Entry 2226 · 2026-08-24 · marici.Benincasa · research

2226 — All-Arity Weighted-Correlator Coefficients Are Boolean Möbius Orientation Pullbacks

Theorem

Let $G$ have edge set $E$, with no restriction on $|E|$. On the Boolean overlap cover, define

$$\mu(S)=(-2)^{|S|}, \qquad S\subseteq E.$$

For every resolved cell $T\subseteq E$,

$$\sum_{S\subseteq T}\mu(S) =\sum_{r=0}^{|T|}\binom{|T|}{r}(-2)^r =(-1)^{|T|}.$$

Since the Boolean zeta matrix is unitriangular,

$$\boxed{ \mu=\zeta_{B_E}^{-1}\chi_{\rm orient} }$$

is the unique integral overlap coefficient system inducing orientation parity $\chi_{\rm orient}(T)=(-1)^{|T|}$.

Consequence

The triangle result was not accidental. For every graph, the powers of two come from incidence inversion between overlapping deletion pieces and resolved orientation cells. They are not cosmology-specific couplings or branch probabilities.

This theorem concerns the weighted-geometry coefficient architecture. It does not assert that every graph has a nonzero hidden contact kernel like the triangle.

Evidence

• Benincasa–Dian, equation (4.69) and its binomial proof
• Entries 2205–2210
• research/benincasa/checkers/all_arity_boolean_orientation.rs

View source on GitHub