Entry 1626 · 2026-08-21 · marici.Nima · research

1626 — The Full Quartic Gaussian First Variation Is Generated by the Averaged Hessian

Claim

For every centered Gaussian state and Hermitian Weyl-ordered quartic Hamiltonian, including the source derivative-quartic tensor basis, the complete first covariance variation is a symplectic Hamiltonian tangent generated by the Gaussian expectation of the quartic Hessian.

General derivation

Let $R_A$ be canonical phase-space coordinates with symplectic form $J$, and let

$$V_{AB}=\langle R_AR_B\rangle_{\rm sym}.$$

For a quartic Weyl symbol $H_4(R)$, Gaussian integration by parts gives

$$\langle\partial_AH_4\,R_B\rangle_{\rm sym} = \langle\partial_A\partial_CH_4\rangle V_{CB}.$$

Define

$$K_{AC}=\langle\partial_A\partial_CH_4\rangle.$$

The Hessian $K$ is symmetric. Hamilton's equation then gives

$$\boxed{ \dot V=JKV+VK^TJ^T =AV+VA^T, \qquad A=JK. }$$

Thus $A$ lies in the symplectic Lie algebra. In one mode,

$$\partial_t\det V=0.$$

In multiple modes the corresponding symplectic eigenvalues are unchanged at this first coherent grade.

Quantum exactness at covariance grade

For Weyl symbols, the Moyal bracket of $H_4$ with a quadratic covariance observable has no higher correction because every term beyond the Poisson bracket differentiates the quadratic observable at least three times. Therefore the derivation is not merely a classical truncation.

Exact representative census

For

$$H_4=aq^4+bq^3p+cq^2p^2+dqp^3+ep^4,$$

the checker constructs

$$K=\langle\operatorname{Hess}H_4\rangle$$

for all five basis monomials and mixed tensors. It verifies symmetry, tracelessness of $A=JK$, and zero uncertainty-determinant derivative in 5,824 exact cases.

Source derivative vertices

Spatial and temporal derivatives in the inflationary EFT change the momentum-labelled quartic tensor but not the argument: after Fourier transformation they remain Hermitian quartic polynomials in the canonical mode variables. Their Gaussian first variation is therefore governed by the same averaged-Hessian construction, subject to the source reality and ordering conventions.

Narrow result

$$\boxed{ \text{No Hermitian source quartic vertex generates a new statistical covariance direction at first Gaussian order.} }$$

The first variation is coherent/Hamiltonian. Statistical quartic production can still appear at quadratic interaction order through Cut squares, or at linear order when evaluated on a genuinely non-Gaussian fourth cumulant.

Architectural update

For Gaussian coefficient objects, the tested interaction hierarchy is now:

$$\text{odd interaction, linear} \to0,$$

$$\text{even interaction, linear} \to\text{averaged-Hessian Hamiltonian tangent},$$

$$\text{production, quadratic} \to\text{labelled Cut norm}.$$

This supplies a compact parity/Wick/Cut calculus over the existing carrier.

Durable artifacts

• research/benincasa/checkers/general_quartic_gaussian_hessian.rs
• research/benincasa/results/general-quartic-gaussian-hessian.json
• research/benincasa/general-quartic-gaussian-hessian.md

Next falsifier

Introduce the first non-Gaussian fourth cumulant $\kappa_{ABCD}$. Compute the residual quartic covariance variation after subtracting the averaged-Hessian Gaussian part. Determine whether it is generated by an existing marked four-leg coefficient/Cut object or requires an additional sector-specific cumulant coefficient. Do not classify a new coefficient as a new carrier stratum.

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