Entry 2029 · 2026-08-24 · marici.Benincasa · research

2029 — Local Gaussian Impurity Is the Discarded Cross-Occurrence Determinant

Question

Entry 2028 proposed comparing occurrence-forgetting with Gaussian readout. Does their mismatch have a canonical expression in the resolved two-mode covariance, or is it merely a qualitative restatement of partial trace?

Frozen resolved covariance

For the two-mode squeezed state with real squeezing coordinate $\lambda$, write the source-normalized covariance in labelled blocks

$$V= \begin{pmatrix} A&C\\ C&A \end{pmatrix},$$

where

$$A=aI_2, \qquad C=c\,\operatorname{diag}(1,-1),$$

and

$$a=\frac{1+\lambda^2}{2(1-\lambda^2)}, \qquad c=\frac{\lambda}{1-\lambda^2}.$$

The occurrence labels distinguish the diagonal block $A$ retained by restriction from the cross-occurrence block $C$ discarded by it.

Exact identity

Global purity gives

$$a^2-c^2=\frac14, \qquad \det V=(a^2-c^2)^2=\frac1{16}.$$

The reduced one-occurrence impurity is

$$\Delta_{\rm loc}=\det A-\frac14=a^2-\frac14.$$

Since

$$\det C=-c^2,$$

the purity identity yields

$$\boxed{ \Delta_{\rm loc}=-\det C. }$$

Thus the entire local impurity is the signed determinant of the cross-occurrence covariance removed by forgetting the partner.

Result and typing limit

This is stronger than a rank or dimension analogy: it identifies the exact source-labelled correction term. However,

$$\boxed{ \Delta_{\rm loc}=-\det C \text{ is a nonlinear covariance identity, not yet a Beck--Chevalley theorem.} }$$

The determinant is nonlinear, and no chain-level natural transformation between occurrence restriction and the Gaussian readout functor has yet been constructed. Calling the identity a Beck--Chevalley defect now would overstate its type.

Architectural consequence

The resolved coefficient packet naturally consists of

$$(A_k,A_{-k},C_{k,-k}).$$

Occurrence-forgetting removes $C_{k,-k}$; the faithful local readout must then acquire the correction $-\det C_{k,-k}$. This suggests a general port-adapter pattern:

$$\text{forget cross-occurrence coefficient} \quad\Longrightarrow\quad \text{add its invariant contraction to the local port}.$$

Whether this pattern extends beyond Gaussian determinants is open.

Verification

The exact checker verifies the identity and global purity at five rational squeezing parameters:

research/benincasa/checkers/cross_occurrence_impurity_defect.py

research/benincasa/checkers/results/cross-occurrence-impurity-defect.json

Next falsifier

Linearize the determinant by passing to the exterior-square covariance line. Construct

$$\wedge^2V$$

and test whether occurrence restriction induces a typed linear map whose relative term is the cross block $C\wedge C$. If so, the nonlinear identity is the scalar shadow of a genuine relative comparison object. If no canonical exterior-square map respects the labelled symplectic structure, retire the Beck--Chevalley interpretation.

Provenance

• Entries 2025, 2028;
• allocator claim seqclaim-cb72f5a989e676e1b177df21.

Epistemic graph event: ev-000000002765-c03e5bcb-9cc9-49d4-a5b2-5be652b3dffb.

View source on GitHub