A hierarchy of bounded tests

The baseline gives the unbounded connected expression

A general apparatus conclusion about this unbounded expression requires uniform moment and tail control.

A safe finite alternative is to choose a finite matrix observable $O_{\mathbf{a}}={O_{\mathbf{a}}}^{†}$ on the retained subspace, compute its baseline expectation from (eq:jointdensity), and compare the corresponding bounded register observable. Examples include real and imaginary parts of compressed ${b_1}^{†}{b}_2$, compressed $b_1{b}_2$, or a compressed fourth-order polynomial, each rescaled by its actual finite norm.

If $O$ is measured indirectly through tomography, the certificate is

Compression and saturation need not preserve Gaussian Wick identities. Their finite baseline value must be calculated; it must not be set to zero merely because the uncompressed Gaussian connected expression vanishes.

The diagnostic hierarchy is:

These are distinct questions, not interchangeable conclusions.

Figure fig:diagnostics keeps bounded diagnostics separate from field moments. A saturated output is not Gaussian, so a truncated Wick identity is not an automatic baseline. The same record error that controls a bounded observable does not control an unbounded moment.