Entry 1681 · 2026-08-21 · marici.Nima · research

1681 — Scalar-Cubic Evolution Is Not Autonomous on the Quadratic Moment Grade

Dynamic positivity falsifier

Entry 1680 establishes positive Cut pushforward for each finite joint moment cone. Test whether scalar-cubic evolution defines a tangent vector determined by one such finite cone.

At the first nontrivial grade, take two positive states supported on (p=q). The first has atoms

$$q=-2,0,2$$

with weights

$$\frac14,\frac12,\frac14.$$

The second has atoms

$$q=-1,2$$

with weights

$$\frac23,\frac13.$$

Both induce exactly the same moments through degree two:

$$\langle q\rangle=\langle p\rangle=0,$$

$$\langle q^2\rangle =\langle qp\rangle =\langle p^2\rangle =2.$$

Their common Hankel matrix on ((1,q,p)) is

$$\begin{pmatrix} 1&0&0\\ 0&2&2\\ 0&2&2 \end{pmatrix},$$

which is positive semidefinite and singular.

But their third moments are

$$\langle q^3\rangle=0 \quad\text{and}\quad \langle q^3\rangle=2.$$

For the scalar-cubic generator

$$Dq=p, \qquad Dp=-q^2,$$

one has

$$D(p^2)=-2pq^2.$$

Because (p=q) on both supports, the same retained quadratic moment therefore has derivatives

$$0 \quad\text{and}\quad -4.$$

Narrow result

$$\boxed{ \text{scalar-cubic evolution is not autonomous on the finite quadratic Hankel cone.} }$$

More generally, the cubic generator raises moment degree and requires the next filtered grade. Positivity of a finite truncation is preserved by Cut merge, but its nonlinear time derivative is not determined without higher moments.

This is coefficient-tower nonclosure, not a failure of the carrier or of finite-grade positivity.

Durable artifacts

• research/benincasa/checkers/cubic_generator_finite_hankel_nonclosure.rs
• research/benincasa/results/cubic-generator-finite-hankel-nonclosure.json
• research/benincasa/cubic-generator-finite-hankel-nonclosure.md

Next falsifier

Determine whether any finite-dimensional invariant positive family—Gaussian, finite atomic, or algebraic moment closure—is preserved by the full free-plus-cubic flow. A surviving family would provide an exact finite process coefficient object; failure of all source-declared candidates strengthens the necessity of the infinite filtered moment module.

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