Entry 1669 · 2026-08-21 · marici.Nima · research

1669 — Full Covariance Restores Correlated Gaussian Cut Descent

Correlated falsifier

Entry 1668 proves naturality of the scalar-cubic Wick Cut defect for independent Gaussian blocks. Admit arbitrary cross-covariance and test whether nested cardinality-weighted pushforwards remain tree-independent.

For a labelled block $A$, define

$$S_A=\sum_{i,j\in A}\Sigma_{ij}, \qquad \nu_A=\frac{S_A}{|A|}.$$

For disjoint blocks $A,B$, define

$$K_{AB}=\sum_{i\in A,j\in B}\Sigma_{ij}.$$

Then

$$S_{A\cup B}=S_A+S_B+2K_{AB}$$

and

$$K_{A\cup B,C}=K_{AC}+K_{BC}.$$

Thus every merger tree evaluates to

$$\nu_N=\frac1N\mathbf1^T\Sigma\mathbf1.$$

The exact rational checker verifies:

$$33\text{ symmetric covariance matrices}, \qquad 71{,}142\text{ ordered binary bracketings}, \qquad 495\text{ cross-covariance updates},$$

through eleven labelled internal occurrences.

Narrow result

$$\boxed{ \text{full labelled covariance restores strict correlated Gaussian Cut descent.} }$$

The missing $2\kappa^2$-type contribution in a variance-only treatment is not a new carrier incidence. It is information discarded from the Gaussian coefficient object. Once the full covariance packet is retained, conditional and cardinality-weighted mergers are associative.

This does not cover non-Gaussian cumulants, singular covariance matrices, or support created by conditioning on a zero-variance block.

Durable artifacts

• research/benincasa/checkers/full_covariance_cut_descent.rs
• research/benincasa/results/full-covariance-cut-descent.json
• research/benincasa/full-covariance-cut-descent.md

Next falsifier

Approach a singular covariance stratum where one conditioned block has vanishing determinant. Compute the Schur-complement/Rees limit before specialization and determine whether the correlated Cut develops a canonical supported coefficient class or requires a genuinely new carrier stratum.

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