Entry 1622 · 2026-08-21 · marici.Nima · research

1622 — The Generic Finite Labelled-Cut Family Generates the Gaussian Second-Rees Cone

Frozen local model

Let one observed oscillator be coupled to finitely many orthogonal environment labels $r$. Freeze the first-order purification

$$|\Psi\rangle = |0;0_E\rangle + \epsilon \left( a|2;0_E\rangle + \sum_r b_r|1;r\rangle \right) +O(\epsilon^2).$$

The amplitude $a$ is the coherent pure-Gaussian squeezing direction. Each $b_r$ is a labelled Cut channel crossing from the observed mode into an orthogonal environment occurrence.

Second normal grade

At the first nontrivial grades,

$$|\beta_1|^2=2|a|^2,$$

while

$$n_2 = 2|a|^2 + \sum_r|b_r|^2.$$

Therefore the second-Rees uncertainty grade is

$$\boxed{ n_2-|\beta_1|^2 = \sum_r|b_r|^2 \geq0. }$$

Equality holds exactly when all environment-crossing Cut amplitudes vanish.

The checker verifies 16,807 exact cases for three independent labelled channels. Orthogonality makes the displayed formula independent of the finite number of labels.

Narrow result

The finite labelled-Cut family realizes the complete local inequality cone

$$n_2\geq|\beta_1|^2$$

for the frozen purification model:

• the boundary is the pure Gaussian/squeezing locus;
• the interior excess is the labelled Cut norm into unobserved channels.

This generalizes Entry 1608's single environment channel and Entry 1610's cubic three-particle channel.

Type qualifications

This does not prove that a particular continuum interaction kernel spans every abstract mixed-Gaussian tangent direction. It proves the coefficient geometry once the environment-crossing amplitudes are source-derived. The continuum measure and interaction selection rules remain separate inputs.

Architectural consequence

The Gaussian state cone is not an additional carrier. Locally it is reconstructed from:

$$\text{marked coherent first jet} + \text{positive labelled Cut pairing} \longrightarrow \text{second-Rees covariance cone}.$$

This is direct evidence for H2: shared labelled Cut carrier and comparison calculus, with a cosmology-specific positive covariance coefficient object.

Durable artifacts

• research/benincasa/checkers/gaussian_mixed_second_rees_cone.rs
• research/benincasa/results/gaussian-mixed-second-rees-cone.json
• research/benincasa/gaussian-mixed-second-rees-cone.md

Next falsifier

Replace the abstract orthogonal labels by the source cubic momentum channels and compute the map

$$C_{p;q,k} \longmapsto \left(x_1(p),y_1(p),n_2(p)\right)$$

with momentum conservation, the symmetric finite-EFT measure, and all occurrence multiplicities retained. Test whether the resulting continuum Cut norm saturates precisely on the source pure-state locus or whether a source interference term survives outside the positive labelled pairing.

View source on GitHub