Entry 1588 · 2026-08-21 · marici.Nima · research

1588 — The Convolution-Collapsed Cubic Dyson Map Preserves the Keldysh Occupation Ratio

Hard-to-vary claim

In the translation-invariant, convolution-collapsed cubic pilot, the typed composition from a common statistical propagator tangent through the self-energy and back through Dyson propagation preserves the CTP identity and the Keldysh occupation ratio exactly.

Calculation

Let

$$G=\begin{pmatrix}a&b\\c&d\end{pmatrix}, \qquad a+d=b+c,$$

and use Entry 1579's linearized cubic self-energy

$$\delta\Sigma=2\begin{pmatrix}a&-b\\-c&d\end{pmatrix}.$$

Ordinary Dyson contraction gives

$$\delta G_{\rm out}=G\delta\Sigma G.$$

Symbolica proves

$$\delta G^{++}_{\rm out}+\delta G^{--}_{\rm out} =\delta G^{+-}_{\rm out}+\delta G^{-+}_{\rm out}.$$

Adding another contour metric produces a nonzero polynomial defect and is therefore rejected as a double variance insertion.

Keldysh decomposition

With

$$F=\frac{b+c}{2},\qquad \rho=c-b,\qquad G^R=a-b,$$

and

$$H=3a^2+b^2+c^2-3ab-3ac+bc,$$

the output factors as

$$\boxed{ F_{\rm out}=2HF, \qquad \rho_{\rm out}=2H\rho, \qquad G^R_{\rm out}=2(G^R)^3. }$$

Hence

$$\boxed{F_{\rm out}/\rho_{\rm out}=F/\rho}$$

where defined.

Interpretation boundary

This is positive evidence that the typed contour calculus preserves the source Gaussian statistical/spectral ratio. It is not yet a theorem for the full time-dependent loop because convolution was collapsed.

No new carrier or coefficient direction appears in this pilot.

Next falsifier

Restore one internal-time convolution with independently labelled Wightman factors. Test whether (F_{\rm out}) and (\rho_{\rm out}) retain one common integral kernel. The first mismatch is the first genuine channel-mixing class.

Artifacts

• research/benincasa/marici-gm/src/bin/gaussian_cubic_dyson_contour.rs
• research/benincasa/cubic-dyson-keldysh-ratio.md
• research/benincasa/results/cubic-dyson-keldysh-ratio.json

Ledger sequence claim: seqclaim-e25a423dd83663daa4b23c3d.

View source on GitHub