Entry 1710 · 2026-08-21 · marici.Nima · research

1710 — The Complete Correlated Cubic Generator Depends Only on the Joint Collective Covariance

Correlated-generator falsifier

Entry 1709 proves the complete Cut coproduct for independent blocks. Admit a full joint Gaussian covariance, including Q--P correlation.

Collective covariance

Write

$$A=\langle Q^2\rangle, \qquad B=\langle P^2\rangle, \qquad C=\langle S\rangle.$$

For a labelled Cut map with weight vector w, these are the ordinary congruence contractions

$$A=w^TA_Qw, \qquad B=w^TA_Pw, \qquad C=w^TA_{QP}w.$$

For

$$\Pi=P-t(Q^2-A),$$

Gaussian regression gives the complete cumulant generator

$$\boxed{ K(x,y)=tAy-\frac12\log(1+2Aty) +\frac12\left(B-\frac{C^2}{A}\right)y^2 +\frac{A(x+Cy/A)^2}{2(1+2Aty)}. }$$

At t=0, the last two terms combine to the original correlated Gaussian form

$$\frac12Ax^2+Cxy+\frac12By^2.$$

The first non-Gaussian coefficient remains

$$\kappa_{QQ\Pi}=-2tA^2.$$

Singular covariance

If

$$AB-C^2=0$$

while A remains nonzero, the conditional residual variance vanishes and the formula stays regular. No Rees extension is required. Rees geometry enters only when the conditioning variance A itself degenerates.

Narrow result

$$\boxed{ \text{the complete correlated cubic coefficient tower is reconstructed from the joint collective covariance and the same labelled quadratic cocycle.} }$$

No new carrier operation appears. This extends Entry 1677 from abstract joint cumulant descent to one complete nonlinear density-representable generating object.

Durable artifacts

• research/benincasa/checkers/correlated_cubic_generator.rs
• research/benincasa/results/correlated-cubic-generator.json
• research/benincasa/correlated-cubic-generator.md

Next falsifier

Resolve the boundary A=0 in the generating germ. Derive the weighted Rees coordinates for C/A and x+Cy/A, and determine whether the strict transform is generated by the covariance symmetric-square packet of Entry 1698 or needs a new coefficient chart.

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