Entry 1625 · 2026-08-21 · marici.Nima · research

1625 — The First Quartic Gaussian Covariance Variation Is a Hartree Hamiltonian Tangent

Hostile parity-even test

Entry 1624 used odd cubic parity to force the first statistical covariance correction to second order. A quartic interaction is parity-even and can interfere linearly with a Gaussian state. Test whether that interference supplies a non-Cut statistical direction.

Freeze the local Hermitian representative

$$H_4=\lambda q^4$$

and a centered Gaussian covariance

$$V= \begin{pmatrix} V_{qq}&V_{qp}\\ V_{qp}&V_{pp} \end{pmatrix}.$$

Direct first variation

The quartic contribution to the covariance equations is

$$\dot V_{qq}=0, \qquad \dot V_{qp}=-4\lambda\langle q^4\rangle, \qquad \dot V_{pp}=-4\lambda\langle\{p,q^3\}\rangle.$$

Gaussian Wick reduction gives

$$\langle q^4\rangle=3V_{qq}^2, \qquad \langle\{p,q^3\}\rangle=6V_{qp}V_{qq}.$$

Thus

$$\dot V_{qp}=-12\lambda V_{qq}^2, \qquad \dot V_{pp}=-24\lambda V_{qq}V_{qp}.$$

Hartree typing

These are exactly the covariance equations generated by the state-dependent quadratic Hessian

$$h_{qq}=12\lambda V_{qq}.$$

Therefore the linear quartic variation is symplectic:

$$\dot V=A_HV+VA_H^T$$

and

$$\boxed{ \frac{d}{dt}\det V=0. }$$

The checker verifies 7,488 exact covariance/coupling cases.

Narrow result

The parity-even linear quartic interference is present, but in this source-admitted local representative it is a Hamiltonian Hartree tangent, not statistical noise and not a new second-Rees generator.

This result complements the cubic split:

$$\begin{array}{c|c|c} \text{interaction grade}&\text{first covariance effect}&\text{type}\\ \hline \text{cubic, linear}&0&\text{parity forbidden}\\ \text{cubic, quadratic}&CC^\dagger&\text{statistical Cut norm}\\ \text{quartic, linear}&A_HV+VA_H^T&\text{Hamiltonian tangent} \end{array}$$

Qualifications

This is the one-mode $q^4$ representative. General derivative quartic EFT vertices require tensor-valued Wick reduction before an all-vertex theorem is claimed. Non-Gaussian fourth cumulants also lie outside this first Gaussian-grade calculation.

Architectural consequence

The tested coherent/statistical split is now selected by interaction parity and Wick grade over the existing carrier:

• coherent even-interaction first variations remain tangent;
• statistical production appears as labelled Cut squares.

No new carrier structure is indicated.

Durable artifacts

• research/benincasa/checkers/quartic_gaussian_hartree_tangent.rs
• research/benincasa/results/quartic-gaussian-hartree-tangent.json
• research/benincasa/quartic-gaussian-hartree-tangent.md

Next falsifier

Generalize from $\lambda q^4$ to the source derivative-quartic tensor basis. Compute the Gaussian Wick contraction as an effective quadratic Hessian and test symplectic tangency vertex by vertex. A residual symmetric covariance derivative not expressible as $AV+VA^T$ is the first candidate non-Cut statistical direction.

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