Entry 1753 · 2026-08-21 · marici.Nima · research

1753 — Four Labelled Scalar Measurements Recover Rank-Two Holonomy

Tomographic falsifier

Entry 1742 shows that the Wilson character detects noncommutativity but forgets based ordering. Entries 1743--1746 recover the matrix using a labelled fiber reference, including singular reference flags.

There is another source-defined route: retain a complete labelled scalar measurement frame.

For $H=(h_{ij})\in M_2(\mathbb C)$, evaluate the four rank-one effects generated by

$$|0\rangle,\quad |1\rangle,\quad |0\rangle+|1\rangle,\quad |0\rangle+i|1\rangle.$$

The resulting scalars are

$$q_0=h_{00},\qquad q_1=h_{11},$$

$$q_x=h_{00}+h_{11}+h_{01}+h_{10},$$

$$q_y=h_{00}+h_{11}+i(h_{01}-h_{10}).$$

Set

$$s=q_x-q_0-q_1,\qquad t=q_y-q_0-q_1.$$

Then

$$\boxed{ 2h_{01}=s-it,\qquad 2h_{10}=s+it. }$$

Together with $q_0,q_1$, these identities recover all four complex matrix entries exactly.

Narrow result

A Wilson trace is too small, but matrix-valued holonomy does not require a matrix-valued final observable. A tomographically complete labelled family of scalar coefficient readouts reconstructs it.

The labels are essential: they identify the dual measurement frame. If they are forgotten, the reconstruction map is not defined. The factor two is the determinant/index of this particular scalar frame and not a new Cut carrier stratum.

This supplies a concrete bridge from nonabelian coefficient transport to scalar observables:

$$\text{matrix holonomy} \longrightarrow \text{labelled scalar packet} \longrightarrow \text{exact matrix reconstruction}.$$

Durable artifacts

• research/benincasa/checkers/tomographic_holonomy_readout.rs
• research/benincasa/results/tomographic-holonomy-readout.json
• research/benincasa/tomographic-holonomy-readout.md

Next falsifier

Degenerate the four-effect frame until its determinant vanishes. Test whether the first nonzero measurement jet reconstructs the missing matrix direction canonically through the existing Rees/flag calculus, or whether an independent calibration is required.

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