E0002 proposition ✓ numeric ✓ formal notation E0001

Boolean Möbius inversion

(1) Proposition. Let \(G\) be an abelian group. For a cube \(a: \KP(k) \to G\), set

\[ \zeta(a; S) := \sum_{T \subseteq S} a(T), \qquad \mu(a; S) := \sum_{T \subseteq S} (-1)^{|S|-|T|}\, a(T). \]

Then \(\zeta\) and \(\mu\) define inverse bijections on \(\mathrm{Map}(\KP(k), G)\).

Proof. Exchange summation order and use \((1-1)^{|S|} = \sum_{T \subseteq S} (-1)^{|S|-|T|} = \delta_{S,\emptyset}\):

\[ \mu(\zeta a; S) = \sum_{T \subseteq S} (-1)^{|S|-|T|} \sum_{R \subseteq T} a(R) = \sum_{R \subseteq S} a(R) \sum_{R \subseteq T \subseteq S} (-1)^{|S|-|T|} = \sum_{R \subseteq S} a(R)\, \delta_{S \setminus R, \emptyset} = a(S), \]

and symmetrically \(\zeta(\mu a; S) = a(S)\).

(2) Validation (AI review, 2026-07-19, claude-fable-5, pass). Summation exchange and \(\delta\)-collapse verified via \(T = R \cup U\), \(U \subseteq S \setminus R\); symmetric direction checked separately; edge cases \(k = 0\), \(S = \emptyset\) (identity maps), signs as \(\IZ\)-action on \(G\). No issues found.

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