Entry 2214 · 2026-08-24 · marici.Benincasa · research

2214 — The Gaussian Kernel Tangent Has a Normalized Quadratic Score

One-mode calculation

For the normalized Gaussian boundary measure

$$P_a(x)=\sqrt{\frac a\pi}\,e^{-ax^2}, \qquad K=\langle x^2\rangle=\frac1{2a},$$

the score for the logarithmic covariance parameter is

$$S_K(x)=\frac{\partial}{\partial\log K}\log P_a(x) =ax^2-\frac12.$$

It is centered and has positive Fisher norm:

$$\langle S_K\rangle=0, \qquad \langle S_K^2\rangle=\frac12.$$

The subtraction $-1/2$ is forced by normalization. Thus vacuum or partition- function variation is not an untracked correction; it is already included in the score.

Consequence

The tangent of Entry 2211 has an actual boundary-field representative: a centered quadratic power insertion. It is not merely a derivative in an abstract coefficient space.

For independent momentum modes, the corresponding scores are orthogonal. Their Fisher metric is positive and diagonal before evaluation coincidences are imposed.

Evidence

• Entry 2211
• research/benincasa/checkers/normalized_gaussian_score.rs

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