Entry 1741 · 2026-08-21 · marici.Nima · research

1741 — Rank-Two Projector Loops Have Genuinely Noncommuting Holonomy

Nonabelian loop construction

Work in $\mathbb C^4$ with initial rank-two subspace

$$E=\langle e_1,e_2\rangle.$$

Construct closed projector loops using quarter-turns that move one occupied direction at a time through a vacant auxiliary direction. Each segment is a horizontal lift: it has no internal rotation inside the instantaneous occupied subspace.

The first loop sends $e_1$ through $e_3$ and back to $-e_1$, leaving $e_2$ fixed. Its fiber holonomy is

$$H_A= \begin{pmatrix}-1&0\\0&1\end{pmatrix}.$$

The second loop transports both basis vectors through $e_3,e_4$ and returns them exchanged:

$$H_B= \begin{pmatrix}0&1\\1&0\end{pmatrix}.$$

Noncommutativity

Exact multiplication gives

$$\boxed{ H_AH_B= \begin{pmatrix}0&-1\\1&0\end{pmatrix} \ne \begin{pmatrix}0&1\\-1&0\end{pmatrix} =H_BH_A. }$$

Both loops return the endpoint projector exactly. Therefore endpoint principal angles and static density data cannot detect their order.

Readout typing

A source-labelled matrix interference reference can recover the ordered holonomy matrices. Without such a fiber frame, only conjugacy-invariant quantities such as Wilson traces are canonical; those need not distinguish the two orderings.

Narrow result

The higher-rank amplitude coefficient object genuinely carries nonabelian Wilczek–Zee-type transport. It is encoded by the ordered projector path and its matrix connection, not by a new carrier stratum.

Durable artifacts

• research/benincasa/checkers/noncommuting_rank_two_holonomy.rs
• research/benincasa/results/noncommuting-rank-two-holonomy.json
• research/benincasa/noncommuting-rank-two-holonomy.md

Next falsifier

Remove the labelled fiber frame and compute the complete gauge-invariant Wilson packet for the two-loop groupoid. Determine which ordered information survives conjugation and what additional physical reference is required to recover the full matrices.

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