Entry 1608 · 2026-08-21 · marici.Nima · research

Entry 1608 — A Cut Norm Supplies the Gaussian Second-Rees Positivity Excess

Hard-to-vary conjecture

The second-Rees positivity completion of the one-loop Gaussian coefficient object is generated by the physical Cut pairing of first-order production amplitudes.

Finite mechanism test

Take two modes and the perturbative state

$$|\Psi\rangle =|0,0\rangle +\varepsilon a|2,0\rangle +\varepsilon b|1,1\rangle +O(\varepsilon^2).$$

For the first observed mode,

$$|\beta_1|^2=2|a|^2, \qquad n_2=2|a|^2+|b|^2.$$

Hence

$$\boxed{ n_2-|\beta_1|^2=|b|^2\ge0. }$$

The $|2,0\rangle$ amplitude is the same-mode coherent pair and supplies the pure Bogoliubov completion. The $|1,1\rangle$ amplitude crosses into the forgotten mode; tracing that mode produces the statistical excess. Its norm is a literal cut square in the finite model.

Architectural consequence

The Gaussian layer now exhibits a typed sequence

$$\text{first phase-space jet} \longrightarrow \text{second Rees uncertainty grade} \longrightarrow \text{Cut norm completion}.$$

This is evidence that the existing Cut calculus can control a nonlinear physical constraint on a sector-specific coefficient object. It does not identify Gaussian covariance with scattering coefficients.

Scope boundary

The finite model proves the mechanism only. The cosmological loop still requires the source-normalized production amplitude, physical Cut measure, and occurrence trace. Positivity must not be inferred from rank matching.

Next falsifier

Extract the first-order production amplitude associated with the cubic cosmological kernel and compare its Cut square with the second-order statistical component of the Dyson covariance. Test equality before momentum integration and before forgetting the $q/k$ occurrence labels.

Artifacts

• research/benincasa/checkers/gaussian_second_rees_cut_completion.rs
• research/benincasa/results/gaussian-second-rees-cut-completion.json
• research/benincasa/gaussian-second-rees-cut-completion.md

Allocator claim: seqclaim-b55b9620a3b5029e74590bea.

View source on GitHub