# Question 22176: Every Divisor Row Recovers Its Original Arithmetic Value [Return to catalogue page 111](https://ocme.evemisslab.com/en/questions/page/111/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations/#q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0051) 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. R0043-22176 · q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0051 ## Original question and givens Use positive integer n=1, exact integer arithmetic function f(k)=a*k+b with a=1, b=-12. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary). - G001 — Use positive integer n=1, exact integer arithmetic function f(k)=a*k+b with a=1, b=-12. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-mobius-inversion-finite-divisors](https://ocme.evemisslab.com/en/mko/mko-mobius-inversion-finite-divisors/) ## Step-by-step derivation S001 · impliesBefore source operation=mobinv-roundtrip-table;original snapshot and mathematics retained After explicit ordered source output fields=[n, roundtrip_rows, all_equal, boundary] Conditions: none Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore Mobius inversion ORIGINAL roundtrip:actual positive input=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, S001 S003 · impliesBefore Mobius roundtrip m=1 forward:actual positive input=1;check complete integer range1..1 After ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n Conditions: none Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction. Step identifiers and dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S002 S004 · calculationBefore Mobius roundtrip m=1 forward f(1) ORIGINAL a times divisor := mul(1, 1) After Mobius roundtrip m=1 forward f(1) ORIGINAL a times divisor = 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, S003 S005 · calculationBefore Mobius roundtrip m=1 forward f(1) plus ORIGINAL b := sum(1, -12) After Mobius roundtrip m=1 forward f(1) plus ORIGINAL b = -11 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Mobius roundtrip m=1 forward full forward divisor sum := sum(-11) After Mobius roundtrip m=1 forward full forward divisor sum = -11 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore Mobius roundtrip m=1 inverse:actual positive input=1;check complete integer range1..1 After ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n Conditions: none Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction. Step identifiers and dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S006 S008 · impliesBefore Mobius roundtrip m=1 inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1 After complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search Conditions: none Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search. Step identifiers and dependenciesRule: mobius_complete_UPF_input_role · Uses: G001, S007 S009 · impliesBefore Mobius roundtrip m=1 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[] After mu=1;empty product case=Yes;repeated prime=No Conditions: none The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities. Step identifiers and dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S008 S010 · calculationBefore Mobius roundtrip m=1 inverse n divided by ORIGINAL divisor1 := div(1, 1) After Mobius roundtrip m=1 inverse n divided by ORIGINAL divisor1 = 1 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore Mobius roundtrip m=1 inverse F(n/1):actual positive input=1;check complete integer range1..1 After ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n Conditions: none Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction. Step identifiers and dependenciesRule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S010 S012 · calculationBefore Mobius roundtrip m=1 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(1, 1) After Mobius roundtrip m=1 inverse F(n/1) f(1) ORIGINAL a times divisor = 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, S011 S013 · calculationBefore Mobius roundtrip m=1 inverse F(n/1) f(1) plus ORIGINAL b := sum(1, -12) After Mobius roundtrip m=1 inverse F(n/1) f(1) plus ORIGINAL b = -11 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Mobius roundtrip m=1 inverse F(n/1) full forward divisor sum := sum(-11) After Mobius roundtrip m=1 inverse F(n/1) full forward divisor sum = -11 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Mobius roundtrip m=1 inverse mu(1) times SAME F(n/1) := mul(1, -11) After Mobius roundtrip m=1 inverse mu(1) times SAME F(n/1) = -11 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Mobius roundtrip m=1 inverse complete inverse-term sum := sum(-11) After Mobius roundtrip m=1 inverse complete inverse-term sum = -11 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Mobius roundtrip m=1 inverse direct ORIGINAL a times n := mul(1, 1) After Mobius roundtrip m=1 inverse direct ORIGINAL a times n = 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, S016 S018 · calculationBefore Mobius roundtrip m=1 inverse direct plus ORIGINAL b := sum(1, -12) After Mobius roundtrip m=1 inverse direct plus ORIGINAL b = -11 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore Mobius roundtrip m=1 inverse:actual n=1;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 1, -11, -11]] After complete reconstructed=-11;direct f(n)=-11;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n) Conditions: none Swap the finite sums over d*e|n.The inner Mobius sum is1 for n/e=1 and0 otherwise by pairing distinct-prime subsets,leaving only e=n.For n1,the d1 term and mu1=1 remain;do not mix indices from the alternate equivalent convention. Step identifiers and dependenciesRule: mobinv_general_finite_double_sum_and_Mobius_cancellation · Uses: G001, S018 ## Reverse checks - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse F(n/1) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse n divided by ORIGINAL divisor1 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute Mobius roundtrip m=1 forward full forward divisor sum with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Mobius roundtrip m=1 forward f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Mobius roundtrip m=1 forward f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (1, [[1, -11, -11, -11]], Yes, mu_one_boundary_preserved) 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: 8acaffb6a85653d72785f3fb0aaf56cb9314eadd2e84186179c6f5239ab3347b - candidate record SHA-256: f4e5ce57ec713ecd48af2098d193b3a000b02263d60c32d1150479f732500472 - 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-0051.json)