Entry 1628 · 2026-08-21 · marici.Nima · research

1628 — Physical Fourth Cumulants Form a Moment Cone Rather Than a Positive Ray

Problem

Entry 1627 finds that the connected fourth-cumulant covariance residual is sign-indefinite. Does either sign imply an unphysical state?

Truncated moment object

For a centered $q$-marginal, define

$$\mu_{2r}=\langle q^{2r}\rangle, \qquad \kappa_4=\mu_4-3\mu_2^2.$$

The degree-eight moment matrix is

$$M_4= \begin{pmatrix} 1&\mu_2&\mu_4\\ \mu_2&\mu_4&\mu_6\\ \mu_4&\mu_6&\mu_8 \end{pmatrix}.$$

Physical measure positivity requires $M_4\succeq0$; it does not require $\kappa_4\geq0$.

Two exact physical witnesses

At fixed variance $\mu_2=1$, the symmetric two-point distribution

$$q=\pm1$$

has

$$(\mu_4,\mu_6,\mu_8)=(1,1,1), \qquad \kappa_4=-2.$$

The symmetric sparse distribution

$$P(0)=\frac34, \qquad P(2)=P(-2)=\frac18$$

has

$$(\mu_4,\mu_6,\mu_8)=(4,16,64), \qquad \kappa_4=1.$$

Both exact moment matrices are positive semidefinite. The checker verifies every principal minor.

Narrow result

$$\boxed{ \text{Both signs of the connected fourth cumulant occur in physical positive moment objects.} }$$

Therefore Entry 1627's sign-indefinite covariance residual is not itself a positivity failure. The physical coefficient object is the coupled moment matrix—or its quantum phase-space refinement—not the isolated scalar $\kappa_4$.

Type qualification

The displayed construction is a classical one-coordinate marginal and hence a necessary slice of the quantum moment cone. The complete quantum object must include:

• mixed $q,p$ moments;
• Weyl ordering;
• canonical-commutator localizing constraints;
• mode and occurrence labels;
• source density-matrix hermiticity.

Architectural consequence

The higher cumulant layer is not an arbitrary coefficient vector. It is constrained by a positive semidefinite moment object containing several normal grades simultaneously:

$$V_2, \kappa_4, M_6, M_8,\ldots$$

This is another instance where a single associated grade is insufficient, while no new carrier stratum is indicated.

Durable artifacts

• research/benincasa/checkers/fourth_cumulant_moment_cone.rs
• research/benincasa/results/fourth-cumulant-moment-cone.json
• research/benincasa/fourth-cumulant-moment-cone.md

Next falsifier

Construct the smallest Weyl-ordered quantum phase-space moment matrix containing $V_{qq},V_{qp},V_{pp},\kappa_{qqqq},\kappa_{pqqq}$. Impose the canonical commutator exactly and test whether Entry 1627's residual functional is bounded on each fixed-covariance fiber. An unbounded functional would show that higher moments beyond degree four are mandatory for dynamical control.

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