Entry 1692 · 2026-08-21 · marici.Nima · research

1692 — The First Cubic Density-State Tangent Is a Third Cumulant

Falsifier

Entry 1691 proves that the source Gaussian moment cone is represented by positive trace-class density operators. Test whether Entry 1630's unbounded degree-four coordinate is the first cubic tangent out of that cone.

Exact density-state orbit

For [Q,P]=2i, freeze the source cubic unitary

$$U_t=\exp(-itQ^3/6).$$

It gives the exact Heisenberg transformation

$$U_t^\dagger P U_t=P-tQ^2.$$

Let the initial state be centered, parity-even, and Gaussian, with

$$\langle Q^2\rangle=a.$$

After recentering momentum,

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

Wick reduction then gives the connected third moment

$$\boxed{ \kappa_{QQ\Pi}(t)=-2ta^2. }$$

It is generically nonzero at first order. Every point on this orbit is a physical density state because

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

Degree-four comparison

For Entry 1630's coordinate

$$Z=\operatorname{Re}\langle Q^2S\rangle, \qquad S=(Q\Pi+\Pi Q)/2,$$

the cubic correction is

$$-t(\langle Q^5\rangle-a\langle Q^3\rangle)=0$$

by parity. The connected fourth-cumulant directions of Entry 1627 also have zero linear term on this orbit.

Narrow result

$$\boxed{ \text{the first source cubic tangent from a centered Gaussian density state is rank three, not [Entry 1630](/research/ledger/20260821-1630-the-full-degree-four-quantum-gram-relaxation-does-not-bound-the-cumulant-residual)'s degree-four }Z\text{ direction.} }$$

This rejects one proposed provenance for the unbounded truncated packet. It does not prove that nonzero-Z packets fail to extend to density operators. That remains a distinct noncommutative moment-extension problem.

Architectural consequence

The source filtration must retain odd cumulant grades before the first non-Hamiltonian fourth-cumulant layer. No new carrier stratum is indicated: the new datum is a sector-specific coefficient grade generated by the exact cubic unitary.

Durable artifacts

• research/benincasa/checkers/gaussian_cubic_first_tangent.rs
• research/benincasa/results/gaussian-cubic-first-tangent.json
• research/benincasa/gaussian-cubic-first-tangent.md

Next falsifier

Compute the second cubic normal grade of the exact Gaussian orbit. Determine which connected fourth cumulants it generates and whether their degree-six moment completion is fixed by the same unitary orbit. Compare those source-derived fourth-order directions with Entry 1630 without identifying their ambient truncated cones.

View source on GitHub