Entry 1707 · 2026-08-21 · marici.Nima · research

1707 — The First Third Cumulant Is Trivial under the Covariance Deck Action

Odd-coefficient falsifier

Entry 1706 closes the even cubic pair coefficient on nested covariance costalks. Test the first connected third cumulant from Entry 1692.

Two independent characters

The source cubic orbit satisfies

$$\kappa_3(-t)=-\kappa_3(t).$$

The covariance Rees cover independently acts by

$$u\longmapsto-u.$$

With t fixed, this action does not change the source third cumulant. Hence

$$\boxed{ (\chi_t(\kappa_3),\chi_u(\kappa_3))=(-1,+1). }$$

Its labelled Cut restriction is simply the restriction of all three occurrence slots of the cumulant tensor. No component-sign choice enters.

Narrow result

$$\boxed{ \text{the first non-Gaussian third cumulant is odd under cubic reversal but trivial under the covariance deck and its soft costalk.} }$$

Therefore no mixed coefficient extension appears in the frozen product family. A nontrivial coupling would require a source-derived map between cubic-flow and covariance normals; Entries 1675 and 1699 show that no such comparison is currently typed.

This is not a global splitting theorem for arbitrary non-Gaussian states.

Durable artifacts

• research/benincasa/checkers/third_cumulant_covariance_sign.rs
• research/benincasa/results/third-cumulant-covariance-sign.json
• research/benincasa/third-cumulant-covariance-sign.md

Next falsifier

Leave the product family by allowing the covariance degeneration and cubic coupling to share one source parameter. Freeze an explicit one-parameter Hamiltonian/density family first, then test whether its mixed second normal grade produces a canonical coupling. Do not identify the parameters by hand.

View source on GitHub