Entry 1652 · 2026-08-21 · marici.Nima · research

1652 — Correlated Nested Cuts Are Associative Only with Conditional Pushforward

Falsifier

Entry 1651 proves nested-Cut coherence for independent Gaussian coefficient states. Test the first correlated internal pair.

Joint Gaussian coefficient

Let

$$\mathbb E[q_3^2]=\nu_3, \qquad \mathbb E[q_4^2]=\nu_4, \qquad \mathbb E[q_3q_4]=\kappa.$$

Wick's theorem gives

$$\boxed{ \mathbb E[q_3^2q_4^2] = \nu_3\nu_4+2\kappa^2. }$$

Naively composing the two marginal Cut functionals gives only $\nu_3\nu_4$. For every nonzero correlation it misses

$$\boxed{2\kappa^2.}$$

Conditional repair derived from the coefficient state

The joint Gaussian state supplies

$$\mathbb E[q_4\mid q_3] = \frac{\kappa}{\nu_3}q_3, \qquad \operatorname{Var}(q_4\mid q_3) = \nu_4-\frac{\kappa^2}{\nu_3}.$$

Conditional pushforward followed by integration over $q_3$ reproduces the full joint fourth moment. Reversing the conditioning order produces the same result.

The checker audits 25,616 positive covariance assignments. All conditional routes match the joint contraction in both orders; all 24,592 nonzero-correlation cases fail under naive marginal multiplication.

Narrow result

$$\boxed{ \text{Correlated nested Cuts are associative and order-independent only after transporting the joint conditional coefficient state.} }$$

The connected term $2\kappa^2$ is not a higher carrier incidence. It is coefficient data of the existing correlated process object. Naive occurrencewise Cut pushforward is mistyped because it silently replaces the joint state by a product state.

This upgrades Entry 1651:

$$\text{independent coefficients}\to\text{ordinary commuting Wick contractions},$$

$$\text{correlated coefficients}\to\text{conditional/process-tensor contractions}.$$

Durable artifacts

• research/benincasa/checkers/correlated_nested_cut_conditioning.rs
• research/benincasa/results/correlated-nested-cut-conditioning.json
• research/benincasa/correlated-nested-cut-conditioning.md

Next falsifier

Lift the conditional contraction from Gaussian moments to the filtered non-Gaussian module generated by cubic evolution. Test whether cumulant-conditioned pushforwards remain associative through the first nonzero third- and fourth-cumulant grades, or require a genuine higher process-tensor coherence.

View source on GitHub