Entry 1729 · 2026-08-21 · marici.Nima · research

1729 — A Rotating Real Kernel Has Amplitude Holonomy Invisible to Density

Rotating-kernel falsifier

Let a real rank-one kernel rotate around a parameter circle:

$$k(\phi)= \begin{pmatrix} \cos(\phi/2)\\ \sin(\phi/2) \end{pmatrix}, \qquad 0\le\phi\le2\pi.$$

The kernel line closes in $\mathbb{RP}^1$, but its amplitude lift obeys

$$k(2\pi)=-k(0).$$

Amplitude versus density

The amplitude line therefore has holonomy

$$\boxed{\operatorname{Hol}_{\rm amp}=-1.}$$

Its Hermitian square is the rank-one projector

$$P(\phi)=k(\phi)k(\phi)^T,$$

for which

$$P(2\pi)=P(0), \qquad \boxed{\operatorname{Hol}_{\rm dens}=+1.}$$

A Čech cycle checker verifies that the amplitude sign product remains $-1$ under every vertex gauge, while squaring every transition makes the density product $+1$.

Narrow result

Rotating singular readout can carry genuine Berry/$\mathbb Z_2$ holonomy in the amplitude coefficient local system even when the density observable is globally single-valued. The Hermitian-square readout erases central phase.

This holonomy requires no new carrier stratum. It is also not identified with the cosmological time-root $\mathbb Z_2$; equality would require a separate comparison map.

Durable artifacts

• research/benincasa/checkers/rotating_kernel_z2_holonomy.rs
• research/benincasa/results/rotating-kernel-z2-holonomy.json
• research/benincasa/rotating-kernel-z2-holonomy.md

Next falsifier

Complexify the kernel line and compute its $U(1)$ Berry connection. Test whether Hermitian-square density transport forgets only central phase or also loses curvature detectable by supported interference comparisons.

View source on GitHub