# Mobius inversion on finite divisor sets

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

If F(n) sums f(d) over every positive divisor of n, Mobius weights exactly recover f(n).

mko-mobius-inversion-finite-divisors · theorem · v0.10.0



## Mathematical statement

For positive integer n, F(n)=sum over d|n of f(d) implies f(n)=sum over d|n of mu(d)*F(n/d). The equivalent indexing mu(n/d)*F(d) is valid but must not be mixed with the first convention. Enumerate every positive divisor and derive mu from complete prime multiplicities. Retain n=1 and mu(1)=1. This is convolution on the divisor poset, not a sum over all d≤n.





## Representative formula (exact TeX)

f(n)=I(F,n)





## Assumptions


- n is a positive safe integer and divisor enumeration is complete.
- Question families use the exact integer arithmetic function f(k)=a*k+b.





## Symbol scope


- f — original arithmetic function
- n — positive integer
- I — Mobius inversion operator
- F — divisor-sum transform





## Intuition, definition, and misconceptions

The forward transform uses all divisors.

Keep d and n/d consistently aligned in the inverse.

The roundtrip includes n=1.

Prime/proper divisors alone are insufficient; d|n does not mean d≤n.





## Adopted derivation guide

[Divisor-poset inversion of an affine function](https://ocme.evemisslab.com/en/derivations/derivation-mko-mobius-inversion-finite-divisors-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Mobius function](https://ocme.evemisslab.com/en/mko/mko-mobius-function/) — Use the mu weights.
- [Natural-number divisibility](https://ocme.evemisslab.com/en/mko/mko-natural-divisibility/) — Enumerate the complete divisor set.
- [Basic counting principles](https://ocme.evemisslab.com/en/mko/mko-counting-principle/) — Regroup the finite sums.





## Verification boundary

Zero, negative inputs and incomplete factorization/divisor lists are outside the positive domain of this slice.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add the Legendre symbol and Euler criterion.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/mobius-inversion-finite-divisors.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-mobius-inversion-finite-divisors/index.md)