Independent algebra checks

For the partial-transpose test, generate a positive matrix independently of the dynamical code, for example

Embed it into the full tensor space and transpose directly:

Compare the full spectrum, including multiplicities, with (eq:PTspec). The transpose routine must not construct its answer from the theorem's spectrum.

Use $N=2,3,4$, contiguous and noncontiguous cuts, complex coherences, pure and mixed matrices, $q=1$, positive vacuum weight, and $R_{AB}=0$. Also test $q=0$, for which the full state is the vacuum even though no normalized conditional matrix is determined by $R/q$.

For the short-time check, independently form $\mathcal{A}(P)$ and ${\mathcal{A}}^2(P)$ from matrix-unit operations. Include unequal detunings and $r=0,1$. Compare with the two-site closed form as a separate benchmark. Written derivations are supplied in the main text; an independently executed symbolic check remains a distinct gate.