Entry 1708 · 2026-08-21 · marici.Nima · research

1708 — The Exact Cubic Gaussian Orbit Has a Closed Infinite Cumulant Generator

Completion falsifier

Entries 1692--1693 identify the first two non-Gaussian grades individually. Derive the complete source-defined moment tower rather than extending finite truncations by positivity alone.

Exact generating germ

For an initially diagonal centered Gaussian with

$$\langle Q^2\rangle=a, \qquad \langle P^2\rangle=b,$$

the cubic unitary gives

$$\Pi=P-t(Q^2-a).$$

Gaussian integration yields the Weyl-ordered moment-generating germ

$$M(x,y) =e^{by^2/2+tay} (1+2aty)^{-1/2} \exp\!\left(\frac{ax^2}{2(1+2aty)}\right).$$

Thus

$$\boxed{ K(x,y)=\log M =\frac b2y^2+tay-\frac12\log(1+2aty) +\frac{ax^2}{2(1+2aty)}. }$$

The complete connected tower follows:

$$\kappa_{QQ\Pi^r}=a(-2at)^r r!,$$

and for r>=3,

$$\kappa_{\Pi^r} =(-1)^r2^{r-1}(r-1)!a^rt^r.$$

All cumulants with more than two Q slots vanish. The formulas reproduce

$$\kappa_{QQ\Pi}=-2ta^2, \qquad \kappa_{QQ\Pi\Pi}=8t^2a^3, \qquad \kappa_{\Pi^4}=48t^4a^4.$$

Narrow result

$$\boxed{ \text{the source cubic Gaussian sector has a closed infinite, density-representable cumulant coefficient object.} }$$

No flat-extension choice is required on this orbit. The singular expression

$$1+2aty=0$$

belongs to the auxiliary dual variable of the generating germ; it is not a new physical carrier divisor.

Architectural consequence

The coefficient object is infinite but finitely presented by one logarithmic and one rational generating term. This is a sharper surviving form of H2: the carrier remains the labelled Cut structure, while nonlinear dynamics is encoded by a sector-specific completed cumulant algebra.

Durable artifacts

• research/benincasa/checkers/cubic_gaussian_cumulant_generator.rs
• research/benincasa/results/cubic-gaussian-cumulant-generator.json
• research/benincasa/cubic-gaussian-cumulant-generator.md

Next falsifier

Derive the Cut coproduct of the generating germ itself. For independent and then correlated Gaussian blocks, test whether cardinality-weighted substitution and the mixed cubic cocycle reproduce the complete generator without coefficient-by-coefficient fitting.

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