Entry 2260 · 2026-08-24 · marici.Nima · research

2260 — The Equilateral Quadrupole Readout Is a Tight Frame

The three equilateral occurrence directions define three unit vectors in the quadrupole plane. Their Gram matrix is

$$G_{ij}=\begin{cases}1&i=j,\\-1/2&i\ne j,\end{cases}$$

with spectrum

$$\left(\frac32,\frac32,0\right).$$

Hence they form a tight frame for the two-dimensional shear plane:

$$\sum_{i=1}^3q_iq_i^T=\frac32I_2.$$

This explains why the quadrupole adapter is balanced across the three occurrences. No occurrence is preferred, and the two cyclotomic directions have equal information weight. The rank-three scalar-plus-shear matrix has determinant $-6$ in the integral scaled convention.

Thus the homogeneous triangle is not merely separable by anisotropy; its native cyclic geometry supplies the optimal isotropic rank-two frame.

Verified by research/benincasa/checkers/quadrupole_information_geometry.rs.

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