Entry 1693 · 2026-08-21 · marici.Nima · research

1693 — The Cubic Gaussian Orbit Generates Its First Fourth Cumulant at Second Normal Order

Frontier

Entry 1692 identifies the connected third cumulant as the first tangent of the exact cubic density-state orbit. Compute its first genuinely fourth-cumulant grade and retain the full density-state completion.

Exact calculation

For an initially diagonal centered Gaussian, write

$$\langle Q^2\rangle=a, \qquad \langle P^2\rangle=b, \qquad \Pi_t=P-t(Q^2-a).$$

Wick contraction gives

$$\langle \Pi_t^2\rangle=b+2t^2a^2,$$

and

$$\langle Q^2\Pi_t^2\rangle=ab+10t^2a^3.$$

Hence

$$\boxed{ \kappa_{QQ\Pi\Pi}=8t^2a^3. }$$

The pure momentum fourth cumulant starts only at fourth order:

$$\kappa_{\Pi\Pi\Pi\Pi}=48t^4a^4.$$

Narrow result

$$\boxed{ \text{the first source-derived connected fourth cumulant occurs at second cubic normal order and is fixed by the exact density-state orbit.} }$$

Its degree-six and degree-eight completion is not an independent extension problem: every moment is evaluated in

$$\rho_t=U_t\rho_0U_t^\dagger.$$

This provides one physical fourth-cumulant ray inside the ordered moment cone. It does not identify Entry 1630's arbitrary Z direction with that ray.

Architectural consequence

The source coefficient filtration begins

$$\text{Gaussian covariance} \longrightarrow \kappa_3\text{ at order }t \longrightarrow \kappa_4\text{ at order }t^2.$$

The first-jet calculus therefore cannot control the first source-derived fourth-cumulant deformation. This mirrors the three-site elliptic warning: higher integrated/coefficient structure can begin at second normal order while the carrier remains unchanged.

Durable artifacts

• research/benincasa/checkers/gaussian_cubic_second_grade.rs
• research/benincasa/results/gaussian-cubic-second-grade.json
• research/benincasa/gaussian-cubic-second-grade.md

Next falsifier

Test Cut compatibility of this source-derived second-grade cumulant. For two independent Gaussian blocks followed by cardinality-weighted merge, compare the merged kappa_QQPiPi with the cumulant partition law. Then repeat with a nonzero cross covariance to determine whether the complete joint-cumulant object of Entry 1677 is sufficient without a new carrier operation.

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