Entry 1637 · 2026-08-21 · marici.Nima · research

1637 — A Split General-Gaussian Internal State Pushes the Cubic Cut to a Positive Scalar Gain

Question

Entry 1636 obtains strict observed filtration only after vacuum pushforward of the internal modes. Does a general Gaussian internal state reintroduce coupling to the full internal moment tower?

Frozen factorization

For a generic labelled channel away from observed/internal coincidence support, write

$$L_{p;qk} = C_{p;qk} a_p^\dagger \otimes (a_q^\dagger a_k^\dagger).$$

Assume the initial density matrix is split between the observed mode and the internal pair. Push the fixed internal Gaussian state forward:

$$\mu_{qk} = \operatorname{Tr} \left[ \rho_{qk} (a_q a_k)(a_q^\dagger a_k^\dagger) \right].$$

For distinct labelled occurrences,

$$\mu_{qk} = \langle(N_q+1)(N_k+1)\rangle.$$

Gaussian Wick reduction

Let

$$n_q=\langle a_q^\dagger a_q\rangle, \qquad c_{qk}=\langle a_q^\dagger a_k\rangle, \qquad \beta_{qk}=\langle a_q a_k\rangle.$$

Then

$$\boxed{ \mu_{qk} =(n_q+1)(n_k+1) +|c_{qk}|^2 +|\beta_{qk}|^2. }$$

This is positive and invariant under $q\leftrightarrow k$.

The checker verifies 194,481 exact Wick packets.

Observed generator

The reduced trace-balanced number-moment generator is

$$\mathcal L_p^\dagger f = \mu_{qk} (N_p+1)[f(N_p+1)-f(N_p)].$$

The internal Gaussian data enter only through a positive scalar coefficient. The observed moment degree remains unchanged.

Narrow result

$$\boxed{ \text{A fixed split Gaussian internal state preserves the strict observed moment tower and q-k occurrence covariance.} }$$

Vacuum is therefore not essential for the generic pushforward; Gaussian Wick closure is sufficient.

Supported qualification

If the source Gaussian covariance correlates $p$ directly with $q$ or $k$, the initial state is not split across the observed/internal partition and the reduced map need not be state-independent or completely positive. Translational invariance confines such correlations to labelled coincidence or opposite-momentum supports. Those supports must be tested separately; they cannot be folded into the generic scalar.

Coincident $q=k$ occurrences likewise carry extra bosonic contraction terms and remain occurrence-resolved.

Architectural consequence

Generic Gaussian internal pushforward is a coefficient evaluation map

$$\mathbb M_{qk}^{\rm Gauss} \longrightarrow \mathbb R_{\ge0}, \qquad \rho_{qk}\mapsto\mu_{qk},$$

followed by the strict observed channel. This fits the existing labelled Cut carrier and sector-specific positive moment coefficients.

Durable artifacts

• research/benincasa/checkers/gaussian_internal_pushforward.rs
• research/benincasa/results/gaussian-internal-pushforward.json
• research/benincasa/gaussian-internal-pushforward.md

Next falsifier

Analyze the first supported observed/internal correlation stratum, e.g. $q=-p$ in a translationally invariant anomalous Gaussian state. Construct the assignment map on that support and test whether a completely positive reduced channel exists after adjoining the correlated partner, or whether the correct object is a non-CP initial-correlation process tensor.

View source on GitHub