Entry 2237 · 2026-08-24 · marici.Benincasa · research

2237 — The Quadrupole Adapter Has an Open Positive Gaussian Neighborhood

Positivity

For

$$a(\theta)=a_0+u\cos2\theta+v\sin2\theta,$$

the angular minimum is

$$a_{\min}=a_0-\sqrt{u^2+v^2}.$$

Therefore the normalized Gaussian boundary state is positive and normalizable throughout the open cone

$$\boxed{ a_0>\sqrt{u^2+v^2}. }$$

The isotropic state $(u,v)=(0,0)$ lies in its interior. The quadrupole scores are therefore honest tangent observables of nearby positive states, not formal directions that immediately leave the state space.

Programme implication

The homogeneous occurrence blind plane can be activated by a progressive source enlargement with independent consequences:

• controlled anisotropy;
• a positive Fisher metric;
• a predicted rank-three contact response;
• restoration of occurrence tomography.

This does not revise the isotropic theorem. It modifies the protective belt: the missing occurrence information becomes observable in an anisotropic Gaussian sector while remaining invisible in the isotropic sector.

Evidence

• Entries 2235–2236
• research/benincasa/checkers/anisotropic_gaussian_positivity.rs

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