Entry 1740 · 2026-08-21 · marici.Nima · research

1740 — Endpoint Principal Angles Do Not Determine Rank-Two Projector Holonomy

Loop falsifier

For $n\in S^2$, let $P_n$ project onto the rank-two plane $n^\perp$. Transport this plane around the geodesic octant triangle

$$e_1\longrightarrow e_2\longrightarrow e_3\longrightarrow e_1$$

using the source-defined minimal quarter-turn rotations $R_{12},R_{23},R_{31}$.

Exact ordered transport

The ordered product

$$H=R_{31}R_{23}R_{12}$$

fixes the normal $e_1$, hence returns the endpoint projector exactly:

$$HP_{e_1}H^T=P_{e_1}.$$

Its action on the endpoint fiber $\langle e_2,e_3\rangle$ is

$$\boxed{ H_{\rm fib}= \begin{pmatrix} 0&-1\\1&0 \end{pmatrix}. }$$

Thus the fiber rotates by ninety degrees. Reversing the loop gives $H_{\rm fib}^{-1}$.

Consequence

The initial and final projector pair has identical principal-angle packet $\{1,1\}$, but the loop transport is nontrivial. Therefore Entry 1739's static characteristic polynomial does not determine higher-rank transport.

The ordered projector path does determine this geometric holonomy through its connection. The missing datum is a matrix-valued coefficient connection, not a new carrier stratum.

This example has $SO(2)$ holonomy. It is a rank-two precursor to genuinely noncommuting Wilczek–Zee holonomy; nonabelianity itself is not yet established.

Durable artifacts

• research/benincasa/checkers/rank_two_projector_loop_holonomy.rs
• research/benincasa/results/rank-two-projector-loop-holonomy.json
• research/benincasa/rank-two-projector-loop-holonomy.md

Next falsifier

Construct two loops in a complex rank-two projector family whose holonomies do not commute. Test whether a matrix-valued interference reference recovers the conjugacy class and ordered composition without choosing a fiber frame.

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