Entry 2101 · 2026-08-24 · marici.Nima · research

2101 — Complex Bogoliubov Holonomy Survives the State Lens but Not the Covariance Lens

Hostile question

Entry 2100 found that a real collision Berry sign disappears from Gaussian covariance coefficients. Test whether complex Bogoliubov transport is erased intrinsically or only by that particular readout.

Frozen loop

Take the normalized squeezed-vacuum family

$$|\zeta\rangle, \qquad \zeta=re^{i\theta}, \qquad 0\le\theta\le2\pi,$$

at the exact rational hyperbolic point

$$\cosh r=\frac53, \qquad \sinh r=\frac43, \qquad \tanh r=\frac45.$$

No support divisor changes around the loop.

State-line transport

In the standard even-Fock expansion, the phase exponent counts squeezed pairs. Hence

$$i\langle\zeta|\partial_\theta\zeta\rangle =-\frac12\sinh^2r =-\frac89.$$

After one turn the Berry phase is

$$\gamma =\int_0^{2\pi}i\langle\zeta|\partial_\theta\zeta\rangle d\theta =-\frac{16\pi}{9},$$

and therefore

$$\boxed{ T_{\rm state}=e^{-16\pi i/9}\ne1. }$$

Covariance transport

The real covariance depends on the squeezing angle only through its quadratic rotation. It returns exactly after $2\pi$, remains pure, and has

$$T_{\rm cov}=I.$$

At the chosen base point,

$$V(0)= \begin{pmatrix} 1/18&0\\ 0&9/2 \end{pmatrix}, \qquad \det V=\frac14.$$

Narrow result

$$\boxed{ \text{the same Carrier loop has trivial covariance transport but nontrivial normalized-state-line transport.} }$$

Complex Bogoliubov holonomy is therefore not universally erased. Whether it survives depends on the coefficient/readout layer:

$$\text{state line}\longrightarrow\text{nontrivial phase}, \qquad \text{covariance lens}\longrightarrow\text{identity}.$$

This strengthens H2. The Carrier supplies the loop, while sector- and layer-specific coefficient objects decide which transport is observable. No new Carrier incidence is implicated.

Next falsifier

Test an interference-sensitive physical pairing of the state line. A global Berry phase of one isolated branch is not itself observable; a source-derived relative phase between two Bogoliubov histories may be. Freeze a two-branch interferometric comparison and determine whether the holonomy survives the physical quotient or again becomes presentation data.

Durable evidence

• research/benincasa/checkers/squeezed_vacuum_berry_vs_covariance.py
• research/benincasa/checkers/results/squeezed-vacuum-berry-vs-covariance.json
• Ledger allocation: seqclaim-b4d4e5f616fd402a28875b49

View source on GitHub