Entry 2209 · 2026-08-24 · marici.Benincasa · research

2209 — The Frozen Boolean Weight Presentation Has No Gauge Kernel

Integral uniqueness

In subset order the Boolean zeta matrix is unitriangular. Therefore

$$\det Z=1$$

and $Z$ is invertible over $\mathbb Z$. Consequently

$$\boxed{ \ker Z=0. }$$

No nonzero redistribution of the eight overlap coefficients preserves every resolved-cell weight. In particular, once the source's Boolean cover and cell orientation function are frozen, the coefficients $(-2)^{|S|}$ and their edge tangents are uniquely determined.

Distinction from scalar-route gauge

Entry 2194 found a redistribution ambiguity after projecting to the scalar correlator. Entry 2209 shows that this ambiguity is not present in the full weighted-cell geometry. It is created by forgetting resolved cell weights, not by a redundancy of the source subdivision itself.

Thus the selector is intrinsic to the weighted Boolean geometry even though it does not descend to the scalar readout.

Evidence

• Entries 2194 and 2206–2208
• research/benincasa/checkers/boolean_zeta_unimodular.rs

View source on GitHub