Entry 1653 · 2026-08-21 · marici.Nima · research

1653 — Non-Gaussian Conditional Cuts Remain Associative but Require Cumulant Grades

Test

Entry 1652 derives conditional nested Cut sewing for correlated Gaussian coefficients. Test whether the first non-Gaussian fourth-cumulant grade creates an associativity obstruction.

Correlated binary coefficient state

Let $X,Y\in\{\pm1\}$ with

$$\Pr(X=x,Y=y)=\frac{1+rxy}{4}, \qquad |r|\le1.$$

Then

$$\mathbb E[X^2]=\mathbb E[Y^2]=1, \qquad \mathbb E[XY]=r, \qquad \mathbb E[X^2Y^2]=1.$$

Gaussian Wick closure predicts

$$1+2r^2.$$

The exact connected fourth cumulant is

$$\boxed{\kappa_{22}=-2r^2,}$$

so

$$1+2r^2+\kappa_{22}=1.$$

Conditional associativity

Because $Y^2=1$ pointwise,

$$\mathbb E\!\left[X^2\mathbb E(Y^2\mid X)\right]=1,$$

and the opposite conditioning order gives the same result. Thus exact conditional pushforward remains associative.

The checker audits 4,224 rational distributions. Gaussian closure fails for all 4,160 nonzero correlations; the fourth cumulant repairs every case, and exact conditional contraction matches every case.

Narrow result

$$\boxed{ \text{Non-Gaussianity enlarges the coefficient filtration but does not obstruct exact conditional Cut associativity.} }$$

The higher cumulant is coefficient data of the same joint process object. Associativity belongs to full conditional pushforward; any finite cumulant truncation is only a declared associated grade and may omit later connected terms.

This strengthens H2:

$$\text{shared Cut/coherence calculus} + \text{sector-specific filtered cumulant coefficients}.$$

No new carrier incidence is indicated.

Durable artifacts

• research/benincasa/checkers/nongaussian_conditional_cut_cumulant.rs
• research/benincasa/results/nongaussian-conditional-cut-cumulant.json
• research/benincasa/nongaussian-conditional-cut-cumulant.md

Next falsifier

Determine whether the cumulant filtration generated by cubic evolution has a finite coalgebraic presentation under conditional Cut sewing. Derive the coproduct/partition rule for connected cumulants and test whether nested pushforwards follow coassociativity without adding grade-specific coherence generators.

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