Entry 1691 · 2026-08-21 · marici.Nima · research

1691 — The Source Gaussian Moment Cone Is Density-Representable

Predeclared-class extension test

Entry 1690 isolates the noncommutative moment-extension problem. Begin with the general Gaussian initial-state class already admitted by the source.

For one centered mode with $[Q,P]=2i$, the covariance packet

$$V=\begin{pmatrix}x&z\\z&y\end{pmatrix}$$

is physical precisely on the Gaussian uncertainty cone

$$V>0,\qquad \det V=xy-z^2\ge1.$$

Williamson decomposition gives

$$\boxed{ V=\nu SS^T,\qquad S\in Sp(2,\mathbb R),\qquad \nu=\sqrt{\det V}\ge1. }$$

The isotropic covariance $\nu I$ is realized by the thermal density operator whose occupation probabilities have geometric ratio

$$r=\frac{\nu-1}{\nu+1},\qquad 0\le r<1.$$

A metaplectic lift of $S$ then realizes $V$. Hence every packet in this predeclared Gaussian cone extends to a positive trace-class density operator.

The exact checker uses rational symplectic matrices

$$S=\begin{pmatrix}1+k\ell&k\\ \ell&1\end{pmatrix}$$

and verifies 3,528 thermal--metaplectic covariances with $\det V=\nu^2$. It also verifies 5,000 cardinality-weighted covariance merges, all remaining in the uncertainty cone.

Narrow result

$$\boxed{ \text{the source one-mode Gaussian moment cone is density-representable and merge-stable.} }$$

This supplies one exact finite representable coefficient class. It does not close the full cosmological process because scalar-cubic evolution generates non-Gaussian cumulants and unbounded Schmidt rank.

Durable artifacts

• research/benincasa/checkers/gaussian_density_representability.rs
• research/benincasa/results/gaussian-density-representability.json
• research/benincasa/gaussian-density-representability.md

Next falsifier

Follow the first cubic tangent out of the Gaussian cone. Determine whether the degree-four Gram packet of Entry 1630 lies in the tangent cone of genuinely density-representable states, rather than merely the truncated positive cone. Use an explicit trace-class perturbation of a thermal--metaplectic state or record the missing extension datum.

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