Entry 1739 · 2026-08-21 · marici.Nima · research

1739 — Higher-Rank Supported Readout Requires the Complete Principal-Angle Packet

Higher-rank falsifier

Let $P$ and $Q$ be state and measurement projectors. The nonzero spectrum of

$$PQP$$

is

$$\{\cos^2\theta_i\},$$

where $\theta_i$ are their principal angles.

Total-overlap failure

The total Born overlap

$$\operatorname{Tr}(PQ)=\sum_i\cos^2\theta_i$$

does not determine the pair. In rank two, the realizable spectra

$$\{0,1\} \qquad\text{and}\qquad \left\{\frac9{25},\frac{16}{25}\right\}$$

both have trace one, but their determinants are

$$0 \qquad\text{and}\qquad \frac{144}{625}.$$

Thus a single summed probability loses invariant higher-rank information.

Complete static packet

The characteristic polynomial

$$\boxed{\chi_{PQP}(\lambda)}$$

equivalently the complete squared principal-angle spectrum, determines all unitary-invariant polynomial readouts of the static projector pair. In rank two, trace and determinant generate the power traces through the exact Newton recurrence.

Narrow result

The rank-one scalar of Entry 1738 generalizes to a finite principal-angle packet, not to one scalar. This is a higher-rank coefficient/readout object over the same labelled state–measurement incidence. No new Cut carrier stratum is required.

This result is static. It does not imply that endpoint principal angles control nonabelian Berry transport along a loop.

Durable artifacts

• research/benincasa/checkers/higher_rank_principal_angle_packet.rs
• research/benincasa/results/higher-rank-principal-angle-packet.json
• research/benincasa/higher-rank-principal-angle-packet.md

Next falsifier

Construct two loops with identical endpoint principal-angle packets but different Wilczek–Zee holonomy. Determine whether the projector path recovers the nonabelian transport and whether physical readout requires a matrix-valued interference reference.

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