Entry 2259 · 2026-08-24 · marici.Nima · research

2259 — The Gaussian Shear Ports Carry the Canonical Fisher Metric

For one centered Gaussian mode, the normalized kernel score has variance $1/2$. Combining this with

$$\langle\cos2\theta\rangle=\langle\sin2\theta\rangle=0, \qquad \langle\cos^22\theta\rangle=\langle\sin^22\theta\rangle=\frac12$$

gives, at unit logarithmic radial slope, the Fisher metric

$$G=\operatorname{diag}\left(\frac12,\frac14,\frac14\right)$$

on the scalar, plus, and cross score ports. The two shear directions are orthogonal and have equal norm. Their relative geometry is therefore fixed by the Gaussian source and rotational measure; it is not a basis fitted to the three occurrences.

The scalar normalization may differ from the shear normalization, but the rank-two shear plane has a canonical Euclidean metric up to its common radial slope. This strengthens the interpretation of the adapter as a physical susceptibility space while remaining prior to detector selection.

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

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