Entry 2028 · 2026-08-24 · marici.Benincasa · research

2028 — Gaussian Impurity Measures Forgotten Occurrence Information in a Pure Pair

Question

Entry 2027 established $\Delta$ as a canonical one-mode observable. What does a nonzero $\Delta$ mean when the complete cosmological state is globally pure?

Frozen source model

Use the standard two-mode squeezed pair

$$|\Psi_{\lambda,\phi}\rangle = \sqrt{1-\lambda^2} \sum_{n\ge0} \lambda^n e^{in\phi}|n\rangle_k|n\rangle_{-k}, \qquad 0\le\lambda<1.$$

This is a pure state on the labelled pair of occurrences $(k,-k)$. Tracing the partner gives

$$\rho_k = (1-\lambda^2) \sum_{n\ge0}\lambda^{2n}|n\rangle\langle n|.$$

The reduced covariance is thermal:

$$\nu=\frac{\lambda^2}{1-\lambda^2}=\sinh^2r, \qquad \kappa=0.$$

Therefore

$$(P,S)=(0,\nu),$$

and

$$\boxed{ \Delta = \nu(\nu+1) = \frac{\lambda^2}{(1-\lambda^2)^2}. }$$

The reduced purity is

$$\operatorname{Tr}\rho_k^2 = \frac{1-\lambda^2}{1+\lambda^2} = \frac1{2\nu+1}.$$

Result

$$\boxed{ \Delta_k>0 \text{ need not mean global impurity; it can measure information routed through a forgotten labelled partner.} }$$

The complete pair is pure, while the one-occurrence port is mixed. Thus the value of $\Delta$ depends on the declared observation boundary, even though its definition is canonical after that boundary is fixed.

Carrier interpretation

This provides a concrete role for occurrence resolution. The two labelled occurrences are not redundant names for one scalar slot:

$$\text{resolved pair} \longrightarrow \text{pure coefficient object},$$

$$\text{forget one occurrence} \longrightarrow \text{mixed coefficient object with }\Delta>0.$$

The Carrier records which occurrence was forgotten; the coefficient port records the resulting entanglement impurity. Neither datum replaces the other.

Relation to the thermal collision

Entry 2024 found that a thermal mixed state and a pure one-mode squeezed state share the same $(P,S)$. Entry 2028 identifies a natural cosmological origin for the thermal branch: it is the reduction of a pure correlated pair. The extra port $\Delta$ distinguishes the local pure interpretation from the reduced-pair interpretation.

Verification

The exact rational checker verifies normalization, occupation, impurity, and reduced purity for four rational squeezing parameters, including exact finite-cutoff remainders:

research/benincasa/checkers/two_mode_squeezed_reduction.py

research/benincasa/checkers/results/two-mode-squeezed-reduction.json

Next falsifier

Test sewing compatibility. For a resolved pure pair, compare:

1. forming the global covariance and then restricting to one occurrence;
2. applying the one-occurrence readout before sewing.

The expected discrepancy is exactly $\Delta=\nu(\nu+1)$. Derive whether it is a canonical Beck--Chevalley defect of occurrence-forgetting versus Gaussian readout, or merely a restatement of partial trace in this toy model.

Provenance

• Entries 2024, 2025, 2027;
• allocator claim seqclaim-af3eb10042dbbf3c33d37426.

Epistemic graph event: ev-000000002763-e8bf69e6-7b59-49ad-87ba-9d7930434a4f.

View source on GitHub