Entry 1624 · 2026-08-21 · marici.Nima · research

1624 — Number Parity Forces the First Cubic Statistical Covariance Grade to Be a Cut Square

Question

With a nonzero Gaussian first jet present, can interference linear in the cubic production amplitude escape the positive labelled Cut pairing in the second covariance grade?

Source-compatible filtration

After imposing the source tadpole/centering condition, a Gaussian state lies in the even number-parity sector. One cubic insertion lies in the odd sector. The centered covariance observables

$$a^\dagger a, \qquad aa, \qquad a^\dagger a^\dagger$$

are parity-even.

Therefore, for every centered Gaussian state and one-cubic state,

$$\boxed{ \langle\Psi_{\rm G}|O_2|\Psi_{\rm cubic}\rangle =0. }$$

Nonzero squeezing changes coefficients inside the even sector but does not change this selection rule.

Consequence

Every covariance contribution linear in the cubic amplitude vanishes. The first statistical cubic grade is quadratic:

$$\delta V_{\rm cubic}^{(2)} = \operatorname{Tr}_{qk} |\Psi_{\rm cubic}\rangle \langle\Psi_{\rm cubic}|,$$

which is exactly the labelled Cut square of Entries 1617--1618 and 1623.

The checker verifies all even/odd matrix-element supports for number states through occupation 13 and audits complex Cut-norm positivity.

Narrow result

$$\boxed{ \text{In the centered Gaussian-plus-cubic sector, no linear cubic interference lies outside the Cut norm.} }$$

Thus the positive second-Rees completion survives a nonzero Gaussian first jet. The cubic statistical correction begins at second normal order for a structural parity reason, not by cancellation fitted to a kernel.

Qualifications

• A nonzero one-point function must first be removed by the source tadpole condition.
• The result concerns centered two-point covariance; parity-odd observables can have linear cubic terms.
• Quartic interactions are parity-even and require a separate test because they can interfere linearly with the Gaussian sector.

Architectural consequence

The second-Rees order is selected jointly by coefficient parity and the existing Cut calculus. No new carrier incidence is needed.

Durable artifacts

• research/benincasa/checkers/gaussian_cubic_parity_cut.rs
• research/benincasa/results/gaussian-cubic-parity-cut.json
• research/benincasa/gaussian-cubic-parity-cut.md

Next hostile falsifier

Repeat the covariance-grade test for a source-admitted quartic interaction. Because a single quartic insertion is parity-even, it can interfere with the Gaussian sector at first order. Determine whether that term is a Hamiltonian tangent, a source-normalization correction, or a genuinely non-Cut statistical direction. Freeze the quartic source and tadpole/normalization prescription before calculation.

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