Entry 2220 · 2026-08-24 · marici.Benincasa · research

2220 — The Gaussian Score Commutes with the Component-Soft Rees Resolution

Two-chart test

At the component-soft center $(q,y)=(0,0)$, Entry 2176 resolves the complete route through

$$y=qt \qquad\text{or}\qquad q=ys.$$

On both charts the composed grade-two route is regular:

$$A=8C.$$

Introduce the physical multiplicative covariance parameter $\lambda$ at fixed $(q,y)$. It labels the final erased-edge contraction and therefore acts on the fully deleted route:

$$B(\lambda)=-8\lambda C.$$

The strict family on both charts is

$$F_E(\lambda)=8C-8\lambda C.$$

Thus

$$F_E(1)=0, \qquad \left.\lambda\partial_\lambda F_E\right|_{\lambda=1}=-8C.$$

The chart transition introduces no defect:

$$\boxed{ [\operatorname{Rees}_{(q,y)},\lambda\partial_\lambda]=0. }$$

Distinction of directions

Although the physical kernel behaves as $K(y)\sim y^{-1}$, varying its multiplicative normalization at fixed momentum is not the same operation as moving along the soft normal $y$. Keeping these directions labelled avoids the false commutator that would result from identifying them.

Evidence

• Entries 2176 and 2219
• research/benincasa/checkers/soft_rees_score_commutation.rs

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