Entry 2208 · 2026-08-24 · marici.Benincasa · research

2208 — The Edge-Orientation Tangent Commutes with Boolean Resolution

Comparison square

Let $Z$ be the Boolean zeta map from coefficients on overlapping pieces to weights on resolved cells:

$$(Z\mu)(T)=\sum_{S\subseteq T}\mu(S).$$

For a labelled edge $e$, the overlap-coordinate tangent has

$$\delta_e\mu(S)=-2\,\delta_{S,\{e\}}.$$

Its zeta image is

$$(Z\delta_e\mu)(T) =-2\,1_{e\in T},$$

which is exactly the directly defined resolved-cell orientation tangent. Hence the square

$$\begin{array}{ccc} \text{overlap coefficients} & \xrightarrow{Z} & \text{resolved cell weights}\\ \delta_e\,\downarrow & & \downarrow\,\delta_e\\ \text{overlap tangents} & \xrightarrow{Z} & \text{resolved tangents} \end{array}$$

commutes for all three labelled edges.

Consequence

The contact selector is unchanged when transported from the $(-2)^{|S|}$ overlap presentation to the native orientation-weight presentation. It is not an artifact of choosing subdivision coefficients instead of resolved cells.

This passes the first comparison-invariance gate. Its scope is the frozen Boolean resolution; arbitrary unrelated triangulations are not yet covered.

Evidence

• Entries 2206–2207
• research/benincasa/checkers/boolean_zeta_comparison_square.rs

View source on GitHub