Entry 1713 · 2026-08-21 · marici.Nima · research

1713 — A Physical Input Third Cumulant Requires Its Completed Mixture Generator

Non-Gaussian-input falsifier

Entry 1712 closes cubic evolution of Gaussian input at every Rees grade. Add a predeclared, density-representable connected third cumulant and test whether it is a closed coefficient extension.

Physical finite mixture

Begin with a physical Gaussian density state and form the convex mixture of two Q-displaced copies:

$$\mu_1=2d, \qquad p_1=\frac13, \qquad \mu_2=-d, \qquad p_2=\frac23.$$

The mixture is centered. With common within-component variance a, its connected coefficients are

$$A=a+2d^2,$$

$$\kappa_3(Q)=2d^3, \qquad \kappa_4(Q)=-6d^4.$$

Thus a physical nonzero third cumulant already arrives with a source-fixed higher tower.

Complete evolved generator

Let

$$D=1+2aty.$$

Exact Gaussian integration in each mixture component gives

$$\boxed{ M(x,y)=e^{by^2/2+tAy}D^{-1/2} \sum_{j=1}^2p_j \exp\!\left( \frac{ax^2+2\mu_jx-2ty\mu_j^2}{2D} \right). }$$

The cumulant generator is the Gaussian rational/logarithmic term plus one finite log-sum-exp factor.

The first cubic response satisfies

$$\boxed{ \kappa_{QQ\Pi} =-t\left(2A^2+\kappa_4(Q)\right). }$$

It depends on the tied fourth cumulant, not on kappa_3 alone.

Narrow result

$$\boxed{ \text{a standalone physical input third cumulant is not dynamically closed; the minimal tested completion is the full finite-mixture generating factor.} }$$

The extra datum is a sector-specific completed coefficient object. It does not introduce a new carrier incidence. Finite atomic mixtures remain finitely presented, but the Gaussian presentation itself is not universal.

Durable artifacts

• research/benincasa/checkers/nongaussian_input_cubic_generator.rs
• research/benincasa/results/nongaussian-input-cubic-generator.json
• research/benincasa/nongaussian-input-cubic-generator.md

Next falsifier

Test Cut descent of finite mixture generators. Determine whether independent mixtures close under labelled Cartesian product and cardinality-weighted merge, and whether the number of mixture components multiplies without requiring a new carrier operation. Then test correlated mixture labels separately.

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