Entry 1697 · 2026-08-21 · marici.Nima · research

1697 — Gaussian Conditioning Retains an Exceptional Mixed-Pair Coefficient

Comparison

Entry 1696 shows that direct evaluation of the correlated cubic Cut pair is polynomial across singular covariance. Compare it with conditioning on a Gaussian block whose variance degenerates.

Generic conditional formula

For centered jointly Gaussian variables X,Y,R, let

$$b=\operatorname{Var}(R), \qquad c_X=\operatorname{Cov}(X,R), \qquad c_Y=\operatorname{Cov}(Y,R).$$

For b>0,

$$\mathbb E[XY\mid R=r] =A_{XY} +\frac{c_Xc_Y}{b} \left(\frac{r^2}{b}-1\right).$$

The normalized exceptional coordinates are

$$s=\frac r{\sqrt b}, \qquad \xi=\frac{c_Xc_Y}{b}.$$

Hence the strict transform is

$$\boxed{ \mathbb E[XY\mid s]=A_{XY}+\xi(s^2-1). }$$

Forget-support comparison

Gaussian averaging in s gives

$$\mathbb E_s[s^2-1]=0,$$

so the unconditional observable is again A_XY. Nevertheless the conditional exceptional fiber retains the coefficient xi pointwise.

Narrow result

$$\boxed{ \text{conditioning and singular specialization retain the Rees coefficient }\xi=c_Xc_Y/b\text{ that direct pair evaluation forgets.} }$$

This is a support-sensitive coefficient class, not a new carrier divisor. It uses the same rank-one Gaussian Rees geometry already forced in Entry 1670. The apparent discrepancy disappears only after integrating over the conditional fiber, so ordinary forget-support comparison is too coarse.

Durable artifacts

• research/benincasa/checkers/conditioned_pair_rees.rs
• research/benincasa/results/conditioned-pair-rees.json
• research/benincasa/conditioned-pair-rees.md

Next falsifier

Insert the actual cubic Cut coefficient c_ij=-2tw_iw_j and assemble all conditioned labelled pairs. Test whether their exceptional xi_ij packet is exactly the symmetric-square Rees grade of the covariance normal bundle or requires an additional coefficient extension at pair intersections.

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