E0003 proposition ✓ formal notation E0001

Binomial Möbius inversion

The Boolean Möbius inversion E0002 lifts to multi-index grids: the binomial zeta and mu are inverse bijections on grid-valued maps, with the Boolean case recovered as the height-one restriction.

(1) Proposition. Let \(G\) be an abelian group and \(S\) a finite set. For a map \(A: \IN_0^S \to G\) and \(\alpha \in \IN_0^S\), let \(A_\alpha = A \circ \nu: \KP(S(\alpha)) \to G\) be the pullback along the fiber measure of the Boolean realization. Set

\[ \zeta(A; \alpha) := \zeta(A_\alpha; S(\alpha)) = \sum_{\beta \leq \alpha} \frac{(\alpha)_\beta}{\beta!}\, A(\beta), \qquad \mu(A; \alpha) := \mu(A_\alpha; S(\alpha)) = \sum_{\beta \leq \alpha} (-1)^{\mathrm{wt}(\alpha-\beta)}\, \frac{(\alpha)_\beta}{\beta!}\, A(\beta). \]

Then \(\zeta\) and \(\mu\) define inverse bijections on \(\mathrm{Map}(\IN_0^S, G)\).

Proof.

The number of subsets \(T \subseteq S(\alpha)\) with \(\nu(T) = \beta\) is \(\frac{(\alpha)_\beta}{\beta!}\), giving the explicit sums.

For the inversion, observe that the cube \(A_\alpha = A \circ \nu\) satisfies \((\zeta A)_\alpha = \zeta(A_\alpha)\): for \(T \subseteq S(\alpha)\) with \(\nu(T) = \beta\), the restriction of \(A_\alpha\) to \(\KP(T)\) is the realization of \(A_\beta\) on \(T\), hence \(\zeta(A_\alpha; T) = \sum_{R \subseteq T} A(\nu(R)) = \zeta(A; \beta)\). Applying Boolean Möbius inversion E0002 to \(A_\alpha\) on \(\KP(S(\alpha))\):

\[ \mu(\zeta A; \alpha) = \mu((\zeta A)_\alpha; S(\alpha)) = \mu(\zeta(A_\alpha); S(\alpha)) = A_\alpha(S(\alpha)) = A(\alpha), \]

and symmetrically for \(\zeta\mu\).

Alternatively: exchange summation order, use \(\binom{\alpha}{\beta}\binom{\beta}{\gamma} = \binom{\alpha}{\gamma}\binom{\alpha-\gamma}{\beta-\gamma}\) to decouple, and apply the multi-index binomial theorem \((1-1)^\alpha = \sum_{\beta \leq \alpha} (-1)^{\mathrm{wt}(\beta)} \binom{\alpha}{\beta} = \delta_{\alpha,0}\).

(2) Validation (AI review, 2026-07-19, claude-fable-5, pass). Counting coefficient \(\binom{\alpha}{\beta}\), the pullback identity \((\zeta A)_\alpha = \zeta(A_\alpha)\) (fiberwise bijection \(T \cong S(\beta)\) over \(S\)), the symmetric direction, and edge cases \(\alpha = 0\), height-one restriction, numeric spot check \(\alpha = 2\). Added the missing hypothesis that \(S\) is finite. No other issues found.

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