Entry 1647 · 2026-08-21 · marici.Nima · research

1647 — Cubic Memory Appears One Grade Before Gaussian Covariance Detects It

Falsifier

Entry 1646 characterizes Markov descent exactly inside the Gaussian category. Apply a cubic interaction and test whether covariance remains a sufficient detector after the state leaves that category.

Cubic kick

For independent centered unit-variance Gaussian quadratures (X,Y,Z), the generator (XYZ) gives

$$P_X'=P_X-gYZ, \qquad P_Y'=P_Y-gXZ.$$

Parity kills all first-order corrections to the linear covariance blocks. A covariance-only audit therefore reports no first-order observed--internal cross block.

First non-Gaussian memory grade

The supported quadratic observable (YZ) gives

$$\operatorname{Cov}(P_X',YZ) = -g\,\mathbb E[Y^2Z^2] =-g.$$

Hence the full state is already nonfactorized at first order.

The internal linear covariance detects the failure only at second order:

$$\operatorname{Var}(P_Y')-operatorname{Var}(P_Y) =g^2\mathbb E[X^2Z^2] =g^2.$$

Meanwhile the displayed linear observed--internal covariance remains zero through this order by odd Gaussian parity.

Narrow result

$$\boxed{ \text{full process memory begins at }g, \qquad \text{Gaussian-covariance memory begins at }g^2. }$$

Thus Entry 1646 remains correct but is insufficient after cubic evolution: covariance sees only a quotient of the process object. The first memory class lives in the mixed linear--quadratic moment layer.

This supplies a concrete cosmological reason for higher moment/Rees grades. It does not require a new carrier stratum; the extra datum is a higher coefficient layer on the same labelled occurrences.

Durable artifacts

• research/benincasa/checkers/cubic_memory_normal_order.rs
• research/benincasa/results/cubic-memory-normal-order.json
• research/benincasa/cubic-memory-normal-order.md

Next falsifier

Construct the minimal degree-two observable extension containing (YZ) and derive its two-step process-tensor closure under the cubic generator. Test whether this finite extension closes or generates an unbounded moment hierarchy. Closure would yield a finite coefficient object; degree growth would force a filtered infinite process module.

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