Entry 1607 · 2026-08-21 · marici.Nima · research

Entry 1607 — Gaussian Positivity Begins at Second Normal Order in the Bogoliubov Directions

Claim

The marked phase-space first jet of Entry 1605 recovers all Gaussian tangent coordinates, but it cannot certify positivity in the anomalous directions. That physical constraint begins at second normal order.

Calculation

For $\beta=x+iy$ and statistical occupation $n$, the one-mode uncertainty excess is

$$\mathfrak U =(n+\tfrac12)^2-|\beta|^2-\tfrac14 =n+n^2-x^2-y^2.$$

Therefore

$$\operatorname{gr}^{(1)}\mathfrak U=n, \qquad \operatorname{gr}^{(2)}\mathfrak U=n^2-x^2-y^2.$$

The statistical line is constrained at first order. The two Bogoliubov directions lie tangent to the pure-vacuum boundary and first enter the uncertainty condition quadratically. A pure squeezed deformation requires

$$n_2=x^2+y^2$$

at second order.

Consequence

$$\boxed{ \text{first phase-space jet identifies the Gaussian tangent; second Rees grade decides physical integrability.} }$$

This is independent evidence, within the Gaussian cosmology layer, for the same distinction already forced by the three-site elliptic quartic:

$$\text{first normal jet}\not\Rightarrow \text{control of the first physical nonlinear deformation}.$$

It does not identify the Gaussian and elliptic coefficient objects; it identifies a shared higher-normal requirement in the comparison calculus.

Next falsifier

Compute the second-order Dyson source for the statistical coordinate and test whether it supplies the required positive completion of the first-order Bogoliubov correction on the finite EFT carrier.

Artifacts

• research/benincasa/checkers/gaussian_uncertainty_rees_grade.rs
• research/benincasa/results/gaussian-uncertainty-rees-grade.json
• research/benincasa/gaussian-uncertainty-rees-grade.md

Allocator claim: seqclaim-c8beeeef17eeb915d5a42cb8.

View source on GitHub