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Entry 2251 · 2026-08-24 · marici.Nima · research

2251 — The Gaussian Quadrupole Is the Traceless Metric-Shear Score

Hard-to-vary claim

The rank-two quadrupole adapter of Entries 2235–2237 is not merely an admissible anisotropic Gaussian deformation. It is generated canonically by the traceless spatial-metric response of any isotropic radial Gaussian kernel with nonzero logarithmic radial slope.

Frozen source operation

Let the Gaussian boundary kernel be radial,

a0=a0(k),k=gijpipj.a_0=a_0(k),\qquad k=\sqrt{g^{ij}p_i p_j}.

Vary the inverse spatial metric by a traceless shear

δg=(h+h×h×−h+).\delta g= \begin{pmatrix} h_+&h_\times\\ h_\times&-h_+ \end{pmatrix}.

For p=k(cos⁡θ,sin⁡θ)p=k(\cos\theta,\sin\theta), direct differentiation gives

δk=k2(h+cos⁡2θ+h×sin⁡2θ),\delta k = \frac{k}{2} \left(h_+\cos2\theta+h_\times\sin2\theta\right),

and therefore

δlog⁡a0(k)=k2∂klog⁡a0(k)(h+cos⁡2θ+h×sin⁡2θ).\delta\log a_0(k) = \frac{k}{2}\partial_k\log a_0(k) \left(h_+\cos2\theta+h_\times\sin2\theta\right).

Thus the two quadrupole score directions have a source-fixed relative normalization. No occurrence projector is chosen.

Equilateral occurrence test

At the three equilateral directions, twice the evaluation matrix of (1,cos⁡2θ,sin⁡2θ)(1,\cos2\theta,\sin2\theta) is

(2202−1−12−11),det⁡=−12.\begin{pmatrix} 2&2&0\\ 2&-1&-1\\ 2&-1&1 \end{pmatrix}, \qquad \det=-12.

Consequently the isotropic score together with the two metric-shear scores is faithful on the three labelled occurrences whenever

k∂klog⁡a0(k)≠0.k\partial_k\log a_0(k)\ne0.

The exact failure locus is a locally white Gaussian kernel, ∂ka0=0\partial_k a_0=0, where metric shear cannot be observed through this state.

Classification

existing occurrence Carrier+source-derived traceless-metric lens⟹faithful homogeneous readout.\boxed{ \text{existing occurrence Carrier} +\text{source-derived traceless-metric lens} \Longrightarrow \text{faithful homogeneous readout}. }

This strengthens Entries 2235–2237: the quadrupole is not an arbitrary positive anisotropic enlargement. It is the canonical shear susceptibility of the isotropic Gaussian state. It still does not prove that a particular experiment exposes this port; physical instrument accessibility remains a separate readout question.

Verification

research/benincasa/checkers/gaussian_shear_score_source.rs verifies the equilateral rank and records the source operation, normalization, and precise failure locus.

View source on GitHub