Entry 2235 · 2026-08-24 · marici.Benincasa · research

2235 — The Minimal Real Homogeneous Occurrence Adapter Is Gaussian Quadrupole Rank Two

Even full-momentum kernel

For a real boundary field, the Gaussian kernel must satisfy

$$a(\vec p)=a(-\vec p).$$

The first anisotropic angular deformation is therefore quadrupolar, not linear. On a fixed momentum-magnitude shell in two dimensions, write

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

The isotropic coefficient $a_0$ supplies the trivial $C_3$ line, while $(u,v)$ span a rank-two angular port.

For the three equilateral directions, a scaled evaluation matrix is

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

Hence

$$\boxed{ \text{isotropic scalar}+\text{Gaussian quadrupole} \text{ resolves all three homogeneous occurrences}. }$$

Minimality

Entry 2234 proved that two additional directions are necessary. The quadrupole supplies exactly two, is even under momentum reversal, and is translation invariant. It is therefore the minimal conventional Gaussian source enlargement with the required representation type.

Evidence

• Entries 2233–2234
• research/benincasa/checkers/quadrupole_gaussian_port.rs

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