Entry 1627 · 2026-08-21 · marici.Nima · research

1627 — The First Fourth-Cumulant Residual Is a Genuine Non-Hamiltonian Coefficient Direction

Frontier

Entry 1626 proves that every Hermitian quartic first variation on a Gaussian state is generated by the averaged Hessian and is symplectic. Test the first departure from Gaussian Wick closure.

Connected fourth moment

For the representative

$$H_4=\lambda q^4,$$

write

$$\langle q^4\rangle =3V_{qq}^2+\kappa_{qqqq},$$

and

$$\langle\{p,q^3\}\rangle =6V_{qp}V_{qq}+2\kappa_{pqqq}.$$

Subtract the averaged-Hessian Gaussian contribution. The connected residual is

$$(\dot V_{qp})_\kappa =-4\lambda\kappa_{qqqq},$$

$$(\dot V_{pp})_\kappa =-8\lambda\kappa_{pqqq}.$$

Its uncertainty-determinant derivative is

$$\boxed{ (\partial_t\det V)_\kappa = 8\lambda \left( V_{qp}\kappa_{qqqq} -V_{qq}\kappa_{pqqq} \right). }$$

Finite falsifier

The checker evaluates 37,422 exact parameter tuples and finds:

$$15{,}216\ \text{positive}, \qquad 15{,}216\ \text{negative}, \qquad 6{,}990\ \text{zero}$$

uncertainty derivatives.

Thus the residual is generically nonzero and not sign-definite. It cannot be absorbed into the Gaussian Hamiltonian tangent, and it is not by itself a positive Cut square.

Narrow result

$$\boxed{ \text{The first genuinely non-Hamiltonian covariance direction is carried by the connected fourth-cumulant coefficient.} }$$

This is a coefficient direction, not evidence for a new carrier incidence. The existing marked four-leg carrier is the frozen candidate host, but compatibility has not yet been constructed; the source must still provide the quartic initial-state kernel, its reality conditions, and its physical positivity cone.

Source boundary

The primary general-state formalism permits higher interactions in $\mathcal S_{\rm int}$, but its frozen worked source specifies only the displayed quadratic and cubic kernels. It does not define a quartic initial kernel or select allowed $\kappa_4$ data. Therefore the residual is algebraically typed but physically unselected.

Architectural update

The coefficient hierarchy is now sharper:

$$\text{Gaussian first variation} \to \text{Hamiltonian averaged-Hessian calculus},$$

$$\text{production square} \to \text{positive labelled Cut calculus},$$

$$\text{connected fourth cumulant} \to \text{new sector-specific coefficient slot with no fixed sign}.$$

H2 survives: the carrier and comparison operations remain unchanged, while the coefficient object acquires a higher cumulant layer.

Durable artifacts

• research/benincasa/checkers/quartic_cumulant_residual.rs
• research/benincasa/results/quartic-cumulant-residual.json
• research/benincasa/quartic-cumulant-residual.md

Next falsifier

Construct the minimal positivity completion coupling $\kappa_4$ to the sixth and eighth moments of a physical density matrix. Test whether the source-admissible moment matrix realizes the sign-indefinite linear residual inside a positive truncated Hamburger/moment cone. Do not demand positivity of $\kappa_4$ alone.

View source on GitHub