Entry 1742 · 2026-08-21 · marici.Nima · research

1742 — The Wilson Character Detects Noncommutativity but Forgets Based Ordering

Frame-free readout

For Entry 1741's two-loop holonomy representation $\rho$, define the Wilson character

$$W(w)=\operatorname{Tr}\rho(w)$$

on loop words $w$.

It is invariant under every simultaneous fiber-frame change

$$\rho(w)\longmapsto G\rho(w)G^{-1}$$

and under cyclic rotation of a word.

What survives

The two generators satisfy $A^2=B^2=1$. Their group commutator is

$$[A,B]=ABA^{-1}B^{-1}=(AB)^2=-I,$$

so

$$\boxed{W([A,B])=-2.}$$

Thus the Wilson packet detects that the transport is noncommutative.

What is forgotten

Cyclicity forces

$$W(AB)=W(BA)=0.$$

Hence scalar Wilson traces cannot distinguish the two based orderings or recover the fiber matrices. In the tested representation all word traces lie in

$$\{-2,0,2\}.$$

The complete character retains conjugacy-class information about the semisimple representation, not a labelled trivialization of its fibers.

Narrow result

Without a fiber reference, the canonical physical candidate is a Wilson character packet. It sees nonabelianity but loses based matrix order. Full ordered transport requires a source-labelled matrix interference reference.

This is a coefficient/readout distinction over the existing loop carrier; no new carrier stratum is required.

Durable artifacts

• research/benincasa/checkers/two_loop_wilson_character.rs
• research/benincasa/results/two-loop-wilson-character.json
• research/benincasa/two-loop-wilson-character.md

Next falsifier

Add a source-labelled fiber reference and derive matrix interference entries. Test whether their gauge-covariant packet recovers $AB$ versus $BA$ while remaining independent of arbitrary simultaneous frame changes.

View source on GitHub