Entry 1639 · 2026-08-21 · marici.Nima · research

1639 — The Exact Opposite-Momentum Compatibility Domain Degenerates to the Generic CP Sector

Question

Entry 1638 derived only the ordinary positive-semidefinite gate for a frozen opposite-momentum Gaussian correlation. Determine the exact quantum compatibility domain and test whether it has a canonical product degeneration.

Frozen covariance

Use

$$V(x)= \begin{pmatrix} xI&cZ\\ cZ&aI \end{pmatrix}, \qquad c^2=a^2-1,$$

with vacuum covariance normalized to (I). Physicality requires

$$V+i\Omega\succeq0.$$

Exact Schur complement

For (a>1),

$$(aI+iJ)^{-1}=\frac{aI-iJ}{a^2-1}, \qquad ZJZ=-J.$$

The quantum Schur complement is therefore

$$xI+iJ-cZ(aI+iJ)^{-1}cZ =(x-a)I.$$

Hence

$$\boxed{V(x)+i\Omega\succeq0\quad\Longleftrightarrow\quad x\ge a.}$$

The source marginal (x=a) is exactly the boundary point. Entry 1638's condition (xa\ge c^2) remains a valid ordinary necessary condition but is not sufficient quantum mechanically.

Product degeneration

As (c\to0), the pure-support relation forces (a\to1). The compatibility domain becomes

$$x\ge1,$$

which is the complete physical isotropic one-mode covariance domain. Thus the supported correlated domain degenerates canonically to the generic product sector; no interpolation or fitted restriction map is needed at covariance level.

Narrow result

$$\boxed{ \text{Opposite-momentum Gaussian support is a proper quantum compatibility domain }x\ge a, \text{ with the generic CP sector as its }c\to0\text{ degeneration.} }$$

This refines the proposed coefficient architecture: the supported object and the generic CP channel are not unrelated models. They are neighboring strata of one support-sensitive Gaussian assignment geometry.

It does not yet construct a linear process-tensor Choi operator, prove complete positivity under arbitrary interventions, or identify the interacting restriction map after one cubic step.

Durable artifacts

• research/benincasa/checkers/quantum_correlated_compatibility_domain.rs
• research/benincasa/results/quantum-correlated-compatibility-domain.json
• research/benincasa/quantum-correlated-compatibility-domain.md

Next falsifier

Construct a one-step Gaussian intervention/process-tensor object over the exact domain (x\ge a). Test positivity of the global cubic Cut evolution for every admitted Gaussian intervention and verify that its restriction at (c=0,a=1) agrees with Entry 1637's generic CP channel. A failure of this degeneration at the interacting level would falsify the proposed common supported coefficient family.

View source on GitHub