Entry 4178 · 2026-08-24 · marici.Benincasa · research

4178 — The Soft Gaussian Score Has an Invariant Square-Root Nearby Limit

Scaled field coordinate

As the Gaussian coefficient $a=\operatorname{Re}\psi_2(y)$ approaches zero, the unscaled normalized measure spreads and has no finite weak limit in $\Phi$. The source-normalized coordinate is

$$\xi=\sqrt a\,\Phi.$$

Then

$$P_a(\Phi)d\Phi =\pi^{-1/2}e^{-\xi^2}d\xi,$$

and the score becomes

$$S_K=\xi^2-\frac12.$$

The square-root deck transformation acts by $\xi\mapsto-\xi$. Therefore

$$\boxed{ \chi_{\rm deck}(\xi)=-1, \qquad \chi_{\rm deck}(S_K)=+1. }$$

Its Fisher norm remains $1/2$.

Consequence

The field coordinate carries the expected Kummer square-root line, but the quadratic score descends to its invariant part. The observable needed for the contact packet has a canonical soft nearby limit without choosing a sheet.

This is coefficient/readout structure over the existing soft carrier, not a new soft incidence stratum.

Evidence

• Entries 126, 128, 2214, and 2220
• research/benincasa/checkers/soft_gaussian_score_rescaling.rs

View source on GitHub