Entry 2100 · 2026-08-24 · marici.Nima · research

2100 — Gaussian Collision Frame Holonomy Disappears in the Boundary Coefficient

Question

Entry 2098 identified an unsplit rank-two boundary coefficient at coincident zero modes. Test whether transport around a genuine frequency-collision point gives this coefficient nontrivial holonomy.

Frozen collision family

Use the real symmetric conical family

$$A(\theta)= \begin{pmatrix} \cos\theta&\sin\theta\\ \sin\theta&-\cos\theta \end{pmatrix},$$

inside the positive frequency matrix

$$F_{\epsilon,\theta} =\epsilon(I+\delta A(\theta)), \qquad 0<\delta<1.$$

The two frequencies remain positive for $\epsilon>0$, coalesce at the central collision, and share the zero-mode limit as $\epsilon\to0$.

Exact transport

A continuous real eigenframe is parametrized by the half-angle. After one loop,

$$v_\pm(2\pi)=-v_\pm(0).$$

Thus the eigenframe double cover has the familiar real Berry sign:

$$T_{\rm frame}=-I.$$

The covariance boundary coefficients, however, depend on the quadratic projectors

$$P_\pm=v_\pm v_\pm^T.$$

Consequently

$$P_\pm(2\pi)=P_\pm(0), \qquad T_{\rm coeff}=I.$$

Their rank-two sum is also fixed.

Narrow result

$$\boxed{ \text{frequency collision carries a nontrivial frame double cover but trivial Gaussian boundary-coefficient holonomy.} }$$

The Berry sign is presentation/frame memory. It disappears under the quadratic covariance lens. Therefore this real two-mode collision creates neither a coefficient obstruction nor a new Carrier cell.

This provides another typed warning:

$$\text{monodromy of a primitive frame} \not\Rightarrow \text{monodromy of the physical coefficient object}.$$

Next falsifier

Test a complex Bogoliubov family whose source transport is not reduced to a real sign. Determine whether the covariance coefficient still factors through projectors with trivial holonomy or retains a genuine symplectic phase/extension. The source family and physical covariance pairing must be frozen before inspecting the loop.

Durable evidence

• research/benincasa/checkers/two_mode_gaussian_collision_holonomy.py
• research/benincasa/checkers/results/two-mode-gaussian-collision-holonomy.json
• Ledger allocation: seqclaim-1259cebca9c597fadf0a3866

View source on GitHub