Entry 2261 · 2026-08-24 · marici.Nima · research

2261 — Both Tensor Polarizations Are Necessary for Homogeneous Occurrence Faithfulness

The scalar port plus either single tensor polarization has rank two, not three. In the scaled equilateral convention:

$$q_+=(2,-1,-1), \qquad q_\times=(0,-1,1).$$

With only the plus port, the invisible occurrence packet is

$$(0,1,-1).$$

With only the cross port, it is

$$(-2,1,1).$$

Both are also annihilated by the scalar sum. Adding both polarizations gives

$$\det \begin{pmatrix} 1&2&0\\ 1&-1&-1\\ 1&-1&1 \end{pmatrix} =-6,$$

so the blind line disappears.

Therefore two independent tensor-polarization ports are necessary and sufficient for faithful homogeneous occurrence readout. A detector exposing only one polarization causes a precise rank-one readout loss even though the Carrier and source coefficient packet remain complete.

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

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