S548 · impliesBefore ORIGINAL positive integer absolute magnitude=84>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, S547 S549 · calculationBefore upf extract divisor2 occurrence0 := div(84, 2) After upf extract divisor2 occurrence0 = 42 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, S548 S550 · calculationBefore upf extract divisor2 occurrence1 := div(42, 2) After upf extract divisor2 occurrence1 = 21 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, S549 S551 · impliesBefore next divisor=2;previous remaining=84;all smaller divisors exhausted After extracted multiplicity=2;next remaining=21;remaining modulo2=1;extracted multiset=[2, 2] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S550 S552 · calculationBefore upf extract divisor3 occurrence2 := div(21, 3) After upf extract divisor3 occurrence2 = 7 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, S551 S553 · impliesBefore next divisor=3;previous remaining=21;all smaller divisors exhausted After extracted multiplicity=1;next remaining=7;remaining modulo3=1;extracted multiset=[2, 2, 3] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S552 S554 · 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, S553 S555 · calculationBefore upf canonical complete product factor1 := mul(2, 2) After upf canonical complete product factor1 = 4 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S554 S556 · calculationBefore upf canonical complete product factor2 := mul(4, 3) After upf canonical complete product factor2 = 12 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, S555 S557 · calculationBefore upf canonical complete product factor3 := mul(12, 7) After upf canonical complete product factor3 = 84 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, S556 S558 · impliesBefore exhausted trial divisors below4;terminal remaining=7;d*d>remaining;ORIGINAL n=84 After complete increasing prime multiset=[2, 2, 3, 7];full product=84;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, S557 S559 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor84:input=84;COMPLETE prime multiset=[2, 2, 3, 7];distinct primes=[2, 3, 7] 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, S558 S560 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor84 := div(420, 84) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor84 = 5 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S559 S561 · impliesBefore Mobius inversion ORIGINAL inverse F(n/84):actual positive input=5;check complete integer range1..5 After ALL positive divisors INCREASING=[1, 5];not only primes/proper divisors or all integers<=n Conditions: none Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction. Step identifiers and dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S560 S562 · calculationBefore Mobius inversion ORIGINAL inverse F(n/84) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/84) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S561 S563 · calculationBefore Mobius inversion ORIGINAL inverse F(n/84) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/84) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S562 S564 · calculationBefore Mobius inversion ORIGINAL inverse F(n/84) f(5) ORIGINAL a times divisor := mul(19, 5) After Mobius inversion ORIGINAL inverse F(n/84) f(5) ORIGINAL a times divisor = 95 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S563 S565 · calculationBefore Mobius inversion ORIGINAL inverse F(n/84) f(5) plus ORIGINAL b := sum(95, 1) After Mobius inversion ORIGINAL inverse F(n/84) f(5) plus ORIGINAL b = 96 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S564 S566 · calculationBefore Mobius inversion ORIGINAL inverse F(n/84) full forward divisor sum := sum(20, 96) After Mobius inversion ORIGINAL inverse F(n/84) full forward divisor sum = 116 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S565 S567 · calculationBefore Mobius inversion ORIGINAL inverse mu(84) times SAME F(n/84) := mul(0, 116) After Mobius inversion ORIGINAL inverse mu(84) times SAME F(n/84) = 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, S566 S568 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor105:actual positive input for mu=105 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, S567 S569 · impliesBefore ORIGINAL positive integer absolute magnitude=105>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, S568 S570 · impliesBefore next divisor=2;previous remaining=105;all smaller divisors exhausted After extracted multiplicity=0;next remaining=105;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, S569 S571 · calculationBefore upf extract divisor3 occurrence0 := div(105, 3) After upf extract divisor3 occurrence0 = 35 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, S570 S572 · impliesBefore next divisor=3;previous remaining=105;all smaller divisors exhausted After extracted multiplicity=1;next remaining=35;remaining modulo3=2;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, S571 S573 · impliesBefore next divisor=4;previous remaining=35;all smaller divisors exhausted After extracted multiplicity=0;next remaining=35;remaining modulo4=3;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, S572 S574 · calculationBefore upf extract divisor5 occurrence1 := div(35, 5) After upf extract divisor5 occurrence1 = 7 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, S573 S575 · impliesBefore next divisor=5;previous remaining=35;all smaller divisors exhausted After extracted multiplicity=1;next remaining=7;remaining modulo5=2;extracted multiset=[3, 5] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S574 S576 · 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, S575 S577 · calculationBefore upf canonical complete product factor1 := mul(3, 5) After upf canonical complete product factor1 = 15 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, S576 S578 · calculationBefore upf canonical complete product factor2 := mul(15, 7) After upf canonical complete product factor2 = 105 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, S577 S579 · impliesBefore exhausted trial divisors below6;terminal remaining=7;d*d>remaining;ORIGINAL n=105 After complete increasing prime multiset=[3, 5, 7];full product=105;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, S578 S580 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor105:input=105;COMPLETE prime multiset=[3, 5, 7];distinct primes=[3, 5, 7] 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, S579 S581 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor105 := div(420, 105) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor105 = 4 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S580 S582 · impliesBefore Mobius inversion ORIGINAL inverse F(n/105):actual positive input=4;check complete integer range1..4 After ALL positive divisors INCREASING=[1, 2, 4];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, S581 S583 · calculationBefore Mobius inversion ORIGINAL inverse F(n/105) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/105) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S582 S584 · calculationBefore Mobius inversion ORIGINAL inverse F(n/105) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/105) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S583 S585 · calculationBefore Mobius inversion ORIGINAL inverse F(n/105) f(2) ORIGINAL a times divisor := mul(19, 2) After Mobius inversion ORIGINAL inverse F(n/105) f(2) ORIGINAL a times divisor = 38 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S584 S586 · calculationBefore Mobius inversion ORIGINAL inverse F(n/105) f(2) plus ORIGINAL b := sum(38, 1) After Mobius inversion ORIGINAL inverse F(n/105) f(2) plus ORIGINAL b = 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, S585 S587 · calculationBefore Mobius inversion ORIGINAL inverse F(n/105) f(4) ORIGINAL a times divisor := mul(19, 4) After Mobius inversion ORIGINAL inverse F(n/105) f(4) ORIGINAL a times divisor = 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, S586 S588 · calculationBefore Mobius inversion ORIGINAL inverse F(n/105) f(4) plus ORIGINAL b := sum(76, 1) After Mobius inversion ORIGINAL inverse F(n/105) f(4) plus ORIGINAL b = 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, S587 S589 · calculationBefore Mobius inversion ORIGINAL inverse F(n/105) full forward divisor sum := sum(20, 39, 77) After Mobius inversion ORIGINAL inverse F(n/105) full forward divisor sum = 136 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S588 S590 · calculationBefore Mobius inversion ORIGINAL inverse mu(105) times SAME F(n/105) := mul(-1, 136) After Mobius inversion ORIGINAL inverse mu(105) times SAME F(n/105) = -136 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S589 S591 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor140:actual positive input for mu=140 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, S590 S592 · impliesBefore ORIGINAL positive integer absolute magnitude=140>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, S591 S593 · calculationBefore upf extract divisor2 occurrence0 := div(140, 2) After upf extract divisor2 occurrence0 = 70 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, S592 S594 · calculationBefore upf extract divisor2 occurrence1 := div(70, 2) After upf extract divisor2 occurrence1 = 35 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, S593 S595 · impliesBefore next divisor=2;previous remaining=140;all smaller divisors exhausted After extracted multiplicity=2;next remaining=35;remaining modulo2=1;extracted multiset=[2, 2] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S594 S596 · impliesBefore next divisor=3;previous remaining=35;all smaller divisors exhausted After extracted multiplicity=0;next remaining=35;remaining modulo3=2;extracted multiset=[2, 2] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S595 S597 · impliesBefore next divisor=4;previous remaining=35;all smaller divisors exhausted After extracted multiplicity=0;next remaining=35;remaining modulo4=3;extracted multiset=[2, 2] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S596 S598 · calculationBefore upf extract divisor5 occurrence2 := div(35, 5) After upf extract divisor5 occurrence2 = 7 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, S597 S599 · impliesBefore next divisor=5;previous remaining=35;all smaller divisors exhausted After extracted multiplicity=1;next remaining=7;remaining modulo5=2;extracted multiset=[2, 2, 5] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S598 S600 · 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, S599 S601 · calculationBefore upf canonical complete product factor1 := mul(2, 2) After upf canonical complete product factor1 = 4 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S600 S602 · calculationBefore upf canonical complete product factor2 := mul(4, 5) After upf canonical complete product factor2 = 20 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, S601 S603 · calculationBefore upf canonical complete product factor3 := mul(20, 7) After upf canonical complete product factor3 = 140 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, S602 S604 · impliesBefore exhausted trial divisors below6;terminal remaining=7;d*d>remaining;ORIGINAL n=140 After complete increasing prime multiset=[2, 2, 5, 7];full product=140;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, S603 S605 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor140:input=140;COMPLETE prime multiset=[2, 2, 5, 7];distinct primes=[2, 5, 7] 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, S604 S606 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor140 := div(420, 140) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor140 = 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, S605 S607 · impliesBefore Mobius inversion ORIGINAL inverse F(n/140):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, S606 S608 · calculationBefore Mobius inversion ORIGINAL inverse F(n/140) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/140) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S607 S609 · calculationBefore Mobius inversion ORIGINAL inverse F(n/140) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/140) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S608 S610 · calculationBefore Mobius inversion ORIGINAL inverse F(n/140) f(3) ORIGINAL a times divisor := mul(19, 3) After Mobius inversion ORIGINAL inverse F(n/140) f(3) ORIGINAL a times divisor = 57 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S609 S611 · calculationBefore Mobius inversion ORIGINAL inverse F(n/140) f(3) plus ORIGINAL b := sum(57, 1) After Mobius inversion ORIGINAL inverse F(n/140) f(3) plus ORIGINAL b = 58 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S610 S612 · calculationBefore Mobius inversion ORIGINAL inverse F(n/140) full forward divisor sum := sum(20, 58) After Mobius inversion ORIGINAL inverse F(n/140) full forward divisor sum = 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, S611 S613 · calculationBefore Mobius inversion ORIGINAL inverse mu(140) times SAME F(n/140) := mul(0, 78) After Mobius inversion ORIGINAL inverse mu(140) times SAME F(n/140) = 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, S612 S614 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor210:actual positive input for mu=210 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, S613 S615 · impliesBefore ORIGINAL positive integer absolute magnitude=210>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, S614 S616 · calculationBefore upf extract divisor2 occurrence0 := div(210, 2) After upf extract divisor2 occurrence0 = 105 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, S615 S617 · impliesBefore next divisor=2;previous remaining=210;all smaller divisors exhausted After extracted multiplicity=1;next remaining=105;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, S616 S618 · calculationBefore upf extract divisor3 occurrence1 := div(105, 3) After upf extract divisor3 occurrence1 = 35 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, S617 S619 · impliesBefore next divisor=3;previous remaining=105;all smaller divisors exhausted After extracted multiplicity=1;next remaining=35;remaining modulo3=2;extracted multiset=[2, 3] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S618 S620 · impliesBefore next divisor=4;previous remaining=35;all smaller divisors exhausted After extracted multiplicity=0;next remaining=35;remaining modulo4=3;extracted multiset=[2, 3] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S619 S621 · calculationBefore upf extract divisor5 occurrence2 := div(35, 5) After upf extract divisor5 occurrence2 = 7 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, S620 S622 · impliesBefore next divisor=5;previous remaining=35;all smaller divisors exhausted After extracted multiplicity=1;next remaining=7;remaining modulo5=2;extracted multiset=[2, 3, 5] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S621 S623 · 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, S622 S624 · 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, S623 S625 · calculationBefore upf canonical complete product factor2 := mul(6, 5) After upf canonical complete product factor2 = 30 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, S624 S626 · calculationBefore upf canonical complete product factor3 := mul(30, 7) After upf canonical complete product factor3 = 210 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, S625 S627 · impliesBefore exhausted trial divisors below6;terminal remaining=7;d*d>remaining;ORIGINAL n=210 After complete increasing prime multiset=[2, 3, 5, 7];full product=210;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, S626 S628 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor210:input=210;COMPLETE prime multiset=[2, 3, 5, 7];distinct primes=[2, 3, 5, 7] 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, S627 S629 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor210 := div(420, 210) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor210 = 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, S628 S630 · impliesBefore Mobius inversion ORIGINAL inverse F(n/210):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, S629 S631 · calculationBefore Mobius inversion ORIGINAL inverse F(n/210) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/210) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S630 S632 · calculationBefore Mobius inversion ORIGINAL inverse F(n/210) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/210) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S631 S633 · calculationBefore Mobius inversion ORIGINAL inverse F(n/210) f(2) ORIGINAL a times divisor := mul(19, 2) After Mobius inversion ORIGINAL inverse F(n/210) f(2) ORIGINAL a times divisor = 38 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S632 S634 · calculationBefore Mobius inversion ORIGINAL inverse F(n/210) f(2) plus ORIGINAL b := sum(38, 1) After Mobius inversion ORIGINAL inverse F(n/210) f(2) plus ORIGINAL b = 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, S633 S635 · calculationBefore Mobius inversion ORIGINAL inverse F(n/210) full forward divisor sum := sum(20, 39) After Mobius inversion ORIGINAL inverse F(n/210) full forward divisor sum = 59 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S634 S636 · calculationBefore Mobius inversion ORIGINAL inverse mu(210) times SAME F(n/210) := mul(1, 59) After Mobius inversion ORIGINAL inverse mu(210) times SAME F(n/210) = 59 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S635 S637 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor420:actual positive input for mu=420 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, S636 S638 · impliesBefore ORIGINAL positive integer absolute magnitude=420>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, S637 S639 · calculationBefore upf extract divisor2 occurrence0 := div(420, 2) After upf extract divisor2 occurrence0 = 210 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, S638 S640 · calculationBefore upf extract divisor2 occurrence1 := div(210, 2) After upf extract divisor2 occurrence1 = 105 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, S639 S641 · impliesBefore next divisor=2;previous remaining=420;all smaller divisors exhausted After extracted multiplicity=2;next remaining=105;remaining modulo2=1;extracted multiset=[2, 2] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S640 S642 · calculationBefore upf extract divisor3 occurrence2 := div(105, 3) After upf extract divisor3 occurrence2 = 35 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, S641 S643 · impliesBefore next divisor=3;previous remaining=105;all smaller divisors exhausted After extracted multiplicity=1;next remaining=35;remaining modulo3=2;extracted multiset=[2, 2, 3] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S642 S644 · impliesBefore next divisor=4;previous remaining=35;all smaller divisors exhausted After extracted multiplicity=0;next remaining=35;remaining modulo4=3;extracted multiset=[2, 2, 3] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S643 S645 · calculationBefore upf extract divisor5 occurrence3 := div(35, 5) After upf extract divisor5 occurrence3 = 7 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, S644 S646 · impliesBefore next divisor=5;previous remaining=35;all smaller divisors exhausted After extracted multiplicity=1;next remaining=7;remaining modulo5=2;extracted multiset=[2, 2, 3, 5] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S645 S647 · 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, S646 S648 · calculationBefore upf canonical complete product factor1 := mul(2, 2) After upf canonical complete product factor1 = 4 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S647 S649 · calculationBefore upf canonical complete product factor2 := mul(4, 3) After upf canonical complete product factor2 = 12 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, S648 S650 · calculationBefore upf canonical complete product factor3 := mul(12, 5) After upf canonical complete product factor3 = 60 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, S649 S651 · calculationBefore upf canonical complete product factor4 := mul(60, 7) After upf canonical complete product factor4 = 420 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, S650 S652 · impliesBefore exhausted trial divisors below6;terminal remaining=7;d*d>remaining;ORIGINAL n=420 After complete increasing prime multiset=[2, 2, 3, 5, 7];full product=420;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, S651 S653 · impliesBefore Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor420:input=420;COMPLETE prime multiset=[2, 2, 3, 5, 7];distinct primes=[2, 3, 5, 7] 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, S652 S654 · calculationBefore Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor420 := div(420, 420) After Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor420 = 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, S653 S655 · impliesBefore Mobius inversion ORIGINAL inverse F(n/420):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, S654 S656 · calculationBefore Mobius inversion ORIGINAL inverse F(n/420) f(1) ORIGINAL a times divisor := mul(19, 1) After Mobius inversion ORIGINAL inverse F(n/420) f(1) ORIGINAL a times divisor = 19 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S655 S657 · calculationBefore Mobius inversion ORIGINAL inverse F(n/420) f(1) plus ORIGINAL b := sum(19, 1) After Mobius inversion ORIGINAL inverse F(n/420) f(1) plus ORIGINAL b = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S656 S658 · calculationBefore Mobius inversion ORIGINAL inverse F(n/420) full forward divisor sum := sum(20) After Mobius inversion ORIGINAL inverse F(n/420) full forward divisor sum = 20 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S657 S659 · calculationBefore Mobius inversion ORIGINAL inverse mu(420) times SAME F(n/420) := mul(0, 20) After Mobius inversion ORIGINAL inverse mu(420) times SAME F(n/420) = 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, S658 S660 · calculationBefore Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(25560, -10960, -6396, 0, -4268, 2744, -3204, 1832, 0, 1376, 1070, 0, 804, 0, -460, 538, -346, 0, -232, 0, -136, 0, 59, 0) After Mobius inversion ORIGINAL inverse complete inverse-term sum = 7981 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S659 S661 · calculationBefore Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(19, 420) After Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 7980 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S660 S662 · calculationBefore Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(7980, 1) After Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 7981 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S661 S663 · impliesBefore Mobius inversion ORIGINAL inverse:actual n=420;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 420, 25560, 25560], [2, -1, 210, 10960, -10960], [3, -1, 140, 6396, -6396], [4, 0, 105, 3656, 0], [5, -1, 84, 4268, -4268], [6, 1, 70, 2744, 2744], [7, -1, 60, 3204, -3204], [10, 1, 42, 1832, 1832], [12, 0, 35, 916, 0], [14, 1, 30, 1376, 1376], [15, 1, 28, 1070, 1070], [20, 0, 21, 612, 0], [21, 1, 20, 804, 804], [28, 0, 15, 460, 0], [30, -1, 14, 460, -460], [35, 1, 12, 538, 538], [42, -1, 10, 346, -346], [60, 0, 7, 154, 0], [70, -1, 6, 232, -232], [84, 0, 5, 116, 0], [105, -1, 4, 136, -136], [140, 0, 3, 78, 0], [210, 1, 2, 59, 59], [420, 0, 1, 20, 0]] After complete reconstructed=7981;direct f(n)=7981;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, S662 S664 · calculationBefore Mobius inversion ORIGINAL omitted-divisor partial sum := sum(-10960, -6396, 0, -4268, 2744, -3204, 1832, 0, 1376, 1070, 0, 804, 0, -460, 538, -346, 0, -232, 0, -136, 0, 59, 0) After Mobius inversion ORIGINAL omitted-divisor partial sum = -17579 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S663 S665 · calculationBefore Mobius inversion complete minus partial := sub(7981, -17579) After Mobius inversion complete minus partial = 25560 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S664 S666 · impliesBefore ORIGINAL full divisors=[1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420] After complete=7981=f(n)=7981;partial=-17579;lost term=25560!=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, S665 ## Reverse checks - S666 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S665 · premise_and_result_audit — Recompute Mobius inversion complete minus partial with exact rational arithmetic, not rounded decimals. - S664 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL omitted-divisor partial sum with exact rational arithmetic, not rounded decimals. - S663 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S662 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S661 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals. - S660 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals. - S659 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(420) times SAME F(n/420) with exact rational arithmetic, not rounded decimals. - S658 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/420) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S657 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/420) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S656 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/420) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S655 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S654 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor420 with exact rational arithmetic, not rounded decimals. - S653 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S652 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S651 · premise_and_result_audit — Recompute upf canonical complete product factor4 with exact rational arithmetic, not rounded decimals. - S650 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals. - S649 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S648 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S647 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S646 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S645 · premise_and_result_audit — Recompute upf extract divisor5 occurrence3 with exact rational arithmetic, not rounded decimals. - S644 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S643 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S642 · premise_and_result_audit — Recompute upf extract divisor3 occurrence2 with exact rational arithmetic, not rounded decimals. - S641 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S640 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals. - S639 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S638 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S637 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S636 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(210) times SAME F(n/210) with exact rational arithmetic, not rounded decimals. - S635 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/210) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S634 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/210) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S633 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/210) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S632 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/210) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S631 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/210) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S630 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S629 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor210 with exact rational arithmetic, not rounded decimals. - S628 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S627 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S626 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals. - S625 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S624 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S623 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S622 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S621 · premise_and_result_audit — Recompute upf extract divisor5 occurrence2 with exact rational arithmetic, not rounded decimals. - S620 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S619 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S618 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals. - S617 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S616 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S615 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S614 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S613 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(140) times SAME F(n/140) with exact rational arithmetic, not rounded decimals. - S612 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/140) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S611 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/140) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S610 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/140) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S609 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/140) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S608 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/140) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S607 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S606 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor140 with exact rational arithmetic, not rounded decimals. - S605 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S604 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S603 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals. - S602 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S601 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S600 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S599 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S598 · premise_and_result_audit — Recompute upf extract divisor5 occurrence2 with exact rational arithmetic, not rounded decimals. - S597 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S596 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S595 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S594 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals. - S593 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S592 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S591 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S590 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(105) times SAME F(n/105) with exact rational arithmetic, not rounded decimals. - S589 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S588 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S587 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S586 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S585 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S584 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S583 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S582 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S581 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor105 with exact rational arithmetic, not rounded decimals. - S580 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S579 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S578 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S577 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S576 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S575 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S574 · premise_and_result_audit — Recompute upf extract divisor5 occurrence1 with exact rational arithmetic, not rounded decimals. - S573 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S572 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S571 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals. - S570 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S569 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S568 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S567 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(84) times SAME F(n/84) with exact rational arithmetic, not rounded decimals. - S566 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/84) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S565 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/84) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S564 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/84) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S563 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/84) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S562 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/84) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S561 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S560 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor84 with exact rational arithmetic, not rounded decimals. - S559 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S558 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S557 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals. - S556 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S555 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S554 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S553 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S552 · premise_and_result_audit — Recompute upf extract divisor3 occurrence2 with exact rational arithmetic, not rounded decimals. - S551 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S550 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals. - S549 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S548 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S547 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S546 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(70) times SAME F(n/70) with exact rational arithmetic, not rounded decimals. - S545 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S544 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S543 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S542 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S541 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S540 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S539 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S538 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S537 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S536 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S535 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor70 with exact rational arithmetic, not rounded decimals. - S534 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S533 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S532 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S531 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S530 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S529 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S528 · premise_and_result_audit — Recompute upf extract divisor5 occurrence1 with exact rational arithmetic, not rounded decimals. - S527 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S526 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S525 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S524 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S523 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S522 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S521 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(60) times SAME F(n/60) with exact rational arithmetic, not rounded decimals. - S520 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/60) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S519 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/60) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S518 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/60) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S517 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/60) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S516 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/60) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S515 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S514 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor60 with exact rational arithmetic, not rounded decimals. - S513 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S512 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S511 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals. - S510 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S509 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S508 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S507 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S506 · premise_and_result_audit — Recompute upf extract divisor3 occurrence2 with exact rational arithmetic, not rounded decimals. - S505 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S504 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals. - S503 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S502 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S501 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S500 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42) with exact rational arithmetic, not rounded decimals. - S499 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S498 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S497 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S496 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S495 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S494 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S493 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S492 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S491 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S490 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S489 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor42 with exact rational arithmetic, not rounded decimals. - S488 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S487 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S486 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S485 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S484 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S483 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S482 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals. - S481 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S480 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S479 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S478 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S477 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(35) times SAME F(n/35) with exact rational arithmetic, not rounded decimals. - S476 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S475 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(12) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S474 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(12) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S473 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S472 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S471 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S470 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S469 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S468 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S467 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S466 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S465 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S464 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S463 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S462 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor35 with exact rational arithmetic, not rounded decimals. - S461 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S460 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S459 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S458 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S457 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S456 · premise_and_result_audit — Recompute upf extract divisor5 occurrence0 with exact rational arithmetic, not rounded decimals. - S455 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S454 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S453 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S452 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S451 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S450 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(30) times SAME F(n/30) with exact rational arithmetic, not rounded decimals. - S449 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S448 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S447 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S446 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S445 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S444 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S443 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S442 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S441 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S440 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S439 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor30 with exact rational arithmetic, not rounded decimals. - S438 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S437 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S436 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S435 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S434 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S433 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S432 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals. - S431 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S430 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S429 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S428 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S427 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(28) times SAME F(n/28) with exact rational arithmetic, not rounded decimals. - S426 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/28) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S425 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/28) f(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S424 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/28) f(15) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S423 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/28) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S422 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/28) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S421 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/28) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S420 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/28) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S419 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/28) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S418 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/28) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S417 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S416 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor28 with exact rational arithmetic, not rounded decimals. - S415 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S414 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S413 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S412 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S411 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S410 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S409 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals. - S408 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S407 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 mu(21) times SAME F(n/21) with exact rational arithmetic, not rounded decimals. - S404 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S403 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(20) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S402 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(20) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S401 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S400 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S399 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S398 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S397 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S396 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S395 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S394 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S393 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S392 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S391 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S390 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S387 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S386 · premise_and_result_audit — Recompute upf canonical complete product factor0 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 divisor3 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S380 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(20) times SAME F(n/20) with exact rational arithmetic, not rounded decimals. - S379 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S378 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S377 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S376 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S375 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S374 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S373 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S372 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S371 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S370 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S369 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor20 with exact rational arithmetic, not rounded decimals. - S368 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S365 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S364 · premise_and_result_audit — Recompute upf canonical complete product factor0 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 — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals. - S361 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S360 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S359 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S358 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(15) times SAME F(n/15) with exact rational arithmetic, not rounded decimals. - S357 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S356 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(28) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S355 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(28) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S354 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S353 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S352 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S351 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S350 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S349 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S348 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S347 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S346 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S345 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S344 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S343 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor15 with exact rational arithmetic, not rounded decimals. - S342 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S339 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S338 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S337 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals. - S336 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S335 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S334 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S333 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) with exact rational arithmetic, not rounded decimals. - S332 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S331 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(30) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S330 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(30) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S329 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S328 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(15) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S327 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S326 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S325 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S324 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S323 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S322 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S321 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S320 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S319 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S318 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S317 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S316 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S315 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S314 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 with exact rational arithmetic, not rounded decimals. - S313 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S312 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S311 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S310 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S309 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S308 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S307 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Recompute Mobius inversion ORIGINAL inverse mu(12) times SAME F(n/12) with exact rational arithmetic, not rounded decimals. - S304 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S303 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(35) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S302 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(35) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S301 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S300 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S299 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S298 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S297 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S296 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S295 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S294 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12 with exact rational arithmetic, not rounded decimals. - S293 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S292 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S291 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S290 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S289 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S288 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S287 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals. - S286 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S285 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S284 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S283 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) with exact rational arithmetic, not rounded decimals. - S282 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S281 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(42) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S280 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(42) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S279 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S278 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S277 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S276 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S275 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S274 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S273 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S272 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S271 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S270 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S269 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S268 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S267 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S266 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S265 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S264 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 with exact rational arithmetic, not rounded decimals. - S263 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S262 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S261 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S260 · premise_and_result_audit — Recompute upf canonical complete product factor0 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 upf extract divisor2 occurrence0 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 Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) with exact rational arithmetic, not rounded decimals. - S254 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S253 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(60) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S252 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(60) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S251 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(30) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S250 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(30) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S249 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(20) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S248 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(20) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S247 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S246 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(15) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S245 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(12) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S244 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(12) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S243 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S242 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S241 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S240 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S239 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S238 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S237 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S236 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S235 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S234 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S233 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S232 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S231 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S230 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S229 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S228 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 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 — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S224 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S223 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S222 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S221 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) with exact rational arithmetic, not rounded decimals. - S220 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S219 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(70) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S218 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(70) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S217 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(35) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S216 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(35) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S215 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S214 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S213 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S212 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S211 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S210 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S209 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S208 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S207 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S206 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S205 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S204 · 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. - S203 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S202 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 with exact rational arithmetic, not rounded decimals. - S201 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S200 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S199 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S198 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S197 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S196 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals. - S195 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S194 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S193 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) with exact rational arithmetic, not rounded decimals. - S192 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S191 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(84) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S190 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(84) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S189 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(42) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S188 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(42) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S187 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(28) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S186 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(28) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S185 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S184 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S183 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S182 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S181 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(12) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S180 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(12) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S179 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S178 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S177 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S176 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S175 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S174 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S173 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S172 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S171 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S170 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S169 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S168 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S167 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S166 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 with exact rational arithmetic, not rounded decimals. - S165 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S164 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S163 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S162 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S159 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) with exact rational arithmetic, not rounded decimals. - S158 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S157 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(105) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S156 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(105) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S155 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(35) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S154 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(35) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S153 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S152 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S151 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S150 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(15) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S149 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S148 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S147 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S146 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S145 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S144 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S143 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S142 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S140 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 with exact rational arithmetic, not rounded decimals. - S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S137 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S136 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S135 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S134 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals. - S133 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 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 Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) with exact rational arithmetic, not rounded decimals. - S129 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S128 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(140) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S127 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(140) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S126 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(70) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S125 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(70) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S124 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(35) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S123 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(35) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S122 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(28) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S121 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(28) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S120 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(20) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S119 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(20) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S118 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S117 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S116 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S115 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S114 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S113 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S112 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S111 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S110 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S109 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S108 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S107 · 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. - S106 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S105 · 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. - S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S103 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S100 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S097 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals. - S096 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S095 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(210) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S094 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(210) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S093 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(105) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S092 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(105) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S091 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(70) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S090 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(70) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S089 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(42) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S088 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(42) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S087 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(35) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S086 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(35) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S085 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(30) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S084 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(30) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S083 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S082 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S081 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S080 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(15) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S079 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S078 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S077 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S075 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S074 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S073 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S070 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S068 · 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. - S067 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S065 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S064 · 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. - S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S062 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S059 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S056 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(420) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(420) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(210) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(210) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(140) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(140) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(105) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(105) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(84) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(84) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(70) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(70) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(60) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(60) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(42) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(42) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(35) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(35) 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(30) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(30) 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(28) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(28) 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(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(21) 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(20) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(20) 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(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(15) 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(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(14) 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(12) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(12) 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(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(10) 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(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(7) 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(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S017 · 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. - S016 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(5) 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(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(4) 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 (420, [1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420], 1, [2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420], 7981, -17579, 7981, 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: 7806e4ff3d8277bd94ac33a0f0e8730891b188bfb72572ca8aafed07af72349e - candidate record SHA-256: 946c79b18fc168292d79eea689bbb5d3f12350e923c73cf96dc22efa7aa05921 - 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-0094.json)